Float Solution Research Articles

Improving Performance of Uncombined PPP-AR Model with Ambiguity Constraints

With the advancement of multi-frequency and multi-constellation GNSS signals and the introduction of observable-specific bias (OSB) products, the uncombined precise point positioning (PPP) model has grown more prevalent. However, this model faces challenges due to the large number of estimated parameters, resulting in strong correlations between state parameters, such as clock errors, ionospheric delays, and hardware biases. This can slow down the convergence time and impede ambiguity resolution. We propose two methods to improve the triple-frequency uncombined PPP-AR model by integrating ambiguity constraints. The first approach makes use of the resolved ambiguities from dual-frequency ionosphere-free combined PPP-AR processing and incorporates them as constraints into triple-frequency uncombined PPP-AR processing. While this approach requires the implementation of two filters, increasing computational demands and thereby limiting its feasibility for real-time applications, it effectively reduces parameter correlations and facilitates ambiguity resolution in post-processing. The second approach incorporates fixed extra-wide-lane (EWL) and wide-lane (WL) ambiguities directly, allowing for rapid convergence, and is well suited for real-time processing. Results show that, compared to the uncombined PPP-AR model, integrating N1 and N2 constraints reduces averaged convergence time from 8.2 to 6.4 min horizontally and 13.9 to 10.7 min vertically in the float solution. On the other hand, integrating EWL and WL ambiguity constraints reduces the horizontal convergence to 5.9 min in the float solution and to 4.6 min for horizontal and 9.7 min for vertical convergence in the fixed solution. Both methods significantly enhance the ambiguity resolution in the uncombined triple-frequency PPP model, increasing the validated fixing rate from approximately 80% to 89%.

A novel cascading partial ambiguity resolution method of BDS triple-frequency with inertial aiding for kinematic-to-kinematic relative positioning

Abstract The integer ambiguity resolution (AR) of the carrier phase is crucial for high-precision positioning. To solve the problem that the three carrier ambiguity resolution method will not be reliable when the observation quality cannot meet the requirement under the kinematic-to-kinematic condition, a novel cascading partial AR method of BeiDou navigation satellite system triple-frequency with inertial aiding for kinematic-to-kinematic relative positioning is proposed. At first, inertial data is utilized to improve the reliability and robustness of the extra wide lane AR in dynamic environments. Meanwhile, to improve the fixed success rate, the wide lane ambiguity is obtained by a linear combination of dual extra wide lane with inertial aiding indirectly. For the most challenging narrow-lane ambiguity, a Geometry-based enhanced mode is individually developed for the model constraint, in which a partial AR method for narrow lane with float solution filtered is designed to improve the fixed success rate under the GB model. Finally, the simulation and practical experiment were conducted to verify the effectiveness of the proposed method. The results show that the proposed method can effectively improve the fixed success rate of ambiguity under the kinematic-to-kinematic conditions, which can reach about 99.54% and achieve cm-level relative positioning in dynamic environments.

Analysis of the gain factors of 5G-assisted BDS RTK positioning in urban environments

The joint utilization of the Fifth Generation Communications Technology (5G) and the Global Navigation Satellite System (GNSS) serves as a promising solution to address the challenges associated with insufficient visible satellites and lower observation quality in urban environments. 5G allows for the angle and distance measurements, augmenting the performance of Real-Time Kinematic (RTK) positioning. To quantify the improvement of 5G observations on RTK positioning, this paper proposes a float solution gain factor and the Ambiguity Dilution of Precision (ADOP) gain factor. Based on these gain factors, the theoretical analysis and simulation are performed. This study designs an extended Kalman filter for 5G-assisted BeiDou Navigation Satellite System (BDS) RTK positioning, employing both the Full Ambiguity Resolution (FAR) and Partial Ambiguity Resolution (PAR) modes. Our experiment verified the effectiveness of 5G-assisted BDS RTK positioning in mitigating outlier occurrences and improving the ambiguity fixing rate as well as the positioning accuracy. In the FAR and PAR modes, the Three-Dimensional (3D) spatial accuracy increased by 48% and 18.8%, respectively, and the results are consistent with theoretical analysis based on gain factors. The fixing rate of RTK increased from 11.11% to 13.93%, while it increased from 32.58% to 44.43% for the PAR mode. The assistance of 5G observations reduced the median error for the FAR mode from over 1.3m to 0.9 m, and the third quartile from 2.1m to 1.05 m. For the PAR mode, the median error decreased from 0.5m to 0.12 m, and the third and fourth quartiles decreased from 0.65m to 0.38 m.

