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Research ArticleOriginal Article
Open Access

Gravity Modeling in GNSS-Aided Inertial Navigation System Safety Certification

Timothy Needham and Michael Braasch
NAVIGATION: Journal of the Institute of Navigation June 2022, 69 (2) navi.520; DOI: https://doi.org/10.33012/navi.520
Timothy Needham
Ohio University
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  • For correspondence: [email protected]
Michael Braasch
Ohio University
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  • FIGURE 1
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    FIGURE 1

    Illustration showing the DOV as the angle of the effective gravity vector relative to the ellipsoidal normal

  • FIGURE 2
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    FIGURE 2

    Trajectory of an RNP 0.3 approach into Runway 02 of the Kahului Airport

  • FIGURE 3
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    FIGURE 3

    North-south DOV for an approach into Runway 02 of Kahului Airport

  • FIGURE 4
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    FIGURE 4

    Simulated approach onto Runway 33 at Ted Stevens Anchorage International Airport

  • FIGURE 5
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    FIGURE 5

    North-south DOV for an approach onto Runway 33 of Ted Stevens Anchorage International Airport

  • FIGURE 6
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    FIGURE 6

    Visualization of gravitational forces acting on Object 1 as it moves on a path offset from Object 2

  • FIGURE 7
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    FIGURE 7

    Normalized along-track and cross-track components of the gravitational force experienced as Object 1 passes Object 2 that is offset from the trajectory

  • FIGURE 8
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    FIGURE 8

    Normalized spatial power spectra demonstrating differences between the along-track and cross-track components of the gravitational forces

  • FIGURE 9
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    FIGURE 9

    Block diagram demonstrating the generation of the total simulated gravity model error

  • FIGURE 10
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    FIGURE 10

    DOV profiles as determined from EGM2008 and DEFLEC data sets for a simulated approach into Ted Stevens International Airport in Anchorage, Alaska

  • FIGURE 11
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    FIGURE 11

    Block diagram outlining the process for using system identification to analyze measured data, identify, and validate representative ARMA models

  • FIGURE 12
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    FIGURE 12

    Example differences between EGM2008 and xDEFLEC19 cross-track components of DOV for select trajectories

  • FIGURE 13
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    FIGURE 13

    Example differences between EGM2008 and XDEFLEC19 along-track components of DOV for select trajectories

  • FIGURE 14
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    FIGURE 14

    Mean of the BIC for each of the selected cross-track runs

  • FIGURE 15
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    FIGURE 15

    Pole-zero plot showing the cross-track and along-track pole results from system identification of each of the trajectories

  • FIGURE 16
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    FIGURE 16

    Comparison of the measured data PSD to the simulated data to show validation of the AR(2) model specified in Equation (5)

  • FIGURE 17
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    FIGURE 17

    Mean of the model order selection criteria (i.e., BIC) over a range of (p,q) combinations up to ARMA(15,15)

  • FIGURE 18
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    FIGURE 18

    Pole-zero plot showing the ARMA(2,2) poles and zeros for model fits in addition to the roots selected for the ARMA(2,2) model used for validation

  • FIGURE 19
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    FIGURE 19

    The measured data PSD with confidence bounds compared to the PSD produced by the ARMA(2,2) model in Equation (7)

  • FIGURE 20
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    FIGURE 20

    Complementary CDF showing that Gaussian distribution bounds the tails of the model errors

  • FIGURE 21
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    FIGURE 21

    Block diagram demonstrating the use of ARMA models to generate simulated reference model

  • FIGURE 22
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    FIGURE 22

    Example simulated reference gravity model errors using the models given in Equations (5) and (7) and a model standard deviation of σx = 16.3 arc-seconds

  • FIGURE 23
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    FIGURE 23

    Block diagram demonstrating an example of low-fidelity architecture

  • FIGURE 24
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    FIGURE 24

    Sample cross-track total DOV for a select trajectory

  • FIGURE 25
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    FIGURE 25

    Sample along-track total DOV for a select trajectory

  • FIGURE 26
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    FIGURE 26

    Mean BIC values for each ARMA(p,q) combination for the cross-track (left) and along-track (right)

