Session B4: GNSS Integrity and Augmentation

Enhancing Availability of SS-RAIM based CDGNSS through Integer Aperture Bootstrapping
Dongchan Min, and Jiyun Lee, Korea Advanced Institute of Science and Technology
Location: Beacon B

Carrier-phase Differential GNSS (CDGNSS) is a promising approach for autonomous vehicle applications that demand highly accurate and safe navigation. The key to achieving high precision in CDGNSS lies in the integer ambiguity resolution of carrier-phase measurements. Successful ambiguity resolution allows carrier-phase measurements to serve as extremely accurate range measurements, enabling centimeter-level position accuracy. However, the resulting position can be biased if the ambiguities are incorrectly resolved due to measurement errors. This problem becomes more severe under fault conditions that introduce biases into GNSS measurements. Faulted measurements increase the likelihood of incorrect ambiguity resolutions, potentially leading to larger biases in positioning errors. Therefore, for safety-critical applications relying on CDGNSS, it is essential to protect users from the impact of measurement errors (including faults as well as nominal errors) on ambiguity resolution and their subsequent effect on positioning errors.
Our team proposed an integrity architecture for CDGNSS by employing Solution Separation Receiver Autonomous Integrity Monitoring (SS-RAIM) at the ION GNSS+ 2023 conference (Min et al., 2023) and submitted a paper on this topic to the Journal of Navigation. Navigation integrity quantifies the system's ability to issue timely warnings when reliance on it should be avoided. Our previous work of (Min et al., 2023) utilizes a monitor that defines its test statistic in the position domain, directly measuring position errors induced by faults. While the monitor identifies potentially unsafe ranging measurements, a statistical bound of position errors is computed to protect users against undetected faults. This statistical bound, termed the Protection Level (PL), corresponds to the integrity risk requirement, defined as the probability of a position error exceeding predefined alert limits. The navigation system is considered available only when the PL is less than the alert limit. Therefore, the availability—defined as the percentage of time during which the navigation system's outputs can be safely utilized—is highly dependent on the PLs.
While ambiguity resolution is crucial for achieving centimeter-level positioning accuracy in CDGNSS, incorrect ambiguity estimation can introduce significant position errors in the ambiguity-resolved position (i.e., the fixed position). In our previous work by Min et al. (2023), the PL of the fixed position was derived as a function of an a priori measure of the probability of incorrect ambiguity estimation to account for its impact on integrity risk. A fundamental assumption in the PL derivation is that position errors caused by any incorrect ambiguity resolution (termed an "incorrect fix") always exceed the PL. Due to this conservative assumption, the Probability of Incorrect Fix (PIF) must be lower than the integrity risk requirement to compute the PL of the fixed position.
Our previous work by Min et al. (2023) utilizes measurements collected over multiple epochs by implementing a Kalman filter to meet the PIF constraint. The rationale behind multi-epoch filtering techniques is that ambiguities remain constant over the filter duration (i.e., no cycle slips occur). Under this assumption, as the filtering time increases, the accuracy of ambiguity estimations improves, leading to a decrease in PIF. Thus, the PIF constraint can be satisfied by allowing a sufficiently long initial filtering period until the PIF becomes smaller than the integrity risk requirement. During this initial filtering period, the float position solution—the position solution before fixing the ambiguity—must be used instead for the integrity purpose. However, since the PL of the float position is much greater than that of the fixed position, the system's availability is degraded during the initial filtering period. This problem is exacerbated in urban areas where filter reinitializations are frequent due to frequent cycle slips.
Determining the use of the fixed position solution based on the PIF is termed "model-based validation." This notation stems from the fact that the PIF is entirely computed from filter covariance matrices, which are derived from statistical models of measurement errors. In other words, the decision to use fixed position solutions is based not on the actual measurements used to estimate the ambiguity solutions, but on a priori information incorporated into the filter. This model-based ambiguity validation method might unnecessarily constrain users to use the float position solution, as the measurement error models do not reflect the quality of current actual measurements. In practice, it might happen that while the quality of current measurements is high enough to yield a very accurate ambiguity solution, but the PIF is still large compared to the integrity risk requirement. Furthermore, in safety-critical systems where PIF must be conservatively assessed, this model-based validation may unnecessarily prolong the initial convergence time due to conservative measurement modeling (i.e., measurement overbounding).
