Design of Estimators with Integrity in the Presence of Error Model Uncertainty
Juan Blanch and Todd Walter, Stanford University
Date/Time: Thursday, Sep. 22, 9:20 a.m.
Fault detection and exclusion are key functions for positioning algorithms meant to provide high integrity error bounds or simply accurate positions in the presence of faulty measurements. It is well known that least square estimators are not optimal when the error model includes the possibility of faults affecting the measurements, as it is the case in RAIM for un-augmented GNSS, or radionavigation signals in cluttered environments. A least square estimator (or even a linear estimator) that includes an unbounded fault will result in an unbounded position error. To limit the growth of the position error, the effect of that measurement must be mitigated. This is the objective of fault detection and exclusion (FDE) algorithms. As indicated by their name, most available FDE algorithms are structured so that [1,2,3,4,5] detection and exclusion are two distinct steps, after which a linear unbiased estimator (most often a least squares filter) is used. This approach works very well in many applications and is the basis for the standard approaches in RAIM and Advanced RAIM. It is however not guaranteed to be the best approach. The purpose of this paper is to develop estimators that are not constrained by this structure and to evaluate their potential benefits.
In our previous work , we described an estimator and its associated protection level that merges the two functions (fault detection and fault exclusion). A key contribution of this work was the derivation of a protection level (PL) with an analytical proof of both integrity and continuity (or probability of alert) for the proposed estimator, which is non-linear. The estimator was applied to GNSS data collected on the ground with artificial faults and evaluated using a service volume model. Preliminary results suggested that this approach may improve upon the classical paradigm of FDE in at least three ways: it provides a better worst-case performance (in terms of protection level), it has a simpler logic, and the position solution tends to be smoother over time. The purpose of this work is to continue the development and evaluation of this new class of estimators as described below.
Lower bounds on achievable estimator performance
One of the methods to evaluate an estimator is by comparing the achieved PLs to those of an optimal estimator, or if not known, to a lower bound on the achievable PLs. We will therefore start by casting the search of the estimator as an optimization problem. This problem appears to be a very complex mini-max problem, for which there are no obvious solutions. (It is precisely this fact that has resulted in a very rich RAIM literature: if there was one known optimal solution, there would be no need to approach the problem from different angles.) We will use this formulation to develop lower bounds on the achievable performance of the optimal estimator. These lower bounds are a function of the geometry of the problem, the nominal error model, and the fault error models. These lower bounds on the PLs will allow us to evaluate the performance of our proposed estimators.
Description of new class of estimators
The key to the approach proposed in  is that rather than focus on a point estimate, we focus on estimating a region in space. First, we need to make sure that the region contains the true position with high probability. This is to guarantee integrity. Then we need to make sure that this region has a predictable size for a given probability of alert. This is to guarantee continuity (or at least an upper limit on the probability of alert). It is relatively straightforward to design a region that contains the true position with high probability by using the principles of solution separation fault detection algorithms [1,3]. The difficult part is to design it such that it does not grow without limits when a fault is present. We show that this can be done by carefully defining the region shape as a function of the measurement residuals. As compared to , we will: 1) generalize the approach to all dimensions, 2) refine the PL computation, 3) clarify the effect of the probability of alert requirement to further improve performance (as opposed to ), 4) analytically compare the PLs to the lower bounds obtained in the previous section.
Applications: ARAIM and PPP autonomous integrity
We will apply the proposed algorithm to a high precision position engine based on PPP using both flight data and ground data. The algorithm will be tested by injecting different types of faults on data collected in flight or on the ground (static and kinematic). For each of these scenarios, we will compare the protection levels obtained with the proposed method to the PLs obtained from a benchmark solution separation algorithm with a standard FDE approach. We will also compare the behavior of the positioning error in both approaches to confirm the benefits already observed in , which included a much smoother behavior of both the position error and the position solution.
We will evaluate this approach as applied to Advanced RAIM for scenarios with the expected constellation Integrity Support Data  using the service volume simulation tool MAAST . Our preliminary results show that PLs can be reduced by 50% compared to ARAIM user algorithms relying on the standard FDE approach.
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 Blanch, J., Walter, T., & Enge, P. (2010). RAIM with optimal integrity and continuity allocations under multiple failures. IEEE Transactions on Aerospace and Electronic Systems, 46(3), 1235–1247. https://doi.org/10.1109/TAES.2010.5545186
 Blanch J., Walter, T., and Enge, P.,” Protection Levels after Fault Exclusion for Advanced RAIM,” NAVIGATION, Journal of The Institute of Navigation, Vol. 64, No. 4, Winter 2017, pp. 505-513.
 Joerger M., Pervan B. Fault detection and exclusion using solution separation and chi-squared ARAIM. IEEE Trans. Aerosp. Electron. Syst. 2016;52:726–742. doi: 10.1109/TAES.2015.140589.
 Blanch, J., Walter, T. “A Fault Detection and Exclusion Estimator for Integrity Monitoring”, 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.
 Fifth Draft of ARAIM SARPs: Baseline Development Standard, ICAO Working Paper, Eighth meeting of the Joint Working Groups, November 2021.
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