SS-RAIM-Based Integrity Architecture for CDGNSSs Against Satellite Measurement Faults

Dongchan Min; Noah Minchan Kim; Gihun Nam; Jiyun Lee,; Sam Pullen

doi:10.33012/navi.718

Abstract

Carrier-phase differential global navigation satellite systems (CDGNSSs) present an attractive option for autonomous vehicles that require accurate and safe navigation. The key to high precision in a CDGNSS is resolving integer ambiguities. However, the discrete nature of ambiguities complicates the analysis of position errors in relation to satellite measurement faults, which poses challenges in protection level (PL) calculation. This paper presents an integrity architecture based on solution separation receiver autonomous integrity monitoring. The test statistic for this monitor is defined in the position domain, directly capturing position errors due to faults. This approach facilitates easier and less conservative evaluations of PLs. This paper provides detailed derivations of PLs and monitor thresholds starting from a common definition of integrity and continuity risk. Additionally, this work presents a method for ensuring that PLs reliably bound actual position errors using a measurement overbounding technique. Simulation results show that the monitor detects most faults and that the PLs bound the position errors from undetected faults.

Keywords

1 INTRODUCTION

Carrier-phase differential global navigation satellite systems (CDGNSSs) are a promising approach for autonomous vehicle applications that demand highly accurate and safe navigation (An & Lass, 2023). The key to achieving high precision in a CDGNSS lies in resolving the integer cycle ambiguity of the carrier-phase measurements (Verhagen, 2005). Successful ambiguity resolution allows carrier-phase measurements to serve as extremely accurate range measurements, enabling centimeter-level position accuracy (Teunissen et al., 1997). However, the resulting position can be biased if the ambiguities are incorrectly resolved because of measurement errors. This problem becomes more severe under fault conditions that introduce biases into global navigation satellite system (GNSS) measurements (Khanafseh & Pervan, 2011). Faulted measurements increase the likelihood of incorrect ambiguity resolutions, potentially leading to larger biases in positioning errors. Therefore, for safety-critical applications relying on a 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.

The primary functions necessary to ensure the integrity of a navigation system are providing warnings of faults and evaluating statistical bounds on position error (Pervan et al., 1998). Significant research efforts in the civil aviation community have yielded various approaches for receiver autonomous integrity monitoring (RAIM) (Walter & Enge, 1995; Blanch et al., 2015) and ground/satellite-based augmentation systems (GBASs and SBASs) (Loh et al., 1995; Enge et al., 1996; Braff, 1997; Lee et al., 2006). However, despite their dedicated efforts, faults of smaller magnitudes may remain undetected because of nominal measurement errors corrupting monitor test statistics. To protect users against these undetected faults, a statistical bound or confidence interval known as a protection level (PL) is computed. PLs are designed to bound not only the position errors caused by nominal measurement errors but also those arising from undetected faults (Joerger et al., 2014). Real-time PLs computed for each position axis are then compared with tolerable error limits (“alert limits”) of specific applications to assess availability, which is defined as the fraction of time during which outputs of the navigation system can be safely used. Thus, integrity depends on deriving PLs that adequately bound position errors caused by undetected faults.

Unlike nominal measurement errors, whose distributions can be reliably modeled via extensive experimental data, faults pose a challenge for observation because of their low probability of occurrence. To avoid assumptions on unknown fault distributions, PLs for aviation are typically evaluated as those corresponding to “worst-case” fault scenarios (Joerger & Pervan, 2013). A worst-case fault is defined as the fault causing the largest position errors among those not detected by the fault monitors. The magnitude of this fault is determined by both the navigation and fault monitor algorithms. Therefore, a comprehensive analysis of how faults affect position errors is essential to identify worst-case faults and to compute PLs accordingly.

Faults in measurements are propagated into position errors through the CDGNSS navigation algorithm. The navigation algorithm consists of three steps (Teunissen, 1997). First, position and ambiguity estimates are obtained by using a conventional estimator, such as a Kalman filter or a weighted least-squares (WLS) filter. These estimates ignore the integer nature of the ambiguities and are referred to as float positions or float ambiguity solutions. Second, float ambiguities (â) are converted into integer values known as fixed ambiguities (ǎ). Various mapping methods, such as integer least squares (Teunissen, 1999a), integer bootstrapping (Teunissen et al., 2021), and integer rounding (Teunissen, 1998), have been introduced in the literature. Owing to the discrete nature of the integer space, this mapping is not a one-to-one process but a many-to-one mapping. Third, the float position is readjusted based on the ambiguity residual (â – ǎ) to obtain more accurate position estimates known as fixed positions. Therefore, faults in measurements sequentially propagate into the float position and fixed ambiguity, and their cumulative effects propagate to the integer-fixed position.

Biases in integer-fixed positions due to faults are calculated as the sum of biases on the float position and biases induced by the ambiguity residual. These biases have distinctly different characteristics in relation to the fault magnitudes. Float position biases are proportional to fault magnitudes because both the Kalman and WLS filters are linear estimators (Verhagen & Teunissen, 2017). In contrast, biases induced by ambiguity residuals are linear but constrained within a specific range (Teunissen, 2002). For instance, for a one-dimensional case in which the integer rounding method is employed for ambiguity mapping, the ambiguity residual consistently remains within a range of [–0.5, 0.5] regardless of how large the fault magnitude is. The ambiguity residual linearly increases within this range as the fault magnitude grows. However, once the ambiguity residual reaches its upper limit, the residual cycles back to its lower limit as the fault magnitude further increases, maintaining its range-bound characteristic. Therefore, as the fault magnitude increases, the integer-fixed position bias globally increases because of the float position bias while locally experiencing periodic rises and falls because of the bias induced by the ambiguity residual. This characteristic of the integer-fixed position bias complicates the determination of the worst-case fault, thereby making PL computation challenging in CDGNSSs.

Khanafseh et al. (2012) addressed this problem in a highly conservative manner. The authors proposed computing the maxima of the float position bias and the bias induced by the ambiguity residual separately. These maximum values are computed by using the maximum fault size that the fault monitor cannot detect. The integer-fixed position bias is then computed as the sum of these maximum biases. This method is applicable in the case of monitors that limit the magnitude of faults in the range domain, such as residual-based RAIM and GBAS/SBAS monitors. However, in practice, the faults that maximize the float position bias and those that induce the maximum ambiguity residual bias differ. Therefore, this method results in a significantly increased conservatism in PL computation, which can potentially reduce system availability.

This limitation can be resolved by utilizing solution separation RAIM (SS-RAIM), which directly captures integer-fixed position errors resulting from faults. SS-RAIM was originally developed for code-based GNSSs (Brenner, 1996; Joerger et al., 2014). If SS-RAIM is applied to carrier-based systems, the detection statistics of SS-RAIM are defined as the differences between fault-free subset integer-fixed positions and the all-in-view integer-fixed position. The all-in-view position is computed by using all available measurements in the main filter. The fault-free subset position under the fault hypothesis in which the i-th satellite is faulted is obtained by the sub-filter that excludes measurements from the i-th satellite. Under fault conditions, the all-in-view integer-fixed position is affected by the fault, whereas the fault-free subset solutions remain unaffected. This feature enables the test statistic of SS-RAIM to directly capture the integer-fixed position bias. When this test statistic exceeds the monitor threshold, the monitor triggers an alert to the user, indicating that the navigation system should not be used. Thus, SS-RAIM ensures that when no alert is triggered by the monitor, the integer-fixed position bias caused by an undetected fault remains smaller than the monitor thresholds. Employing these thresholds enables less conservative and simpler PL computations compared with methods based upon range-domain monitoring.

Khanafseh et al. (2013) and Khanafseh and Pervan (2011) utilized the solution separation (SS) method to detect reference receiver faults and evaluate integrity and continuity risks for carrier-phase-based GBAS-like systems. They presented a method that incorporates the effect of incorrect integer fixes in PLs and monitors threshold computation by assuming that incorrect fixes always lead to integrity or continuity loss. In prior work by El-Mowafy and Kubo (2017, 2018), the SS method was applied to an integrated system comprised of real-time kinematics, an inertial measurement unit, and an odometer. However, the authors did not include the effect of incorrect fixes in their monitor threshold calculations. Furthermore, prior work by Wang et al. (2020) and Wang et al. (2022) assumed that the probability of incorrect ambiguity resolution is negligible after internal ambiguity validation tests (e.g., ratio tests) have been passed. This assumption is not strictly true. The ratio test only assesses the closeness of a float solution to its nearest integer, not the accuracy of the integer ambiguity (Teunissen & Verhagen, 2009). Although Wang et al. (2023) quantified the impact of incorrect fixes due to nominal measurement errors in their PL equation, they left a consideration of this impact on test statistics to future work.

The other key concern for CDGNSS integrity is incorrect ambiguity resolution caused by nominal measurement errors, which are typically modeled by zero-mean normal distributions. Khanafseh and Pervan (2010), Khanafseh and Langel (2011), Pervan et al. (2003), and Green and Humphreys (2019) proposed integrity algorithms to mitigate incorrect ambiguity resolution under fault-free conditions. These approaches ensure integrity by bounding the probability of an incorrect fix to the required integrity risk or less. This bounding is achieved by either extending the filter time or employing partial fixing methods.

This paper presents an SS-RAIM-based integrity architecture for CDGNSSs against satellite measurement faults. This work addresses the concerns arising from ambiguity resolution, emphasizing three key aspects. First, a complete step-by-step derivation of the PL, which is currently missing in the literature, is provided, starting from a common definition of the integrity risk. Second, the distribution of SS-based test statistics is investigated to derive their mean and variance, which are used to determine the monitor threshold in a practical manner. This paper demonstrates that the conditional distribution of these test statistics, given fixed ambiguities, follows a normal distribution and that their variances can be calculated as in code-based SS-RAIM systems. Third, a method for ensuring that the PL reliably bounds the actual position error is presented. PLs are computed from the filter covariance matrices of position and ambiguity errors. These filter covariance matrices do not correspond to true covariances; thus, the use of the filter covariances might underestimate the integrity risk. This problem arises in all aforementioned previous work on CDGNSS integrity. Langel et al. (2016) addressed this problem through an optimization framework, focusing only on fault-free integrity risk without considering measurement fault scenarios. However, this approach is computationally intensive to use in the SS-RAIM architecture because the optimization problem should be implemented in fault hypotheses as well as the fault-free hypothesis. This limitation is resolved in the present work by utilizing an overbounding method originally developed for code-based systems and by assuming that any incorrect fixes result in integrity losses.

Section 2 provides a background on integer-fixed position, with a review of its error distribution. Section 3 introduces the integrity architecture with a derivation of PLs. Section 4 examines the distribution of the monitor test statistic and derives the monitor threshold from the continuity risk equation. Section 5 presents an integrity risk bounding method for reliable PL computation. In Section 6, simulation results show the monitor’s ability to detect faults and the PL’s ability to bound position errors, even in the presence of undetected faults. Section 7 concludes with findings and recommendations for future research.

2 INTEGER-FIXED POSITION ESTIMATION

2.1 Navigation Algorithm

Figure 1 presents a navigation algorithm for a CDGNSS. The raw measurements of the user and reference receivers are fed into the navigation filter. In this study, a Kalman filter is used as the navigation filter. The code and carrier measurement are processed to compute the float baseline solution $({\hat{b}}_{0} (k))$ and float double-differenced ambiguity solution $({\hat{a}}_{0} (k))$ between the user and the reference station, along with their covariance matrices, $Q_{{\hat{b}}_{0}} (k)$ and $Q_{{\hat{a}}_{0}} (k)$ , respectively, at epoch k Teunissen, 2017). The subscript of 0 in these estimates denotes computation using measurements from all available satellites. The navigation filter estimates the float ambiguity at each epoch, and the updated float ambiguity is fixed in an ambiguity mapping step, as shown in Figure 1. To prevent potential error propagation due to incorrect ambiguity resolution, the fixed ambiguity is not fed back into the navigation filter. For simplicity, the epoch notation k is omitted throughout the remainder of this paper unless explicitly specified. Once the position of the reference receiver is known, the baseline vector can be transformed into the user’s position. These estimates are the solutions obtained by ignoring the integer nature of the ambiguities. The integer constraints are exploited during the ambiguity resolution step to obtain a better estimate of the position. This step involves mapping ambiguity solutions from the space of real numbers to the space of integer numbers, converting ${\hat{a}}_{0}$ to corresponding integer values $({\overset{ˇ}{a}}_{0})$ . Several integer mapping methods exist, including integer rounding (Teunissen, 1998) and integer least squares (Teunissen, 1999a), but this work uses integer bootstrapping (IB). IB is a popular method that combines conditional least-squares estimation with rounding to sequentially estimate the components of the integer vector (Teunissen, 2021). IB is straightforward to implement and has a closed-form expression for the probability of successful integer resolution. Once ${\overset{ˇ}{a}}_{0}$ is obtained, the ambiguity residual (i.e., ${\hat{a}}_{0} - {\overset{ˇ}{a}}_{0}$ ) is used to readjust ${\hat{b}}_{0}$ and obtain the so-called fixed position $({\overset{ˇ}{b}}_{0})$ solution (Teunissen, 2017):

FIGURE 1

Schematic diagram of CDGNSS navigation algorithm

${\overset{ˇ}{b}}_{0} = {\hat{b}}_{0} - Q_{{\hat{b}}_{0} {\hat{a}}_{0}} Q_{{\hat{a}}_{0}}^{- 1} ({\hat{a}}_{0} - {\overset{ˇ}{a}}_{0})$ 1

where $Q_{{\hat{b}}_{0} {\hat{a}}_{0}}$ is a correlation matrix between ${\hat{a}}_{0}$ and ${\hat{b}}_{0}$ . The quality of ${\overset{ˇ}{b}}_{0}$ aligns with the high precision of the carrier-phase data, provided that the probability of ${\overset{ˇ}{a}}_{0}$ being the correct integer is sufficiently high (Teunissen, 2017). Finally, ${\overset{ˇ}{b}}_{0}$ is used as the navigation solution unless a fault is detected by the fault monitor. The fault monitor utilizes SS-based detection statistics as detailed in Section 3.

