Improving the Accuracy of Vehicle Position in an Urban Environment Using the Outlier Mitigation Algorithm Based on GNSS Multi-Position Clustering

Abstract

In this paper, we propose a multi-position cluster-based weighted position estimation method that minimizes the influence of multipath (MP)/non-line-of-sight (NLOS) signals using a global navigation satellite system (GNSS) receiver. The proposed method is suitable for positioning passenger cars, particularly in urban driving environments. Density-based spatial clustering of applications with noise (DBSCAN)-based clustering is performed by generating multi-position data through a subset of observable satellites and analyzing the density characteristics of position data generated by line-of-sight (LOS) satellite signals from the generated multi-position data. To estimate the optimal position through clustered data, we propose a method by constructing a weighted model through Doppler-based velocity measurements, which is robust to MP delay signals compared to code-based pseudorange measurements. In addition, to prevent unnecessary clustering points from having weights, the predicted range is selected based on the nonholonomic constraint of the vehicle. The proposed algorithm was quantitatively validated by selecting a region in an actual urban environment where the MP/NLOS error could occur significantly. It was confirmed that the accuracy of vehicle position was improved by approximately 24% by the proposed method compared to existing positioning solutions.

1. Introduction

The global navigation satellite system (GNSS), which was initially confined to aircraft, ships, military, weapon systems, and surveying, has recently been used in personal, corporate, and public domains with the development of artificial intelligence, big data, and information communication technology. In particular, the scope of application of GNSS is expanding in various forms in many fields, such as simple location checks of pedestrians and vehicles, road guidance services, logistics, road control services, and shared mobility services [1,2,3,4]. Recently, interest in autonomous driving has increased, owing to the development of computing infrastructure (cloud and central processing unit). As autonomous driving systems are gradually introduced into real driving environments, the accurate positioning of the vehicle is considered an important requirement to reach a destination without driver intervention and with a high degree of autonomy. However, in general, the performance of low-cost GNSS receivers installed in vehicles cannot compensate for signal-blocking factors such as multipath (MP)/non-line-of-sight (NLOS) signals when the reception environment is poor. Therefore, positioning is impossible, or its accuracy is significantly reduced. The limited accuracy and unavailability of MP/NLOS-contaminated positioning is recognized as a major limitation of its use in urban areas because it cannot meet the requirements of some safety-critical applications. Nevertheless, the satellite-based positioning method is still considered one of the most valuable components of an autonomous driving system because it has a higher accuracy compared to other positioning technologies and does not require additional infrastructure. In addition, because the satellite-based positioning method is integrated with inertial navigation systems, network-based wireless positioning systems, and other auxiliary sensors, it is essential to provide basic information for various positioning services [5]. Therefore, reducing the GPS positioning error in an urban environment is an important step for improving the position solution provided by the integrated navigation system. There is need for technology that is accurate and reliable in determining the position by discriminating GNSS signal types.

Various methods have been proposed to reduce position errors by predicting MP/NLOS signals [6,7,8,9,10]. One of the most representative techniques is the removal of outliers based on observation data obtained from satellites. In the case of an urban environment, because the quality of observation data deteriorates owing to signal shielding factors, the position is estimated by selecting observation data based on the dilution of precision (DOP), signal-to-noise ratio (SNR), and elevation angle to construct a weighted model. Li et al. proposed a weighted model to reduce the MP error in an urban environment based on SNR [11]. The basic principle is to reduce the weight of the observation satellite as the SNR signal, which is the strength of the satellite signal. This technique is used for the estimation of position by minimizing the influence of the MP signal. Tay et al. proposed a weighted model to reduce the MP/NLOS error by using the SNR and elevation angle to complement the model that used only the elevation angle [12]. The basic principles of the model are as follows. As the elevation angle of the satellite decreases, the signal from the satellite is susceptible to the influence of ionospheric and tropospheric errors as well as the signal shielding factor in the urban environment. In addition, in an urban environment, even if the elevation angle is high, the SNR can be lowered by signal diffraction, MP/NLOS effects, and receiver characteristics. However, there is a limit to predicting the MP/NLOS signal using only observational data, because the strength of the MP/NLOS signal may be stronger than that of the line-of-sight (LOS) signal in an urban environment [13]. To compensate for this, an antenna with good performance and multiple receivers may be used, but additional costs are incurred. Thus, the implementation of the navigation system may be limited in terms of cost-effectiveness.

To overcome this limitation, derivative methods for processing MP/NLOS signals using 3D-matching technology have been studied [14,15,16]. They are based on the principle of more accurately estimating the position using a 3D building model to predict LOS and NLOS signals according to the elevation and azimuth of the building boundary. Despite the effectiveness of this position estimation method, it is expensive to build and maintain an accurate 3D building model, and it is necessary to separately build a database with a large amount of building information for each position. Therefore, in this study, a multi-position clustering-based weighted position estimation method was proposed for a vehicle-based navigation system to implement cost-effective solutions while limiting the use of additional infrastructure. This method distinguishes between MP/NLOS and LOS signals for position estimation.

As the existing pseudorange-based positioning technique in an urban environment cannot predict the satellite signal contaminated with MP/NLOS, it generates multi-position data by using a combination of observed satellites with various signal characteristics. Accordingly, we propose a weighted position estimation method based on Doppler velocity to predict position data composed of LOS signals from the generated position data. First, multi-position data are constructed by positioning a reception point through a subset of observable satellites, and the consistency of position data composed of various signal characteristics is analyzed. By performing unsupervised learning and density-based spatial clustering of applications with noise (DBSCAN) [17] using the analyzed consistency, position data composed of LOS signals can be clustered. The cluster speed was increased by defining two parameters required for DBSCAN considering the characteristics of position data composed of LOS to overcome the processing delay of the existing clustering algorithm. We propose a weighted model using Doppler-based velocity measurements using cluster data that are robust against MP errors. We present a method for estimating the position by selecting the prediction range that meets the nonholonomic constraint condition of the vehicle with the velocity measurement value and assigning weight to the average position of each cluster within the selected range. The proposed algorithm can overcome the limitations of the position distortion of the weighted least-squares method (WLSM), which is known as a single satellite-based positioning method in an urban environment. In addition, it is possible to provide a more optimal measurement signal to realize an integrated vehicle navigation system with improved position accuracy and cost-effectiveness in an urban environment.

The article’s contributions are summarized as follows:

• Generate position data for a receiving point based on a subset of observable satellites and calculate the position of the receiving point without relying on observation data. There is no need to take into account errors in position estimation caused by distortions in the observation data;
• In order to overcome the positioning error caused by observation data, we have utilized a density-based clustering technique, and also, by defining the initial input parameters for density-based clustering, we can effectively identify position data that are combined with LOS satellites. This approach not only addresses the drawback of processing speed commonly associated with clustering methods but also mitigates the inaccuracies resulting from observation data;
• To identify position data combined with LOS satellites from various clustered data points, we utilize the concept of nonholonomic constraints based on Doppler velocity measurements. This allows us to set a prediction range for improved position estimation. Furthermore, we incorporated weighted factors into the extracted clustering data and defined a probability density function based on exponential function.

