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Abstract

The need for high-accuracy and high-integrity positioning using global navigation satellite system (GNSS) sensors is increasing, given that GNSSs are the only observation systems capable of delivering absolute positioning information. Nevertheless, all GNSS positioning techniques are susceptible to environmental conditions, presenting a significant challenge in meeting the localization requirements for autonomous vehicles, especially in dense urban environments.

This paper presents a novel approach that combines map information with robust estimation techniques to provide high-accuracy and high-integrity localization of vehicles – even in dense urban environments. The proposed adapted robust GNSS real-time kinematic positioning method includes a newly developed robust estimator – the HG-estimator – that retains the advantages of existing robust loss functions and mitigates their drawbacks. This strategy was evaluated and validated through two kinematic automotive experiments situated within medium and deep urban trenches. The application of the HG-estimator incorporating map information resulted in improvements of 54% and 60% in horizontal accuracy compared with a C/N₀ weight model for the medium and deep urban trenches, respectively. Consequently, lane-keeping and lane determination accuracy requirements were met.

Keywords

1 INTRODUCTION

Accurate localization in urban environments is crucial for various intelligent transportation system (ITS) applications, including autonomous driving. The demand for high-accuracy and high-integrity positioning using global navigation satellite system (GNSS) sensors is increasing, as GNSSs are the only observation systems capable of providing absolute positioning information. To satisfy the stringent localization requirements of autonomous vehicle operations, i.e., a horizontal 95% accuracy requirement of 0.21 m and a corresponding alert limit of 0.62 m (Reid et al., 2019), carrier-phase-based positioning techniques are essential. High-precision GNSS positioning strategies, such as real-time kinematic (RTK) methods, enable positioning accuracy in the centimeter range. However, the performance of all GNSS positioning strategies is dependent on the signal quality, posing a significant challenge in meeting the localization requirements for autonomous vehicles in dense urban settings. The principal source of error in GNSS-based vehicle positioning arises from the reception of multipath signals, which are a mixture of direct and reflected signals, and non-line-of-sight (NLOS) signals, which consist solely of reflected signals reaching the antenna. This type of signal reception can induce substantial inaccuracies in the vehicle’s position estimation, regardless of the high-precision GNSS positioning strategy employed. The resulting ranging errors due to reflection at buildings can reach several hundred meters (Braasch, 2017; McGraw et al., 2021).

For static applications, advanced multipath mitigation techniques are applied, such as those based on multipath stacking maps (Fuhrmann et al., 2014) and multipath hemispherical maps (Dong et al., 2015). These methods rely on the repeatability of satellite ground tracks and, consequently, the reflection process. For kinematic applications, however, the magnitude and occurrence of multipath effects and NLOS signals exhibit a complex spatiotemporal behavior, owing to user movement and the variability of urban landscapes with diverse reflecting surfaces. As a result, numerous studies have focused on multipath mitigation approaches that employ ray-tracing algorithms in conjunction with three-dimensional (3D) city models. These 3D mapping-aided (3DMA) ranging techniques anticipate line-of-sight (LOS) and NLOS conditions for satellites, enabling the exclusion of NLOS satellites from the estimation process to improve positioning accuracy (Hsu et al., 2015; Obst et al., 2012; Peyraud et al., 2013). The execution of ray tracing for each epoch to eliminate NLOS satellites is computationally demanding and requires specialized computational methods for real-time application, as well as access to high-definition and up-to-date 3D city models (O’Connor et al., 2021). To enhance positioning in urban canyons, Lee et al. (2023) proposed dynamic multipath maps generated via machine learning. These maps are constructed from short-term data collected by moving platforms and operate independently of supplementary sensors or 3D models. This approach predicts multipath errors for current observations based on previously learned spatial patterns, allowing for direct correction of pseudorange measurements in the estimation process.

When the measurement distribution precisely adheres to Gaussian requirements, with a perfectly known covariance matrix, a maximum-likelihood estimator (MLE) for the normal model will provide an optimal solution (Kim & Shevlyakov, 2008). In practice, however, the underlying assumptions regarding the probability distribution of observations are not perfectly satisfied. As a result, even slight deviations from the assumed model, such as those caused by outliers or contaminated observations, can compromise the guaranteed optimality of the solution. Therefore, a necessary alternative approach for mitigating errors from multipath and NLOS signal reception is the implementation of robust estimation strategies. A study by Knight and Wang (2009) provided a comprehensive summary of robust estimation methods for Global Positioning System (GPS) positioning. In recent years, there has been an increase in research on the application of robust estimators to GNSS positioning in challenging environments. This research spans various fields, including inland waterborne applications (Medina et al., 2016; Pozo-Pérez et al., 2017), ground vehicles (Akram et al., 2018; Crespillo et al., 2020), sensor fusion algorithms (Crespillo et al., 2018), least-squares estimation models (Gaglione et al., 2017; Medina et al., 2019), and carrier-phase-based Kalman filters for RTK positioning (Li et al., 2019; Medina et al., 2021; Tan et al., 2020).

These techniques significantly enhance the accuracy of the resulting GNSS positioning solution, as long as the number of faulty satellites remains moderate. Robust estimation methods incorporate all observations while substantially reducing the influence of erroneous measurements on the positioning solution through the application of various robust loss function designs, e.g., as reported by Huber (1964) and Tukey (1960). However, if the number of measurement errors surpasses the number of nominal observations, the calculation of an optimal solution becomes impossible.

In recent studies, we have introduced the concept of a GNSS feature map, which effectively captures the complex spatiotemporal variability of urban GNSS error sources using an azimuth-elevation grid along the trajectory. We have employed this map for correcting NLOS pseudorange biases in the observation domain (Ruwisch & Schön, 2022a). One of our prior studies (Ruwisch & Schön, 2022b) involved the generation of a detailed GNSS feature map incorporating ray classification data from ray-tracing simulations, which was shown to yield performance comparable to that of epoch-wise ray tracing at the true position. Moreover, we have comprehensively described the methodology for generating a fully populated GNSS feature map utilizing real GNSS data (Ruwisch & Schön, 2023). Additionally, we have demonstrated the map’s applicability to dynamic GNSS positioning using a straightforward and general weighting model based on the GNSS feature map. In a recent study (Ruwisch & Schön, 2025), we combined map information with the concept of robust estimation and achieved an improvement in positioning accuracy when using existing robust loss functions.

The main contributions in this article are as follows:

The concept of robust statistics is thoroughly described, with its mathematical foundations, and a review of the literature for robust estimators in GNSS-based localization, which precisely demonstrates the research gap, is presented.
A new robust estimator, the HG-estimator, is introduced. This estimator retains the advantages of existing robust loss functions and mitigates their drawbacks. A performance simulation is presented, along with the integration of this estimator into GNSS feature map-aided estimation.
A detailed evaluation of the accuracy and integrity of the proposed adapted robust GNSS RTK positioning method is provided, based on two case studies in medium and deep urban trench situations.

The remainder of the paper is structured as follows. The concept and mathematical foundation of robust statistics are summarized in Section 2, and Section 3 provides a comprehensive literature review on robust estimation techniques in GNSS-based localization. Then, in Section 4, the adapted robust weight model is defined, its performance with respect to existing robust estimators is evaluated by a simulation study, and its integration into GNSS feature map-aided estimation is described. The proposed method is evaluated through an analysis of the accuracy performance and integrity based on real data for two vehicle test drives in Section 5, followed by a conclusion in Section 6.

2 RECALLING ROBUST STATISTICS

When the distribution of measurements strictly adheres to the Gaussian criteria with a precisely defined covariance matrix, an MLE for the normal distribution model yields an optimal solution (Kim & Shevlyakov, 2008). In practice, however, the assumptions regarding the probability distribution of observations are not satisfied. Even slight deviations from the presumed model, for instance, due to outliers or contaminated observations, will compromise optimality. Studies in many engineering disciplines have demonstrated that measurements often include outliers, resulting in heavy-tailed distributions, as noted in work by, e.g., Blankenship et al. (1997), Abramovich and Turcaj (1999), Middleton (1999), and Etter (2003), which can cause estimators to become biased or even fail (Zoubir et al., 2012). This issue is also prevalent in GNSS-based applications, where true Gaussianity is challenging to ensure, particularly in environments affected by multipath and NLOS conditions (Medina, 2021), such as urban environments. Unlike an MLE, robust statistics focuses on developing estimators that can provide nearly optimal solutions not only for Gaussian data distributions but also when deviations from such models occur (Huber & Ronchetti, 2009). The theoretical foundation of robust statistics was established by the pioneering work of Tukey (1960), Huber (1964), and Hampel (1971), who laid the groundwork for its application across various technological and engineering domains. For further details on the following mathematical relations, the reader is referred to the relevant literature, including the works of Medina (2021) and Ruwisch (2025).