2.2 Distribution of the Integer-Fixed Position Solution

In this section, a brief review of the distribution of the integer-fixed position solution is given based on work by Teunissen (2002). In most GNSS applications utilizing the integer-fixed position, it is commonly assumed that the fixed ambiguity is deterministic and that the integer-fixed position follows a normal distribution. Theoretically, this is not correct, as noted by Teunissen (2002). The fixed ambiguity is not deterministic; rather, it is random, as it is derived from measurements containing random errors. Assigning a normal distribution for the integer-fixed position solution, in fact, neglects the random nature of the fixed ambiguity. In practice, because of the randomness of the fixed ambiguity, the integer-fixed position solution follows a multi-modal distribution. Teunissen (2002) derived the probability distribution function (PDF) of the integer-fixed position $(f_{{\hat{b}}_{0}})$ as follows:

$f_{{\overset{ˇ}{b}}_{0}} (x) = \sum_{z_{0} \in ℤ^{n_{a_{0}}}} f_{{\hat{b}}_{0} ({\hat{a}}_{0})} (x (z_{0})) \cdot P ({\overset{ˇ}{a}}_{0} = z_{0})$ 2

where $f_{{\hat{b}}_{0} | {\hat{a}}_{0}} (x | z_{0})$ is a conditional PDF of ${\hat{b}}_{0} = x$ given ${\hat{a}}_{0} = z_{0}$ . $n_{a_{0}}$ is a dimension of ${\overset{ˇ}{a}}_{0}$ and ${\hat{a}}_{0}$ . $ℤ^{n_{a_{0}}}$ is an n_a₀ -dimensional space of integers, and z₀ is an arbitrary integer in $ℤ^{n_{a_{0}}}$ . P(·) is a probability of an event. $P ({\overset{ˇ}{a}}_{0} = z_{0})$ is a probability of ${\overset{ˇ}{a}}_{0} = z_{0}$ , as detailed by Teunissen (2001). Because the float position and float ambiguity are jointly and normally distributed, $f_{{\hat{b}}_{0} | {\hat{a}}_{0}}$ is a normal distribution with the following mean $(μ_{{\hat{b}}_{0} | {\hat{a}}_{0} = z_{0}})$ and covariance matrix $(Q_{{\hat{b}}_{0} | {\hat{a}}_{0}})$ (Teunissen, 2002):

$\begin{matrix} μ_{{\hat{b}}_{0} ({\hat{a}}_{0} = z_{0})} = b + Q_{{\hat{b}}_{0} {\hat{a}}_{0}} Q_{{\hat{a}}_{0}}^{- 1} (z_{0} - a_{0}) \\ Q_{{\hat{b}}_{0} ({\hat{a}}_{0})} = Q_{{\hat{b}}_{0}} - Q_{{\hat{b}}_{0} {\hat{a}}_{0}} Q_{{\hat{a}}_{0}}^{- 1} Q_{{\hat{b}}_{0} {\hat{a}}_{0}}^{T} \end{matrix}$ 3

where b is a true position vector and a₀ is a true ambiguity vector. $μ_{{\hat{b}}_{0} | {\hat{a}}_{0} = z_{0}}$ depends on z₀ whereas $Q_{{\hat{b}}_{0} | {\hat{a}}_{0}}$ does not. Equations (2) and (3) demonstrate that the multi-modal nature of the marginal ${\overset{ˇ}{b}}_{0}$ can be interpreted as a result of the potential biases in ${\overset{ˇ}{b}}_{0}$ when ${\overset{ˇ}{a}}_{0}$ is incorrectly fixed. The magnitudes of these biases are dependent on the discrete errors in ${\overset{ˇ}{a}}_{0}$ .

3 INTEGRITY ARCHITECTURE

3.1 Fault Monitoring Using SS-RAIM

A fault is defined as a deterministic but unknown bias on measurements arising from various causes, including signal-in-space malfunctions or irregular atmospheric behavior. These satellite measurement faults result in frequent incorrect ambiguity resolutions and lead to biases on ${\overset{ˇ}{b}}_{0}$ . To guarantee the integrity of ${\overset{ˇ}{b}}_{0}$ against these faults, a fault list to be monitored should be established. This list consists of H_i hypotheses for which the i-th subset of all-in-view satellites is faulted, where i ∈ {0, …, h}, determined based on a prior probability of satellite fault and an integrity risk requirement (Blanch et al., 2015). The subscript 0 is used to designate the fault-free hypothesis (H₀). This paper uses SS-based detection statistics (Δ_i) to detect the H_i_≠0 faults, which are defined as follows:

$Δ_{i} = {\overset{ˇ}{b}}_{i} - {\overset{ˇ}{b}}_{0}$ 4

The subscript i identifies the fault hypothesis of H_i and designates its corresponding fault-free solution $({\overset{ˇ}{b}}_{i})$ . ${\overset{ˇ}{b}}_{i}$ is obtained from a sub-Kalman filter that does not use the measurements of the i-th subset satellite, as shown in Figure 2. Although all ${\hat{b}}_{i}$ for i ∈ {0, 1, …, h} are estimating the same true position (b), they differ in which measurements they use. It is important to note that the subscript 0 indicates the fault-free hypothesis (H₀) as well as the all-in-view navigation solution $({\overset{ˇ}{b}}_{0})$ . ${\overset{ˇ}{b}}_{0}$ is computed under all hypotheses and is not fault-free under H_i for i ≠ 0. Therefore, under H_i for i ≠ 0, Δ_i directly captures the fault-induced position bias, which includes the impact of incorrect fixes due to the fault. The monitor issues an alert when any Δ_i for i ∈ {0, 1, …, h} exceeds its corresponding predefined threshold (T_i). T_i is determined from the continuity risk allocated to the H₀ hypothesis detailed in Section 4.

FIGURE 2

Parallel filters for SS-RAIM

Although the monitor is carefully designed, it occasionally fails to detect faults due to nominal measurement errors, such as thermal noise, even when faults are present in the measurements. These undetected faults might introduce significant ambiguity resolution errors, resulting in substantial position errors and ultimately posing a threat to integrity. Therefore, a PL for the all-in-view navigation solution is computed to protect users against these undetected faults. The impact of an undetected fault on ambiguity resolution should be considered in the PL computation. However, quantifying this impact without specific knowledge of the fault (e.g., its magnitude and direction) is complex because of the discrete nature of the ambiguity resolution process. Thus, this paper derives the PL using the fault-free subset solution $({\hat{b}}_{i})$ , which remains unaffected by the fault under the H_i hypothesis, and using T_i, which is the threshold used to detect the fault, to avoid directly quantifying the impact of undetected faults on ${\hat{b}}_{0}$ . Further details are provided in the following section.

3.2 Integrity Risk and PL

Integrity is related to the trust that can be placed in the information provided by the navigation system. The integrity risk is defined as the probability that the position error exceeds an error bound without a notice from the fault monitor (Pullen, 2011). The error bound that satisfies the integrity risk requirement is referred to as the PL. In the proposed architecture, when none of the Δ_is for i ∈ {0, 1, …, h} exceed the thresholds, no alert is issued by the monitor. Thus, the vertical PL (VPL) is defined as follows (Joerger et al., 2014):

$\sum_{i = 0}^{h} P (({\overset{ˇ}{b}}_{0, v} - b_{v}) > VPL, \cap_{j = 1}^{h} (Δ_{j, v}) < T_{j, v} (H_{i})) \cdot P_{H_{i}} = I_{req, v}^{*} - P_{NM} = I_{req, v}$ 5

where the subscript v represents the vertical component; for instance b_v is a true vertical position. P_{H_i} is the a priori probability of $H_{i} . I_{req, v}^{*}$ is the integrity risk requirement allocated to the vertical direction. P_NM is the summation of the prior probabilities of the fault modes that are “not monitored” (i.e., not included in the fault list {1, …, h}). This paper focuses on the vertical direction, but this approach can be easily extended to the horizontal direction.

Directly computing the VPL satisfying I_req,v in Equation (5) is challenging because of the complex distributions of ${\overset{ˇ}{b}}_{0, v}$ and Δ_j,v. The remainder of this section derives an equation that enables a practical evaluation of Equation (5), although it is conservative in nature. This approach is achieved by employing a triangular inequality commonly used for code-based systems and by making conservative assumptions regarding the impact of incorrect fixes.

Equation (5) necessitates a calculation of the integrity risk under both fault-free and fault conditions. Evaluating the integrity risk under fault conditions presents a challenge due to insufficient information about the fault. The given condition H_i provides information about which satellites are faulted, but does not specify the magnitude and direction of the fault. This limitation introduces complications in determining the distribution of ${\overset{ˇ}{b}}_{0, v}$ and Δ_j,v. Thus, an upper bound is used for the term inside the summation operator in Equation (5). This upper bound is derived via a triangle inequality (Joerger et al., 2014):

$P (({\overset{ˇ}{b}}_{0, v} - b_{v}) > VPL, \cap_{j = 1}^{h} (Δ_{j, v}) < T_{j, v} (H_{i})) < (P ({\overset{ˇ}{b}}_{i, v} - b_{v}) + T_{i, v} > VPL (H_{i}))$ 6

The upper bound probability can be computed without a knowledge of the fault because ${\overset{ˇ}{b}}_{i}$ is a fault-free solution under the H_i hypothesis. This upper bound is established without any assumptions on the distribution of navigation solutions and detection statistics.

As demonstrated in Section 2.2, ${\overset{ˇ}{b}}_{i}$ follows a multi-modal distribution, which makes the direct evaluation of Equation (6) challenging. Teunissen (2002) derived the distribution of ${\overset{ˇ}{b}}_{i}$ as a weighted sum of the conditional distributions of the float position solutions, as given in Equation (2). Therefore, Equation (6) can be expanded as follows:

$P (({\overset{ˇ}{b}}_{i, v} - b_{v}) + T_{i, v} > (VPL) H_{i}) = \sum_{z_{i} \in ℤ^{n_{a_{i}}}} P (({\hat{b}}_{i, v} - b_{v}) + T_{i, v} > VPL ({\hat{a}}_{i}) = z_{i}, H_{i}) \cdot P ({\overset{ˇ}{a}}_{i} = z_{i} {(H)}_{i})$ 7

where n_{a_i} is the number of elements of ${\overset{ˇ}{a}}_{i}$ . The event in which z_i equals the true ambiguity (a_i) refers to a correct fix, and $P ({\overset{ˇ}{a}}_{i} = a_{i} | H_{i})$ represents the probability of a correct fix.

In Equation (7), because the set of all possible z_i constitutes a countably infinite set, exact evaluation is impossible. Therefore, the conservative approach used by Khanafseh et al. (2013), where any incorrect ambiguity resolution results in a loss of integrity, is applied, which leads to $P (({\hat{b}}_{i, v} - b_{v}) + T_{i, v} > VPL ({\hat{a}}_{i} = z_{i}, H_{i})) = 1$ for z_i ≠ a_i in Equation (7). This approach allows us to account for the impact of incorrect fixes on integrity risk in the most conservative manner, as some z_i (≠ a_i) may induce a small bias on ${\overset{ˇ}{b}}_{i}$ without any severe impact on integrity risk. From this conservative assumption, the following upper bound of Equation (7) can be used to evaluate the integrity risk:

$P (({\overset{ˇ}{b}}_{i, v} - b_{v}) + T_{i, v} > (VPL) H_{i}) < P (({\hat{b}}_{i, v} - b_{v}) + T_{i, v} > VPL ({\hat{a}}_{i}) = a_{i}, H_{i}) \cdot P ({\overset{ˇ}{a}}_{i} = a_{i} {(H)}_{i}) + (1 - P ({\overset{ˇ}{a}}_{i} = a_{i} (H_{i})))$ 8

This upper bound is obtained from $\sum_{z_{i} \in ℤ^{n_{a_{i}}}} P ({\overset{ˇ}{a}}_{i} = z_{i}) = 1$ . The use of the upper bound equation in the integrity risk evaluation results in a larger VPL compared with its exact value.

The purpose behind this conservative assumption is not just to achieve computational efficiency but, more importantly, to establish an upper bound for the integrity risk, even in the absence of precise knowledge about the statistical characteristics of the measurement errors. The integrity risk expressed as Equation (8) is computed by using the filter covariance matrices. To ensure that the integrity risk calculation is reliable, these filter covariance matrices should precisely represent the true covariances. However, a precise representation is impossible because the filter covariance matrices are derived from statistical models of the measurement errors, which are not exactly known either. Therefore, the use of the measurement error model without thorough consideration may lead to an optimistic assessment of integrity risk, potentially resulting in severe consequences in safety. This paper demonstrates that, even without perfect knowledge of the measurement model, the conservative assumption used for Equation (8), when combined with the measurement overbounding method, can provide an upper bound for the integrity risk, as further detailed in Section 5.