The reminder of this paper is organized as follows. Section 2 details previous studies that are similar to and different from our study. Section 3 describes the process of obtaining position data through a subset of observable satellites used in this algorithm and analyzes the consistency of position data. Based on this, Section 4 presents the DBSCAN initial parameter selection value for clustering position data composed of clean LOS, and proposes a weighted position estimation method based on speed measurement. In Section 5, we verify and analyze the performance of the proposed algorithm by comparing it with existing methods using experimental data. Finally, in Section 6, we present the conclusions.

2. Related Work

Recently, a data-mining-based positioning estimation method that detects outliers in a data set to reduce the operating cost and improve the usability of the 3D building matching-based position estimation method was presented [18]. The proposed method consists of a matching relationship between the GNSS signal categories and multiple feature parameters that can be directly or indirectly extracted from a RINEX file format obtained from a GNSS device. A method for identifying MP/NLOS signals through a data-mining algorithm using machine learning or deep learning from the constructed relationship was proposed.

Classification algorithms for data mining such as decision tree [19], support vector machine [20], and convolutional neural networks [21] are commonly utilized and studied. However, classification methods based on supervised learning require accurate pre-labeling of training samples to represent all types of states to estimate position effectively. In addition, because the currently employed labeling method uses data based on a 3D building model, the operating costs for maintaining the model are incurred. There are limitations in using observational data and 3D building models owing to inaccuracy of building models, SNR distortion caused by building materials, different SNR characteristics for each GNSS, and factors that interfere with the reception of various satellite signals other than those by buildings.

Unsupervised learning-based position estimation algorithms have gained attention in recent research as they offer improvements over the drawbacks of supervised learning-based methods and enhance the accuracy of traditional single-receiver positioning algorithms.

Table 1. A comparative evaluation of previous research in urban positioning based on unsupervised learning.

Authors	Identification Technique	Contribution	Algorithm Verification	Note
Y. Xia et al. [9]	Chi-square and HDBSCAN	The authors present the HDBSCAN algorithm to label an offline dataset as normal based on observation data, effectively contributing to the estimation of position through a hybrid machine learning framework designed for GNSS anomaly detection.	Urban scenario	Need training set for labeling
R. Yozevitch et al. [18]	Decision tree and expectation maximization	The authors present the classification of LOS and NLOS scenarios using satellite signal strength. This classification is achieved through the application of both supervised and unsupervised learning technique.	Specific section and small scenario in an urban environment	Need training set for labeling
Uaratanawong et al. [23]	K-means	The authors focused on identifying LOS and NLOS signal using K-means based on SNR residual obtained through a classical noise model for carrier phase.	Specific points	Two location points: surrounded by buildings and rooftop of the building
Hao Wang et al. [25]	K-means	The authors considered different feature parameters derived from observation data to effectively classify GNSS signals: pseudorange, carrier phase, SNR, doppler frequency, etc.	Urban scenario	The author focused on multipath/NLOS detection
Luo et al. [26]	DBSCAN	The authors considered the problem by assigning a large weight to low mobility and overcoming the distortion of observation data based on the position trajectory without using observation data.	Stop section	MP/NLOS signals were not considered in an urban environment, and they focused only on the stop section.
M. R. Mosavi [27]	Ant colony optimization	The author present satellites geometry clustering for good navigation satellites subset selection.	Simulation

lists the comparative evaluation of previous research in urban positioning based on unsupervised learning. These algorithms leverage clustering through unsupervised learning to estimate positions. They employ similarity calculations of position trajectory data or observation data to classify multiple data points into a single cluster. Uaratanawong et al proposed a technique for selecting NLOS and LOS signals using the K-means [22] algorithm based on SNR residual that can be obtained through the classical noise model for the carrier phase. Thus, the positioning accuracy was efficiently improved in the MP section [23]. However, when implementing the K-means algorithm, if there is an outlier in the SNR residual, a position estimate distorted by the K-means set value K is obtained. The proposed algorithm’s validity is challenging to evaluate due to the limited verification conducted at only two specific points. Therefore, it is difficult to prove efficiency in an urban environment. Research on position estimation methods based on K-means is actively being conducted. However, in scenarios where there are a larger number of NLOS satellites, there are significant distortions in the observed data and there is a possibility of erroneous estimations due to incorrect initial configuration. As a result, additional infrastructure is often required to be implemented and utilized to address these challenges [24,25]. Luo et al. proposed a clustering technique based on position trajectory density [26]. Mobility was considered by assigning a large weight to low mobility and overcoming the distortion of observation data based on the location trajectory without using observation data. However, MP/NLOS signals were not considered in an urban environment, and the study focused only on the stop section. In addition, this trajectory data-based position estimation technique has limitations in estimating positions for non-empirical path data because similar trajectories are clustered from a temporal or spatial point of view, and a large amount of trajectory data are required to estimate the position accurately. Although prior studies have been conducted on cluster-based localization methods using the similarity of observation data, there are few studies considering MP/NLOS signals in urban environments through position trajectories and position measurements [27,28,29,30,31,32]. In addition, it is difficult to prove the efficiency of these urban environments position methods solely through simulators. Furthermore, a hybrid machine learning framework that combines supervised and unsupervised learning has been proposed as a method for GNSS anomaly detection [9,18]. This approach utilizes recent observation data and online/offline navigation datasets. However, this method ultimately requires a training phase for label assignment and may be vulnerable to estimating positions for unknown paths.

Although mean shift [33] and Gaussian mixer models (GMM) [34] are widely used clustering algorithms, these algorithms have the disadvantage of being able to misestimate clustering region owing to incorrect initial setting values. Consequently, these algorithms are not commonly utilized in approaches that rely on position data and observation data. The use of mean shift and GMM methods can lead to incorrect group selection owing to differences in distribution with normal position data, especially in urban environments with many MP/NLOS signals, However, the cluster selection method through DBSCAN is adopted because it has the advantage of being able to cluster with a desired distribution even with a very small amount of data.

3. Positioning Method and Consistency Analysis Based on a Subset of Observable Satellites

In general, many prior studies use a position estimation technique based on code pseudorange by constructing a weighted model based on observation data to compensate for the distorted satellite signal. However, because it is impossible to predict satellite signals, including MP/NLOS, using only observation data in extreme urban environments, this study proposes a method for estimating position data with various signal characteristics. Accordingly, in the positioning method, the receiving point is positioned through a least-square method (LSM) solution that enables simple and fast position estimation rather than a weighted model that can distort the positioning for the consistency of position data.