2.1 Maximum Likelihood Estimation for One-Dimensional Data

In the case of one-dimensional (1D) location estimation, we have a vector of n measurements y^T =[y₁,…y_n] available, where each element y_i is scattered around the location μ with independent and identical distributed noise v_i, that follows a probability distribution F₀:

$\begin{array}{l} y_{i} = μ + v_{i} \end{array}$ 1

The likelihood function of the observations is defined as follows:

$\begin{array}{l} L (y ∣ μ) = \prod_{i = 1}^{n} f (y_{i} - μ) \end{array}$ 2

where f(⋅) is the probability density function (PDF) corresponding to the distribution F₀. By maximizing the likelihood, the MLE yields an estimate $\hat{μ}$ for the location μ:

$\begin{array}{l} \hat{μ} (y) = \arg max_{μ} L (y_{1}, \dots, y_{n} ∣ μ) \end{array}$ 3

In the case that the measurement distribution is $F_{0} = N (μ, σ^{2})$ , i.e., the Gaussian model, the PDF for the measurements is given by the following:

$\begin{array}{l} f (y_{i} ∣ μ, σ^{2}) = \frac{1}{\sqrt{2 π σ^{2}}} \exp (- \frac{{(y_{i} - μ)}^{2}}{2 σ^{2}}) \end{array}$ 4

Because L(y∣μ) is positive in all cases, the maximization problem outlined in Equation (2) for the Gaussian model can be equivalently expressed as the following minimization:

$\begin{array}{l} {\hat{μ}}_{mean} = \arg min_{μ} \sum_{i = 1}^{n} {(r_{i})}^{2} \end{array}$ 5

where the i-th residual r_i = y_i − μ is the difference between the i-th observation and the estimate. Equation (5) describes the minimization of the l₂ norm corresponding to a least-squares adjustment (LSA), and it is obvious that the sample mean aligns with the MLE for the normal model.

Alternatively, if the measurements follow the Laplace distribution, the PDF for the measurements is given by the following:

$\begin{array}{l} f (y_{i} ∣ μ, σ^{2}) = \frac{1}{\sqrt{2} σ} \exp (- \frac{\sqrt{2} | y_{i} - μ |}{σ^{2}}) \end{array}$ 6

The corresponding MLE solves the following minimization:

$\begin{array}{l} {\hat{μ}}_{median} = \arg min_{μ} \sum_{i = 1}^{n} | r_{i} | \end{array}$ 7

Equation (7) describes the minimization of the l₁ norm or the least absolute deviation (LAD) adjustment, and it is obvious that the sample median aligns with the MLE for the Laplace distribution.

2.2 M-Estimator

M-estimators are a generalization of MLEs introduced by Huber (1964) that can resist outliers by solving the following minimization:

$\begin{array}{l} {\hat{μ}}_{M} = min_{μ} \sum_{i = 1}^{n} ρ (\frac{r_{i}}{σ}) \end{array}$ 8

where ρ(⋅) denotes the loss function. For symmetric and differentiable loss functions, the score function is defined as the loss function’s derivative:

$\begin{array}{l} ψ (x) = \frac{d ρ (x)}{d x} \end{array}$ 9

which can be solved as follows:

$\begin{array}{l} \sum_{i = 1}^{n} ψ (\frac{y_{i} - μ}{σ}) = 0 \end{array}$ 10

Intuitively, M-estimators for 1D location estimation can also be interpreted as a weighted average, with weights given by the following weighting function:

$\begin{array}{l} w (x) = {\begin{cases} \frac{ψ (x)}{x}, & if x \neq 0 \\ 1, & if x = 0 \end{cases} \end{array}$ 11

For robust M-estimators, the weighting scheme generally assigns weights of one (or near one) to inliers, whereas outliers are down-weighted based on the score function. Numerous score functions, denoted as ψ(x), have been designed to mitigate the influence of outliers in measurement data, thereby achieving high efficiency under the normal model and ensuring robustness against outliers with a high breakdown point. The breakdown point signifies the maximum proportion of outliers that an estimator can tolerate before yielding a biased result. M-estimators are typically categorized into monotone and redescending types, with the latter being particularly effective for data sets containing extreme outliers.

Table 1 presents the loss functions ρ(x), score functions ψ(x), and weighting functions w(x) for a typical monotone M-estimator from the Huber family (Huber, 1964), a common redescending M-estimator from Tukey’s bisquare family (Tukey, 1960), and the functions associated with the MLE. The parameters c_H and c_T are selected to control the robustness and efficiency of the M-estimators. When these parameters approach $c_{H}, c_{T} \to \infty$ or $c_{H}, c_{T} \to 0$ , the M-estimators approximate the sample mean or sample median, respectively. These parameters are typically chosen to achieve 95% relative efficiency under the normal model, which corresponds to c_H = 1.345 and c_T = 4.685. Another robust family of functions, initially developed for image analysis, is based on the Geman–McClure loss (Barron, 2019; Geman & McClure, 1985). This redescending estimator differs by using continuous functions that further damp very large values without relying on residual-based decisions. The Geman–McClure function’s tuning parameter c_G is typically set to unity. Near zero, this function behaves similarly to the MLE loss, but it becomes significantly bounded, providing robustness against large outliers.

View this table:

TABLE 1 Loss Functions ρ(x), Score Functions ψ(x), and Weighting Functions w(x) for Various Estimators

Monotone M-estimators yield convex optimization problems for Equation (8), ensuring uniqueness of the solutions, whereas redescending M-estimators are fully bounded, offering superior quantitative robustness at the cost of non-convex optimization for Equation (8) and decreased efficiency under the normal model. The functions for the Huber, Tukey, and Geman–McClure estimators and the MLE for the normal model are depicted in Figure 1. For inliers, the Huber-based M-estimator is expected to demonstrate higher efficiency in the normal model relative to redescending alternatives, as its loss, score, and weighting functions align with those of the MLE for nominal data. A single sufficiently large outlier can cause the MLE for the normal model to break down. The impact of large residuals is evident in the score and weighting functions: the monotone Huber-based M-estimator reduces the influence of such outliers, the redescending Tukey-based M-estimator completely eliminates their effect on the solution, and the Geman–McClure-based M-estimator significantly diminishes their impact by assigning weights close to zero. Robust M-estimators are typically computed in an iteratively reweighted least-squares (IRLS) adjustment with an initially computed robust scale estimate using the median absolute deviation:

FIGURE 1

Loss functions ρ(x), score functions ψ(x), and weighting functions w(x) of the MLE for the normal model and M-estimators based on the Huber, Tukey, and Geman–McClure family of functions

The control parameters are set to c_H = 1.345, c_T = 4.685 for 95% relative efficiency for the normal model and to c_G = 1.

$\begin{array}{l} \hat{σ} (x) = 1.4826 \cdot median (| x - median (x) |) \end{array}$ 12

Thereby, the normalization factor 1.4826 corresponds to the standard deviation for the normal model. The procedure of the IRLS adjustment is displayed in Algorithm 1. Given a set of measurements y and predictors A, the first step is to compute an initial solution $\hat{μ}$ and scale $\hat{σ}$ using an LSA. In the initialization step, the weighting matrix W is typically configured to assign unit weights to the measurements. Subsequently, the weighting matrix is adapted through the use of robust weighting functions (such as the functions presented in Table 1) until the convergence criterion is met. Concurrently, the location parameter $\hat{μ}$ is iteratively estimated.

ALGORITHM 1

IRLS Adjustment Using M-Estimators

Estimate the initial $\hat{σ}$ and $\hat{μ}$

while $\frac{{\hat{μ}}^{(t + 1)} - {\hat{μ}}^{(t)}}{\hat{σ}} > \in$ do

Compute the weighting matrix according to the weighting function:

$W = diag (w_{1}, \dots, w_{n}) with w_{i} = w (\frac{r_{i}^{(t)}}{\hat{σ}})$

Perform weighted LSA:

${\hat{μ}}^{(t + 1)} = {(A^{T} W A)}^{- 1} A^{T} W y$

end while

When M-estimation is extended beyond 1D location estimation to applications such as linear regression (referred to as generalized M [GM] estimation), a key limitation of generalized M-estimators is that they generally exhibit a low breakdown point, at most 1/(p + 1) where p is the number of parameters to be estimated. Depending on the specific context, these estimators may not even achieve this upper limit (Maronna et al., 1979).

2.3 S-Estimator

The class of S-estimators is designed to have a high breakdown point, based on the least median of squares (LMS):

$\begin{array}{l} {\hat{μ}}_{LMS} = \arg min_{μ} median (r) \end{array}$ 13

and the least trimmed squares (LTS):

$\begin{array}{l} {\hat{μ}}_{LTS} = \arg min_{μ} \sum_{i = 1}^{h} {(r_{i})}^{2} \end{array}$ 14

where the residuals in the LTS are sorted in ascending order such that $r_{1}^{2} \leq \dots r_{n}^{2}$ and h is chosen to attain a high breakdown point, e.g., h = n/2 yields a breakdown point of 50% (Rousseeuw, 1984). The S-estimator, i.e., the minimization of a robust scale estimate for the residuals, is then defined as follows:

$\begin{array}{l} {\hat{μ}}_{S} = \arg min_{μ} \hat{σ} (r) \end{array}$ 15

and the scale estimate is the solution of the following:

$\begin{array}{l} \frac{1}{n} \sum_{i = 1}^{n} ρ (\frac{r_{i}}{\hat{σ}}) = δ \end{array}$ 16

where δ balances the consistency at a particular distribution and the breakdown point, e.g., for the maximum breakdown point δ = 0.5(1 − n/p). The S-estimator typically considers the use of redescending score functions. A high breakdown point of up to 50% can be achieved, although the corresponding drawback is a relatively low efficiency for the normal model.

