Vertical Integrity Monitoring with Direct Positioning
Arthur Hsi-Ping Chu and Grace Xingxin Gao, University of Illinois at Urbana-Champaign
An effective integrity monitoring scheme is paramount for GPS receivers in safety-of-life applications. Such a scheme protects the users from Hazardously Misleading Information (HMI) [1, 2]. This protection is facilitated by an error model that overbounds the actual positioning error, reflecting the trustworthiness of the navigation solutions. Existing algorithms generally assume the pseudorange measurement errors to have zero-mean Gaussian distributions [3, 4]. The position error distribution is then derived by convolving these Gaussian distributions based on the satellite geometry.
We propose a novel integrity monitoring scheme based on Direct Position Estimation (DPE). DPE is a GPS receiver architecture that replaces the pseudorange measurements with a grid of candidates in the position-velocity-time (PVT) domain [5, 6, 7], with each candidate representing a unique PVT coordinate. DPE then synthesizes an expected signal reception for every candidate, and computes the correlation between the expected reception and the raw signal. We use the resulting correlation value as the likelihood at which the candidate may be the navigation solution. The navigation solution is subsequently derived from these likelihood values. Collectively, these values form a likelihood manifold in the PVT domain.
The goal of this work is to present that, from the DPE likelihood manifold, a probability bound for the position error distribution can be effectively established to provide integrity monitoring service. Our proposed algorithm is distinctive from, and more advantageous than, the conventional approaches in the following aspects:
1. Our integrity monitoring scheme directly derives an overbounding distribution in the position domain using the DPE likelihood manifold. The existing pseudorange-based schemes, meanwhile, assumes Gaussian errors for the pseudorange measurements. This assumption is fundamentally incompatible with the pseudorange-free DPE receiver.
2. As the DPE likelihood manifold also contains receiver-specific information, our DPE-based integrity monitoring scheme monitors possible receiver faults along with the signal-in-space ones. Conventional integrity schemes, in contrast, typically focused on the signal-in-space integrity risks (e.g. ionospheric scintillation, excessive drifts in the satellite clock) , while remaining largely oblivious to the factors on the receiver end, such as dense multipath.
3. For a DPE-based scheme, we directly model the error in the position domain based on the DPE likelihood manifold, eliminating the possibility for a convolution-originated HMI. In contrast, for conventional schemes, an asymmetric or biased error distribution in one of the pseudorange measurements might cause an underbounding in the position domain and precipitate HMI [3, 9].
Furthermore, we factor the DPE-based error bound to compute the Protection Level (PL), which is the foundation for many safety-of-life applications . The magnitude of the factoring is a function to a predetermined tolerance for HMI likelihood. As an initial investigation, we primarily focus on the integrity monitoring for the vertical axis (i.e. altitude), instead of a full 3D-monitoring. This is consistent with the principal concern from
the aviation community for a reliable Vertical Protection Level (VPL) estimation.
We integrate the proposed integrity monitoring scheme into the DPE receiver architecture, which then operates iteratively as follows:
1. Populate a grid of PVT candidates based on an initializing coordinate. This initialization ensures the navigation solution to correctly fall within the range of the candidates. The accuracy of the initialization is not of pressing concern, and can be obtained from any source: previous iterations, manual inputs, external navigation sources (e.g. a scalar tracking receivers), etc.
2. For each PVT candidate, generate an expected signal reception based on the candidate's PVT coordinate as well as the PVT coordinates of the visible satellites.
3. For each PVT candidate, correlate the expected reception with the raw signal captured by the antenna. The correlation values across the candidate grid collectively form a likelihood manifold. This manifold is typically unimodal, with the peak situating at the candidate closest to the navigation solution.
4. We generate the optimized navigation solution through the interpolation of the PVT coordinates of the candidates, since the navigation solution may be surrounded by a few candidates, rather than directly overlapped by one. The interpolation is achieved through the weighted average of the candidates, with the correlation values as weights.
5. As we are addressing the VPL in this work, we extract all candidates that share the horizontal coordinates with the navigation solution. We then populate a one-dimensional vector with each element corresponding to the likelihood value of one candidate. Collectively, these likelihood values form a unimodal curve.
6. Normalize the vertical likelihood curve from the previous step such as their sum becomes one. The normalization renders this likelihood curve a probability distribution function (PDF). We then convert the PDF into a CDF and search for an overbounding zero-mean Gaussian distribution. Note that the variance of the overbounding distribution is to be the smallest-possible (i.e. tightest) value satisfying the overbounding criterion prescribed in .
7. We conclude the iteration by factoring the variance of the overbounding distribution, which produces the VPL. We empirically establish this factor by collecting the variance values of the overbounding distribution over a significant number of iterations (e.g. 10 million). We then find the proper scaling constant such that the occurrence of HMI in this data set is sufficiently rare, depending on the vertical guidance standard
 being implemented.
We collect L1 GPS raw samples with a commercial off-the-shelf (COTS) radio-frequency front-end, clocked by a chip-scale atomic clock (CSAC) to ensure minimal clock drift at the receiver. We process the raw data samples with a PyGNSS-implemented DPE receiver. PyGNSS is our software-defined radio (SDR) research suite, based on Python 2. The proposed integrity monitoring scheme is incorporated into the DPE receiver.
To validate our algorithm, we conducted experiments at a static, surveyed location and collected continuous data for 24 hours. The antenna was mounted on top of an urban, multi-story building. The surrounding structures created multipath propagation, or increased the likelihood thereof, given the satellite geometry at different times throughout the day. We computed the absolute errors by comparing the surveyed coordinate with the outputs of the DPE receiver. The errors were compared against the variance of the overbounding distribution.
To summarize, we proposed a DPE-based integrity monitoring scheme and integrated this scheme with the existing DPE architecture. We derived the DPE likelihood manifold to generate an overbounding, Gaussian error model directly in the position domain. We implemented this algorithm using a software-defined radio. We experimentally validated that the Vertical Protection Level (VPL) was effectively established by factoring the variance of the DPE-derived error model.
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