3.1. Receiver Observation Model

A positioning technique based on the LSM consists of the estimation of the receiver position from a signal transmitted by an observable satellite. The distance between the satellite and receiver can be represented by the geometric relationship between the satellite and receiver positions, including clock bias and other propagation delays. By measuring the distance between the observable satellite and receiver, the pseudorange can be calculated as follows [35]:

$${\rho }_u=\sqrt{{\left(x-x_u\right)}^2+{\left(y-y_u\right)}^2+{\left(z-z_u\right)}^2}+c·∆b_u+\epsilon ,$$

(1)

where

• ${\rho }_u$: pseudorange from the satellite to the receiver;
• $x_u={\left[x_u y_u z_u\right]}^T$: receiver position in the Earth-centered, Earth-fixed (ECEF) coordinate system;
• $x={\left[x y z\right]}^T$: satellite position in ECEF coordinates;
• $c$: speed of light;
• $∆b_u$: receiver clock bias;
• $\epsilon$: error term associated with the $i$-th propagation channel (ionospheric delay, tropospheric delay, satellite clock bias, satellite position uncertainty, and MP/NLOS error).

Equation (1) is linearized based on the nominal point and arranged for the selected $i$ satellites, which can be expressed as follows:

$$∆\rho =H_i∆x_u+{\epsilon }^i,$$

(2)

where

• $∆\rho ={\rho }_u^i-{\rho }_{u,0}^i\in {ℝ}^N$, ${\rho }_{u,0}^i=\Vert x^i-x_u\Vert +c∆b_u+\epsilon$;
• $∆x_u={\left[∆x_u ∆y_u ∆z_u c∆b_u\right]}^T\in {ℝ}^4$;
• ${\epsilon }^i\in {ℝ}^N$ is an error term owing to the possible presence of MP affecting the pseudoranges (we assume that all errors except MP have been corrected);
• $i$: number of selected satellites from at least four to $N$;
• $H_i=\left[\begin{array}{cccc}-\frac{x^1-x_0}{{\rho }_{u,0}^1}& -\frac{y^1-y_0}{{\rho }_{u,0}^1}& -\frac{z^1-z_0}{{\rho }_{u,0}^1}& 1\\ -\frac{x^2-x_0}{{\rho }_{u,0}^2}& -\frac{y^2-y_0}{{\rho }_{u,0}^2}& -\frac{z^2-z_0}{{\rho }_{u,0}^2}& 1\\ ⋮& ⋮& ⋮& 1\\ -\frac{x^N-x_0}{{\rho }_{u,0}^N}& -\frac{y^N-y_0}{{\rho }_{u,0}^N}& -\frac{z^N-z_0}{{\rho }_{u,0}^N}& 1\end{array}\right]\in {ℝ}^{N\times 4}$, the Jacobian matrix associated with the linearized system.

The method of generating multi-position data with a subset of observable satellites is based on the fact that the position of the receiver must be calculated from at least four or more satellites. When a total of N satellites are observed at a reception point at specific moment, multi-position data are generated by estimating the position of the reception point from a combination of _nC_r $\left(4\le r\le n\right)$ satellite signals. For example, if nine satellites are observed, the maximum of 382 position data points can be obtained. Accordingly, N selected satellites are determined by a subset of observable satellites, and the number of selected satellites and the maximum number of position data vary according to the masking values for the SNR, elevation angle, and DOP.

Table 2. A set of position data calculated based on a subset of observation satellites.

Number of Observable Satellites	Maximum Number of Position Data for all Combination of Satellites	Matrix H
4	4C4	H4
5	5C4 + 5C5	H4,  H5
$⋮$	⋮	⋮
N	NC4 + NC5 + $\cdots$ + NCN	H4,  H5, $\cdots ,$HN

lists the size and maximum position data of matrix $H_i$ for a subset of satellites according to the number of observable satellites.

Through the LSM solution [36], the position estimation error and the estimated position of the receiver can be defined as in (3) and (4), respectively, and multi-position data are generated through a subset of observable satellites.

$$∆{\hat{x}}_u=inv\left(H_i^T{H}_i\right)H_i^T∆\rho ,$$

(3)

$${\hat{x}}_u=x_{u,0}+∆{\hat{x}}_u$$

(4)

3.2. Consistency Analysis of Position Data

An error analysis of the position of the receiver obtained from a subset of satellites must first be performed to accurately estimate the position of the reception point using the generated multi-position data. The position error of a receiver mounted on a vehicle varies depending on the DOP, which indicates the accuracy of the determined position of the satellite’s relative geometry, and the user equivalent range error (UERE), which is a pseudorange measurement error. In addition, because the vehicle does not move vertically, the horizontal position error of the receiver can be considered the main focus.

The horizontal position error can be expressed by the relational expression of DOP $D$ and UERE ${\sigma }_{UERE}$ from the error covariance calculated in the LSM solution, as follows [37]:

$$\begin{array}{cc}\hfill \mathrm{E}\left[∆{\hat{x}}_u∆{\hat{x}}_u^T\right]& ={\left(H_i^T{H}_i\right)}^{-1}{H}_i^{T}E\left[∆\rho ∆{\rho }^T\right]H_i{\left(H_i^T{H}_i\right)}^{-1}\hfill \\ & ={\left(H_i^T{H}_i\right)}^{-1}{H}_i^T\left[{\sigma }_{UERE}^2{I}_{i\times i}\right]H_i{\left(H_i^T{H}_i\right)}^{-1}\hfill \\ & ={\sigma }_{UERE}^2{\left(H_i^T{H}_i\right)}^{-1}\hfill \\ & ={\sigma }_{UERE}^{2}D\hfill \end{array}$$

(5)

$$\mathrm{Cov}\left[∆{\hat{x}}_u\right]={\sigma }_{UERE}^2\left[\begin{array}{cccc}{D}_{11}& D_{12}& D_{13}& D_{14}\\ D_{21}& D_{22}& D_{23}& D_{24}\\ D_{31}& D_{32}& D_{33}& D_{34}\\ D_{41}& D_{42}& D_{43}& D_{44}\end{array}\right],$$

(6)

where

• $D={\left(H_i^T{H}_i\right)}^{-1}=inv\left(H_i^T{H}_i\right)$: position precision by the placement of satellites;
• $D_{ij}$: the i-th row and j-th column diagonal element of matrix $D$;
• ${\sigma }_{UERE}^2$: total error affecting a pseudorange from the user’s point of view; user equivalent range error.