2.4 MM-Estimator

The MM-estimator proposed by Yohai (1987) applies a two-step approach to achieve both a high breakdown point and a high relative efficiency. First, an initial estimate $\hat{μ} = {\hat{μ}}_{S}$ and its scale ${\hat{σ}}_{1}$ are computed via S-estimation using a redescending function ρ₁ tuned for a high breakdown point. The MM-estimator is then computed using an IRLS adjustment by minimizing the following equation using a second redescending function such that $ρ_{2} \leq ρ_{1}$ :

$\begin{array}{l} {\hat{μ}}_{M M} = \arg min_{μ} \sum_{i = 1}^{n} ρ_{2} (\frac{r_{i} ({\hat{μ}}_{S})}{{\hat{σ}}_{1}}) \end{array}$ 17

3 REVIEW OF THE LITERATURE ON ROBUST ESTIMATORS IN GNSS-BASED LOCALIZATION

To ensure that the stochastic model accurately reflects the GNSS measurement noise in challenging environments, several weighting schemes have evolved. Elevation-dependent models assign lower weights to satellites at low elevation angles to account for increased atmospheric errors and multipath (Euler & Goad, 1991; Rothacher & Beutler, 1998). C/N₀-dependent models use the signal-to-noise-power-density ratio as a direct indicator of signal quality, allowing for a more realistic evaluation of non-ideal observations (Hartinger & Brunner, 1999; Luo et al., 2009; Wieser & Brunner, 2000). Hybrid models combine C/N₀ and elevation data with external information, such as 3D city models, to explicitly scale uncertainties for LOS and NLOS conditions (Xin et al., 2022; Zhu, 2018).

In contrast, robust estimators are able to provide solutions that remain close to optimal in the presence of outliers by identifying these outliers and reducing their influence in the estimation process. The mathematical principles underlying robust estimators have been described in Section 2. This section discusses recent advancements in the field of robust estimation, particularly within the context of GNSS positioning in challenging environments. The demand for robust estimators for GNSS positioning has increased owing to the increasing probability of multiple outliers resulting from the growing number of satellites with fully functional systems and the application of GNSS sensors to ITSs, particularly in difficult settings such as urban areas. Robust methods retain all observations but down-weight suspicious measurements or minimize alternatives to the sum of squared residuals. Recent contributions in this research domain can be categorized into two main areas: the first involves pure performance simulations of various methods using only simulated observations, and the second involves evaluations based on actual data measurements, applicable to inland waterborne or urban vehicular contexts. A selection of the most noteworthy studies is compiled in Table 2. Simulation studies have demonstrated the efficacy of robust least-squares estimators, such as the LMS, LTS, modified LST (MLTS), greedy search (GS), position and time variation (PTV), R-, M-, GM-, S-, and MM-estimators, in effectively detecting error-prone observations and mitigating their impact relative to the standard LSA (Knight & Wang, 2009; Medina et al., 2019; Wendel, 2022). Further simulation research suggests that robust estimation theory can improve the Kalman filter update step. This method has proven effective in integrating GNSS/inertial navigation system (INS) measurements via an extended Kalman filter (EKF) using pseudorange observations with the M-estimator (Crespillo et al., 2018) and the variational inference method, specifically the variational Bayesian Kalman filter (VB-KF), for RTK positioning as outlined by Li et al. (2019). The impact of heavy-tailed measurement distributions on position distortion can be mitigated, as noted by Medina et al. (2021).

View this table:

TABLE 2

Literature Review for Robust GNSS Positioning

Estimators: LMS, LTS, MLTS, GS, PTV, R, M, GM, S, MM, EKF+M, cubature quadrature Kalman filter (CQKF), VB-KF, GM-KF, RIF, I-VKF, S-VKF, context-adaptive robust (CAR) EKF, random sample consensus (RANSAC)

Observations: Pseudorange (PR), carrier phase (CP), INS

The simulation outcomes are highly promising for robust GNSS positioning, prompting ongoing research to apply robust estimation theory to real measurements in GNSS-challenged environments. Numerous studies have shown that employing robust estimation techniques in pseudorange-based GNSS positioning can improve accuracy in both inland waterborne scenarios (Medina et al., 2016; Pozo-Pérez et al., 2017) and urban vehicular applications (Akram et al., 2018; Crespillo et al., 2020; Ding et al., 2023; Gaglione et al., 2017). However, there is a notable lack of analyses evaluating the performance of robust carrier-phase-based GNSS positioning techniques using real measurements in challenging conditions. In a recent investigation, Medina et al. (2021) expanded the exploration of robust filtering techniques for RTK positioning. The performance of four algorithms - namely, the generalized M-estimator Kalman filter (GM-KF), robust information filter (RIF), independent indicator variational Bayesian Kalman filter (I-VKF), and scalar variational Bayesian Kalman filter (S-VKF) - was evaluated in a demanding waterborne scenario. These robust filtering techniques are capable of effectively mitigating most significant positioning errors. R. Wang et al. (2023) demonstrated that integrating an EKF with M-estimation theory can significantly enhance the ambiguity resolution ratio for RTK positioning in an urban vehicular context.

A common limitation of the aforementioned algorithms is that their robustness is only assured up to their respective breakdown points (see Section 2). If the number of measurement errors surpasses the number of available nominal observations, computing an optimal solution becomes infeasible. Consequently, the subsequent sections of this paper introduce a method that ensures a near-optimal GNSS positioning solution even under adverse propagation conditions in which the breakdown point is exceeded.

4 THE HG-ESTIMATOR - AN ADAPTED ROBUST WEIGHT MODEL

4.1 Definition

The literature review on urban GNSS positioning indicates that when a sufficient number of healthy satellites are available, excluding biased observations using building models and ray-tracing methods significantly improves accuracy. Reformulating the NLOS exclusion approach as a weighting model involves retaining LOS satellites and assigning near-zero weights to NLOS satellites, which are typically associated with larger residuals. Upon revisiting existing robust methods, it becomes clear that the Huber loss and Geman–McClure loss each possess limitations. The Huber loss assigns full weights to moderate residuals (normally between –1.345 and +1.345) but only slightly reduces large residual impacts. Conversely, the Geman–McClure loss markedly diminishes the effect of increasing residuals, thus aligning with the NLOS exclusion method, yet the weights are already substantially reduced for relatively small residuals.

To address these issues, the HG-estimator – the combination of Huber and Geman–McClure models – is developed, integrating the strengths of both robust functions and mitigating their weaknesses. This new model effectively capitalizes on observations with small residuals while strongly suppressing those with larger residuals. Consequently, the fusion of the Huber and Geman–McClure losses yields a robust model akin to the NLOS exclusion strategy. Table 3 presents the corresponding equations for the loss function (ρ(x)), score function (ψ(x)), and weighting function (w(x)). Further illustration of the novel approach is provided by the graphs in Figure 2, demonstrating adherence of the HG function to the Huber definition for residuals within ± 1.345 and its alignment with the Geman–McClure loss, score, and weighting function for larger residuals. The HG-estimator results in a non-convex optimization of Equation (8) and thus requires accurate initialization to avoid ending up at a local minimum. The non-continuity of the function causes issues with classical iterative approaches for weight determination. In Section 4.3, a method is defined that provides the necessary initialization values, making iteration obsolete and thus rendering the HG-estimator a suitable function for improving the robustness of the estimation.

View this table:

TABLE 3

Loss Function ρ(x), Score Function ψ(x), and Weighting Function w(x) of the Proposed Adapted Robust HG-Estimator

The parameters are set to c_H = 1.345 and c_G = 1.

FIGURE 2

Loss function, score function, and weighting function of the proposed adapted robust HG-estimator

The MLE for the normal model and M-estimators based on the Huber, Tukey, and Geman–McClure family of functions are also illustrated. The control parameters are set to c_H = 1.345, c_T = 4.685 for 95% relative efficiency for the normal model and to c_G = 1.

4.2 Performance Simulation of Robust Estimators

To didactically demonstrate the necessity of an adapted robust estimator, a comparative simulation study was conducted to evaluate the performance of existing robust estimators, the adapted HG-estimator, and an MLE within the context of simple epoch-wise single-point positioning (SPP) (Odijk, 2017). A real multi-GNSS satellite constellation (GPS, GLONASS, Galileo, and BeiDou) is identified for a specific geographic location and time, utilizing final satellite orbit data from the Center for Orbit Determination in Europe (CODE) (Dach et al., 2024). The standard observation errors are simulated following a normal distribution with a mean of zero and a standard deviation of σ = 1 m, whereas the contaminated observations follow a uniform distribution, with biases randomly selected from a range of 20–100 m. The multi-constellation configuration involves a total of 36 satellites. Details of all configuration parameters are outlined in Table 4. The multi-GNSS satellite constellation of the simulation configuration is shown in Figure 3, where the left skyplot shows the observation errors for the 50% contamination proportion case and the right skyplot represents the observation errors for the 70% contamination proportion case. The observation errors are categorized into two classes: nominal observations with magnitudes near 1 m and contaminated observations with magnitudes larger than 20 m.

View this table:

TABLE 4 Simulation Configuration

FIGURE 3

Multi-GNSS satellite constellation for the simulation configuration

The randomly generated observation errors are categorized in blue ( $l_{in} \sim N (0, 1)$ ) and red ( $l_{out} \sim U (20, 100)$ ) for 50% outliers (left) and 70% contaminated observations (right).