Improving the ambiguity resolution with the consideration of unmodeled errors in GNSS medium and long baselines

Abstract Ambiguity resolution (AR) is fundamental to achieve high-precision solution in GNSS (Global Navigation Satellite System) relative positioning. Extensive research has shown that systematic errors are associated with the performance of AR. However, due to the physical complexity, some systematic errors would inevitably remain in the observation equations even after processed with some popular models and parameterization. In the medium and long baselines, these unmodeled errors are the leading cause of the slow or even incorrect fixation of ambiguity. Therefore, to improve the AR performance in the medium and long baselines, we present a procedure with the careful consideration of unmodeled errors. At first, we develop a method to estimate the unmodeled errors based on the float ambiguity bias. Then, the overall procedure and key steps to fix the float solutions corrected by the unmodeled error estimate is designed. Finally, some real-measured baselines (from 68 km to 120 km) are utilized to validate the proposed procedure. The experimental results are analyzed and discussed from the aspects of AR and positioning, respectively. For the AR performance, the time required for the first fixation have been reduced by about 41.58% to 83.51%, from 12 to 100 min. Besides, 12.72% to 48.59% and 2.96% to 36.28% improvements of the ambiguity-fixed rate and the ambiguity-correct rate can be respectively obtained in the four baselines. As for the positioning performance, the mean values and RMSEs have improved by 0.2 to 4.8 cm (1.63% to 22.43%) and 0.2 to 2.8 cm (1.47% to 10.57%), respectively.

Accelerometer static state detection (SSD)-assisted GNSS/accelerometer bridge monitoring algorithm

Abstract In the field of structural health monitoring (SHM), a loosely coupled (LC) Kalman filtering algorithm that accounts for baseline drift errors is commonly used to integrate GNSS data with accelerometer data. In the LC algorithm, the baseline drift errors are considered unknown parameters that need to be estimated. In scenario of continuous float solutions, the estimation of baseline drift error is often inaccurate, leading to the divergence of monitoring results. Theoretically, as a type of motion sensor, accelerometers are expected to qualitatively determine the priori state of bridges, whether dynamic or static. Utilizing the inherent characteristics of accelerometers and the principle of zero-velocity detection in integrated navigation, we originally propose a bridge static state detection (SSD) method based on low-cost accelerometer, and introduces this prior SSD information as a constraint in GNSS/accelerometer LC algorithm, called SSD-LC bridge monitoring algorithm. Through a simulation platform and real-world bridge monitored tests, the effectiveness of our proposed SSD method has been verified. Furthermore, our proposed SSD-LC bridge monitoring algorithm can effectively mitigate the divergence problem in baseline drift estimation that occurs with continuous GNSS float solutions in traditional algorithms, which can effectively avoid misjudgments and false alarms in bridge monitoring during GNSS anomalies.

Multiple integer candidates ambiguity resolution: a unification ambiguity resolution algorithm

Among all the ambiguity resolution techniques, the Full Ambiguity Resolution (FAR), Partial Ambiguity Resolution (PAR) and Best Integer Equivariant (BIE) estimator are widely used. Although the researches have been done on the different classes of ambiguity resolution, we still hope to find the relationships among these specific algorithms. In this work, we unify the PAR and FAR algorithms under a whole framework of BIE by applying multiple integer candidates. A concise estimation formula of the variance of Gaussian BIE estimator based on the variance of float solution and the probability distribution of the candidates is first derived. Then, we propose an algorithm named Multiple Integer Candidates Ambiguity Resolution (MICAR) to discover as many ambiguities in the BIE as possible that can be estimated more precisely by PAR (FAR) algorithm instead of BIE. In the experiments, we utilize the simulated data of GPS (Global Positioning System) + BDS (BeiDou Navigation Satellite System) + Galileo (Galileo navigation satellite system) to contrast the effects of MICAR and single candidate estimator, i.e., FAR. By taking the threshold of 5 cm at 95% confidence level as an example, MICAR accelerates the convergence process by about 3.0 min. When the positioning sequence converges, MICAR reduces the root mean square of the positioning error by 9.8% in horizontal directions and 3.5% in vertical direction, which is attributed to more fixed NL.