By substituting Equation (8) into Equation (5), the final integrity risk equation can be obtained:

$\sum_{i = 0}^{h} (P (({\hat{b}}_{i, v} - b_{v}) + T_{i, v} > (VPL) {\hat{a}}_{i} = a_{i}, H_{i}) \cdot P ({\overset{ˇ}{a}}_{i} = a_{i} (H_{i})) + (1 - P ({\overset{ˇ}{a}}_{i} = a_{i} | H_{i}))) \cdot P_{H_{i}} = I_{req, v}$ 9

where T₀,_v is zero. The term of $1 - P ({\overset{ˇ}{a}}_{i} = a_{i})$ represents the probability of an incorrect fix of ${\overset{ˇ}{a}}_{i}$ . Equation (9) is more computationally tractable than Equation (5) because, given ${\hat{a}}_{i} = a_{i}$ and H_i, the conditional ${\hat{b}}_{i, v}$ follows an unbiased normal distribution. Although several methods are available for identifying the VPL satisfying I_req,_v in Equation (9), this study employs the half-interval search algorithm proposed by Blanch et al. (2015).

In Equation (9), it is important to note that the VPL can be determined only when the following constraint is satisfied:

$\sum_{i = 0}^{h} (1 - P ({\overset{ˇ}{a}}_{i} = a_{i} (H_{i}))) \cdot P_{H_{i}} < I_{req, v}$ 10

This constraint requires that the total incorrect fix probability (which is the sum of contributions of each sub-filter) remains below the integrity risk requirement. Because the incorrect fix probability decreases as the precision of ${\hat{a}}_{i}$ improves, the constraint can be met when the filtering time is sufficiently long. Until this incorrect fix probability decreases sufficiently to satisfy the condition of Equation (10), the user must employ the float solution to ensure integrity.

4 MONITOR DESIGN

4.1 False Alert Risk and Detection Threshold

Continuity risk is defined as the probability of an unexpected interruption in navigation after initiation of an operation. These interruptions can arise from two categories of events: alerts under the fault-free hypothesis (H₀) and alerts under fault hypotheses (H_i_≠0). The probability of alerts under fault hypotheses is primarily determined by the a priori probability of the fault and is, therefore, not controllable. In contrast, the probability of alerts under H₀, known as “false alerts,” can be controlled by adjusting the detection threshold. This section demonstrates how the detection threshold is determined by focusing on the H₀ hypothesis. As with the integrity risk computation approach, it is assumed that any incorrect fixes result in a loss of continuity. This section begins by providing a general definition of false alert risk and subsequently illustrates the influence of these conservative assumptions on the thresholds.

The detection threshold, T_i, is set based on a continuity risk requirement allocated to the fault-free hypothesis (C_req_,0) in order to limit the likelihood of a false alert. In our proposed system, an alert is triggered whenever any Δ_i exceeds its corresponding T_i. Therefore, in the absence of the fault-exclusion algorithm, T_i can be defined from the following false alert risk equation (Joerger et al., 2014):

$P (\cup_{i = 1}^{h} (Δ_{i, v}) > T_{i, v} (H_{0})) \cdot P_{H_{0}} = C_{req, 0, v}$ 11

where Δ_i_,_v is a vertical component of Δ_i, P_H₀ is the a priori probability of H₀, and C_req,0,v is a vertical portion of C_req,0. Directly evaluating the false alert risk poses a challenge because of the correlation among Δ_i for i ∈ {1, …, h}. In practice, Equation (12), which provides an upper bound for Equation (11), is used to determine T_i,_v (Joerger et al., 2014):

$\sum_{i = 1}^{h} P ((Δ_{i, v}) > T_{i, v} (H_{0})) \cdot P_{H_{0}} = C_{req, 0, v}$ 12

This bound allows an individual evaluation of the false alert risk for each test statistic.

To compute the T_i,v that satisfies Equation (12), distributions of Δ_i should be determined. In a CDGNSS, Δ_i follows a multi-modal distribution because both ${\overset{ˇ}{b}}_{i}$ and ${\overset{ˇ}{b}}_{0}$ follow multi-modal distributions. Thus, Equation (12) can be expanded by using the law of total probability, akin to the approach used in Equation (7):

$P ((Δ_{i, v}) > T_{i, v} (H_{0})) = \sum_{z_{i} \in ℤ^{n_{a_{i}}}} \sum_{z_{0} \in ℤ^{n_{a_{0}}}} P ((Δ_{i, v}) > T_{i, v} ({\overset{ˇ}{a}}_{i}) = z_{i}, {\overset{ˇ}{a}}_{0} = z_{0}, H_{0}) \cdot P ({\overset{ˇ}{a}}_{i} = z_{i}, {\overset{ˇ}{a}}_{0} = z_{0} (H_{0}))$ 13

Under the H₀ hypothesis, the conditional Δi, given ${\overset{ˇ}{a}}_{i} = z_{i}, {\overset{ˇ}{a}}_{0} = z_{0}$ , follows a normal distribution with a mean of $μ_{Δ_{i} | {\hat{a}}_{i} = z_{i}, {\hat{a}}_{0} = z_{0}}$ and a covariance matrix of $Q_{Δ_{i} | {\hat{a}}_{i}, {\hat{a}}_{0}}$ . When z_i = a_i and z₀ = a₀, $μ_{Δ_{i} | {\hat{a}}_{i} = z_{i}, {\hat{a}}_{0} = z_{0}}$ is zero. The distribution of the conditional Δi will be detailed in Section 4.2.

An exact computation of Equation (13) is impossible because of the countably infinite set of z_i and z₀. Thus, for a practical evaluation, it is conservatively assumed that any incorrect fix on ${\overset{ˇ}{a}}_{i}$ or ${\overset{ˇ}{a}}_{0}$ results in continuity loss (i.e., $P ((Δ_{i, v}) > T_{i, v} ({\overset{ˇ}{a}}_{i} = z_{i}, {\overset{ˇ}{a}}_{0} = z_{0}, H_{0})) = 1$ for z_i ≠ a_i or z₀ ≠ a₀). This assumption conservatively accounts for the impact of incorrect fixes on continuity risk. From this assumption, an upper bound of Equation (13) is derived by using $\sum_{z_{i} \in ℤ^{n_{a_{i}}}} \sum_{z_{0} \in ℤ^{n_{a_{0}}}} P ({\overset{ˇ}{a}}_{i} = z_{i}, {\overset{ˇ}{a}}_{0} = z_{0} | H_{0}) = 1$ :

$P ((Δ_{i, v}) > T_{i, v} (H_{0})) < P ((Δ_{i, v})) > T_{i, v} ({\overset{ˇ}{a}}_{i}) = a_{i}, {\overset{ˇ}{a}}_{0} = a_{0}, (H_{0}) \cdot P ({\overset{ˇ}{a}}_{i} = a_{i}, {\overset{ˇ}{a}}_{0} = a_{0} (H_{0})) + (1 - P ({\overset{ˇ}{a}}_{i} = a_{i}, {\overset{ˇ}{a}}_{0} = a_{0} (H_{0})))$ 14

By substituting Equation (14) into Equation (12) and considering the fact that the conditional Δ_i given ${\overset{ˇ}{a}}_{i} = a_{i}, {\overset{ˇ}{a}}_{0} = a_{0}$ follows a zero-mean normal distribution under the H₀ hypothesis, T_i,v is determined as follows:

$T_{i, v} = Φ^{- 1} (\frac{\frac{C_{req, 0, i, v}}{P_{H_{0}}} - (1 - P ({\overset{ˇ}{a}}_{i} = a_{i}, {\overset{ˇ}{a}}_{0} = a_{0} (H_{0})))}{P ({\overset{ˇ}{a}}_{i} = a_{i}, {\overset{ˇ}{a}}_{0} = a_{0} (H_{0}))}) \cdot σ_{Δ_{i, v} ({\hat{a}}_{i}, {\hat{a}}_{0})}$ 15

where Φ^-1(·) is the inverse tail probability distribution of the two-tailed standard normal distribution (i.e., Φ(·) = 1 - normcdf(·), where normcdf(·) is the standard normal cumulative distribution function). $σ_{Δ_{i, v} | {\hat{a}}_{i}, {\hat{a}}_{0}}$ represents a standard deviation of the conditional Δ_i,v and is computed as the square root of the vertical component of $Q_{Δ_{i} | {\hat{a}}_{i}, {\hat{a}}_{0}}$ . $C_{req, 0, i, v}$ is an allocated probability of C_req,0,v for T_i,v. In this paper, we equally allocate C_req,0,v to each T_i,v (i.e., $C_{req, 0, i, v} = C_{req, 0, v} / h$ ), although any other arbitrary allocation that ensures that the continuity risk requirement is met can be chosen. It is important to note from Equation (15) that T_i,v can be determined only when the term of $1 - P ({\overset{ˇ}{a}}_{i} = a_{i}, {\overset{ˇ}{a}}_{0} = a_{0} | H_{0})$ is smaller than $C_{req, 0, i, v} / P_{H_{0}}$ . Therefore, as was done in Equation (10), the user must employ the float position solution until this probability decreases sufficiently to be smaller than $C_{req, 0, i, v} / P_{H_{0}}$ .

In Equation (15), $P ({\overset{ˇ}{a}}_{i} = a_{i}, {\overset{ˇ}{a}}_{0} = a_{0} | H_{0})$ should be computed from the joint distribution of ${\hat{a}}_{i}$ and ${\hat{a}}_{0}$ to consider their correlation. This calculation, however, may be cumbersome. Therefore, a lower bound for $P ({\overset{ˇ}{a}}_{i} = a_{i}, {\overset{ˇ}{a}}_{0} = a_{0} | H_{0})$ is used for the computation of this joint probability (Khanafseh et al., 2013):

$P ({\overset{ˇ}{a}}_{i} = a_{i}, {\overset{ˇ}{a}}_{0} = a_{0} (H_{0})) \geq P ({\overset{ˇ}{a}}_{i} = a_{i} (H_{0})) \cdot P ({\overset{ˇ}{a}}_{0} = a_{0} (H_{0}))$ 16

The use of this lower bound results in a conservative assessment of continuity risk, leading to a larger T_i.

4.2 Distribution of Conditional Δ_i Under H₀

This section demonstrates that, under the H₀ hypothesis, the conditional Δ_i given ${\overset{ˇ}{a}}_{i} = a_{i}$ and ${\overset{ˇ}{a}}_{0} = a_{0}$ follows a zero-mean normal distribution and that its covariance matrix can be simply computed from the covariance matrices of the conditional all-in-view float position and the conditional fault-free float position. For a general demonstration, a distribution of the conditional Δ_i given ${\overset{ˇ}{a}}_{i} = z_{i}$ and ${\overset{ˇ}{a}}_{0} = z_{0}$ , where z_i and z₀ are arbitrary integer values, is derived.

The conditional Δ_i given ${\overset{ˇ}{a}}_{i} = z_{i}$ and ${\overset{ˇ}{a}}_{0} = z_{0}$ is defined as the difference between the conditional fixed position solutions. Teunissen (2002) has shown that the distribution of a conditional ${\overset{ˇ}{b}}_{i}$ given ${\overset{ˇ}{a}}_{i} = z_{i}$ for i ∈ {0, 1, … h} is identical to that of a conditional ${\hat{b}}_{i}$ given ${\hat{a}}_{i} = z_{i}$ . Therefore, the conditional Δ_i can be expressed as follows:

$\begin{matrix} Δ_{i} |_{{\overset{ˇ}{a}}_{i} = z_{i}, {\overset{ˇ}{a}}_{0} = z_{0}} & = & ({\overset{ˇ}{b}}_{i} - {\overset{ˇ}{b}}_{0}) |_{\overset{ˇ}{a} = z_{i}, {\overset{ˇ}{a}}_{0} = z_{0}} \\ = & ({\hat{b}}_{i} - {\hat{b}}_{0}) |_{\hat{a} = z_{i}, {\hat{a}}_{0} = z_{0}} \end{matrix}$ 17

The conditional ${\hat{b}}_{i}$ . given ${\hat{a}}_{i} = z_{i}$ for i ∈ {0, 1, … h} follows a normal distribution with a mean of $b + Q_{{\hat{b}}_{i} {\hat{a}}_{i}} Q_{{\hat{a}}_{i}}^{- 1} (z_{i} - a_{i})$ under the H₀ hypothesis. Note that the means of all ${\hat{b}}_{i}$ s are identical to b under the H₀ hypothesis. The conditional Δ_i also follows a normal distribution with the following mean:

$Under H_{0} hypothesis μ_{Δ_{i} ({\overset{ˇ}{a}}_{i} = z_{i}, {\overset{ˇ}{a}}_{0} = z_{0})} = Q_{{\hat{b}}_{i} {\hat{a}}_{i}} Q_{{\hat{a}}_{i}}^{- 1} (z_{i} - a_{i}) - Q_{{\hat{b}}_{0} {\hat{a}}_{0}} Q_{{\hat{a}}_{0}}^{- 1} (z_{0} - a_{0})$ 18

$μ_{Δ_{i} | {\overset{ˇ}{a}}_{i} = z_{i}, {\overset{ˇ}{a}}_{0} = z_{0}}$ is zero only when z_i = a_i and z₀ = a₀. The conditional Δ_i given ${\overset{ˇ}{a}}_{i} = a_{i}$ and ${\overset{ˇ}{a}}_{0} = a_{0}$ follows a zero-mean normal distribution under the H₀ hypothesis.