Figure 4 presents the loss functions for various estimators – namely, the MLE for the normal model (first column), Huber estimator (c_H = 1.345, second column), Geman–McClure estimator (c_G = 1, third column), and HG-estimator (c_H = 1.345, fourth column) – applied to SPP in the north-east position domain utilizing the simulated observation errors. The top row of the figure presents contour plots for a contamination level of 50%, whereas the lower row corresponds to a contamination level of 70%. Each global minimum is indicated by a red cross, while the ground truth is marked by a green diamond.

FIGURE 4

Loss functions of the MLE (first column), Huber (c_H = 1.345, second column), Geman–McClure (c_G = 1, third column), and HG (c_H = 1.345, c_G = 1, fourth column) estimators for the SPP formulation using the simulated observation errors depicted in Figure 3

The top and bottom rows represent the loss function as a contour plot for a contamination proportion of 50% and 70%, respectively. The green diamond represents the ground truth at [0, 0] m, and the red cross depicts the respective minimum of the loss functions.

As anticipated, the MLE and Huber estimators exhibit a conventional convex optimization profile, characterized by a single minimum that ensures uniqueness and stability. Nonetheless, the minimum tends to form a flattened conical shape, especially evident at a 70% contamination level. Minimizing the MLE loss function leads to a bias in the estimation for both contamination levels, as even one outlier can compromise the estimator’s optimal efficiency. When the Huber estimator is used, the loss function’s minimization yields a position estimate near the ground truth at a 50% contamination level. However, contamination at 70%, exceeding the breakdown point of 50%, introduces positional bias.

The Geman–McClure estimator demonstrates characteristics indicative of non-convex optimization. It exhibits a sharp peak at the global minimum but also presents numerous potential solutions due to the presence of local minima. The need for accurate initialization of the non-convex solver becomes more pronounced as contamination increases, because an elevated likelihood exists for convergence to a local minimum, thereby inducing positional bias. Nonetheless, if the initial estimate is sufficiently close to the ground truth, the Geman–McClure estimator can still achieve global convergence, providing a position estimate proximate to the true location. This result suggests that a redescending-type robust estimator is advantageous for mitigating the impact of outliers, particularly when high contamination is present and accurate initial estimates are available.

For the HG-estimator, the global minimum of its loss function demonstrates minimal deviation from the ground truth, a finding consistent with the Geman–McClure results. However, the presence of local minima is reduced, indicating an enhancement in the overall performance. Consequently, the HG-estimator is distinguished by its robust nature, which enables it to effectively mitigate the impact of outliers in the presence of high data contamination. It is evident that, owing to its redescending characteristic, the HG-estimator still generates some local minima, which necessitates precise initial estimates to ensure convergence towards the global minimum.

4.3 GNSS Feature Map-Aided Robust Estimation

The concept of GNSS feature map-aided robust estimation aims to deliver precise estimates of the true residuals to robust weighting schemes, thereby addressing the challenge of robustness when dealing with highly contaminated data. Figure 5 illustrates the fundamental principle of employing GNSS feature map information (Ruwisch & Schön, 2022a, 2022b, 2023) for the localization of automated vehicles. Automated vehicles navigating through urban trenches can access stored map data when situated within a specified proximity to a corresponding map point. The map information provided is not limited to predicted pseudorange residual data (as shown in the right skyplot of Figure 5); it can also encompass satellite visibility classification information (depicted in the left skyplot of Figure 5), as well as other spatiotemporal-dependent features. These features may include data derived from fish-eye cameras offering visibility information or light detection and ranging (lidar) sensors, which provide details about building wall structures. In the literature, these sensors are already used to aid GNSS position estimation (Sánchez et al., 2016; Wen & Hsu, 2021).

FIGURE 5

Conceptual application of GNSS feature maps for localization of automated vehicles

The shaded polygons represent the boxes of the map.

Unlike traditional GNSS shadow matching, which primarily utilizes 3D building geometry to exclude NLOS signals (Groves, 2011; Groves et al., 2015; Wang et al., 2015), the approach applied in this study utilizes the GNSS feature map to provide a prior on the expected residual magnitude. These residuals contain all of the remaining sources of error, which primarily consist of multipath and NLOS ranging errors. This information is utilized for adapting the weight model, leading to the designation of the method as GNSS feature map-aided weighting (GNSS FMA-W). The process of generating this map has been thoroughly discussed by Ruwisch and Schön (2023, 2025). The map resolution is specified as 5 m in the longitudinal direction and 4 m in the lateral direction. This resolution is equivalent to the accuracy requirement of the a priori position estimate, the purpose of which is to retrieve the correct map information. The map functions as a look-up table indexed by the a priori position. The GNSS FMA-W approach is applicable to any estimator, provided that a preliminary predicted user location is available. Within a standard EKF framework, GNSS FMA-W is implemented after the time update step of the EKF. The predicted user position is then calculated, and it is determined whether the predicted position is located within one of the predefined box polygons of the map. If this condition is met, meaning that the predicted position aligns with a specific map grid, the pseudorange residual information corresponding to that particular box is retrieved from the GNSS feature map. Then, the respective predicted residuals are normalized by the a priori pseudorange measurement noise $σ_{ρ_{0}}$ as follows:

$\begin{array}{l} {\overset{―}{v}}^{FM} = \frac{v^{FM}}{σ_{ρ_{0}}} \end{array}$ 18

where v^FM is s vector of predicted pseudorange residuals retrieved from map information, which is used for computing robust weights based on a defined robust loss function. The minimization formulation in Equation (8) can be re-formulated as minimization of the predicted pseudorange residuals around the location μ as follows:

$\begin{array}{l} {\hat{μ}}_{FM} = min_{μ} \sum_{i = 1}^{n} ρ (\frac{v_{i}^{FM}}{σ_{ρ_{0}}}) \end{array}$ 19

where ρ(⋅) denotes one of the robust loss functions and i = 1,…, n is the observation index. Based on the results of Equation (19), the variance-covariance matrix is subject to modification during the update step of the EKF, thereby ensuring that all observations are assigned with an appropriate variance. If no map information is available, a standard weighting approach must be applied. In this work, we applied C/N₀-based weighting (Luo et al., 2009). The algorithmic realization is depicted in Algorithm 2. Please note that no iteration of the estimation is required because the obtained predicted residuals are taken as true values and are thus expected to solve the minimization in Equation (19), which yields an efficient and inexpensive robust estimator.

ALGORITHM 2

GNSS Feature Map-Aided Weighting

for every epoch do

Calculate predicted user position

Calculate satellite positions

if Predicted position is inside any box polygon then

retrieve feature map residual information v^FM

Normalize $v^{FM} to {\bar{v}}^{FM} = \frac{v^{FM}}{σ_{ρ_{0}}}$

Adapt weights based on robust loss function ρ(∙)

else

Use conventional weighting

end if

Update user position

end for

5 APPLICATION EXAMPLES AND CASE STUDIES

The core of our analyses includes two vehicle test drives, where the impact of the proposed adapted robust estimation is investigated by GNSS RTK positioning. These kinematic case studies are further distinguished by their environmental situation, which encompasses two distinct scenarios: moderate signal reception conditions in a medium urban trench and harsh signal reception conditions in a deep urban trench. Static phases at parking areas are omitted from the analyses because, in non-dynamic and nearly open-sky environments, all methods achieve accuracy levels that are within given alert limits, and the position errors remain within acceptable boundaries. Hence, the evaluation accurately reflects the actual conditions, free from any augmentation from the static segments of the respective experiment. The observation data used for these analyses, given in the receiver independent exchange format (RINEX) notation, are indicated in Table 5.

View this table:

TABLE 5 Observation Data Used in the GNSS RTK Algorithm, Given in RINEX Notation

5.1 Automotive Experiment in a Medium Urban Trench

To evaluate the performance of the proposed approaches for urban GNSS navigation, a kinematic experiment was conducted in a residential area in the city of Hannover, Germany, on day of year (DOY) 235 in 2023 (August 23, 2023).

The ground truth of the trajectory, together with the institute’s reference station EE01, is depicted in Figure 6(a). The trajectory begins and ends with a static phase on a parking space with few obstacles; in between the beginning and ending points, the rectangular section was driven ten times. The kinematic part of the route passes through a residential area in the city of Hannover, where the streets are approximately 5 m wide and the surrounding houses are approximately 20 m high. A typical situation of this trajectory is shown in Figure 6(b), where the projected grids of the generated map are displayed with white boxes.

