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### Session A5: Aviation and Aeronautics

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**Epistemic Positioning Uncertainty Bounding Methods Using Zonotopes and Non Least Squares Estimators**

*Mathieu Joerger and Danielle Racelis, Virginia Tech; Jingyao Su and Steffen Schön, Leibniz Universität Hannover*

**Date/Time:** Friday, Sep. 20, 8:35 a.m.

In this paper, we derive, analyze, and evaluate two methods to reduce the impact of epistemic measurement uncertainty on GNSS-based positioning integrity bounds. Unlike aleatory (or random) errors that can be accounted for using probabilistic models, epistemic uncertainty, or uncompensated deterministic errors, are unknown or impractical to calibrate. A GNSS ranging measurement’s epistemic uncertainty can be treated as a bounded bias (or nominal bias), i.e., can be confined to a predefined interval with known minimum and maximum values such as the nominal bias in Advanced Receiver Autonomous Integrity Monitoring (ARAIM) [1-3]. This paper evaluates the impacts of measurement-level intervals on position-level error bounds. First, we show that zonotope-based methods help tighten predicted positioning error bounds as compared to conventional methods in [4,5], and we quantify this reduction in an example application of horizontal ARAIM (H-ARAIM). Second, we design a modified non least squares estimator that reduces the impact of nominal biases on fault-free positioning error bounds.

The high-integrity GPS and Galileo ranging error models used in ARAIM account for (a) random errors overbounded using Gaussian functions [6-8], and (b) nominal measurement biases. The latter include overbounding mean-error bounds and systematic, satellite-specific, and user antenna/receiver-specific signal deformation biases [2,3]. Error calibration is not scalable across equipment manufacturers and models.

This two-part paper aims at reducing the impacts of epistemic uncertainty to tighten horizontal positioning error bounds as compared to conventional methods in [5]. The first approach exploits the horizontal measurement error interval geometry. The second method is the design of an estimator that not only minimizes the positioning error variance but also considers epistemic uncertainty.

(1) Using zonotopes to derive tighter horizontal positioning bias bounds

To bound the epistemic ranging uncertainty, interval mathematics and set theory have been investigated in [9,10]. For linear estimators, measurement-level intervals have an impact on estimation states that can be expressed as a set of inequalities. For example, a least squares horizontal position estimator produces predictive error bounds represented by zonotopes in [10].

A reduction in horizontal bias bound is obtained by deriving a single radial positioning error bound instead of root-sum-squaring the individual East and North bias bounds as in [5]. The radial error is bounded by the zonotope vertex farthest from its center, which can be efficiently computed using the method in [11,12].

We evaluated the impact of the zonotope-based bias bounding method in an example GPS/Galileo HARAIM application, using the nominal error model parameter values and constellation setting in [5], at an example location over 24 hours. The new method is computationally more expensive than that in [5]. However, under fault-free conditions, it reduces the nominal positioning error bias bound under fault-free conditions by more than 12%.

The method is not as effective when considering fault hypotheses in HARAIM. In this case, the bound on the impacts of undetected faults contributes to epistemic uncertainty. When using solution separation, this bound is derived in the position domain and cannot be reduced using the zonotope approach. For nominal HARAIM parameter, this term is significantly larger than the nominal bias bound, which attenuates the improvement brought about by zonotopes. The paper identifies other cases where the zonotope-based bound is effective.

(2) Non least squares estimation to reduce horizontal positioning error bounds

Non least squares estimators were developed in ARAIM to minimize the integrity risk at the cost of lower accuracy [13,14]. The optimization process was carried out over multiple fault hypotheses. In contrast, this paper focuses on fault-free conditions and addresses the fact that a weighted least squares (WLS) estimator weighs measurements only considering random-error models regardless of epistemic uncertainty, which does not minimize positioning error bounds.