The covariance matrix of the conditional Δ_i should be derived by quantifying the correlation between the conditional ${\hat{b}}_{i}$ and ${\hat{b}}_{0}$ . In code-based systems, it has been proven that the covariance matrix of an SS-based statistic is calculated by subtracting the covariance matrix of ${\hat{b}}_{0}$ from that of ${\hat{b}}_{i}$ . However, this relationship cannot be directly applied to a CDGNSS without mathematical validation for two primary reasons. First, this covariance matrix computation relationship for code-based systems was established for a marginal Δ_i, which follows a normal distribution. In contrast, the covariance matrix of a conditional Δ_i, whose marginal distribution is multi-modal, is needed to compute the threshold for a CDGNSS from Equation (15). Second, CDGNSSs utilize the pivot satellite concept for measurement double-differencing. As the selection criterion for the pivot satellite is at the user’s choice, we choose the satellite with the highest elevation angle. If the pivot satellite used in the main filter is assumed to be faulted and excluded in the sub-filter, an alternative pivot satellite—the one with the second-highest elevation angle—is selected for the sub-filter. This change of the pivot satellite in the sub-filter affects the covariance matrix computation because it changes the measurement combinations. Therefore, the CDGNSS necessitates a new derivation for the covariance matrix of a conditional Δ_i. Appendix A provides a detailed derivation for the following result:

$Q_{Δ_{i} ({\hat{a}}_{i}, {\hat{a}}_{i})} = Q_{{\hat{b}}_{i} ({\hat{a}}_{i})} - Q_{{\hat{b}}_{0} ({\hat{a}}_{0})}$ 19

The derivation proves that the covariance matrix of the conditional Δ_i is computed by subtracting the covariance matrix of the conditional ${\hat{b}}_{0}$ from that of the conditional ${\hat{b}}_{i}$ .

5 POSITION AND INTEGER AMBIGUITY ERROR BOUNDING

The integrity risk, as defined in Section 3.2, is calculated from the filter covariance matrices of the position and ambiguity solutions. These filter covariance matrices are derived from the statistical models of the measurement errors. Thus, the measurement error models must be accurately characterized for a reliable evaluation of the integrity risk. If the measurement models are approximated from the collected data without a careful consideration of their tail probabilities, the integrity risk assessment may be overly optimistic. This optimistic integrity risk assessment may result in an unreliable PL that cannot bound the position errors with the required integrity risk, potentially leading to hazardous consequences in safety-critical applications. This section demonstrates a reliable PL computation method leveraging overbounding methods that have been proposed for code-based GNSSs.

5.1 Review of Overbounding Methods

This subsection reviews overbounding methods developed for code-based systems. The integrity risk in code-based systems is calculated by using a one-dimensional integral of position estimates over a fixed range, where the position error exceeds the PL. For instance, when SS-RAIM is used in a code-based system, the integrity risk under the H_i hypothesis is computed from the following:

$P (({\hat{b}}_{c o d e, i, v} - b_{v}) + T_{c o d e, i, v} > VPL (H_{i}))$ 20

where ${\hat{b}}_{c o d e, i}$ represents a code-based fault-free subset solution under the H_i hypothesis and T_code,i is a threshold of its detection statistic. This probability can only be exactly evaluated when the covariance matrix of ${\hat{b}}_{c o d e, i}$ is precisely known. However, a perfect knowledge of its true covariance matrix is unavailable in practice. In this context, several measurement overbounding methods have been developed to ensure that the evaluated integrity risk from the filter covariance matrices is greater than the exact integrity risk (Rife & Gebre-Egziabher, 2007; Langel et al., 2014; Langel et al., 2020, 2021). This approach requires that the filter covariance matrix of position estimate errors (Q) be greater than the true covariance matrix (∑) in a positive semi-definite sense, i.e., $\sum \underline{≺} Q$ . Langel et al. (2014) derived a condition for $\sum \underline{≺} Q$ , stating that a covariance matrix model of measurement errors $(\bar{R})$ used in the filter should be greater than a true covariance matrix of errors (R), i.e., $\bar{R} \underline{≻} R$ . Furthermore, Langel et al. (2021) proposed a method for determining $\bar{R}$ by investigating measurement errors in the frequency domain, which is referred to frequency domain overbounding. Therefore, a reliable PL computation can be achieved by the use of $\bar{R}$ in the filter, which leads to $\sum \underline{≺} Q$ .

Unlike code-based systems that require the one-dimensional integral only, integrity risk evaluation in a CDGNSS necessitates a multidimensional integral to compute the probability of correct fixes. Moreover, the multidimensional integration region is not constant but is a function of the float ambiguity covariance matrix. These complexities prevent a straightforward interpretation of whether the overbounding method facilitates a conservative integrity risk evaluation in a CDGNSS. From Equation (9), it can be observed that an upper bound for $P (({\hat{b}}_{i, v} - b_{v}) + T_{i, v} > VPL ({\hat{a}}_{i} = z_{i}, H_{i}))$ and a lower bound for $P ({\overset{ˇ}{a}}_{i} = a_{i} | H_{i})$ are needed to provide an upper bound on the integrity risk. The following section demonstrates that the overbounding methods developed for code-based systems ensure upper and lower bounds for each term.

Upper Bound on $P (({\hat{b}}_{i, v} - b_{v}) + T_{i, v} > VPL ({\hat{a}}_{i} = z_{i}, H_{i}))$

In a CDGNSS, the GNSS measurement model is as follows:

$y_{i} = G_{i} b + A_{i} a_{i} + ε_{i}$ 21

where y_i is a fault-free measurement vector under the H_i hypothesis. G_i and A_i are corresponding design matrices, and ε_i is a nominal measurement error vector. The conditional ${\hat{b}}_{i}$ given ${\hat{a}}_{i} = z_{i}$ is obtained by solving a constrained version of this model. Therefore, by constraining the ambiguity vector a_i to z_i, the conditional ${\hat{b}}_{i}$ given ${\hat{a}}_{i} = z_{i}$ can be simply expressed as follows (Teunissen, 1999b):

${\hat{b}}_{i} |_{{\hat{a}}_{i} = z_{i}} = {(G_{i}^{T} Q_{y_{i}}^{- 1} G_{i})}^{- 1} G_{i}^{T} Q_{y_{i}}^{- 1} (y_{i} - A_{i} z_{i})$ 22

where $Q_{y_{i}}^{- 1}$ is a covariance matrix of y_i. A generalized derivation for a Kalman filter can be found in Equation (A–7) in Appendix A.

The form of the conditional ${\hat{b}}_{i}$ in Equation (22) is identical to that of the position solution of code-based systems, except that the measurements are adjusted by z_i. Therefore, once the overbounding method developed for code-based systems is applied in a CDGNSS, the following relationship holds:

$P (({\hat{b}}_{i, v} - b_{v}) + T_{i, v} > VPL {({\hat{a}}_{i} = z)}_{i}, H_{i}) < P_{O B} (({\hat{b}}_{i, v} - b_{v}) + T_{i, v} > VPL {({\hat{a}}_{i} = z_{i}, H)}_{i})$ 23

where P_OB(·) is the probability computed via the covariance matrix derived from the overbounded “code and carrier-phase” measurement error models. Any overbounding technique that guarantees that the filter covariance matrix is larger than the true covariance matrix in a positive semi-definite sense can be used for Equation (23).

5.3 Lower Bound on $P ({\overset{ˇ}{a}}_{i} = a_{i} | H_{i})$

This paper utilizes the IB method to convert the float ambiguity solution into an integer value. This method provides a closed-form expression for the probability of the ambiguity correct fix, $P ({\overset{ˇ}{a}}_{i} = a_{i} | H_{i})$ . This probability is computed by integrating the PDF of the float ambiguity solution over the correct fix pull-in region $(B_{Q_{{\hat{a}}_{i}}, a_{i}})$ (Teunissen, 2006):

$B_{Q_{{\hat{a}}_{i}}, a_{i}} = ((x \in ℝ^{n_{a_{i}}}) (c_{j}^{T} \cdot L_{Q_{{\hat{a}}_{i}}}^{- 1} \cdot (x - a_{i})) \leq \frac{1}{2} f o r j = 1, \dots, n_{a_{i}})$ 24

where $ℝ^{n_{a_{i}}}$ is an $n_{a_{i}}$ -dimensional space of real numbers. $L_{Q_{{\hat{a}}_{i}}}$ is a lower unit triangular matrix, resulting from the LDL decomposition of $Q_{{\hat{a}}_{i}}$ , and c_j is a column vector whose elements are all zero except for the j-th element, which is one. $P ({\overset{ˇ}{a}}_{i} = a_{i} | H_{i})$ can be computed as follows (Teunissen, 2006):

$P ({\overset{ˇ}{a}}_{i} = a_{i} (H_{i})) = \int_{B_{Q_{{\hat{a}}_{i}}, a_{i}}} f_{{\hat{a}}_{i}} (x) d x = \int_{B_{Q_{{\hat{a}}_{i}}, 0}} f_{{\hat{a}}_{i}} (x + a) d x$ 25

where $f_{{\hat{a}}_{i}} (x)$ represents the PDF of ${\hat{a}}_{i}$ under the H_i hypothesis and is a multivariate normal distribution with a mean of a_i and covariance matrix of $Q_{{\hat{a}}_{i}}$ . $f_{{\hat{a}}_{i}} (x + a)$ is the distribution of $f_{{\hat{a}}_{i}} (x)$ shifted by -a_i, resulting in a zero-mean multivariate normal distribution with a covariance matrix of $Q_{{\hat{a}}_{i}}$ . $B_{Q_{{\hat{a}}_{i}}, 0}$ represents a pull-in region centered at a zero vector, determined by substituting a zero vector instead of a_i in Equation (24). Therefore, the correct fix probability can be computed without knowledge of a_i.

In practice, the true covariance matrix of ${\hat{a}}_{i}$ is unknown; only the filter covariance matrix $Q_{{\hat{a}}_{i}}$ is known. In this situation, ${\hat{a}}_{i}$ follows a normal distribution with a true covariance matrix of $\sum_{{\hat{a}}_{i}}$ (unknown), and the pull-in region is determined based on $Q_{{\hat{a}}_{i}}$ . Therefore, the real $P ({\overset{ˇ}{a}}_{i} = a_{i} | H_{i})$ can be expressed as follows (Teunissen, 2006):

$P ({\overset{ˇ}{a}}_{i} = a_{i} (H_{i})) = \int_{B_{Q_{{\hat{a}}_{i}}, 0}} N (0, \sum_{{\hat{a}}_{i}}) d x$ 26

where $N (0, \sum_{{\hat{a}}_{i}})$ is a PDF of a normal distribution with zero mean and a covariance matrix of $\sum_{{\hat{a}}_{i}}$ .

Figure 3 illustrates the difference between the theoretical correct fix probability (i.e., Equation (25)) and the real correct fix probability (i.e., Equation (26)) in the two-dimensional case, taken as an example. Each axis represents each float ambiguity, and the red parallelogram depicts the correct fix pull-in region. The ellipses illustrate the PDF of ${\hat{a}}_{i}$ . It is important to note that although the integrands of Equations (25) and (26) are different, the integration regions are the same, owing to the fact that both regions are defined by the filter covariance matrix $Q_{{\hat{a}}_{i}}$ , as the true covariance matrix $\sum_{{\hat{a}}_{i}}$ is unknown.

FIGURE 3

A theoretical correct fix probability and real correct fix probability

In general, the relation between $Q_{{\hat{a}}_{i}}$ and $\sum_{{\hat{a}}_{i}}$ remains undetermined. However, overbounded measurement methods proposed by Langel et al. (2020, 2021) ensure the relationship of $\sum_{{\hat{a}}_{i}} \underline{≺} Q_{{\hat{a}}_{i}}$ . In this context, Appendix B proves that when $\sum_{{\hat{a}}_{i}} \underline{≺} Q_{{\hat{a}}_{i}}$ , the theoretical correct fix probability is lower than the real probability. Therefore, the use of overbounded measurement error models in the filter guarantees the lower bound of the correct fix probability.

It is essential to note that for z_i where z_i ≠ a_i, the inequalities between the theoretical $P ({\overset{ˇ}{a}}_{i} = z_{i} | H_{i})$ and real $P ({\overset{ˇ}{a}}_{i} = z_{i} | H_{i})$ cannot be established. Appendix B utilizes Anderson’s theorem to prove the inequality between the correct fix probabilities. This theorem is applicable only if the integration region is convex and symmetric around the origin. However, the pull-in regions of z_i where z_i ≠ a_i do not meet the symmetric condition, and the inequalities between the theoretical $P ({\overset{ˇ}{a}}_{i} = z_{i} | H_{i})$ and real $P ({\overset{ˇ}{a}}_{i} = z_{i} | H_{i})$ cannot be established. In this context, the integrity risk under ${\overset{ˇ}{a}}_{i} = z_{i}$ where z_i ≠ a_i must be assumed as one, as was done in Equation (8), to upper bound the integrity risk without an exact knowledge of the measurement error models.