FIGURE 6

Kinematic experiment setup of the automotive experiment in a medium urban trench: (a) ground truth of the repeatedly driven trajectory, (b) typical environmental situation with map grids depicted as white polygons, (c) measurement configuration, and (d) multi-GNSS satellite visibility information based on ray-tracing results (BLK: blocked; MP: multipath)

The measurement configuration depicted in Figure 6(c) consists of one Septentrio PolaRx5e receiver connected to a Tallysman TW7972 patch antenna mounted on the roof of the test vehicle. The receiver collected raw multi-GNSS data at a sampling rate of 1 Hz. In addition, a high-quality INS (iMAR iNAT-RQT-4003) was connected to a NαvXperience NAX3G+C antenna collecting multi-GNSS data at a sampling rate of 1 Hz and inertial measurement unit (IMU) data at a sampling rate of 400 Hz. The ground truth of the trajectories is computed by combining the GNSS carrier-phase and Doppler observations with the IMU data in a tightly coupled relative positioning that was computed in post-processing using the commercial software TerraPOS (Kjørsvik et al., 2009). In post-processing, the data are further evaluated with regard to the multi-GNSS satellite visibility condition based on ray-tracing classification results. All observed satellite signals are classified as LOS, multipath, NLOS, and blocked. The number of available satellites per ray condition is illustrated in Figure 6(d) (left), together with the percentage of LOS satellites (right). The data set is classified as a medium urban trench, as the LOS satellite availability is below 50% (minimum 20% LOS satellites) throughout most of the trajectory, with the availability recovering between these parts and reaching up to 70%.

5.1.1 Accuracy

The position solution of the GNSS RTK algorithm with respect to the reference trajectory is shown in Figure 7. Two approaches from the existing literature, i.e., C/N₀ weighting and 3DMA NLOS exclusion (3DMA-NE), are compared with the developed GNSS feature map-aided approaches, i.e., GNSS FMA-W using the Huber loss function, GNSS FMA-W using the Geman–McClure loss function, and GNSS FMA-W using the HG loss function.

FIGURE 7

HPEs of the different approaches (a) versus time and (b) as a cumulative frequency diagram

Figure 7 presents the horizontal position error (HPE) relative to the ground truth, both as a function of time and as a cumulative distribution. The analysis excludes the static phases in the parking area as, in this non-dynamic, nearly open-sky environment, all methods exhibit deviations of a few centimeters, typical for RTK solutions. During the test drive through the medium urban trench, the horizontal deviations vary between the centimeter and decimeter range. Nonetheless, significant differences are observable among the various estimation approaches.

The NLOS exclusion strategy employing ray-tracing (3DMA-NE) showed a transition from an RTK fixed solution to a differential GNSS (DGNSS) solution at approximately 9.83 h and 9.86 h, attributed to the lack of LOS carrier-phase availability, which led to a maximum HPE of approximately 8 m. In contrast, all other strategies achieve centimeter-to-decimeter accuracy throughout the trajectory, although the LOS observation availability drops to as low as 20%.

The GNSS FMA-W methods demonstrate similar performance, notably reducing the HPE for epochs in which the C/N₀ weighting solution exhibits deviations exceeding 40 cm. This similarity is further highlighted in the cumulative distribution, where the yellow, purple, and green plots overlap. The plots initially follow the 3DMA-NE graph at minor deviations but diverge with a steeper slope as deviations increase, indicating a mitigation of larger errors.

Table 6 provides a comprehensive summary of the characteristic values associated with the horizontal position accuracy of the different methodologies. The minor differences in position errors at specified percentiles illustrate the similar performance levels of the three GNSS FMA-W approaches.

View this table:

TABLE 6 Horizontal Position Accuracy of the Different Approaches, Given in Percentiles

Referring to the automotive application performance specifications outlined by Reid et al. (2023), it is observed that the C/N₀ weighting method does not satisfy the 95% accuracy requirement in the horizontal direction necessary for lane-keeping applications. An analysis of the overall root mean square (RMS) errors reveals that the GNSS FMA-W methods offer improved accuracy relative to the C/N₀ weighting approach. Specifically, the integration of map information combined with the Huber, Geman–McClure, and HG loss functions leads to a reduction in the HPE RMS by 58%, 54%, and 54%, respectively. Conversely, the 3DMA-NE method negatively impacts the overall performance, owing to its transition from an RTK fixed solution to a DGNSS solution at certain epochs.

5.1.2 Attainable Integrity

The integrity of the various positioning solutions was assessed using Stanford diagrams (Toissant et al., 2006) for the horizontal component. The protection levels associated with these solutions were derived from the corresponding covariance matrix values, assuming a probability of failure of 10⁻⁸. The horizontal alert limit (HAL) is calculated as HAL $= \sqrt{{0.44}^{2} + {0.44}^{2}} = 0.622 [m]$ (see the work by Reid et al. (2023) for details on the localization requirement definitions). The horizontal protection level (HPL) is computed as follows:

$\begin{array}{l} HPL = κ_{HPL} σ_{0} \sqrt{\frac{q_{n n} + q_{e e}}{2} + \sqrt{{(\frac{q_{n n} - q_{e e}}{2})}^{2} + q_{n e}^{2}}} = κ_{HPL} σ_{0} \sqrt{λ_{m a x}} \end{array}$ 20

where λ_max is the maximum eigenvalue of the covariance sub-matrix of horizontal positions and q_nn, q_ee, q_uu and q_ne are elements of the rotated covariance matrix of the estimated parameters in the local frame:

$\begin{array}{l} Q_{x} = R Q_{X} R^{T} = [\begin{array}{lll} q_{n n} & q_{n e} & q_{n u} \\ q_{e n} & q_{e e} & q_{e u} \\ q_{n u} & q_{e u} & q_{u u} \end{array}] \end{array}$ 21

Here, R is a rotation matrix from the global to the local coordinate system. The a priori variance factor σ₀ and the κ-factor scale the HPL to a level compatible with the integrity requirement of the particular application. In this study, the values are set to σ₀ = 1 and κ_HPL = 6.18.

The resulting Stanford diagrams are depicted in Figure 8. When the C/N₀ weighting method is employed, the system functions in nominal mode for 88.2% of cases. However, in 10.2% of epochs, misleading information is provided, as the protection level is insufficient relative to the position error, thereby inadequately constraining the error. Furthermore, a small fraction (1.6%) of epochs present hazardous misleading information, which is the most critical condition. With the application of the 3DMA-NE method, these instances of hazardous misleading information are reduced to just one epoch (0.1%), albeit with an increase in system unavailability instances and a slight decrease in the share of nominal operation epochs, which constitute 87.2% of the total. All three GNSS feature map-aided robust estimation methods enhance the overall integrity of the system, with only minor differences observed in the proportion of nominal operation epochs among them. Of the GNSS FMA-W Huber, GNSS FMA-W Geman–McClure, and GNSS FMA-W HG methods, the GNSS FMA-W HG method has the highest number of nominal operating modes and the fewest misleading information epochs, although the differences are minimal. More substantial improvements are demonstrated in more challenging environments in the next section.

Integrity evaluation of the horizontal component for the different methods (GMC: Geman–McClure). (a) C/N0, (b) 3DMA-NE, (c) GNSS FMA-W Huber, (d) GNSS FMA-W GMC, (e) GNSS FMA-W HG.

FIGURE 8

Integrity evaluation of the horizontal component for the different methods (GMC: Geman–McClure). (a) C/N₀, (b) 3DMA-NE, (c) GNSS FMA-W Huber, (d) GNSS FMA-W GMC, (e) GNSS FMA-W HG.

5.2 Automotive Experiment in a Deep Urban Trench

Another kinematic experiment was conducted in a similar residential area in the city of Hannover, Germany, on DOY 44 in 2024 (February 13, 2024). The trajectory’s ground truth in conjunction with the institute’s reference station, denoted as EE01, is illustrated in Figure 9(a). The trajectory begins and concludes with a static phase in a parking space characterized by minimal obstructions. In between the beginning and ending points, a triangular segment was traversed eight times. The dynamic segment of the trajectory cuts through a residential area in Hannover, where street widths are approximately 5 m, bordered by structures approximately 22 m in height. A typical scenario along this trajectory is depicted in Figure 9(b), featuring the projected grids of the generated map delineated with white boxes. The measurement configuration is the same as in the previous experiment, i.e., a Septentrio PolaRx5e receiver is connected to a Tallysman TW7972 antenna.

FIGURE 9

Kinematic experiment setup of the automotive experiment in a deep urban trench: (a) ground truth of the repeatedly driven trajectory, (b) typical environmental situation with map grids depicted as white polygons, (c) test vehicle, (d) multi-GNSS satellite visibility information based on ray-tracing results

Furthermore, a high-quality INS (iMAR iNAT-RQT-4003) was utilized, which was connected to the same antenna and collected multi-GNSS data at a sampling rate of 1 Hz and IMU data at a sampling rate of 400 Hz. Similar to the previous experiment, the trajectory’s ground truth was determined by integrating GNSS carrier-phase and Doppler observations with IMU data through a tightly coupled relative positioning approach, processed post-experiment using the commercial software TerraPOS (Kjørsvik et al., 2009).

Figure 9(d) categorizes this data set as a deep urban trench. This classification is derived from the characteristics of the received GNSS satellite signals, as analyzed by ray tracing. The analysis indicates that in certain instances, the number of obstructed signals surpasses that of LOS signals. This trend is evident in the percentage availability of LOS signals, which falls below 50% throughout nearly the entire kinematic phase (from 13.43 h to 13.65 h). In several instances, data from less than 15% of LOS satellites are available in a single epoch. In contrast to the preceding data set, the maximum percentage of LOS satellite availability during this period is only approximately 50%. The challenging reception characteristics of this environment pose substantial difficulties for positioning algorithms. Consequently, this deep urban trench data set serves as an exemplary test case for assessing the effectiveness of the proposed methodologies under highly challenging conditions, as well as for identifying their limitations.