Through a series of processes, the $D_{\mathit{HDOP}}$ can be expressed as (7). As UERE ${\sigma }_{UERE}$ cannot be predicted by practically representing each error component, it can be defined as the root-mean-square (RMS) value of the pseudorange residual for all satellites in the section where the navigation solution converges, as shown in (8).

$$D_{HDOP}=\sqrt{D_{11}+D_{22}}$$

(7)

$${\sigma }_{UERE}=\sqrt{\frac{1}{N·K}{\displaystyle \sum }_{t=1}^K\left({\displaystyle \sum }_{i=1}^N{\left(∆{\rho }^i\left(t\right)\right)}^2\right)},$$

(8)

where

• $∆{\rho }^i\left(t\right)$: pseudorange error value of the $i$-th satellite computed at time $t$;
• $N$: the number of visible satellites;
• $K$: total number of epochs used to calculate the UERE.

From $D_{HDOP}$ and ${\sigma }_{UERE}$ obtained by (7) and (8), the horizontal position error ${\sigma }_{hori}$ of the receiver can be expressed as follows:

$${\sigma }_{hori}=D_{HDOP}{\sigma }_{UERE}$$

(9)

In (9), $D_{HDOP}$ can select a masking value according to a user’s specification in the process of forming position data so that the reception point whose horizontal position error of the receiver determined by the UERE is positioned.

The horizontal position error of reception points in an urban environment can be predicted using combined satellite signal characteristics, and crowding can be performed using the predicted reception points. Therefore, in the case of urban environments, satellite signal characteristics need to be classified in order to classify the characteristics of combined position data. The position data were assumed and classified based on satellite signal errors, which have been extensively studied in previous research.

• Position data consisting only of the LOS signal;
• Position data composed of signals with large MP/NLOS delays;
• Position data composed of signals with various error characteristics.

To analyze and verify the cases, it was evaluated using the differential GNSS (DGNSS) as a reference. The GNSS receiver used in the experiment was Ublox’s ZED-F9P-04B and ANN-MB-00. The masking values based on observation data used in the experiment are listed in

Table 3. GNSS setting and masking value used in the experiment.

Ublox-ZED-F9P-04B
GNSS	GPS, QZSS
Number of observable satellites	$6~12$
Frequency	5 Hz
DOP	$<3$
Elevation angle	<15
CEP	1.5 (in case of DOP = 1)
Speed	Doppler shift

. The dataset was made public at https://github.com/kakusang2020/iXR_GNSS-IMU_TightlyCouplingProgram/tree/master/Data (accessed on 7 September 2021).

Case 1 indicates open areas where there satellite signals are not contaminated with MP/NLOS. To estimate the UERE consisting of LOS signals, the pseudorange error for each observed satellite must be analyzed, which can be estimated using the positional accuracy of the GNSS receiver used in this study.

On average, in rehabilitation settings, GNSS receivers have horizontal position accuracies of 1.5–2.0 m circular error probable (CEP) [38,39]. CEP implies that there is a 50% probability that the true horizontal position coordinates exist within the radius $R$ of the receiving point and follows a bivariate normal distribution. Therefore, if the random variables for the horizontal position error in the horizontal reference x- and y-directions are $x$ and $y$, the two random variables are given as follows [40,41]:

$$p\left(x,y\right)=\frac{1}{2\pi {\sigma }_{GNSS}^2}{e}^{\left(-\frac{\left(x^2+y^2\right)}{2{\sigma }_{GNSS}^2}\right)},$$

(10)

where

• ${\sigma }_{GNSS}^2={\sigma }_x{\sigma }_y$: multiplication of the standard deviations for discrepancies between observation points in each direction.

The probability that an observation point is in the draft of radius $R_{0.5}$ with 50% probability from the position is

$$\begin{array}{cc}\hfill P\left(R_{0.5}\right)& ={{\displaystyle \iint }}_{x^2+y^2\le {\left(R_{0.5}\right)}^2}p\left(x,y\right)dxdy=0.5\hfill \\ & =1-e^{-\frac{{\left(R_{0.5}\right)}^2}{2{\sigma }_{GNSS}^2}}=0.5\hfill \end{array}$$

(11)

CEP is expressed in (12), and the horizontal position error ${\sigma }_{GNSS}$ can be obtained using an inexpensive receiver performance index.

$$R_{CEP}=1.1774{\sigma }_{GNSS}$$

(12)

As the open-field environment has an ideal satellite arrangement $D_{HDOP}\approx 1$, the horizontal position error of the GNSS receiver ${\sigma }_{GNSS}$ obtained using (12) can be defined as equal to the pseudorange error $∆{\rho }_{LOS}\left(t\right)$ of the GNSS receiver.

$${\sigma }_{GNSS}\approx 1\times {\sigma }_{GL}$$

(13)

$$\begin{array}{cc}\hfill {\sigma }_{GL}& \approx \sqrt{\frac{1}{N·K}{\displaystyle {\displaystyle \sum }_{t=1}^K}\left({\displaystyle {\displaystyle \sum }_{i=1}^N}{\left(∆{\rho }_{LOS}^i\left(t\right)\right)}^2\right)}\hfill \\ & \approx \sqrt{\frac{1}{N}{\displaystyle {\displaystyle \sum }_{i=1}^N}{\left(∆{\rho }_{LOS}^i\right)}^2}\hfill \\ & \approx ∆{\rho }_{LOS}\left(t\right)\hfill \end{array}$$

(14)

In case 1, a small number of LOS signal satellites and a constellation of satellites limited to an open area depending on the combination state can generate position data with various position errors owing to the large fluctuation range of $D_{HDOP}$. Therefore, the precision of position data can be defined by selecting the masking value for $D_{HDOP}$ as follows:

$${\sigma }_{case1}=D_{HDOP}{\sigma }_{GNSS}$$

(15)

Figure 1 shows the result of outputting multi-position data through a combination of satellite signals observed in an open area. As shown in Figure 1, the distribution of position data consisting of only LOS should be represented in the form of a bivariate normal distribution with a standard deviation of ${\sigma }_{LOS}$ for each direction based on the true value, which is similar to the precision of the GNSS receiver. Therefore, the position data distribution characteristics shown in Figure 1 are that of an open area, and all satellite signals can be regarded as LOS signals.

In case 2, the pseudorange error ${\sigma }_{MP,NLOS}$ can contain $k$ large MP/NLOS delay errors $∆{\rho }_{MP,NLOS}^i\left(t\right)$ and follows:

$${\sigma }_{MP,NLOS}=\sqrt{ \frac{1}{N}\left({\displaystyle \sum }_{i=1}^{N-k}{\left(∆{\rho }_{LOS}^i\right)}^2+{\displaystyle \sum }_{i=1}^k{\left(∆{\rho }_{MP,NLOS}^i\left(t\right)\right)}^2\right)}$$

(16)

where

• $N$: number of selected satellites based on the subset;
• $k$: number of MP/NLOS satellites; $N\ge k$;
• $∆{\rho }_{MP,NLOS}^i\left(t\right)$: $i$-th satellite signal with a large MP/NLOS error.