6 PERFORMANCE EVALUATION

6.1 Simulation Conditions

Performance analyses for an illustrative example of a dual-constellation, dual-frequency navigation system have been carried out to evaluate the SS-RAIM-based architecture for a CDGNSS. The integrity risk and fault-free continuity risk requirements are set at 10^-7 and 10^-5, respectively, and an a priori satellite fault probability of 10^-5/satellite is used. The multipath errors for code and carrier-phase measurements are modeled as first-order Gauss–Markov processes. For signals coming from the zenith direction, the standard deviations (σ) of nominal code and carrier multipath errors are set at 1 m and 2 cm, respectively, whereas the standard deviations are set at 2 m and 4 cm for those from an elevation angle of 10º (the mask angle – the minimum elevation allowed). These maximum standard deviations are conservatively set to be greater than the values determined by applying paired Gaussian overbounding methods, as reported by Khanafseh et al. (2018). For elevation angles between 10º and 90º, the standard deviations are generated via the elevation-dependent term of the GBAS Ground Accuracy Designator-B model reported by McGraw et al. (2000). The time constants (τ) for code and carrier multipath errors are set to 30 s, which fall within the range presented by Khanafseh et al. (2018). The ionospheric spatial decorrelation error, proposed to be 4 mm/km for mid-latitude regions by Lee et al. (2012), is applied. The differential tropospheric delays are negligible, based on the assumption that the difference in altitude between the reference station and the user is small. In simulations, it is assumed that the user is moving at a speed of 60 km/h in the Daejeon area of South Korea, located at 36º12’ N latitude and 127º5’ E longitude. Table 1 summarizes the navigation requirements and the simulation condition.

View this table:

TABLE 1

Simulation Parameters

6.2 PL and Monitor Performance

To validate the algorithm developed in this work, simulations were conducted under two scenarios: 1) the fault-free scenario and 2) a fault scenario. Under these two scenarios, the PLs were compared with the all-in view position errors, and the detectability of the monitor was analyzed. The baseline distance between the reference receiver and the user was assumed to be 5 km.

6.2.1 Fault-Free Scenario

As demonstrated in Equations (10) and (15), the user employs the float position solution until the incorrect fix probabilities are sufficiently small to meet the requirements of these equations. Prior to the transition to the fixed ambiguity solution, the VPLs for the float positions were computed by using the SS-RAIM algorithm presented by Joerger et al. (2014). The bottom plot in Figure 4 illustrates the decrease in the incorrect fix probability of ${\overset{ˇ}{a}}_{0}$ over time, resulting from the convergence of the Kalman filter. Once the incorrect fix probabilities satisfy the conditions of Equations (10) and (15), the fixed position VPL is calculated using the algorithm developed in this work. The top plot in Figure 4 illustrates the fixed position VPLs in blue and the float VPLs in red.

FIGURE 4

Simulation results from fault-free scenario. The top plot shows the vertical estimation errors and protection level. The bottom plot shows the incorrect fix probability

To quantitatively validate these VPLs, a Monte Carlo simulation was performed. The simulation consisted of 10,000 runs, each spanning 1,000 s, generating a total of 10 million all-in-view position error samples at a rate of 1 Hz. These simulated vertical position errors are depicted as light gray curves in the top plot of Figure 4. The results demonstrate that none of the vertical position error samples surpassed the VPL, confirming that the derived VPL satisfies the integrity risk requirement.

6.2.2 Fault Scenario

To assess the detectability of the SS-RAIM, we simulated a scenario in which a pivot satellite of the Global Positioning System (GPS) constellation experiences an ephemeris fault. A step-type bias of 2 m was introduced into all measurements (dual-frequency code and carrier phase) from the affected satellite starting at the 400-s mark and continuing until the end of the simulation. Figure 5 illustrates the response of the fixed position solution and the detection statistics under this injected fault condition. The top plot presents the position errors as a dashed black curve and the VPL as a solid blue curve. The bottom plot displays the ratios of detection statistics (Δ_i) to their corresponding thresholds (T_i), with different colors representing each i ∈ {1, …, h}. The ratio corresponding to the faulty satellite is shown by a thick black line. When any curve in the bottom plot exceeds one, the monitor triggers an alert, notifying the user of a fault. Upon injection of the fault, the position errors exhibited a dramatic increase, which manifested as jumps. This behavior is attributed to the discrete nature of mapping float ambiguity to fixed ambiguity. Small differences in float ambiguity can result in substantial differences in fixed ambiguity, leading to step-like increments in position errors. Several detection statistic ratios exceeded unity concurrently with the increase in position errors, prompting the activation of monitor alerts. The instances in which the monitor alert was triggered are represented by red circles in the top plot. Despite the large position errors compared with fault-free cases, the user remains protected from these errors by the prompt monitor alerts.

FIGURE 5

Simulation results when the fault size is 2 m. The top plot shows the vertical estimation errors and protection levels. The bottom plot shows the ratio of detection statistic to corresponding threshold.

Figure 6 illustrates the simulation results obtained when a smaller fault of 20 cm was injected. The other simulation parameters remained the same as those used in Figure 5. Owing to the smaller magnitude of the fault, the position errors and detection statistic ratios increased, but not as dramatically as observed in Figure 5. In some instances, the monitor accurately detected the injected faults; however, in other cases, the monitor failed to detect these faults. In these instances of monitoring failure, the detection statistics did not exceed the thresholds, and the faulted measurements were still utilized in the position estimation. These cases, referred to as missed detections, are indicated by green circles in the top plot of Figure 6. The occurrence of missed detections can be attributed to two primary factors. The first factor is the recursive nature of the Kalman filter. When a fault is initially injected, its initial impact on the Kalman filter estimates may be relatively small. Consequently, the fault might not be immediately detected, resulting in missed detections during the early stages of the fault’s presence. The second factor is the presence of nominal measurement errors. In certain cases, the magnitude of the fault may be comparable to the nominal measurement errors but in the opposite direction. Under such circumstances, the fault’s impact can be partially obscured or attenuated by the inherent noise in the measurements. As a result, distinguishing the fault from nominal measurement errors becomes challenging, leading to missed detections. However, it is important to note that even for instances in which the monitor failed to detect the faults, the integrity of the system was consistently maintained because the PLs bounded the position errors associated with these missed detection cases.

FIGURE 6

Simulation results when the fault size is 20 cm. The top plot shows the vertical estimation errors and protection levels. The bottom plot shows the ratio of detection statistic to corresponding threshold.

7 CONCLUSION

In this paper, an SS-RAIM-based integrity architecture for a CDGNSS was introduced. A PL equation was derived for integer-fixed position solutions that follow a multi-modal distribution. The PL equation was formulated by using conditional float position solutions that follow normal distributions and using correct fix probabilities, assuming that any incorrect fixes lead to a loss of integrity loss. Subsequently, the SS-based monitor was designed, and the distributions of the monitor test statistics were demonstrated. The results show that the variances of these test statistics can be computed by subtracting the covariance matrix of the all-in-view conditional float position from that of the fault-free conditional float position, enabling real-time threshold determination. This study also demonstrated that the integrity risk can be conservatively assessed by using the measurement overbounding method. This approach facilitates a reliable PL computation that conservatively bounds the actual position errors, without requiring a precise knowledge of the measurement error models. The simulation results show that the monitor detects most faults and that the PLs effectively bound the position errors from undetected faults. However, the proposed method requires an initial filtering time to utilize the fixed position solution, as the incorrect fix probabilities must be decreased to satisfy the integrity and continuity risk requirements. Thus, there is a need for future research focused on reducing this initial filtering time by controlling the incorrect fix probabilities through the adoption of data-driven ambiguity validation methods.

HOW TO CITE THIS ARTICLE

Min, D., Kim, N. M., Nam, G., Lee, J., & Pullen, S. (2025). SS-RAIM-Based integrity architecture for CDGNSSs against satellite measurement faults. NAVIGATION, 72(4). https://doi.org/10.33012/navi.718

ACKNOWLEDGMENTS

This paper is a revised version of a previous publication (Min et al., 2023) in the proceedings of the 36th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2023). This research was supported by the Unmanned Vehicles Advanced Core Technology Research and Development Program through the National Research Foundation of Korea (NRF) and the Unmanned Vehicle Advanced Research Center funded by the Ministry of Science and ICT, the Republic of Korea (No. 2020M3C1C1A01086407). This work was also supported by an NRF grant funded by the Korea government (MSIT) (No. RS-2024-00354326).

Appendix A

Derivation Of The Covariance Matrix Of Conditional Δ_i

This appendix provides a derivation of Equation (19). For simplicity of notation, we assume a single constellation and single frequency in this derivation, but the approach can be easily extended to multi-constellation and multi-frequency cases. Additionally, it was assumed that all cycle slips are detected, and upon detection, the filter is re-initialized. In this work, we assumed the use of a Kalman filter for float ambiguity and position estimations. However, the derived equation is also applicable to cases in which batch least-squares estimation methods are used instead of a Kalman filter, because the current time state estimates are identical for a Kalman filter and a batch least-squares estimator, given a time sequence of measurements and state dynamics. This property is used in this appendix to derive Equation (19).

A1 Conditional Float Position Solution Given ${\hat{a}}_{i} = z_{i}$

In this subsection, a conditional float position solution is derived as a function of code and carrier measurements in a closed-form expression. Let ρ_* and ϕ_* be m × 1 vectors of single-differenced (between the receivers) code and carrier measurements, respectively. These measurement vectors are assumed to be normally distributed with covariance matrices of V_ρ* and V_ϕ*. ρ* and ϕ* are pre-multiplied by $V_{ρ *}^{- 1 / 2}$ and $V_{ϕ *}^{- 1 / 2}$ to obtain “normalized” measurements of ρ and ϕ, respectively. Then, for any discrete time k of a time sequence, a linear dynamic system is described by a dynamic equation and a measurement equation:

Dynamic Equation:

$\begin{matrix} b_{k} = b_{k - 1} + w_{b, k} \\ a_{i, k} = a_{i, k - 1} \end{matrix}$ A-1

Measurement Equation:

$\begin{matrix} D_{i}^{T} N_{i}^{T} ρ_{k} = D_{i}^{T} N_{i}^{T} B_{k} b_{k} + D_{i}^{T} N_{i}^{T} ε_{ρ, k} \\ D_{i}^{T} N_{i}^{T} ϕ_{k} = D_{i}^{T} N_{i}^{T} B_{k} b_{k} + Λ_{k} a_{i, k} + D_{i}^{T} N_{i}^{T} ε_{ϕ, k} \end{matrix}$ A-2

where the subscript i represents the fault hypothesis H_i. b is a 3 × 1 position vector. a_i is an (m – n_i – 1) × 1 double-differenced ambiguity vector of the i-th sub-filter, where n_i is the number of faulted satellites under the H_i hypothesis. w_b is a 3 × 1 vector assumed to be normally distributed with a zero mean and a covariance matrix of W_b. W_b is assumed to be a diagonal matrix whose diagonals are infinite, as there is no information for the propagation from b_k–1 to b_k. Therefore, although the constant dynamic equation in Equation (A-1) is utilized, the equation accounts for both dynamic and static users.

ρ and ϕ are m × 1 normalized measurement vectors, and ε_ρ and ε_ϕ are corresponding m × 1 normalized error vectors. These error vectors are normally distributed with zero mean and covariance matrices of I_m, which is an m-dimensional identity matrix. B is a normalized satellite geometry matrix, and Λ is a normalized wavelength matrix. $D_{i}^{T}$ is an (m–n_i – 1) × (m–n_i ) double-differencing matrix utilized for the i-th sub-filter, defined as follows (Teunissen, 1997):

$D_{i}^{T} = (\begin{matrix} I_{p_{i} - 1} & - 1_{(p_{i} - 1) \times 1} & 0_{(p_{i} - 1) \times (m - n_{i} - p_{i})} \\ 0_{(m - n_{i} - p_{i}) \times (p_{i} - 1)} & - 1_{(m - n_{i} - p_{i}) \times 1} & I_{m - n_{i} - p_{i}} \end{matrix})$ A-3

when the p_i -th satellite is used as a pivot satellite. 1_{n × m} and 0_{n × m} are n × m. matrices, with all elements being ones and zeros, respectively. Note that the reason for the inclusion of $D_{i}^{T}$ in Equation (A-2) lies in the fact that it directly shows the impact of the pivot satellite on the position solution. $N_{i}^{T}$ is a partitioning matrix that isolates the fault-free subset of measurements from the entirety of measurements under the H_i hypothesis. Without a loss of generality (given that the order in which measurements are stacked in ρ and ϕ is arbitrary), it is assumed that, under the H_i hypothesis, the faulty measurements are the first n_i elements of ρ and ϕ. Thus, $N_{i}^{T}$ can be expressed as follows:

$N_{i}^{T} = (\begin{matrix} 0_{(m - n_{i}) \times n_{i}} & I_{m - n_{i}} \end{matrix})$ A-4

Given a time sequence of measurements and state dynamics, the current time state estimates of the Kalman filter are identical to those of a batch least-squares filter. The measurements stacked in a batch can be expressed in a single equation, which is much easier to analyze than a Kalman filter:

$(\begin{matrix} 0 \\ 0 \\ D_{i}^{T} N_{i}^{T} ρ_{1} \\ D_{i}^{T} N_{i}^{T} ϕ_{1} \\ ⋮ \\ 0 \\ 0 \\ D_{i}^{T} N_{i}^{T} ρ_{k} \\ D_{i}^{T} N_{i}^{T} ϕ_{1} \end{matrix}) = (\begin{matrix} I & 0 & \dots & 0 & 0 \\ 0 & 0 & \dots & 0 & 0 \\ D_{i}^{T} N_{i}^{T} B_{1} & 0 & \dots & 0 & 0 \\ D_{i}^{T} N_{i}^{T} B_{1} & 0 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & \dots & - I & I \\ 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & \dots & 0 & D_{i}^{T} N_{i}^{T} B_{k} \\ 0 & 0 & \dots & 0 & D_{i}^{T} N_{i}^{T} B_{k} \end{matrix}) (\begin{matrix} b_{1} \\ ⋮ \\ b_{k - 1} \\ b_{k} \end{matrix}) + (\begin{matrix} 0 & 0 & \dots & 0 & 0 \\ I & 0 & \dots & 0 & 0 \\ 0 & 0 & \dots & 0 & 0 \\ Λ_{1} & 0 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & \dots & - I & I \\ 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & \dots & 0 & Λ_{k} \end{matrix}) (\begin{matrix} a_{i, 1} \\ ⋮ \\ a_{i, k - 1} \\ a_{i, k} \end{matrix}) + (\begin{matrix} w_{b, 1} \\ 0 \\ D_{i}^{T} N_{i}^{T} ε_{ρ, 1} \\ D_{i}^{T} N_{i}^{T} ε_{ϕ, 1} \\ ⋮ \\ w_{b, k} \\ 0 \\ D_{i}^{T} N_{i}^{T} ε_{ρ, k} \\ D_{i}^{T} N_{i}^{T} ε_{ϕ, k} \end{matrix})$ A-5

where I and 0, whose dimensions are not unspecified, represent identity and zero matrices of appropriate dimensions, respectively.