5.2.1 Accuracy

First, the accuracy of the position solution computed by the GNSS RTK algorithm with different error mitigation strategies was evaluated. Figure 10 presents the HPE with respect to time, as well as a cumulative distribution diagram. Please note that for enhanced visual clarity, the y-axis in the time series figures has been truncated. To ensure completeness, the maximum errors, marked by diamond indicators, can be derived from the cumulative distribution diagram.

FIGURE 10

HPEs of the different approaches (a) versus time and (b) as a cumulative distribution diagram

From the time series, it is clear that the solution employing C/N₀ weighting suffers from degraded performance, particularly between 13.46 h and 13.57 h. During this period, the initial centimeter-level deviations could not be sustained, resulting in HPEs ranging from 1 m to 4 m.

The 3DMA-NE solution reveals that LOS carrier-phase measurements were not consistently available. This unavailability led to numerous DGNSS epochs (11%) due to the exclusion of NLOS satellites, causing the HPE to increase to 10 m. However, when more than four LOS carrier-phase measurements were available, the estimation strategy quickly achieved a solution with centimeter-level deviations. All other methods provide a 100% RTK fix rate; however, it is important to note that the position deviations clearly indicate many false fix events. A more detailed ambiguity resolution analysis has been presented by Ruwisch (2025).

The GNSS FMA-W Huber method provided robust results for the medium urban trench data set; however, its improvement with respect to the C/N₀ weighting method under highly challenging signal reception conditions is relatively limited. The time series shows similar deviations despite the integration of additional feature map information in the estimation process. This limitation is likely due to the Huber loss function’s insufficient attenuation of erroneous observations during the estimation.

The GNSS FMA-W Geman–McClure method offers stable results within the most challenging parts of the trajectory. Similar performance is observed for the GNSS FMA-W HG method. Following the sudden increase in position deviations during transitions of the 3DMA-NE method from RTK fix mode to DGNSS mode, both estimation strategies demonstrate the capability to quickly return to centimeter-level deviations. A comparative analysis between the two methods shows that the HG loss function is more effective in mitigating gross errors and provides more stable results in less challenging scenarios.

With regard to the performance specifications for automotive applications, as discussed by Reid et al. (2023), and based on the characteristic performance metrics in Table 7, it is evident that none of the approaches meet the 95% accuracy requirement for lane-keeping applications in the horizontal direction. This shortcoming is primarily due to the challenging signal propagation conditions in the deep urban trench environment.

View this table:

TABLE 7 Horizontal Position Accuracy of the Different Approaches, Given in Percentiles

To assess the capability for lane determination, the HPE is further divided into longitudinal and lateral position errors, which are depicted as cumulative distribution diagrams in Figure 11. Diamond markers denote 95% values for the various estimation strategies, while the black dashed line indicates the lane determination accuracy requirement at the same percentile. It is clear that the GNSS FMA-W HG method is the only one capable of achieving satisfactory accuracy to fulfill the specified lane determination requirements in both longitudinal and lateral directions. The horizontal position accuracy at the 95% level is 0.91 m for the longitudinal direction and 0.28 m for the lateral direction. In this experiment, the longitudinal error is observed to be greater than the lateral error. This trend is due to the challenging and short north–south-oriented street of the repeated trajectory, where a suboptimal satellite constellation results in significant north errors, i.e., in the longitudinal direction.

FIGURE 11

Longitudinal and lateral position errors of the different approaches as a cumulative distribution diagram

The black dashed line depicts the 95% accuracy requirement for lane determination applications according to Reid et al. (2023).

A comparison of the overall RMS errors indicates that all three GNSS FMA-W methods enhance accuracy compared with the C/N₀ weighting approach. The integration of map information with the Huber, Geman–McClure, and HG loss functions improves the HPE RMS by 17%, 17%, and 60%, respectively. In contrast, the 3DMA-NE method negatively affects the overall results, owing to its transition from an RTK fixed solution to a DGNSS solution at certain epochs.

5.2.2 Attainable Integrity

The analyses of the accuracy for the deep urban trench data show that the requirements for lane-keeping applications are not met. However, to investigate how far the provided solutions are from these requirements, the integrity was evaluated based on the corresponding alert limit specified in the previous section. The resulting Stanford diagrams are shown in Figure 12. Please note that data points that exceed the axis limits of the position error or protection level are displayed at the respective border of the figure.

Integrity evaluation of the horizontal component for the different methods. (a) C/N0, (b) 3DMA-NE, (c) GNSS FMA-W Huber, (d) GNSS FMA-W GMC, (e) GNSS FMA-W HG.

FIGURE 12

Integrity evaluation of the horizontal component for the different methods. (a) C/N₀, (b) 3DMA-NE, (c) GNSS FMA-W Huber, (d) GNSS FMA-W GMC, (e) GNSS FMA-W HG.

The system operates in nominal mode in 50.1% of cases when the C/N₀ weighting method is employed. However, a significant number of hazardous misleading information epochs are present, amounting to 29.4%, which constitutes the most critical condition. Additionally, 10.2% of epochs provide misleading information, owing to the protection level being smaller than the position error, thus inadequately constraining the error.

The application of the 3DMA-NE approach reduces the most critical hazardous misleading information instances to 0.5%. However, this reduction comes with an increased number of system unavailability instances (29.3%) and a slight decrease in nominal operation epochs, which comprise 48.8% of the total. This result suggests that although the method accurately bounds the position errors, the increased protection level values result in the system not operating within nominal mode under the given specifications.

As previously noted, the GNSS FMA-W Huber method does not improve the solution. Its integrity measures are comparable to those of the C/N₀ weighting solution, resulting in many instances of hazardous misleading information (28.5%).

In contrast, the GNSS FMA-W Geman–McClure and GNSS FMA-W HG methods enhance the system’s overall integrity, showing substantial improvements in nominal operations (64.1% and 69.7%, respectively). Among these two methods, the combination of map information with the HG loss function yields the best performance compared with the Geman–McClure loss function, with reduced system unavailability, misleading information, and hazardous misleading information epochs.

6 CONCLUSION

In this paper, we presented a novel approach for high-accuracy and high-integrity autonomous vehicle localization. The proposed adapted robust GNSS RTK positioning algorithm combines map information, i.e., pseudorange residual information for all satellite positions in a regular grid, with robust estimation techniques to maintain localization requirements, even under harsh signal propagation conditions. First, we provided a detailed overview of the definition of robust estimators from the literature, as well as an extensive review of current research activities for robust estimation in GNSS-based localization, followed by a definition of the proposed novel HG-estimator. The theoretical performance of robust estimators was evaluated by a simulation study, emphasizing the limitations for highly contaminated data with more than 50% outliers. However, the redescending-type estimators (e.g., Geman–McClure loss and HG loss) were still capable of providing a global minimum close to the ground truth with the requirement of an accurate initial estimation.

The adapted robust GNSS RTK positioning algorithm was further evaluated based on two automotive experiments in environments classified as a medium and deep urban trench with LOS satellite availabilities below 50%. All robust GNSS feature map-aided approaches effectively mitigated gross errors and improved the overall position accuracy, compared with existing methods from the literature, i.e., C/N₀ weighting and 3DMA-NE. Under medium multipath conditions, combining the GNSS feature map with any robust estimator yields comparable performance. Of these methods, combining map information with the novel HG-estimator achieves the greatest accuracy in deep urban trenches, improving the HPE RMS by 60% compared with C/N₀ weighting. Thus, the GNSS FMA-W using the HG-estimator is the only method capable of providing localization results in the deep urban trench scenario that meet the requirements for lane determination applications of autonomous vehicles. In terms of integrity, the localization errors are adequately bounded when prior information on the potential ranging errors is utilized, yielding an increased number of nominal operations (GNSS FMA-W HG: 69.7% and 97.4% compared with C/N₀ weighting: 50.1% and 88.2%) and a reduced number of epochs of hazardous misleading information (GNSS FMA-W HG: 0.1% and 3.6% compared with C/N₀ weighting: 1.6% and 29.4%).

HOW TO CITE THIS ARTICLE:

Ruwisch, F., & Schön, S. (2026). Adapted robust GNSS RTK positioning: Improving accuracy and integrity for autonomous vehicle localization in urban trenches. NAVIGATION, 73. https://doi.org/10.33012/navi.779

ACKNOWLEDGMENTS

The work was carried out in the framework of the KOMET project, which is managed by TÜV-Rheinland (PT-TÜV) under grant 19A20002C and is funded by the Federal Ministry for Economic Affairs and Climate Action (BMWK), based on a resolution of the German Bundestag, and in the framework of the i.c.sens research training group funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grant GRK2159. The ground truth positioning system is part of the Major Research Instrumentation funded by the DFG - 436385244.