The Modified Ambiguity Function Approach with regularization for instantaneous precise GNSS positioning

Abstract The Modified Ambiguity Function Approach (MAFA) implicitly conducts the search procedure of carrier phase GNSS integer ambiguity resolution (IAR) in the coordinate domain using the integer least squares (ILS) principle, i.e. MAFA-ILS. One of the still open scientific problems is an accurate definition of the search region, especially in the context of instantaneous IAR. In doing so, the float solution results, which encompass float position (FP) and its variance-covariance (VC) matrix, must be improved as these are necessary for defining the search region. For this reason, the ambiguity parameters are separately regularized, and then the baseline parameters are conditioned on regularized float ambiguities. The conditional-regularized estimation is thus designed, obtaining the regularized FP (RFP) and its VC-matrix. This solution is promising because its accuracy is enhanced in the sense of mean squared error (MSE) thanks to the improved precision at the cost of regularized bias. The optimal regularization parameter (RP) values obtained for ambiguity parameters balance the contributions of improved precision and bias in the regularized float baseline solution’s MSE. Therefore, the regularized search region is defined accurately in the coordinate domain to contain such approximate coordinates that more frequently give the correct ILS solution. It also contains fewer MAFA-ILS candidates, improving the search procedure’s numerical efficiency. The regularized ILS estimator performs well with the presence of bias, increasing the probability of correct IAR in the coordinate domain.

UPDs estimation and ambiguity resolution performance evaluation of LEO navigation system

Low-Earth orbit (LEO) satellites have fast motion and thus provide rapid geometric change relation to the ground stations in a short period of time, which are expected to improve the positioning performance. In this study, a LEO constellation of 160 LEO satellites was simulated and simulated LEO observation data at global, European, and American stations were used to estimate the uncalibrated phase delays (UPDs) and achieve static and kinematic precise point positioning (PPP) ambiguity resolution (AR). The quality of the estimated UPD products were evaluated. More than 89 % and 96 % of the wide- and narrow-lane UPDs had a posteriori residual of less than 0.15 and 0.25 cycles, respectively. In static PPP, all ambiguity fixing rates were larger than 63 %. Compared with float solutions, the average convergence time of the fix solutions at the global, European and American stations was accelerated by 8 %, 6 % and 7 %, respectively; the average root-mean-square errors (RMSEs) in the east, north, and up components at the global, European and American stations were decreased by (22 %, 20 %, 15 %), (36 %, 34 %, 28 %) and (44 %, 31 %, 29 %), respectively. In kinematic PPP, all ambiguity fixing rates were larger than 40 %. Compared with float solutions, the average convergence time of the fix solutions at the global, European and American stations was accelerated by 4 %, 19 % and 14 %, respectively; the average RMSEs in the east, north, and up components at the global, European and American stations were decreased by (10 %, 10 %, 7 %), (27 %, 22 %, 11 %) and (32 %, 20 %, 10 %), respectively. In both static and kinematic PPP, AR accelerated the convergence time and improved the positioning accuracy.

An improved wide-lane ambiguity resolution method for kinematic smartphone positioning

Abstract The release of GNSS raw data on Android smartphones provides the potential for high-precision smartphone positioning using multi-constellation and multi-frequency signals. However, severe multipath and low observation quality in kinematic environments make double-differenced uncombined ambiguities difficult to resolve reliably. To address this, the paper proposes an improved wide-lane (WL) integer ambiguity resolution (IAR) method that combines integer rounding and the Least-Square AMBiguity Decorrelation Adjustment (LAMBDA) methods. The proposed method achieved fix rates of 57% to 70% in challenging environments, with an average improvement of 7 · 7% in horizontal positioning accuracy compared to the float solution. The traditional partial integer rounding method only improved accuracy by 1 · 1%.

Improved Information Fusion for Agricultural Machinery Navigation Based on Context-Constrained Kalman Filter and Dual-Antenna RTK

Automatic navigation based on dual-antenna real-time kinematic (RTK) positioning has been widely employed for unmanned agricultural machinery, whereas GNSS inevitably suffers from signal blocking and electromagnetic interference. In order to improve the reliability of an RTK-based navigation system in a GNSS-challenged environment, an integrated navigation system is preferred for autonomous navigation, which increases the complexity and cost of the navigation system. The information fusion of integrated navigation has been dominated by Kalman filter (KF) for several decades, but the KF cannot assimilate the known knowledge of the navigation context efficiently. In this paper, the geometric characteristics of the straight path and path-tracking error were employed to formulate the constraint measurement model, which suppresses the position error in the case of RTK-degraded scenarios. The pseudo-measurements were then imported into the KF framework, and the smoothed navigation state was generated as a byproduct, which improves the reliability of the RTK positioning without external sensors. The experiment result of the mobile vehicle automatic navigation indicates that the tracking error-constrained KF (EC-KF) outperforms the trajectory-constrained KF (TC-KF) and KF when the RTK system outputs a float or single-point position (SPP) solution. In the case where the duration of the SPP solution was 20 s, the positioning errors of the EC-KF and TC-KF were reduced by 38.50% and 24.04%, respectively, compared with those of the KF.