In Equation (A-5), because W_b (which is a covariance matrix of w_b) is an infinite covariance matrix, the process equation of b_k does not affect the filter estimate. Furthermore, all a_i,t for t ∈ {1, …, k} are identical. From these two facts, Equation (A-5) can be simplified and rearranged as follows:

$(\begin{matrix} {\bar{D}}_{i}^{T} {\bar{N}}_{i}^{T} \bar{ρ} \\ {\bar{D}}_{i}^{T} {\bar{N}}_{i}^{T} \bar{ϕ} \end{matrix}) = (\begin{matrix} {\bar{D}}_{i}^{T} {\bar{N}}_{i}^{T} \bar{B} \\ {\bar{D}}_{i}^{T} {\bar{N}}_{i}^{T} \bar{B} \end{matrix}) \bar{b} + (\begin{matrix} 0 \\ \bar{Λ} \end{matrix}) a_{i} + (\begin{matrix} {\bar{D}}_{i}^{T} {\bar{N}}_{i}^{T} {\bar{ε}}_{ρ} \\ {\bar{D}}_{i}^{T} {\bar{N}}_{i}^{T} {\bar{ε}}_{ϕ} \end{matrix})$ A-6

where ${\bar{D}}_{i}^{T}$ is blkdiag $(D_{i}^{T})$ , which is a block diagonal matrix in which all diagonals equal $D_{i}^{T}$ . ${\bar{N}}_{i}^{T}$ and ${\bar{B}}_{i}^{T}$ represent blkdiag $(N_{i}^{T})$ and blkdiag $(B_{i}^{T})$ , respectively. $\bar{Λ}$ is ${(\begin{matrix} Λ_{1}^{T} & \dots & Λ_{k}^{T} \end{matrix})}^{T}$ , and $\bar{b}$ is ${(\begin{matrix} b_{1}^{T} & , \dots, & b_{k}^{T} \end{matrix})}^{T}$ . $\bar{ρ}$ and $\bar{ϕ}$ represent ${(\begin{matrix} ρ_{1}^{T} & , \dots, & ρ_{k}^{T} \end{matrix})}^{T}$ and ${(\begin{matrix} ϕ_{1}^{T} & , \dots, & ϕ_{k}^{T} \end{matrix})}^{T}$ , and their corresponding errors are denoted as ${\bar{ε}}_{ρ}$ and ${\bar{ε}}_{ϕ}$ .

The conditional float position solution is identical to the solution obtained by solving Equation (A-6) under the constraint that a_i is set equal to z_i (Teunissen, 1999b). For a given constraint of ${\hat{a}}_{i} = z_{i}$ , the conditional float position solution is as follows:

${\hat{\bar{b}}}_{i} |_{{\hat{a}}_{i} = z_{i}} = Q_{{\hat{\bar{b}}}_{i} | {\hat{a}}_{i}} {(\begin{matrix} {\bar{D}}_{i}^{T} {\bar{N}}_{i}^{T} \bar{B} \\ {\bar{D}}_{i}^{T} {\bar{N}}_{i}^{T} \bar{B} \end{matrix})}^{T} Q_{ε_{i}}^{- 1} ((\begin{matrix} {\bar{D}}_{i}^{T} {\bar{N}}_{i}^{T} \bar{ρ} \\ {\bar{D}}_{i}^{T} {\bar{N}}_{i}^{T} \bar{ϕ} \end{matrix}) - (\begin{matrix} 0 \\ \bar{Λ} \end{matrix}) z_{i})$ A-7

where $Q_{{\bar{ε}}_{i}}^{- 1}$ is the inverse of the covariance matrix of ${\bar{ε}}_{i} = {(\begin{matrix} {\bar{D}}_{i} {\bar{N}}_{i} {\bar{ε}}_{ρ}^{T} & {\bar{D}}_{i} {\bar{N}}_{i} {\bar{ε}}_{ϕ}^{T} \end{matrix})}^{T}$ :

$Q_{{\bar{ε}}_{i}}^{- 1} = (\begin{matrix} {({\bar{D}}_{i}^{T} {\bar{D}}_{i})}^{- 1} & 0 \\ 0 & {({\bar{D}}_{i}^{T} {\bar{D}}_{i})}^{- 1} \end{matrix})$ A-8

The terms regarding $N_{i}^{T}$ are eliminated because $N_{i}^{T}$ . $N_{i}$ is an identity matrix based on the definition of $N_{i}^{T}$ . $Q_{{\hat{\bar{b}}}_{i} ({\hat{a}}_{i})}$ is a covariance matrix of the conditional ${\hat{\bar{b}}}_{i}$ given ${\hat{a}}_{i} = z_{i}$ , expressed as follows:

$Q_{{\hat{\bar{b}}}_{i} ({\hat{a}}_{i})} {({(\begin{matrix} {\bar{D}}_{i}^{T} {\bar{N}}_{i}^{T} \bar{B} \\ {\bar{D}}_{i}^{T} {\bar{N}}_{i}^{T} \bar{B} \end{matrix})}^{T} Q_{ε_{i}}^{- 1} (\begin{matrix} {\bar{D}}_{i}^{T} {\bar{N}}_{i}^{T} \bar{B} \\ {\bar{D}}_{i}^{T} {\bar{N}}_{i}^{T} \bar{B} \end{matrix}))}^{- 1} = \frac{1}{2} {({\bar{B}}^{T} {\bar{N}}_{i} {\bar{P}}_{i} {\bar{N}}_{i}^{T} \bar{B})}^{- 1}$ A-9

where ${\bar{P}}_{i}$ is ${\bar{D}}_{i} {({\bar{D}}_{i}^{T} {\bar{D}}_{i})}^{- 1} {\bar{D}}_{i}^{T}$ . When i = 0, $Q_{{\hat{\bar{b}}}_{i} ({\hat{a}}_{i})}$ can be simplified as $\frac{1}{2} {({\bar{B}}^{T} {\bar{P}}_{i} \bar{B})}^{- 1}$ because ${\bar{N}}_{0} = I$ . Note that from the definition of ${\bar{D}}_{i}$ , ${\bar{P}}_{i}$ can be represented as $I - 1_{n_{{\bar{P}}_{i}} \times 1} {(1_{n_{{\bar{P}}_{i}} \times 1}^{T} \cdot 1_{n_{{\bar{P}}_{i}} \times 1})}^{- 1} 1_{n_{{\bar{P}}_{i}} \times 1}^{T}$ , where $n_{{\bar{P}}_{i}}$ is a dimension of ${\bar{P}}_{i}$ . Thus, ${\bar{P}}_{i}$ is independent of the pivot satellite, even though ${\bar{P}}_{i}$ is defined from ${\bar{D}}_{i}$ .

By substituting Equations (A-8) and (A-9) into Equation (A-7) and rearranging the equation, the error of the conditional ${\hat{\bar{b}}}_{i}$ can be expressed as follows:

$\hat{\bar{b}} |_{{\hat{a}}_{1} = z_{1}} - \bar{b} = \frac{1}{2} S_{i} ({\bar{ε}}_{ρ} + {\bar{ε}}_{ϕ}) + {({\bar{B}}^{T} {\bar{N}}_{i} {\bar{P}}_{i} {\bar{N}}_{i}^{T} \bar{B})}^{- 1} {\bar{B}}^{T} {\bar{N}}_{i} {\bar{P}}_{i} \bar{Λ} (a_{i} - z_{i})$ A-10

where $S_{i} = {({\bar{B}}^{T} {\bar{N}}_{i} {\bar{P}}_{i} {\bar{N}}_{i}^{T} \bar{B})}^{- 1} {\bar{B}}^{T} {\bar{N}}_{i} {\bar{P}}_{i} {\bar{N}}_{i}^{T}$ . In this equation, the first term of the right-hand side, which results from the measurement errors, is a random variate assumed to be normally distributed. The second term, which is induced by the constraint of ${\hat{a}}_{i} = z_{i}$ , is a bias and is zero only when $z_{i} = a_{i}$ (i.e., correct fix). Note that ${\hat{\bar{b}}}_{i}$ is a fault-free float subset solution of $\bar{b}$ under the H_i hypothesis, which comprises the position vectors for times 1 through k. Thus, at time epoch k, the last element of ${\hat{\bar{b}}}_{i}$ is our main focus. It is also important to note that Equations (A-9) and (A-10) show that the conditional ${\hat{\bar{b}}}_{i}$ and its covariance matrix are independent of the choice of pivot satellite.

A2 Covariance Matrix of Conditional Δ_i Given ${\hat{a}}_{i}$ and ${\hat{a}}_{0}$

The conditional SS $({\bar{Δ}}_{i})$ given ${\hat{a}}_{i} = a_{i}$ and ${\hat{a}}_{0} = a_{0}$ can be expressed as follows:

${\bar{Δ}}_{i} |_{{\hat{a}}_{i} = a_{i}, {\hat{a}}_{0} = a_{0}} = {\hat{\bar{b}}}_{i} |_{{\hat{a}}_{i} = a_{i}} - {\hat{\bar{b}}}_{i} |_{{\hat{a}}_{0} = a_{0}} = \frac{1}{2} (S_{i} - S_{0}) ({\bar{ε}}_{ρ} + {\bar{ε}}_{ϕ})$ A-11

Its covariance matrix is as follows:

$Q_{\bar{Δ} ({\hat{a}}_{i}, {\hat{a}}_{0})} = \frac{1}{2} (S_{0} S_{0}^{T} - S_{0} S_{i}^{T} - S_{i} S_{0}^{T} + S_{i} S_{i}^{T})$ A-12

This equation can be obtained by using the fact that ε_ρ and ε_ϕ follow a zero-mean normal distribution with identity covariance matrices.

The first term in Equation (A-12) can be written as follows:

$\begin{matrix} S_{0} S_{0}^{T} & = {({\bar{B}}^{T} {\bar{N}}_{0} {\bar{P}}_{0} {\bar{N}}_{0}^{T} \bar{B})}^{- 1} {\bar{B}}^{T} {\bar{N}}_{0} {\bar{P}}_{0} {\bar{N}}_{0}^{T} \cdot {\bar{N}}_{0} {\bar{P}}_{0}^{T} {\bar{N}}_{0}^{T} \bar{B} {({\bar{B}}^{T} {\bar{N}}_{0} {\bar{P}}_{0} {\bar{N}}_{0}^{T} \bar{B})}^{- T} \\ = {({\bar{B}}^{T} {\bar{P}}_{0} \bar{B})}^{- 1} {\bar{B}}^{T} {\bar{P}}_{0} \cdot {\bar{P}}_{0}^{T} \bar{B} {({\bar{B}}^{T} {\bar{P}}_{0} \bar{B})}^{- T} \\ = {({\bar{B}}^{T} {\bar{P}}_{0} \bar{B})}^{- 1} {\bar{B}}^{T} {\bar{P}}_{0} \bar{B} {({\bar{B}}^{T} {\bar{P}}_{0} \bar{B})}^{- T} \\ = {({\bar{B}}^{T} {\bar{P}}_{0} \bar{B})}^{- 1} \\ = 2 Q_{{\hat{\bar{b}}}_{0} ({\hat{a}}_{0})} \end{matrix}$ A-13

The first equality follows from the definition of S₀, whereas the second follows from ${\bar{N}}_{0} = I$ . The third line follows from ${\bar{P}}_{0} {\bar{P}}_{0}^{T} = {\bar{P}}_{0}$ , which can be easily derived from the definition of ${\bar{P}}_{0}$ . The last equality follows from Equation (A-9). Similarly, the last term in Equation (A-12) is as follows:

$S_{i} S_{i}^{T} = 2 Q_{{\hat{\bar{b}}}_{i} ({\hat{a}}_{i})}$ A-14

The second term in Equation (A-12) can be written as follows:

$\begin{matrix} S_{0} S_{i}^{T} & = {({\bar{B}}^{T} {\bar{P}}_{0} \bar{B})}^{- 1} {\bar{B}}^{T} {\bar{P}}_{0} \cdot {\bar{N}}_{i} {\bar{P}}_{i}^{T} {\bar{N}}_{i}^{T} \bar{B} {({\bar{B}}^{T} {\bar{N}}_{i} {\bar{P}}_{i} {\bar{N}}_{i}^{T} \bar{B})}^{- T} \\ = {({\bar{B}}^{T} {\bar{P}}_{0} \bar{B})}^{- 1} {\bar{B}}^{T} {\bar{N}}_{i} {\bar{P}}_{i} {\bar{N}}_{i}^{T} \bar{B} {({\bar{B}}^{T} {\bar{N}}_{i} {\bar{P}}_{i} {\bar{N}}_{i}^{T} \bar{B})}^{- T} \\ = {({\bar{B}}^{T} {\bar{P}}_{0} \bar{B})}^{- 1} \\ = 2 Q_{{\hat{\bar{b}}}_{0} ({\hat{a}}_{0})} \end{matrix}$ A-15

The first equality follows from the definitions of S₀ and S_i, whereas the second equality follows from ${\bar{P}}_{0} {\bar{N}}_{i} {\bar{P}}_{i}^{T} = {\bar{N}}_{i} {\bar{P}}_{i}$ , which is derived in Appendix A3. The last equality follows from Equation (A-9).