Case 2 shows a bivariate normal distribution with a position error of ${\sigma }_{case2}$ in each direction from the average position to which the large MP/NLOS delay signal is applied according to (17) as follows:

$${\sigma }_{case2}=D_{HDOP}{\sigma }_{MP,NLOS}$$

(17)

Theoretically, in the case of a GPS L1 signal with an interval of approximately 300m per chip, the pseudorange is approximately 70m or more owing to the MP/NLOS delay signal caused by the urban environment [42]. This characteristic is due to signal delays caused by large terrain features such as large buildings and steel tower structures commonly found in urban environments. As the number of observation satellites with extreme MP/NLOS delayed signals increases, the position error of the receiving point increases significantly. For example, assuming that only one satellite out of 10 observation satellites has a delay signal of $70 \mathrm{m}$, a position error of approximately $9 \mathrm{m}$ is obtained according to (16). Therefore, even if only one extreme MP/NLOS delay signal occurs, it causes a large position error. As shown in Figure 2, there is a large difference between position data consisting of only LOS and position data, including extreme MP/NLOS delay signals.

In Figure 3, position data shows a relatively wide distribution compared to Figure 2, as it contains pseudorange errors with various signal combinations and varying delay characteristics. In general, the pseudorange caused by the MP/NLOS delayed signal generated in an urban environment results in an error of at least ten to several tens of meters [43,44]. Accordingly, in case 3, an increase in the number of MP/NLOS satellites leads to a separation of multi-position data, resulting in position errors in the receiver. Therefore, it is necessary to understand the characteristics between position data composed of LOS signals and MP/NLOS errors obtained from multi-position data with various delay errors in the pseudorange signals.

Based on numerous experimental results in urban environments, many studies have defined the position error of MP signals at the 1-sigma level as $20 \mathrm{m}$ [45,46,47]. Therefore, through previous research indicators, the position error can be analyzed, and even if a delayed signal from one satellite is received, it can cause a minimum error of 6.6m. It can be observed that this has a larger error range than the maximum error (≤$95\%$) of $3.12 \mathrm{m}$ in position data generated in an LOS environment. As a result, MP/NLOS data have various distribution characteristics because they are composed of various signal characteristics, and a greater number of position data composed of small delay signals can be formed compared to position data composed of LOS (Figure 3). As the number of MP/NLOS satellites increases, the distance between position data composed of LOS signals in the multi-position data also increases.

4. Weighted Position Estimation Method through LOS Satellite-Based Position Data Clustering

4.1. DBSCAN Parameter Definition

The DBSCAN algorithm quantitively classifies clustering and noise based on the density of points. The noise and density output by Eps, the size of radius to define the density region from the reference point, and Minpts, the minimum number of data to define the minimum cluster size, are different. Therefore, to find the optimal densities of a plurality of positions generated from satellites with LOS signals, a process for finding Eps and Minpts values (two parameters used in DBSCAN) is required. However, this is considered a major disadvantage of clustering algorithms because a considerable processing delay occurs in the process of selecting clusters. Therefore, in this study, the processing speed was improved by defining the parameters required by the DBSCAN to overcome the disadvantages.

To predict the plurality of position data generated from the LOS signal, the maximum horizontal position error of the receiving point positioned with the LOS signal must first be defined. The maximum horizontal error of the plurality of position data generated from the LOS signal must be smaller than that of position data generated by MP/NLOS so that the distribution of the plurality of position data can be divided into LOS signals and MP/NLOS signals. For example, the horizontal position error may be relatively large because a high $D_{HDOP}$ may be formed by a narrow arrangement of satellites despite estimating a position composed of LOS signals. This causes horizontal position errors, such as position data generated by MP/NLOS, making clustering impossible. As a result, the maximum horizontal position error is considered by selecting the masking value of $D_{HDOP}$ because it is impossible to cluster a plurality of position data generated by the correct LOS. $D_{HDOP}$ can be considered from (9). The reception point of the LOS signal increase in proportion to the horizontal dilution of precision because UERE can be estimated through the performance index of a low-cost receiver in (9).

It is ideal to have a horizontal position error that includes the length of the primary lane and the radius of the vehicle’s overall length to meet the requirements of a vehicle navigation system by estimating a reliable position in narrow alleys and intersections in an urban environment, as shown in Figure 4.

Globally, the length of a primary lane is regulated to be $3.0\text{~}3.5 \mathrm{m}$ on average and considering the average overall length of a vehicle (about $5 \mathrm{m}$), the maximum horizontal error can be assumed to be 2× distance root-mean-square (2DRMS). The masking value of $D_{HDOP}$ can be estimated as follows through the assumed maximum horizontal error:

$$\frac{2DRMS }{2.45{\sigma }_{GNSS}}\ge D_{HDOP},$$

(18)

where

• $2DRMS$: maximum horizontal position error ≈ 6.1 m;
• $2.45{\sigma }_{GNSS}$: Low GNSS receiver horizontal position error (95% accuracy) ≈ 3.12 m.

The maximum horizontal position error when the $D_{HDOP}$ is maximum from (18) is shown if Figure 5. By defining maximum $D_{HDOP}$ based on the length of the vehicle and the lane, we can predict the maximum size of CEP composed of LOS signals. With this, effective clustering of position data composed of LOS signals can be achieved by defining parameters for clustering.

To cluster multi-position data generated by the LOS signal, the size Eps of the radius is defined as 3${\sigma }_{noise}$ of the generally known receiver noise (${\sigma }_{noise}=0.4$) [48,49]. This can obtain an area with a small number of data, and Minpts can be defined as five, considering the minimum number of combinations of minimum observable satellites. Multi-position data generated by the LOS satellite signals may be clustered only when the position error between position data generated by LOS satellite signal and MP/NLOS is greater than or equal to the defined parameter. For example, assuming case 2, which can be composed of the above-described position data, clustering is possible because one satellite signal is affected by a large MP/NLOS signal and has a position difference greater than Eps.

4.2. Weighted Position Estimation Method

When applying the DBSCAN algorithm, a cluster area that satisfies the defined parameter values is estimated. Accordingly, multi-position data that form one cluster and have a range within the maximum error can be determined as an open-area section. However, multi-position data can be generated because the LOS signal can be clustered by satisfying the DBSCAN parameter, including the MP/NLOS signal. Therefore, multi-position data generated as LOS satellite signals were determined by considering the pseudorandom noise (PRN) code combination matching rate with the previous time within data constituting the cluster. PRN code combination matching rate means a rate that matches PRN combinations of each position data value used for previous position estimation and PRN combinations in each position data to be used for current position estimation. We propose a position estimation method by defining a weighted model using Doppler-based velocity measurements when the PRN coincidence rate is below a certain level.