Orbit and clock products for quad-system satellites with undifferenced ambiguity fixing approach

Integer Ambiguity Resolution (IAR) can significantly improve the accuracy of GNSS Precise Orbit Determination (POD). Traditionally, the IAR in POD is achieved at the Double Differenced (DD) level. In this contribution, we develop an Un-Differenced (UD) IAR method for Global Positioning System (GPS)+ BeiDou Navigation Satellite System (BDS) + Galileo navigation satellite system (Galileo)+ Global'naya Navigatsionnaya Sputnikovaya Sistema (GLONASS) quad-system POD by calibrating UD ambiguities in the raw carrier phase and generating the so-called carrier range. Based on this method, we generate the UD ambiguity-fixed orbit and clock products for the Wuhan Innovation Application Center (IAC) of the International GNSS Monitoring and Assessment System (iGMAS). One-year observations in 2020 from 150 stations are employed to investigate performance of orbit and clock products. Notably, the UD Ambiguity Resolution (AR) yields more resolved integer ambiguities than the traditional DD AR, scaling up to 9%, attributable to its avoidance of station baseline formation. Benefiting from the removal of ambiguity parameters, the computational efficiency of parameter estimation undergoes a substantial 70% improvement. Compared with the float solution, the orbit consistencies of UD AR solution achieve the accuracy of 1.9, 5.2, 2.8, 2.1, and 2.7 cm for GPS, BeiDou-2 Navigation Satellite System (BDS-2), BeiDou-3 Navigation Satellite System (BDS-3), Galileo, and GLONASS satellites respectively, reflecting enhancements of 40%, 24%, 54%, 34%, and 42%. Moreover, the standard deviations of Satellite Laser Ranging (SLR) residuals are spanning 2.5–3.5 cm, underscoring a comparable accuracy to the DD AR solution, with discrepancies below 5%. A notable advantage of UD AR lies in its capability to produce the Integer Recovered Clock (IRC), facilitating Precise Point Positioning (PPP) AR without requiring additional Uncalibrated Phase Delay (UPD) products. To assess the performance of quad-system kinematic PPP based on IRC, a network comprising 120 stations is utilized. In comparison to the float solution, the IRC-based PPP AR accelerates convergence time by 31% and enhance positioning accuracy in the east component by 54%.

Contribution of BDS-3 observations to the precise orbit determination of LEO satellites: a case study of TJU-01

The precise orbit determination (POD) of scientific low Earth orbit (LEO) satellites is a prerequisite for the successful implementation of scientific missions. In recent years, global navigation satellite systems have become the main means of determining the orbits of LEO satellites. The global navigation satellite system receiver onboard the Tianjin University No. 1 (TJU-01) satellite receives both GPS and BDS-2/3 signals, with the addition of BDS-2/3 observations playing an important role in improving the POD of LEO satellites. This study comprehensively analyzes the spaceborne GPS/BDS data quality, including BDS-2/3 and GPS code multipath errors. Appreciable code multipath errors are found for the B1I signal of BDS-2 medium Earth orbit (MEO) satellites at elevations higher than 40°, whereas slight near-field relevant multipath errors of both frequencies are found for GPS and BDS-3 MEO satellites. The GPS and BDS-2/3 code multipath errors are estimated through elevation/azimuth-relevant piece-wise modeling and applied in the POD calculations. Several schemes, namely GPS-based, BDS-based, BDS-based without geo-synchronous (GEO) satellites, and GPS/BDS combined schemes, are designed to evaluate the POD performance. Fourteen days of data are calculated and the average three-dimensional (3D) orbital root mean square (RMS) of orbit overlapping differences obtained from GPS-based and BDS-based POD (without GEO satellites) solutions are 37.4 and 27.1 mm, respectively. The BDS-based solutions are obviously better than the GPS-based solutions, mainly owing to better data availability. The GPS/BDS combined solutions have the best accuracy, with a 3D RMS value of 20.6 mm. In addition, when BDS GEO satellites are included, the 3D RMS of the overlapping orbit differences reduces to 32.9 and 27.4 mm for BDS-based and GPS/BDS combined solutions, respectively. Double-difference (DD) and single-difference (SD) integer ambiguity resolution (IAR) are adopted to further improve the POD performance. The fixed orbit of the TJU-01 satellite is solved through DD IAR and SD IAR, and the contribution of the TJU-01 satellite to ambiguity fixing is analyzed. Relative to the float solution, the improvements made using the two ambiguity fixing approaches are equivalent, both being approximately 13%. The importance of this research is not only the precise determination of the orbit of TJU-01 for occultation service but also the demonstration of the contribution of BDS observations to the performance of the POD of LEO satellites.