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

↵
Abramovich, Y. I., & Turcaj, P. (1999). Impulsive noise mitigation in spatial and temporal domains for surface-wave over-the-horizon radar (tech. rep.). DTIC Document. Cooperative Research Centre for Sensor Signal and Information Processing.
Google Scholar
↵
Akram, M., Liu, P., Wang, Y., & Qian, J. (2018). GNSS positioning accuracy enhancement based on robust statistical MM estimation theory for ground vehicles in challenging environments. Applied Sciences, 8(6), 876. https://doi.org/10.3390/app8060876
Google Scholar OpenURL
↵
Barron, J. T. (2019, June). A general and adaptive robust loss function. In 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 4326–4334). https://doi.org/10.1109/cvpr.2019.00446
Google Scholar
↵
Blankenship, T. K., Kriztman, D. M., & Rappaport, T. S. (1997, May). Measurements and simulation of radio frequency impulsive noise in hospitals and clinics. In 1997 IEEE 47th Vehicular Technology Conference. Technology in Motion (pp. 1942–1946). https://doi.org/10.1109/vetec.1997.605897
Google Scholar
↵
Braasch, M. S. (2017). Multipath. In P. J. G. Teunissen & O. Montenbruck (Eds.), Springer handbook of global navigation satellite systems (pp. 443–468). Springer International Publishing.
Google Scholar
↵
Crespillo, O. G., Andreetti, A., & Grosch, A. (2020, January). Design and evaluation of robust M-estimators for GNSS positioning in urban environments. In Proceedings of the 2020 International Technical Meeting of the Institute of Navigation (pp. 750–762). https://doi.org/10.33012/2020.17211
Google Scholar
↵
Crespillo, O. G., Medina, D., Skaloud, J., & Meurer, M. (2018, April). Tightly coupled GNSS/INS integration based on robust M-estimators. In 2018 IEEE/ION Position, Location and Navigation Symposium (PLANS) (pp. 1554–1561). https://doi.org/10.1109/plans.2018.8373551
Google Scholar
↵
Dach, R., Schär, S., Arnold, D., Brockmann, E., Kalarus, M. S., Lasser, M., Stebler, P., & Jäggi, A. (2024). CODE final product series for the IGS. Astronomical Institute, University of Bern. https://doi.org/10.48350/197025
Google Scholar
↵
Ding, Y., Feriol, F., Watanabe, Y., Asseman, P., Pages, G., & Vivet, D. (2023, January). Adaptive robust-statistics GNSS navigation based on environmental context detection. In Proceedings of the 2023 International Technical Meeting of the Institute of Navigation (pp. 138–152). https://doi.org/10.33012/2023.18636
Google Scholar
↵
Dong, D., Wang, M., Chen, W., Zeng, Z., Song, L., Zhang, Q., Cai, M., Cheng, Y., & Lv, J. (2015). Mitigation of multipath effect in GNSS short baseline positioning by the multipath hemispherical map. Journal of Geodesy, 90(3), 255–262. https://doi.org/10.1007/s00190-015-0870-9
Google Scholar OpenURL
↵
Etter, P. C. (2003). Underwater acoustic modelling and simulation. CRC Press. https://doi.org/10.1201/9781482295146
Google Scholar
↵
Euler, H.-J., & Goad, C. C. (1991). On optimal filtering of GPS dual frequency observations without using orbit information. Bulletin Géodésique, 65(2), 130–143. https://doi.org/10.1007/bf00806368
Google Scholar OpenURL
↵
Fuhrmann, T., Luo, X., Knöpfler, A., & Mayer, M. (2014). Generating statistically robust multipath stacking maps using congruent cells. GPS Solutions, 19(1), 83–92. https://doi.org/10.1007/s10291-014-0367-7
Google Scholar OpenURL
↵
Gaglione, S., Innac, A., Carbone, S. P., Troisi, S., & Angrisano, A. (2017). Robust estimation methods applied to GPS in harsh environments. In 2017 European Navigation Conference (ENC) (pp. 14–25). https://doi.org/10.1109/euronav.2017.7954169
Google Scholar
↵
Geman, S., & McClure, D. E. (1985). Bayesian image analysis: An application to single photon emission tomography. Proceedings of the American Statistical Association (pp. 12–18).
Google Scholar
↵
Groves, P. D. (2011). Shadow matching: A new GNSS positioning technique for urban canyons. Journal of Navigation, 64(3), 417–430. https://doi.org/10.1017/s0373463311000087
Google Scholar OpenURL
↵
Groves, P. D., Wang, L., Adjrad, M., & Ellul, C. (2015, September). GNSS shadow matching: The challenges ahead. In Proceedings of the 28th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+) (pp. 2421–2443). https://www.ion.org/publications/abstract.cfm?articleID=12866
Google Scholar
↵
Hampel, F. R. (1971). A general qualitative definition of robustness. The Annals of Mathematical Statistics, 42(6), 1887–1896. https://doi.org/10.1214/aoms/1177693054
Google Scholar OpenURL
↵
Hartinger, H., & Brunner, F. K. (1999). Variances of GPS phase observations: The SIGMA-ε model. GPS Solutions, 2(4), 35–43. https://doi.org/10.1007/pl00012765
Google Scholar OpenURL
↵
Hsu, L.-T., Gu, Y., & Kamijo, S. (2015). NLOS correction/exclusion for GNSS measurement using RAIM and city building models. Sensors, 15(7), 17329–17349. https://doi.org/10.3390/s150717329
Google Scholar OpenURL
↵
Huber, P. J. (1964). Robust estimation of a location parameter. The Annals of Mathematical Statistics, 35(1), 73–101.
Google Scholar OpenURL
↵
Huber, P. J., & Ronchetti, E. M. (2009). Robust statistics (2nd ed.). Wiley.
Google Scholar
↵
Kim, K., & Shevlyakov, G. (2008). Why gaussianity?. IEEE Signal Processing Magazine, 25(2), 102–113. https://doi.org/10.1109/msp.2007.913700
Google Scholar OpenURL
↵
Kjørsvik, N. S., Øvstedal, O., & Gjevestad, J. G. O. (2009). Kinematic precise point positioning during marginal satellite availability. In M. G. Sideris (Ed.), Observing our changing earth. International association of geodesy symposia (Vol. 133, pp. 691–699). Springer, Berlin, Heidelberg.
Google Scholar OpenURL
↵
Knight, N. L., & Wang, J. (2009). A comparison of outlier detection procedures and robust estimation methods in GPS positioning. Journal of Navigation, 62(4), 699–709. https://doi.org/10.1017/s0373463309990142
Google Scholar OpenURL
↵
Lee, Y., Wang, P., & Park, B. (2023). Nonlinear regression-based GNSS multipath dynamic map construction and its application in deep urban areas. IEEE Transactions on Intelligent Transportation Systems, pp. 1–12. https://doi.org/10.1109/tits.2023.3246493
Google Scholar
↵
Li, H., Medina, D., Vilà-Valls, J., & Closas, P. (2019, September). Robust Kalman filter for RTK positioning under signal-degraded scenarios. In Proceedings of the 32nd International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+ 2019) (pp. 3717–3729). https://doi.org/10.33012/2019.17021
Google Scholar
↵
Luo, X., Mayer, M., & Heck, B. (2009). Improving the stochastic model of GNSS observations by means of SNR-based weighting. In M. G. Sideris (Ed.), Observing our changing earth. International association of geodesy symposia (Vol. 133, pp. 725–734). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85426-5_83
Google Scholar OpenURL
↵
Maronna, R., Bustos, O., & Yohai, V. (1979). Bias- and efficiency-robustness of general M-estimators for regression with random carriers. In T. Gasser & M. Rosenblatt (Eds.), Smoothing techniques for curve estimation: Proceedings of a workshop held in Heidelberg (pp. 91–116). Springer Berlin, Heidelberg. https://doi.org/10.1007/BFb0098486
Google Scholar
↵
McGraw, G. A., Groves, P. D., & Ashman, B. W. (2021). Robust positioning in the presence of multipath and NLOS GNSS signals. In Y. J. Morton, F. van Diggelen, J. J. Spilker Jr., & B. W. Parkinson (Eds.), Position, navigation, and timing technologies in the 21st century: Integrated satellite navigation, sensor systems, and civil applications (pp. 551–589). John Wiley & Sons, Inc., Hoboken, New Jersey.
Google Scholar
↵
Medina, D. (2021). Robust GNSS carrier phase-based position and attitude estimation [Doctoral dissertation, Universidad Carlos III. de Madrid].
Google Scholar
↵
Medina, D., Li, H., Vila-Valls, J., & Closas, P. (2019, October). On robust statistics for GNSS single point positioning. In 2019 IEEE Intelligent Transportation Systems Conference (ITSC) (pp. 3281–3287). https://doi.org/10.1109/itsc.2019.8917484
Google Scholar
↵
Medina, D., Li, H., Vilà-Valls, J., & Closas, P. (2021). Robust filtering techniques for RTK positioning in harsh propagation environments. Sensors, 21(4), 1250. https://doi.org/10.3390/s21041250
Google Scholar OpenURL
↵
Medina, D., Romanovas, M., Herrera-Pinzon, I., & Ziebold, R. (2016, April). Robust position and velocity estimation methods in integrated navigation systems for inland water applications. In 2016 IEEE/ION Position, Location and Navigation Symposium (PLANS) (pp. 491–501). https://doi.org/10.1109/plans.2016.7479737
Google Scholar
↵