First, despite the short time rate of change, the LOS satellite signal may be contaminated with MP/NLOS by buildings and other factors, resulting in signal delay errors. Theoretically, pseudorange errors of several tens of meters can be generated by an MP delay factor. However, in the case of a carrier having a wavelength of approximately $19 \mathrm{cm}$, an error of approximately $5 \mathrm{cm}$ occurs, therefore, Doppler-based velocity measurements are relatively robust to MP errors rather than to code information. Accordingly, when weighted position estimation is performed through correlation between times, the position error may increase because of unnecessary clustering points caused by polluted satellite signals. Therefore, to eliminate unnecessary clustering points, a prediction range is determined based on the motion constraints of the vehicle. Thus, a position estimation method for the weight model based on Doppler measurements within the determined range is presented.

Figure 6 shows the predicted angle and predicted range for GNSS position estimation by selecting the range in which the vehicle can move as its speed changes, considering the nonholonomic constraints of the vehicle.

The range of the look-ahead angle can be selected by obtaining the estimated heading direction $\hat{\theta }$ through the minimum turning radius so that the vehicle cannot turn within the turning radius. The minimum turning radius is an applied rule for road structure facilities, and the minimum radius of a passenger car can be defined as $6 \mathrm{m}$. Accordingly, the prediction angle, ${\alpha }_{angle}$ of the vehicle based on the direction angle, ${\psi }_{G,\mathit{car}}$ is according to (19) as follows:

$${\psi }_{G,car}-\hat{\theta } \le {\alpha }_{angle}\le {\psi }_{G,car}+\hat{\theta },$$

(19)

where

• ${\psi }_{G,car}={\tan }^{-1}\frac{L_{E,t-1}}{L_{N,t-1}}$: GNSS course angle at time $t-1$, [$L_{N,t-1} L_{E,t-1} L_{D,t-1}$] is the displacement vector of the vehicle in the north-east-down coordinate system;
• $\hat{\theta }=\frac{{360}^{°}d}{2\pi R}$: estimated angle of the vehicle’s maximum turning radius;
• $R$: turning radius of the vehicle;
• $d$: distance traveled, calculated based on speed measurement.

In the case of the prediction distance, the maximum selection distance $6d$, which is six times the distance $d$ obtained through the speed measurement, was selected by considering the range of the prediction angle $\hat{\theta }$ estimated based on the traveling direction $x$ and error range of the speed measurement. In addition, considering the GNSS position error, the prediction range of ${\sigma }_{GNSS}$ is selected based on the previously estimated position. The prediction range is used to preferentially remove outlier; if cluster data within the prediction range do not exist, the position within the entire range is estimated.

To estimate the position based on the speed measurement value, the average position value of each data cluster, which is the sample extracted within the selected range, is defined as follows:

$${\beta }_s^M=\frac{1}{N_M}{{\displaystyle \sum }}_{i=1}^{N_M}{x}_{i,M},$$

(20)

where

• ${\beta }_s=\left\{{\beta }_s^1,{\beta }_s^2,\cdots , {\beta }_s^M\right\}$: samples for the mean positions in each cluster;
• $M=\left\{1,2,\cdots ,j\right\}$: j-th selected clusters;
• $i$: i-th clustered position of data.

The weighted value for the distance difference between the distance measurement value obtained through the speed measurement and the average position value was selected, as shown in (21).

$$\tilde{w}\left({\beta }_s^j\right)=e^{-\left(\int S_{dop,t} dt-\left|\left|{\hat{p}}_{t-1}-{\beta }_s^j\right|\right|\right)},$$

(21)

where

• $j$: j-th selected clusters;
• $S_{dop,t}$: speed-measurement based on doppler shift;
• ${\hat{p}}_{t-1}$: result of previously estimated position.

Owing to the nature of the exponential function, the larger the difference between the speed measurement and the average position of the cluster, the more rapidly the weight decreases, thereby minimizing the error for the distorted cluster.

The normalized weighted probability density function obtained by applying a weight to the extracted average position data is as follows:

$$w\left({\beta }_s^j\right)=\frac{\tilde{w}\left({\beta }_s^j\right)}{{{\displaystyle \sum }}_{j=1}^M\left(\tilde{w}\left({\beta }_s^j\right)\right)},$$

(22)

The weighted estimation position based on the prediction range can be expressed as the sum of the weighted samples as follows:

$${\hat{p}}_w={{\displaystyle \sum }}_{j=1}^M{\beta }_s^j·w\left({\beta }_s^j\right),$$

(23)

Assuming that the pseudorange change rate for each LOS satellite is the same in the instantaneous change rate for a short time, the cluster data generated by the LOS satellite signals at the previous and present times have the same PRN signal combination rate. That is, if the PRN signal combination ratio between the single-cluster data finally selected in the previous time and the clustered data in the current time are identical, it can be determined as the cluster data by the LOS satellite signal. However, when the LOS satellite signal is delayed, cluster data with the same PRN signal combination rate as the previous time is not available (

Table 4. Maximum position involvement rate according to satellite signal change (%).

LOS→MP	$\mathbf{Number} \mathbf{of} \mathbf{LOS} \mathbf{Satellites} \mathbf{at} \mathit{t}-1$
	6	7	8	9	10	11
1	72	65	60.74	57.33	54.95	53.3
2	95.45	90.62	68.50	83.25	80.79	78.96
3	-	98.44	83.86	94.24	82.45	91.02
4	-	-	99.39	98.69	87.41	96.48
5	-	-	-	-	99.29	98.79

). The selected single-cluster data were defined as the closet to the estimated position.

Here, it is shown that the NLOS satellites observed at time $t-1$ change to $k$ MP/NLOS signals at time $t$. It is assumed that $k$ MP/NLOS signal satellites are equal to or less than $N-k$ LOS satellites. In addition, the delay error was assumed to be within the predicted range and forms clusters. However, as the ratio of the LOS satellite signal to the MP/NLOS signal is similar, the position error of the receiving point increases rapidly; therefore, it is not involved in the position estimation.

As shown in Algorithm 1, if the PRN signal combination rate between the previous time and the current time matches 99% or more (Table 4), it can be considered that there is no effect on MP/NLOS. This definition can be considered as having the same environment as the previous time, and the position can be estimated with the average value for the corresponding cluster. As MP/NLOS satellites can change to LOS signals, if the PRN code agreement rate does not exhibit the same level, the position cannot be estimated based on it. Therefore, a weighted model is selected based on the speed measurement and the position is estimated as follow:

$${\hat{p}}^G=\left\{\begin{array}{c}\frac{1}{N_{LOS}}{\displaystyle \sum }_{i=1}^{N_{LOS}}{x}_i^{LOS}, if PRN rate>99\%\\ {\hat{p}}_w, else \end{array}\right..$$

(24)