We therefore design a modified non least squares (NLS) estimator matrix, which we express as the sum of the WLS matrix and of a matrix that projects measurements in the left-null space of the observation matrix. The expression ensure that the NLS estimator is unbiased. Both numerical and analytical methods are explored to reduce a probabilistic bound on the positioning error in a computationally efficient algorithm. Performance is evaluated for GNSS satellite geometries simulated at an example location over 24 hours.

References

[1] J. Blanch, T. Walter, G. Berz, J. Burns, B. Clark, M. Joerger, M. Mabilleau, I. Martini, C. Milner, B. Pervan et al., “Development of Advanced RAIM minimum operational performance standards,” in Proceedings of the 32nd International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2019), Miami, Florida, September 2019, pp. 1381–1391.

[2] Phelts, R. Eric, Altshuler, Eric, Walter, Todd, Enge, Per, "Validating Nominal Bias Error Limits Using 4 years of WAAS Signal Quality Monitoring Data," Proceedings of the ION 2015 Pacific PNT Meeting, Honolulu, Hawaii, April 2015, pp. 956-963.

[3] Macabiau, C., Milner, C., Chabory, A., Suard, N., Rodriguez, C., Mabilleau, M., Vuillaume, J., Hegron, S., "Nominal Bias Analysis for ARAIM User," Proceedings of the 2015 International Technical Meeting of The Institute of Navigation, Dana Point, California, January 2015, pp. 713-732.

[4] Blanch, J., M. Joerger, C. Milner, D. Bouvier, B. Pervan, and T. Walter. “Baseline Advanced RAIM User Algorithm: Proposed Updates.” Proceedings of the Institute of Navigation International Technical Meeting (ITM 2022). Long Beach, CA. (2022).

[5] She, Jianming, Misovec, Kathleen, Blanch, Juan, Caccioppoli, Natali, Duchet, David, Tijero, Enrique Domínguez, Liu, Fan, Racelis, Danielle, Joerger, Mathieu, Sgammini, Matteo, "Implementation of the Reference Advanced RAIM User Algorithm," Proceedings of the 36th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2023), Denver, Colorado, September 2023, pp. 1099-1127.

[6] B. DeCleene, "Defining pseudorange integrity-overbounding." Proceedings of the 13th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GPS 2000). 2000.

[7] J. Rife, et al. "Paired overbounding and application to GPS augmentation." PLANS 2004. Position Location and Navigation Symposium (IEEE Cat. No. 04CH37556). IEEE, 2004.

[8] J. Blanch, T. Walter and P. Enge, “Gaussian Bounds of Sample Distributions for Integrity Analysis,” IEEE Transactions on Aerospace and Electronic Systems, vol. 55, no. 4, pp. 1806-1815, Aug. 2019, doi: 10.1109/TAES.2018.2876583.

[9] S. Schön and H. Kutterer, “Uncertainty in GPS networks due to remaining systematic errors: The interval approach,” Journal of Geodesy, vol. 80, no. 3, pp. 150–162, 2006.

[10] S. Schön and H. Kutterer, “Using zonotopes for overestimation-free interval least-squares–some geodetic applications,” Reliable Computing, vol. 11, no. 2, pp. 137–155, 2005.

[11] J. Su and S. Schön, “Improved observation interval bounding for multi-GNSS integrity monitoring in urban navigation,” in Proceedings of the 34th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2021), St. Louis, Missouri, September 2021, pp. 4141–4156.

[12] J. Su and S. Schön. “Advances in Deterministic Approaches for Bounding Uncertainty and Integrity Monitoring of Autonomous Navigation,” In Proceedings of the 35th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2022), Denver, Colorado, September, 2022, pp. 1442-1454.

[13] Joerger, M., S. Langel, and B. Pervan. “Integrity Risk Minimization in RAIM, Part 2: Optimal Estimator Design.” Journal of Navigation of the RIN. 69.4. (2016).

[14] J. Blanch and T. Walter, “A fault detection and exclusion estimator designed for integrity,” in Proceedings of the 34th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2021), St. Louis, Missouri, September 2021, pp. 1672–1686

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