Middleton, D. (1999). Non-Gaussian noise models in signal processing for telecommunications: New methods an results for class A and class B noise models. IEEE Transactions on Information Theory, 45(4), 1129–1149. https://doi.org/10.1109/18.761256
Google Scholar OpenURL
↵
Obst, M., Bauer, S., & Wanielik, G. (2012, April). Urban multipath detection and mitigation with dynamic 3D maps for reliable land vehicle localization. In Proceedings of the 2012 IEEE/ION Position, Location and Navigation Symposium (PLANS) (pp. 685–691). https://doi.org/10.1109/plans.2012.6236944
Google Scholar
↵
O’Connor, M., Ruwisch, F., Kersten, T., Skupin, C., Ren, L., Wübbena, T., & Schön, S. (2021, September). Low-latency GNSS multipath simulator for real-time applications in autonomous driving. In 2021 IEEE/ACM 25th International Symposium on Distributed Simulation and Real Time Applications (DS-RT) (pp. 1–9). https://doi.org/10.1109/ds-rt52167.2021.9576146
Google Scholar
↵
Odijk, D. (2017). Positioning model. In P. J. G. Teunissen & O. Montenbruck (Eds.) , Springer handbook of global navigation satellite systems (pp. 605–638). Springer International Publishing.
Google Scholar
↵
Peyraud, S., Bétaille, D., Renault, S., Ortiz, M., Mougel, F., Meizel, D., & Peyret, F. (2013). About non-line-of-sight satellite detection and exclusion in a 3D map-aided localization algorithm. Sensors, 13(1), 829–847. https://doi.org/10.3390/s130100829
Google Scholar OpenURL
↵
Pozo-Pérez, J. A., Medina, D., Herrera-Pinzón, I. D., Heßelbarth, A., & Ziebold, R. (2017, January). Robust outlier mitigation in multi-constellation GNSS-based positioning for waterborne applications. In Proceedings of the 2017 International Technical Meeting of the Institute of Navigation (pp. 1330–1343). https://doi.org/10.33012/2017.14936
Google Scholar
↵
Reid, T. G. R., Houts, S. E., Cammarata, R., Mills, G., Agarwal, S., Vora, A., & Pandey, G. (2019). Localization requirements for autonomous vehicles. SAE International Journal of Connected and Automated Vehicles, 2(3), 173–190. https://doi.org/10.4271/12-02-03-0012
Google Scholar OpenURL
↵
Reid, T. G. R., Neish, A., & Manning, B. (2023, January). Localization & mapping requirements for level 2+ autonomous vehicles. In Proceedings of the 2023 International Technical Meeting of the Institute of Navigation (pp. 107–123). https://doi.org/10.33012/2023.18634
Google Scholar
↵
Rothacher, M., & Beutler, G. (1998). The role of GPS in the study of global change. Physics and Chemistry of the Earth, 23(9–10), 1029–1040. https://doi.org/10.1016/s0079-1946(98)00143-8
Google Scholar OpenURL CrossRef
↵
Rousseeuw, P. J. (1984). Least median of squares regression. Journal of the American Statistical Association, 79(388), 871–880. https://doi.org/10.1080/01621459.1984.10477105
Google Scholar OpenURL CrossRef Web of Science
↵
Ruwisch, F. (2025). GNSS feature maps — robust lane-level accurate GNSS navigation in urban trenches [Doctoral dissertation, Leibniz Universität Hannover]. Ausschuss Geodäsie der Bayerischen Akademie der Wissenschaften, Reihe C ; 967. https://doi.org/10.15488/19718
Google Scholar
↵
Ruwisch, F., & Schön, S. (2022a, January). GNSS feature map: Representation of signal propagation-related features in urban trenches. In The International Technical Meeting of the Institute of Navigation (pp. 701–711). https://doi.org/10.33012/2022.18171
Google Scholar
↵
Ruwisch, F., & Schön, S. (2022b, September). Performance assessment of GNSS RTK positioning in urban environments: Outlier detection versus 3DMA-FDE. In Proceedings of the 35th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+) (pp. 2649–2663). https://doi.org/10.33012/2022.18510
Google Scholar
↵
Ruwisch, F., & Schön, S. (2023, September). GNSS feature map aided RTK positioning in urban trenches. In 2023 IEEE 26th International Conference on Intelligent Transportation Systems (ITSC): iLoc - High-integrity Localization for Automated Vehicles (pp. 5798–5804). https://doi.org/10.1109/itsc57777.2023.10422176
Google Scholar
↵
Ruwisch, F., & Schön, S. (2025). Feature map aided robust high precision GNSS positioning in harsh urban environments. IEEE Transactions on Intelligent Transportation Systems, 26(9), 13721–13733. https://doi.org/10.1109/TITS.2025.3569975
Google Scholar OpenURL
↵
Sánchez, J. S., Gerhmann, A., Thevenon, P., Brocard, P., Afia, A. B., & Julien, O. (2016, January). Use of a fisheye camera for GNSS NLOS exclusion and characterization in urban environments. In Proceedings of the 2016 International Technical Meeting of the Institute of Navigation (pp. 283–292). https://doi.org/10.33012/2016.13404
Google Scholar
↵
Tan, T.-N., Khenchaf, A., Comblet, F., Franck, P., Champeyroux, J.-M., & Reichert, O. (2020). Robust-extended Kalman filter and long short-term memory combination to enhance the quality of single point positioning. Applied Sciences, 10(12), 4335. https://doi.org/10.3390/app10124335
Google Scholar
↵
Toissant, M., Samson, J., Toran, F., Ventura-Traveset, J., Sanz, J., Hernandez-Pajares, M., & Juan, J. M. (2006, September). The Stanford - ESA integrity diagram: Focusing on SBAS integrity. In Proceedings of the 19th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+) (pp. 894–905). https://www.ion.org/publications/pdf.cfm?articleID=6963
Google Scholar
↵
Tukey, J. W. (1960). A survey of sampling from contaminated distributions. In I. Olkin, S. G. Ghurye, W. Hoeffding, W. G. Madow, & H. B. Mann (Eds.) , Contributions to probability and statistics: Essays in honor of Harold Hotelling (pp. 448–485). Stanford University Press.
Google Scholar
↵
Wang, L., Groves, P. D., & Ziebart, M. K. (2015). Smartphone shadow matching for better cross-street GNSS positioning in urban environments. Journal of Navigation, 68(3), 411–433. https://doi.org/10.1017/S0373463314000836
Google Scholar OpenURL
↵
Wang, R., Becker, D., & Hobiger, T. (2023). Stochastic modeling with robust Kalman filter for real-time kinematic GPS single-frequency positioning. GPS Solutions, 27(3). https://doi.org/10.1007/s10291-023-01479-5
Google Scholar
↵
Wen, W., & Hsu, L.-T. (2021, September). 3D LiDAR aided GNSS real-time kinematic positioning. In Proceedings of the 34th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+) (pp. 2212–2220). https://doi.org/10.33012/2021.18072
Google Scholar
↵
Wendel, J. (2022, September). GNSS pseudorange fault detection and exclusion with multiple outliers. In Proceedings of the 35th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+) (pp. 1481–1495). https://doi.org/10.33012/2022.18421
Google Scholar
↵
Wieser, A., & Brunner, F. K. (2000). An extended weight model for GPS phase observations. Earth, Planets and Space, 52(10), 777–782. https://doi.org/10.1186/bf03352281
Google Scholar OpenURL
↵
Xin, S., Geng, J., Zhang, G., Ng, H.-F., Guo, J., & Hsu, L.-T. (2022). 3D-mapping-aided PPP-RTK aiming at deep urban canyons. Journal of Geodesy, 96, 78. https://doi.org/10.1007/s00190-022-01666-1
Google Scholar
↵
Yohai, V. J. (1987). High breakdown-point and high efficiency robust estimates for regression. The Annals of Statistics, 15(2). https://doi.org/10.1214/aos/1176350366
Google Scholar
↵
Zhu, N. (2018). GNSS propagation channel modeling in constrained environments: Contribution to the improvement of the geolocation service quality [Doctoral dissertation, Université de Lille].
Google Scholar
↵
Zoubir, A. M., Koivunen, V., Chakhchoukh, Y., & Muma, M. (2012). Robust estimation in signal processing: A tutorial-style treatment of fundamental concepts. IEEE Signal Processing Magazine, 29(4), 61–80. https://doi.org/10.1109/msp.2012.2183773
Google Scholar OpenURL

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Adapted Robust GNSS RTK Positioning: Improving Accuracy and Integrity for Autonomous Vehicle Localization in Urban Trenches

Abstract

1 INTRODUCTION

2 RECALLING ROBUST STATISTICS

2.1 Maximum Likelihood Estimation for One-Dimensional Data

2.2 M-Estimator

ALGORITHM 1

2.3 S-Estimator

2.4 MM-Estimator

3 REVIEW OF THE LITERATURE ON ROBUST ESTIMATORS IN GNSS-BASED LOCALIZATION

4 THE HG-ESTIMATOR - AN ADAPTED ROBUST WEIGHT MODEL

4.1 Definition

4.2 Performance Simulation of Robust Estimators

4.3 GNSS Feature Map-Aided Robust Estimation

ALGORITHM 2

5 APPLICATION EXAMPLES AND CASE STUDIES

5.1 Automotive Experiment in a Medium Urban Trench

5.1.1 Accuracy

5.1.2 Attainable Integrity

5.2 Automotive Experiment in a Deep Urban Trench

5.2.1 Accuracy

5.2.2 Attainable Integrity

6 CONCLUSION

HOW TO CITE THIS ARTICLE:

ACKNOWLEDGMENTS

REFERENCES