In this context, this paper aims to reduce the initial filtering duration required for the fixed position in our SS-RAIM-based CDGNSS architecture. We employ a "measurement-based validation" test, such as a ratio test, to determine the use of the fixed position. These tests base their decisions on computed ambiguity estimates rather than filter covariance matrices. They formulate test statistics from float-valued and integer-valued ambiguity estimates and compare these statistics against corresponding thresholds. By adjusting the threshold sizes, these validation tests can control the PIF for the test-passed fixed ambiguity. In this study, the threshold is set to meet the PIF constraint once the ambiguity estimates successfully pass the validation test. Therefore, once the measurement-based validation test is passed, the fixed ambiguity solution satisfies the PIF constraint, allowing the PL of the fixed position solution to be computed. This approach provides an opportunity to utilize the integrity-assured fixed position solution earlier than with the model-based validation test.
Several studies have proposed this class of measurement-based validation tests. The most widely adopted approach is the ratio test, which examines the ratio between two differences: (1) the float-valued ambiguity solution minus the most likely integer-valued ambiguity solution, and (2) the float-valued ambiguity solution minus the second most likely integer-valued ambiguity solution. However, the complexity of this test statistic precludes analytical computation of the PIF, making it challenging to determine a threshold that ensures the PIF constraint is met. In response to this limitation, the Integer Aperture Bootstrapping (IAB) test is a viable option for SS-RAIM-based CDGNSS. The IAB test provides a closed-form expression for the PIF of the test-passed ambiguity solution, enabling exact determination of the threshold that satisfies the PIF constraint. This test evaluates the difference between the decorrelated float ambiguity vector and the integer ambiguity vector. The decorrelated float ambiguity vector comprises components that are fully decorrelated from each other. This decorrelated nature facilitates the analytical computation of the PIF for the ambiguity vector as the product of the PIF of each ambiguity component.
One of the critical distinctions between the model-based and the measurement-based validation tests lies in their impact on the probability distributions of the float and fixed position solutions. In the model-based validation, these distributions remain unchanged before and after the test because the test does not consider the actual current measurements used to estimate the position solutions. In contrast, the measurement-based validation directly evaluates the measurement quality through the ambiguity estimates, consequently affecting the distributions based on test outcomes. This paper derives the changed distributions of the float and fixed position solutions when the test fails and passes, respectively, using Bayes' theorem. The results show that the distribution of the float position solution, given the test fails, is expressed as a weighted sum of normal distributions, whereas it is a unimodal normal distribution before the test. These derived distributions are subsequently utilized in computing the PLs for both float and fixed position solutions in the SS-RAIM-based CDGNSS system.
Performance analyses of the proposed SS-RAIM-based CDGNSS utilizing the IAB test have been conducted. The results indicate that with the IAB tests, the fixed position solution becomes available depending on the ambiguity estimate quality during periods when it would not be possible using the model-based validation test. The opportunity to utilize the fixed position solution increases as the filtering time extends, approaching the initial time required for model-based validation. This is due to the initially poor accuracy of the ambiguity estimates, which improves as measurements continue to be filtered. The proposed algorithm can be used for applications that require high integrity and accuracy, particularly in urban environments.

References
Min, D., Kim, N. M., Kim, J., Lee, J. & Pullen, S. (2023). SS-RAIM Based Integrity Architecture for CDGNSS Systems Against Satellite Measurement Faults. Proceedings of the 36th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2023), Denver, Colorado, September 2023