In a similar fashion, the third term in Equation (A-12) becomes the following:

$S_{i} S_{0}^{T} = 2 Q_{{\hat{\bar{b}}}_{0} ({\hat{a}}_{0})}$ A-16

Finally, by substituting Equations (A-13), (A-14), (A-15), and (A-16) into Equation (A-12), we obtain the following relation:

$Q_{{\bar{Δ}}_{i} ({\hat{a}}_{i}, {\hat{a}}_{0})} = Q_{{\hat{\bar{b}}}_{i} ({\hat{a}}_{i})} - Q_{{\hat{\bar{b}}}_{0} ({\hat{a}}_{0})}$ A-17

Because ${\bar{Δ}}_{i}$ is defined by the difference between ${\hat{\bar{b}}}_{i}$ and ${\hat{\bar{b}}}_{0}$ , it contains all SSs for times 1 through k. Δ_i at the current time epoch k is the last element of ${\bar{Δ}}_{i}$ . Thus, based on Equation (A-17), $Q_{Δ_{i} ({\hat{a}}_{i}, {\hat{a}}_{0})} = Q_{{\hat{b}}_{i} ({\hat{a}}_{i})} - Q_{{\hat{b}}_{0} ({\hat{a}}_{0})}$ remains valid.

A3 ${\bar{P}}_{0} {\bar{N}}_{i} {\bar{P}}_{i}^{T} = {\bar{N}}_{i} {\bar{P}}_{i}$

Because ${\bar{P}}_{0}$ , ${\bar{P}}_{i}$ , and ${\bar{N}}_{i}$ are the block diagonal matrices of $P_{0}$ , $P_{i}$ , and $N_{i}$ , respectively, ${\bar{P}}_{0} {\bar{N}}_{i} {\bar{P}}_{i}^{T}$ is blkdiag $(P_{0}, N_{i}, P_{i}^{T})$ ) and ${\bar{N}}_{i} {\bar{P}}_{i}$ is blkdiag $(N_{i}, P_{i})$ . Therefore, we will show that $P_{0} N_{i} P_{i}^{T} = N_{i} P_{i}$ to prove ${\bar{P}}_{0} {\bar{N}}_{i} {\bar{P}}_{i}^{T} = {\bar{N}}_{i} {\bar{P}}_{i}$ .

Based on its definition, P₀ can be expressed as follows:

$\begin{matrix} P_{0} = I - 1_{n_{P_{0}} \times 1} {(1_{n_{P_{0}} \times 1}^{T} 1_{n_{P_{0}} \times 1})}^{- 1} & 1_{n_{P_{0}} \times 1}^{T} = I - \frac{1}{n_{P_{0}}} 1_{n_{P_{0}} \times n_{P_{0}}} \end{matrix}$ A-18

where $n_{P_{0}}$ is the dimension of $P_{0}$ . Note that because I and $1_{n_{p_{0}} \times n_{p_{0}}}$ are symmetric matrices, $P_{0}$ is also a symmetric matrix. With the use of Equation (A-18), $P_{0} N_{i} P_{i}^{T}$ can be expressed as follows:

$\begin{matrix} P_{0} N_{i} P_{i}^{T} & = & (I - \frac{1}{n_{p_{0}}} 1_{n_{p_{0}} \times n_{p_{0}}}) N_{i} P_{i} \\ = & N_{i} P_{i} - \frac{1}{n_{p_{0}}} 1_{n_{p_{0}} \times n_{p_{0}}} N_{i} P_{i} \\ = & N_{i} P_{i} - \frac{1}{n_{p_{0}}} (\begin{matrix} s u m r o w (N_{i} P_{i}) \\ ⋮ \\ s u m r o w (N_{i} P_{i}) \end{matrix}) \end{matrix}$ A-19

where sumrow (A) is a row vector whose i-th element is $\sum_{j = 1}^{n_{A}} A_{j i}$ . Here, n_A is the number of rows of A , and A_ji is the element of A at position (j, i).

From the definition of N_i , N_iP_i can be expressed as follows:

$N_{i} P_{i} = (\begin{matrix} 0_{n_{i} \times n_{P_{i}}} \\ P_{i} \end{matrix})$ A-20

where $n_{p_{i}}$ is the dimension of P_i. Because $P_{i} = I - \frac{1}{n_{P_{i}}} 1_{n_{p_{i}} \times n_{P_{i}}}$ , we can easily see that sumrow (N_i P_i) is a zero vector. Therefore, $P_{0} N_{i} P_{i}^{T} = N_{i} P_{i}$ based on Equation (A-19).

Appendix B

Derivation of the Lower Bound on Correct Fix Probability

This appendix demonstrates that when $Q_{{\hat{a}}_{i}}$ is greater than $\sum_{{\hat{a}}_{i}}$ in a positive semi-definite sense, the theoretical correct fix probability $P ({\overset{ˇ}{a}}_{i} = a_{i} (H_{i}))$ is smaller than the real probability.

From the corollary of Anderson’s theorem (Anderson, 1955), the following statement holds:

if 1) $\sum_{{\hat{a}}_{i}} \leq Q_{{\hat{a}}_{i}}$ in a positive semi – definite sense and

2) $B_{Q_{a_{i}}, 0}$ is a convex set, symmetric about the origin,

$t h e n \int_{B_{Q_{{\hat{a}}_{i}}, 0}} N (0, Q_{{\hat{a}}_{i}}) d x \leq \int_{B_{Q_{{\hat{a}}_{i}}, 0}} N (0, \sum_{{\hat{a}}_{i}}) d x$ B-1

The left term of Equation (B-1) is the theoretical $P ({\overset{ˇ}{a}}_{i} = a_{i} (H_{i}))$ , which is computed from the filter covariance matrix $Q_{{\hat{a}}_{i}}$ , whereas the right term is the real $P ({\overset{ˇ}{a}}_{i} = a_{i} (H_{i}))$ . The remainder of this appendix is dedicated to demonstrating the convexity and symmetry of $B_{Q_{{\hat{a}}_{i}}, 0}$ . Let us recall the definition of $B_{Q_{{\hat{a}}_{i}}, 0}$ in Equation (25).

- Convexity of $B_{Q_{{\hat{a}}_{i}}, 0}$ :

Let us pick two arbitrary points $\begin{matrix} x_{1}, & x_{2} \in B_{Q_{{\hat{a}}_{1}}, 0} \end{matrix}$ and define an interior point $x_{3} = α \cdot x_{1} + (1 - α) \cdot x_{2}$ where α ∈ [0,1]. Then, we have the following:

$\begin{matrix} (c_{j}^{T} \cdot L_{Q_{{\hat{a}}_{i}}}^{- 1} \cdot x_{3}) & = (α \cdot c_{j}^{T} \cdot L_{Q_{{\hat{a}}_{i}}}^{- 1} \cdot x_{1} + (1 - α) \cdot c_{j}^{T} \cdot L_{Q_{{\hat{a}}_{i}}}^{- 1} \cdot x_{2}) \\ \leq α (c_{j}^{T} \cdot L_{Q_{{\hat{a}}_{i}}}^{- 1} \cdot x_{1}) + (1 - α) (c_{j}^{T} \cdot L_{Q_{{\hat{a}}_{i}}}^{- 1} \cdot x_{2}) \\ \leq α \cdot \frac{1}{2} + \frac{(1 - α) 1}{2} \\ \leq \frac{1}{2} \end{matrix}$ B-2

The first equality originates from the definition of x₃. The second inequality follows from the Cauchy-Schwarz inequality, and the third follows from the definition of $\begin{matrix} x_{1}, & x_{2} \in B_{Q_{{\hat{a}}_{1}}, 0} \end{matrix}$ . Therefore, x₃ is in $B_{Q_{{\hat{a}}_{i}}, 0}$ , which implies that $B_{Q_{{\hat{a}}_{i}}, 0}$ is a convex set.

- Symmetry of $B_{Q_{{\hat{a}}_{i}}, 0}$ :

Let us pick an arbitrary point $x \in B_{Q_{{\hat{a}}_{i}}, 0}$ . Then, we have the following:

$\begin{matrix} (c_{j}^{T} \cdot L_{Q_{{\hat{a}}_{i}}}^{- 1} \cdot (- x)) = (c_{j}^{T} \cdot L_{Q_{{\hat{a}}_{i}}}^{- 1} \cdot x) \\ \leq \frac{1}{2} \end{matrix}$ B-3

The second inequality follows from the definition of $x \in B_{Q_{{\hat{a}}_{i}}, 0}$ . Therefore, –x is in $B_{Q_{{\hat{a}}_{i}}, 0}$ , which implies that $B_{Q_{{\hat{a}}_{i}}, 0}$ is symmetric about the origin.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