  • FIGURE 27
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    FIGURE 27

    Pole-zero plots showing roots of the ARMA(1,1) fits to the low-order cross-track and along-track

  • FIGURE 28
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    FIGURE 28

    PSDs of the measured data compared to the spectrum of the simulated data generated by Equation X and Equation Y

  • FIGURE 29
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    FIGURE 29

    DOV for a trajectory over Puerto Rico showing the large low-frequency effects and gravitational effects of the trench

  • FIGURE 30
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    FIGURE 30

    Mean BIC for the cross-track low-fidelity residuals from areas near Hawaii and Puerto Rico

  • FIGURE 31
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    FIGURE 31

    Pole-zero plot demonstrating the results of the system identification on cross-track DOV of trajectories near Hawaii and the Puerto Rico trench using AR(2)

  • FIGURE 32
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    FIGURE 32

    Pole-zero plot demonstrating the results of the system identification on DOV of trajectories near Hawaii and the Puerto Rico trench using AR(3)

  • FIGURE 33
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    FIGURE 33

    Low-fidelity Geographic Area #2 along-track

  • FIGURE 34
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    FIGURE 34

    Pole-zero plot demonstrating the results of the system identification on along-track DOV of trajectories near Hawaii and the Puerto Rico trench using AR(2)

  • FIGURE 35
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    FIGURE 35

    Low-fidelity Geographic Area #2 along-track

  • FIGURE 36
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    FIGURE 36

    The measured data PSD with confidence bounds compared to the PSD produced by AR(2) cross-track model in Equation (10)

  • FIGURE 37
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    FIGURE 37

    The measured data PSD with confidence bounds compared to the PSD produced by AR(2) along-track model in Equation (11)

  • FIGURE 38
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    FIGURE 38

    Complementary CDF showing that Gaussian distribution bounds the tails of the low-fidelity model errors

  • FIGURE 39
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    FIGURE 39

    Example cross-track and along-track data generated using the model as specified in Equation (10) and Equation (11)

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NAVIGATION: Journal of the Institute of Navigation: 69 (2)
NAVIGATION: Journal of the Institute of Navigation
Vol. 69, Issue 2
Summer 2022
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Gravity Modeling in GNSS-Aided Inertial Navigation System Safety Certification
Timothy Needham, Michael Braasch
NAVIGATION: Journal of the Institute of Navigation Jun 2022, 69 (2) navi.520; DOI: 10.33012/navi.520

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Gravity Modeling in GNSS-Aided Inertial Navigation System Safety Certification
Timothy Needham, Michael Braasch
NAVIGATION: Journal of the Institute of Navigation Jun 2022, 69 (2) navi.520; DOI: 10.33012/navi.520
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  • Article
    • Abstract
    • 1 INTRODUCTION
    • 2 GRAVITY COMPENSATION AND MIS-MODELING
    • 3 EFFECTS OF A STATIONARY MASS ON A MOVING OBJECT
    • 4 HIGH-FIDELITY GRAVITY SIMULATION ARCHITECTURE
    • 5 HIGH-FIDELITY STOCHASTIC MODEL DEVELOPMENT APPROACH
    • 6 EXAMPLE HIGH-FIDELITY STOCHASTIC MODEL
    • 7 LOW-FIDELITY STOCHASTIC MODEL DEVELOPMENT APPROACH AND EXAMPLE
    • 8 CONCLUSION AND RECOMMENDATIONS
    • HOW TO CITE THIS ARTICLE
    • ACKNOWLEDGEMENTS
    • REFERENCES
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  • Supplemental
  • References
  • Info & Metrics
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Keywords

  • gravity modeling
  • GNSS/INS
  • inertial coasting
  • inertial navigation
  • integrity
  • RTCA

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