Algorithm 1. Outlier mitigation algorithm based on multi-position clustering
Input:		Observation satellite info: PRN, SNR, $S_{dop,t}$, Ele
Input:		Masking value of SNR, Ele: ${\beta }_S$, ${\beta }_{Ele}$
Input:		masking value of HDOP for optimal clustering: $D_{HDOP}$
Input:		Subset of observable satellites for each epoch: $G=\left\{P_1, P_2, \cdots ,P_t\right\}$
Input:		Define DBSCAN input parameter for Eps, Minpts: $\epsilon$, Γ
Output:		Estimated position: ${\hat{p}}^G$
Output:		Set of utilized PRN satellites in selected position data: $ob{s}_t$
1:		for t = 1: time
2:		for j = 1: $length\left(G\right)$
3:		if SNR $<{\beta }_S$ & ELE $<{\beta }_{Ele}$ & HDOP $<D_{HDOP}$
4:		$x^t$=Positioning($P_t$)
5:		End
6:		position data = {$x_1^t$, $x_2^t$, $\cdots$, $x_u^t$}
7:		End
8:		idx = dbscan($\left\{x_1^t,x_2^t, \cdots x_u^t\right\}, \epsilon$,Γ)
9:		$\left\{x_1^t,x_2^t,\cdots ,x_k^t\right\}=delete\left(\left\{x_1^t,x_2^t, \cdots x_u^t\right\},idx==-1\right)$
10:		for i = 1:k
11:		if $x_k^t$ <= $\left|\mathrm{nhc}\left({\psi }_{G,car},\hat{\theta }\right)\right|$
12:		Continue
13:		Else
14:		$delete x_k^t$
15:		End
16:		End
17:		for M = 1:Max(idx)
18:		${\beta }_s^M=\frac{1}{N_M}{\displaystyle \sum }_{i=1}^{N_M}{x}_{i,M}^t$
19:		end
20:		$if PRN rate>99\%$
21:		$\left\{x_1^{LOS},x_2^{LOS},\cdots ,x_i^{LOS}\right\}$ = select($ob{s}_{t-1}$)
22:		${\hat{p}}^G=\frac{1}{N_{LOS}}{\displaystyle \sum }_{i=1}^{N_{LOS}}{x}_i^{LOS}$
23:		$ob{s}_t=\left\{{\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_i\right\}$
24:		Else
25:		$\tilde{w}\left({\beta }_s^j\right)=e^{-\left(\int S_{dop,t} dt-\left|\right|{\hat{p}}_{t-1}-{\beta }_s^j\left|\right|\right)}$
26:		$w\left({\beta }_s^j\right)=\frac{\tilde{w}\left({\beta }_s^j\right)}{{{\displaystyle \sum }}_{j=1}^M\left(\tilde{w}\left({\beta }_s^j\right)\right)}$
27:		${\hat{p}}^G={\displaystyle \sum }_{j=1}^M{\beta }_s^j·w\left({\beta }_s^j\right)$
28:		$\left\{{\gamma }_1^{idx},{\gamma }_2^{idx},\cdots ,{\gamma }_i^{idx}\right\}$ = idx($\min (\int S_{dop,t} dt-\left|\left|{\hat{p}}_{t-1}-{\beta }_s^j\right|\right|$)
29:		$ob{s}_t=\left\{{\gamma }_1^{idx},{\gamma }_2^{idx},\cdots ,{\gamma }_i^{idx}\right\}$
30:		End
31:		End

Figure 7 shows the structure of the proposed outlier mitigation based on GNSS multi-position clustering. An algorithm for generating multi-position data was added to the existing algorithm, and Doppler-based velocity measurements were used in weighted position estimation to effectively estimate the vehicle’s position from position data generated by LOS signals through density-based clustering of multi-position data.

5. Performance Verification

To analyze and verify the performance of the proposed algorithm, an open dataset was used for driving experiments in Tsukishima, a commercial district in Tokyo. The test trajectory is illustrated in Figure 8. The dataset was made public at https://github.com/kakusang2020/iXR_GNSS-IMU_TightlyCouplingProgram/tree/master/Data (accessed on 7 September 2021).

It includes the central building district where skyscrapers are concentrated, and it has an urban scenario environment with many extreme MP/NLOS influences, including various urban environment such as narrow alleys, intersections, and bridges.

To analyze and verify the algorithm, the receiver specifications and masking values based on observation data are listed in Table 3.

5.1. Algorithm Verification

First, to analyze the performance of the algorithm, we selected the largest error of the existing predecessor algorithm. A quantitative evaluation was performed on the clustering performance of the proposed algorithm by comparing the position and error of the existing WLSM algorithm [12]. Additionally, to ensure the reliability of the analysis, we compared and verified the correlation about the difference between the Doppler velocity and the position difference results, and the PRN code combination rate used in the proposed algorithm.

Figure 9 shows the results of the clustering data obtained using the general WLSM. For the weight model, the model analyzed in previous studies was used, and the commonly used thresholds of each observation indicator of SNR > 28 dB-Hz and elevation angle > 15 were used as the masking value. Owing to the nature of the exponential function, the weight was inversely proportional to ${\sin }^2\left({\alpha }_{ele}\right)$ as the elevation angle increased. Accordingly, data results of the weight model show that multi-position data with errors of approximately $20 \mathrm{m}$ or more were generated based on the reference point owing to the characteristics of observation data, which are different for each GNSS satellite, and the distortion of observation data. The error of the positioning result is very large owing to the positioning result obtained through the weight model and the threshold value selected from the observation index, which shows that there is a limit to predicting the MP/NLOS signal through the observation index.

Figure 10 shows the result of clustering position data using LS, which is a method proposed for the consistency of position data. In the case of position data generated through LS, a cluster result of multi-position data generated from LOS satellite signals can be seen with consistent characteristics through a code pseudorange signal combination without correlation of observation indicators.

Figure 11 shows the result of outputting only the finally selected cluster data using the proposed algorithm only for the section where the position estimation result of the existing LS is large. The blue rectangle represents the position estimation results when using the existing algorithm, while cluster 1 to 13 represent the results of the proposed DBSCAN algorithm, which groups the position data composed of LOS connections. We can observe cluster 13, which consists of LOS data points. As a result, it can be confirmed that the proposed algorithm has fewer errors and forms a smoother position compared to the position estimation using the existing algorithm.

Table 5. Comparison of position error of the proposed algorithm.

Time	WLSM [12]	Proposed Position Error (m)	Match Rate of Current and Previous PRN Data of Selected Cluster (%)	Speed Match Rate (%)
$t$	$6.0032$	$1.6608$	$100$	$96.78$
$t+1$	$18.4571$	$1.8397$	$99.38$	$95.52$
$t+2$	$27.0649$	$4.0043$	$59.6$	$83.54$
$t+3$	$31.7745$	$3.6203$	$100$	$95.19$

shows the results on the experimental verification, as shown in Figure 10. When the PRN signal combination rate between the current time point and selected cluster data is $99\%$ or more, the speed estimated through the position difference and the Doppler-based speed measurement has an average concordance rate of $95.83\%.$ A speed estimation result similar to the Doppler-based speed measurement shows a highly improved positioning result. In addition, when the PRN combination rate between the current time and the previous time selected cluster data is lowered owing to the delay of several satellite signals, it shows a distance similar to the Doppler-based speed measurement through the weighted model. As a result, the proposed algorithm shows that the position error is a significantly improved compared to the existing algorithm, decreasing by $88.74\%$ on average compared to the existing algorithm.