REFERENCES

↵
An, X., Ziebold, R., & Lass, C. (2023). From RTK to PPP–RTK: Towards real-time kinematic precise point positioning to support autonomous driving of inland waterway vessels. GPS Solutions, 27(2), 86. https://doi.org/10.1007/s10291-023-01428-2
Google Scholar
↵
Anderson, T. W. (1955). The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proceedings of the American Mathematical Society, 6(2), 170–176. https://doi.org/10.2307/2032333
Google Scholar CrossRef
↵
Blanch, J., Walker, T., Enge, P., Lee, Y., Pervan, B., Rippl, M., Spletter, A., & Kropp, V. (2015). Baseline advanced RAIM user algorithm and possible improvements. IEEE Transactions on Aerospace and Electronic Systems, 51(1), 713–732. https://doi.org/10.1109/TAES.2014.130739
Google Scholar
↵
Braff, R. (1997). Description of the FAA’s local area augmentation system (LAAS). NAVIGATION, 44(4), 411–424. https://doi.org/10.1002/j.2161-4296.1997.tb02357.x
Google Scholar
↵
Brenner, M. (1996). Integrated GPS/inertial fault detection availability. NAVIGATION, 43(2), 111–130. https://doi.org/10.1002/j.2161-4296.1996.tb01920.x
Google Scholar
↵
El-Mowafy, A., & Kubo, N. (2017). Integrity monitoring of vehicle positioning in urban environment using RTK-GNSS, IMU and speedometer. Measurement Science and Technology, 28(5), 055102. https://doi.org/10.1088/1361-6501/aa5c66
Google Scholar
↵
El-Mowafy, A., & Kubo, N. (2018). Integrity monitoring for positioning of intelligent transport systems using integrated RTK‐GNSS, IMU and vehicle odometer. IET Intelligent Transport Systems, 12(8), 901–908. https://doi.org/10.1049/iet-its.2018.0106
Google Scholar
↵
Enge, P., Walter, T., Pullen, S., Kee, C., Chao, Y. C., & Tsai, Y. J. (1996). Wide area augmentation of the Global Positioning System. Proceedings of the IEEE, 84(8), 1063–1088. https://doi.org/10.1109/5.533954
Google Scholar
↵
Green, G. N., & Humphreys, T. E. (2019). Data-driven generalized integer aperture bootstrapping for high-integrity positioning. IEEE Transactions on Aerospace and Electronic Systems, 55(2), 757–768. https://doi.org/10.1109/TAES.2018.2864770
Google Scholar
↵
Joerger, M., Chan, F. C., & Pervan, B. (2014). Solution separation versus residual-based RAIM. NAVIGATION, 61(4), 273–291. https://doi.org/10.1002/navi.71
Google Scholar
↵
Joerger, M., & Pervan, B. (2013). Kalman filter-based integrity monitoring against sensor faults. Journal of Guidance, Control, and Dynamics, 36(2), 349–361. https://doi.org/10.2514/1.59480
Google Scholar
↵
Khanafseh, S., Joerger, M., & Pervan, B. (2012). Integrity risk of cycle resolution in the presence of bounded faults. Proc. of the 2012 IEEE/ION Position, Location and Navigation Symposium, Myrtle Beach, SC, 664–672. https://doi.org/10.1109/PLANS.2012.6236941
Google Scholar
↵
Khanafseh, S., Kujur, B., Joerger, M., Walter, T., Pullen, S., Blanch, J., Doherty, K., Norman, L., de Groot, L., & Pervan, B. (2018). GNSS multipath error modelling for automotive applications. Proc. of the 31st International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+ 2018), Miami, FL, 1573–1589. https://doi.org/10.33012/2018.16107
Google Scholar
↵
Khanafseh, S., & Langel, S. (2011). Implementation and experimental validation of cycle ambiguity resolution with position domain integrity risk constraints. NAVIGATION, 58(1), 45–58. https://doi.org/10.1002/j.2161-4296.2011.tb01791.x
Google Scholar
↵
Khanafseh, S., Langel, S., Joerger, M., & Pervan, B. (2013). Evaluating integrity and continuity risks of cycle resolution in the presence of receiver faults. Proc. of the 26th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+ 2013), Nashville, TN, 2583–2591. https://www.ion.org/publications/abstract.cfm?articleID=11231
Google Scholar
↵
Khanafseh, S., & Pervan, B. (2010). New approach for calculating position domain integrity risk for cycle resolution in carrier phase navigation systems. IEEE Transactions on Aerospace and Electronic Systems, 46(1), 296–307. https://doi.org/10.1109/TAES.2010.5417163
Google Scholar
↵
Khanafseh, S., & Pervan, B. (2011). Detection and mitigation of reference receiver faults in differential carrier phase navigation systems. IEEE Transactions on Aerospace and Electronic Systems, 47(4), 2391–2404. https://doi.org/10.1109/TAES.2011.6034640
Google Scholar
↵
Langel, S., Crespillo, O. G., & Joerger, M. (2020). A new approach for modeling correlated Gaussian errors using frequency domain overbounding. Proc. of the 2020 IEEE/ION Position, Location and Navigation Symposium (PLANS), Portland, OR, 868–876. https://doi.org/10.1109/PLANS46316.2020.9110192
Google Scholar
↵
Langel, S., Crespillo, O. G., & Joerger, M. (2021). Overbounding the effect of uncertain Gauss-Markov noise in Kalman filtering. NAVIGATION, 68(2), 259–276. https://doi.org/10.1002/navi.419
Google Scholar
↵
Langel, S., Khanafseh, S., & Pervan, B. (2014). Bounding integrity risk for sequential state estimators with stochastic modeling uncertainty. Journal of Guidance, Control, and Dynamics, 37(1), 36–46. https://doi.org/10.2514/1.62056
Google Scholar
↵
Langel, S., Khanafseh, S., & Pervan, B. (2016). Bounding the integer bootstrapped GNSS baseline’s tail probability in the presence of stochastic uncertainty. Journal of Geodesy, 90(11), 1293–1305. https://doi.org/10.1007/s00190-016-0923-8
Google Scholar
↵
Lee, J., Pullen, S., Datta-Barua, S., & Enge, P. (2012). Assessment of ionosphere spatial decorrelation for Global Positioning System-based aircraft landing systems. Journal of Aircraft, 44(5), 1662–1669. https://doi.org/10.2514/1.28199
Google Scholar
↵
Lee, J., Pullen, S., Enge, P., Pervan, B., & Gratton, L. (2006). Monitoring Global Positioning System satellite orbit errors for aircraft landing systems. Journal of Aircraft, 43(3), 799–808. https://doi.org/10.2514/1.17339
Google Scholar
↵
Loh, R., Wullschleger, V., Elrod, B., Lage, M., & Haas, F. (1995). The U.S. wide-area augmentation system (WAAS). NAVIGATION, 42(3), 432–466.
Google Scholar
↵
McGraw, G. A., Murphy, T., Brenner, M., Pullen, S., & Van Dierendonck, A. J. (2000). Development of the LAAS accuracy models. Proc. of the 13th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GPS 2000), Salt Lake City, UT, 1212–1223. https://www.ion.org/publications/abstract.cfm?articleID=1523
Google Scholar
↵
Min, D., Kim, N. M., Kim., J., Lee, J., & Pullen, S. (2023). SS-RAIM based integrity architecture for CDGNSS systems against satellite measurement faults. Proc. of the 36th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+ 2023), Denver, CO, 2550–2569. https://doi.org/10.33012/2023.19219
Google Scholar
↵
Pervan, B., Chan, F. C., Gebre-Egziabher, D., Pullen, S., Enge, P., & Colby, G. (2003). Performance analysis of carrier-phase DGPS navigation for shipboard landing of aircraft. NAVIGATION, 50(3), 181–191. https://doi.org/10.1002/j.2161-4296.2003.tb00328.x
Google Scholar
↵
Pervan, B. S., Pullen, S. P., & Christie, J. R. (1998). A multiple hypothesis approach to satellite navigation integrity. NAVIGATION, 45(1), 61–84. https://doi.org/10.1002/j.2161-4296.1998.tb02372.x
Google Scholar
↵
Pullen, S. (2011). Augmented GNSS: Fundamentals and keys to integrity and continuity [Conference tutorial]. Proc. of the 24th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS 2011), Portland, OR, 1–60.
Google Scholar
↵
Rife, J., & Gebre-Egziabher, D. (2007). Symmetric overbounding of correlated errors. NAVIGATION, 54(2), 109–124. https://doi.org/10.1002/j.2161-4296.2007.tb00398.x
Google Scholar
↵
Teunissen, P. (1997). A canonical theory for short GPS baselines. Part I: The baseline precision. Journal of Geodesy, 71, 513–525. https://doi.org/10.1007/s001900050119
Google Scholar
↵
Teunissen, P. (1998). Success probability of integer GPS ambiguity rounding and bootstrapping. Journal of Geodesy, 72, 606–612. https://doi.org/10.1007/s001900050199
Google Scholar
↵
Teunissen, P. (1999a). An optimality property of the integer least-squares estimator. Journal of Geodesy, 73, 587–593. https://doi.org/10.1007/s001900050269
Google Scholar
↵
Teunissen, P. (1999b). The probability distribution of the GPS baseline for a class of integer ambiguity estimators. Journal of Geodesy, 73, 275–284. https://doi.org/10.1007/s001900050244
Google Scholar
↵
Teunissen, P. (2001). The probability distribution of the ambiguity bootstrapped GNSS baseline. Journal of Geodesy, 75, 267–275. https://doi.org/10.1007/s001900100172
Google Scholar
↵
Teunissen, P. (2002). The parameter distributions of the integer GPS model. Journal of Geodesy, 76, 41–48. https://doi.org/10.1007/s001900100223
Google Scholar
↵
Teunissen, P. (2006). Influence of ambiguity precision on the success rate of GNSS integer ambiguity bootstrapping. Journal of Geodesy, 81, 351–358. https://doi.org/10.1007/s00190-006-0111-3
Google Scholar
↵
Teunissen, P., Massarweh, L., & Verhagen, S. (2021). Vectorial integer bootstrapping: Flexible integer estimation with application to GNSS. Journal of Geodesy, 95(9), 99. https://doi.org/10.1007/s00190-021-01552-2
Google Scholar
↵
Teunissen, P., & Verhagen, S. (2009). The GNSS ambiguity ratio-test revisited: A better way of using it. Survey Review, 41(312), 138–151. https://doi.org/10.1179/003962609x390058
Google Scholar
↵
Teunissen, P. J., De Jonge, P. J., & Tiberius, C. C. J. M. (1997). Performance of the LAMBDA method for fast GPS ambiguity resolution. NAVIGATION, 44(3), 373–400. https://doi.org/10.1002/j.2161-4296.1997.tb02355.x
Google Scholar
↵
Teunissen, P. J. G. (2017). Carrier phase integer ambiguity resolution. In P. J. G. Teunissen & O. Montenbruck (Eds.), Springer handbook of global navigation satellite systems (pp. 661–685). Springer.
Google Scholar
↵
Verhagen, S. (2005). On the reliability of integer ambiguity resolution. NAVIGATION, 52(2), 99–110. https://doi.org/10.1002/j.2161-4296.2005.tb01736.x
Google Scholar
↵
Verhagen, S., & Teunissen, P. J. G. (2017). Least-squares estimation and Kalman filtering. In P. J. G. Teunissen & O. Montenbruck (Eds.), Springer handbook of global navigation satellite systems (pp. 639–660). Springer.
Google Scholar
↵
Walter, T., & Enge, P. (1995). Weighted RAIM for precision approach. Proc. of the 8th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GPS 1995), Palm Springs, CA, 1995–2004. https://www.ion.org/publications/abstract.cfm?articleID=2524
Google Scholar
↵
Wang, K., El-Mowafy, A., Qin, W., & Yang, X. (2022). Integrity monitoring of PPP-RTK positioning; Part I: GNSS-based IM procedure. Remote Sensing, 14(1), 44. https://doi.org/10.3390/rs14010044
Google Scholar
↵
Wang, K., El-Mowafy, A., Rizos, C., & Wang, J. (2020). Integrity monitoring for horizontal RTK positioning: New weighting model and overbounding CDF in open-sky and suburban scenarios. Remote Sensing, 12(7), 1173. https://doi.org/10.3390/rs12071173
Google Scholar
↵
Wang, S., Zhan, X., Zhai, Y., & Gao, Z. (2023). Solution separation-based integrity monitoring for integer ambiguity resolution-enabled GNSS positioning. Proc. of the 2023 International Technical Meeting of the Institute of Navigation, Long Beach, CA, 492–513. https://doi.org/10.33012/2023.18607
Google Scholar

PDF

Previous Next

SS-RAIM-Based Integrity Architecture for CDGNSSs Against Satellite Measurement Faults

Abstract

1 INTRODUCTION

2 INTEGER-FIXED POSITION ESTIMATION

2.1 Navigation Algorithm

2.2 Distribution of the Integer-Fixed Position Solution

3 INTEGRITY ARCHITECTURE

3.1 Fault Monitoring Using SS-RAIM

3.2 Integrity Risk and PL

4 MONITOR DESIGN

4.1 False Alert Risk and Detection Threshold

4.2 Distribution of Conditional Δ_i Under H₀

5 POSITION AND INTEGER AMBIGUITY ERROR BOUNDING

5.1 Review of Overbounding Methods

Upper Bound on $P (({\hat{b}}_{i, v} - b_{v}) + T_{i, v} > VPL ({\hat{a}}_{i} = z_{i}, H_{i}))$

5.3 Lower Bound on $P ({\overset{ˇ}{a}}_{i} = a_{i} | H_{i})$

6 PERFORMANCE EVALUATION

6.1 Simulation Conditions

6.2 PL and Monitor Performance

6.2.1 Fault-Free Scenario

6.2.2 Fault Scenario

7 CONCLUSION

HOW TO CITE THIS ARTICLE

ACKNOWLEDGMENTS

Appendix A

Derivation Of The Covariance Matrix Of Conditional Δ_i

A1 Conditional Float Position Solution Given ${\hat{a}}_{i} = z_{i}$

A2 Covariance Matrix of Conditional Δ_i Given ${\hat{a}}_{i}$ and ${\hat{a}}_{0}$

A3 ${\bar{P}}_{0} {\bar{N}}_{i} {\bar{P}}_{i}^{T} = {\bar{N}}_{i} {\bar{P}}_{i}$

Appendix B

Derivation of the Lower Bound on Correct Fix Probability

REFERENCES

In this Issue

NAVIGATION: Journal of the Institute of Navigation

Jump to section

Keywords

SS-RAIM-Based Integrity Architecture for CDGNSSs Against Satellite Measurement Faults

Abstract

1 INTRODUCTION

2 INTEGER-FIXED POSITION ESTIMATION

2.1 Navigation Algorithm

2.2 Distribution of the Integer-Fixed Position Solution

3 INTEGRITY ARCHITECTURE

3.1 Fault Monitoring Using SS-RAIM

3.2 Integrity Risk and PL

4 MONITOR DESIGN

4.1 False Alert Risk and Detection Threshold

4.2 Distribution of Conditional Δi Under H0

5 POSITION AND INTEGER AMBIGUITY ERROR BOUNDING

5.1 Review of Overbounding Methods

Upper Bound on Pb^i,v−bv+Ti,v>VPLa^i=zi,Hi

5.3 Lower Bound on Paˇi=ai|Hi

6 PERFORMANCE EVALUATION

6.1 Simulation Conditions

6.2 PL and Monitor Performance

6.2.1 Fault-Free Scenario

6.2.2 Fault Scenario

7 CONCLUSION

HOW TO CITE THIS ARTICLE

ACKNOWLEDGMENTS

Appendix A

Derivation Of The Covariance Matrix Of Conditional Δi

A1 Conditional Float Position Solution Givena^i=zi

A2 Covariance Matrix of Conditional Δi Given a^i and a^0

A3 P¯0N¯iP¯iT=N¯iP¯i

Appendix B

Derivation of the Lower Bound on Correct Fix Probability

REFERENCES

In this Issue

NAVIGATION: Journal of the Institute of Navigation

Alerts for this Article

Email This Article

Citation Tools

SS-RAIM-Based Integrity Architecture for CDGNSSs Against Satellite Measurement Faults

Citation Manager Formats

Share This Article

SS-RAIM-Based Integrity Architecture for CDGNSSs Against Satellite Measurement Faults

SAVE ITEM IN SELECTED FOLDER.

Jump to section

Related Articles

Cited By...

More in this TOC Section

Similar Articles

Keywords

4.2 Distribution of Conditional Δ_i Under H₀

Upper Bound on $P (({\hat{b}}_{i, v} - b_{v}) + T_{i, v} > VPL ({\hat{a}}_{i} = z_{i}, H_{i}))$

5.3 Lower Bound on $P ({\overset{ˇ}{a}}_{i} = a_{i} | H_{i})$

Derivation Of The Covariance Matrix Of Conditional Δ_i

A1 Conditional Float Position Solution Given ${\hat{a}}_{i} = z_{i}$

A2 Covariance Matrix of Conditional Δ_i Given ${\hat{a}}_{i}$ and ${\hat{a}}_{0}$

A3 ${\bar{P}}_{0} {\bar{N}}_{i} {\bar{P}}_{i}^{T} = {\bar{N}}_{i} {\bar{P}}_{i}$