5.2. Analysis of Result

The DGNSS was selected as a reference value to quantitatively evaluate and verify the performance of the proposed algorithm in an urban environment. The algorithm was verified by classifying the driving trajectories into three areas: narrow roads (alleys and intersections) with severe satellite delay errors, bridge areas, and building-centered districts with different driving speeds. The performance of the algorithm was compared and evaluated with four existing methods that have been verified and widely used: LSM [36]. SNR–WLSM [11], SNR–ELE–WLSM [12], and FWLSM [50].

Figure 12 shows the estimated position trajectories of narrow alleys and intersections. In such a narrow area, not only is the geometric arrangement of satellites narrow, but the estimated position is also easily distorted by signal shielding elements such as buildings and trees in the urban environment, resulting in a large position error as in the existing algorithm. As relatively accurate position estimation is often required in narrow alleys and intersections, the use of existing algorithms is limited. It can be confirmed that the proposed algorithm creates a trajectory that fits the reference trajectory better than existing algorithms.

Figure 13 shows the estimated position trajectories for the low- and high-speed driving sections in a densely populated building area. The densely populated building area has the largest delay factor because of the signal delay caused by buildings. In particular, in the case of Figure 13a, the positioning result of the existing algorithm shows that the MP/NLOS delay signal caused by the building is continuously provided in a specific stop section, resulting in a position error of more than $40 \mathrm{m}$ from the receiving point. In the case of the proposed algorithm, all satellite signals observed by skyscrapers include MP/NLOS delay signals; therefore, errors continuously occur in a certain section. Overall, a non-smooth position prediction trajectory is observed, but compared to the existing algorithm, a plurality of position data containing large MP/NLOS delay signals is removed, resulting in greatly improved performance. In the case of Figure 13b, the MP/NLOS delayed signal is received and has positioning information with a position error of more than $100 \mathrm{m}$ from the receiving point. In the proposed algorithm, it can be observed that the error increases in a specific section. In this case, it is determined that a positioning error occurs because the delay error is included during positioning because the assumption proposed in this paper that there must be five or more observation satellites without MP/NLOS delay signals is incorrect. However, unlike the position estimation result in the low-speed driving section, the optimal position estimation from the MP/NLOS delay section is achieved owing to the speed of the receiver with improved performance compared to the existing algorithm.

Figure 14 shows the position trajectory for the driving environment on the bridge. As position errors can occur around bridges due to pylons and cables, it is necessary to verify the algorithm around bridges in an urban environment. In the case of the driving trajectory in Figure 14, there may be a position error due to the steel structure on the bridge, but it has a relatively open section compared to other urban environments. In addition, since the steel structure is placed at low position and is located on the road, the error of the MP delay signal is not significantly affected, and, hence, the existing algorithm also has a low position error. However, compared to the existing algorithm, the proposed algorithm shows a predicted trajectory that fits well with the reference trajectory with a reduced position error.

The estimations of the proposed algorithm for each urban environment trajectory are presented in

Table 6. Experimental results for specific urban areas.

Trajectory	RMSE (m)		Max Error (m) General/Proposed Algorithm
	General Algorithm	Proposed Algorithm
Central building district 1 (low speed)	36.6700	4.4640	128.3053/23.4324
Central building district 2 (high speed)	33.2654	3.8837	307.6143/14.5854
Bridge	3.6038	1.0857	7.5718/2.8457
Narrow road (alley and intersection)	18.9787	1.8719	80.3850/4.3433

. The average root-mean-square error (RMSE) is reduced by more than $80\%$, and the maximum position error is reduced by approximately $95.26\%$, particularly in dense building areas, including areas for high-speed driving.

Figure 15 and Figure 16 show the estimated position trajectories in dense urban areas for the various position estimation algorithms. Compared to the existing algorithms, including SNR–WLSM, SNR–ELE–WLSM, and FWLSM, it can be seen that the proposed algorithm exhibits a smoother position trajectory and lower position error.

Experimental results for the entire trajectory are shown in Figure 17, and the RMSE values of existing position estimation methods are shown in

Table 7. RMSE values of experimental results of estimated position [m].

Data	LSM [36]	SNR–WLSM [11]	SNR–ELE–WLSM [12]	FWLSM [50]	Proposed
Total trajectory	25.483	17.390	6.785	5.7656	4.355

. Table 7 lists the RMSE results of various localization methods for the entire driving trajectory. It can be confirmed that the existing algorithms have limitations in extreme urban environments. In particular, for FWLSM, the RMSE for the entire trajectory shows improved position estimation results compared to other previous studies by incorporating fuzzy logic for satellite signal observations. However, for certain epochs, it can be observed that the position error significantly diverges due to incorrect weight application. The proposed algorithm shows a relatively stable estimated position compared to those of existing algorithms, with significantly improved performance in a severe urban environment.

6. Conclusions

In this paper, we propose an algorithm for estimating the position of vehicles using data from a subset of observable satellites. The consistency of position data from various satellite signals was analyzed using the proposed algorithm and it was confirmed the experimental data composed of clean satellite signals can be clustered using actual position data.

An urban-environment-based positioning technique was proposed by configuring a weighted position estimation method based on speed measurements and specifying the prediction range. To verify the reliability of the proposed algorithm, the rate of speed fit was determined by comparing the agreement rate of the PRN combination used in the proposed algorithm with position difference and speed measurement.

The performance of the proposed algorithm was verified through experimental data in an actual urban environment. The estimation performance of the proposed algorithm was improved compared to representative existing algorithms in an urban environment. In addition, it was confirmed that the RMSE decreases by approximately $24\%$ or more in the experiment over the entire driving trajectory compared to the weighted least-square algorithm, which removes outliers based on observed indicators. Results confirm that the proposed algorithm in an urban environment with a large MP/NLOS error is more effective for position estimation compared to existing algorithms by overcoming their limitations.

The advantage of our approach is its applicability not only to code-based, carrier-phase methods based on multi-constellation but also to other trilateral positioning methods, such as UWB, Wi-Fi, and various techniques. Although the proposed algorithm can be used on various platforms based on triangulation methods or significantly improves the positioning error of GNSS receivers in extreme MP environments, it has some limitations. As the proposed algorithm is based on specific vehicle, additional research is needed to select the prediction range for the airframe and analyze fixed parameters for DBSCAN for application to other systems such as large cars, drones, or robots. For future research, we aim to enhance navigation performance by developing the proposed method further through carrier-phase-based GNSS positioning or sensor fusion algorithms based on trilateral positioning methods.