A flexible method for BDS-3 multi-frequency phase observable-specific biases estimation and its PPP analysis with different ambiguity resolution strategies

With the rapid progress of Global Navigation Satellite Systems (GNSS), integer ambiguity resolution has gradually become one of the key factors that affecting Precise Point Positioning (PPP) to achieve high-precision positioning results and fast convergence. Fortunately, the proposal of observable-specific code/phase biases (OSB) has greatly alleviated this problem tremendously. Different from obtaining OSB by dual-frequency linear combinations, this research proposed BeiDou Navigation Satellite System (BDS) OSB estimation method from raw observations. This method allows for more flexible estimation in a global network, without the need to compose the complex linear combinations when involving multi-frequency measurements processing. The BDS-3 observation data collected from 112 stations over one month (October 2022), containing B1C/B1I/B2a/B2b/B3I frequencies, were used for five-frequency phase OSBs estimation. Subsequently, static and kinematic PPP were executed utilizing two different ambiguity resolution strategies to corroborate the efficacy of this method. The results revealed that the BDS-3 five-frequency phase OSBs possessed good monthly stability, with standard deviation less than 0.3 cycle. High position accuracy was achieved in the static mode by fixing the ambiguity directly, with root mean square (RMS) of 3.0, 4.8 and 9.9 mm. Meanwhile, the ambiguity fixed rate was maintained at a high level around 96 % in multi-frequency tests. Simultaneously, the kinematic mode also presented the centimeter-level dynamic positioning results and the positioning accuracy improved by 55, 31 and 17 % in the East, North and Up components, respectively. Compared to the float solutions, the adoption of multi-frequency OSBs significantly accelerated the convergence speed by 55 % on average.

BDS-3/Galileo multi-frequency phase-only model for long baseline with best integer equivariant estimation

Currently, long baselines generally adopt phase-code models that include both phase and pseudorange observations, such as dual-frequency ionosphere-free (PC-DFIF) and multi-frequency uncombined (PC-MFUC) models. The more significant unmodeled errors in pseudorange are likely to reduce ambiguity resolution (AR) performance, and the phase-only model with only phase observations involved is expected to avoid this problem. In view of the fact that existing phase-only research is mainly focuses on GPS/BDS-2 triple-frequency signals, a BDS-3/Galileo multi-frequency phase-only (MFPO) model is proposed in this paper. Four linearly independent extra-wide-lane (EWL) phase ranges (i.e. IR1, IR2, EWL3 and EWL4) are determined stepwise with a sliding window, among which the first two belong to ionosphere-reduced (IR) combination. In the determination of EWL3/EWL4, in addition to theoretical precision, the amplified phase multipath is also considered. The aforementioned four EWL phase ranges are then equivalently transformed to wide-lane (WL) combinations with smaller coefficients, and further incorporated with narrow-lane (NL) equation to construct MFPO model. Meanwhile, the best integer equivariant (BIE) is introduced to improve AR performance. The MFPO model is evaluated with real-tracked long baseline data. The performance of EWL phase ranges can be significantly improved after phase multipath is considered. The float solution, fixed solution and BIE solution of MFPO model all achieve centimeter-level accuracy. However, during convergence stage, incorrect fixing might exist in fixed solution, while BIE solution can improve this situation. The MFPO model is further compared with commonly used PC-DFIF and PC-MFUC models. The accuracy of the three models after convergence is basically the same, while the MFPO model has higher accuracy during convergence stage.

High-rate GNSS multi-frequency uncombined PPP-AR for dynamic deformation monitoring

Precise point positioning (PPP) is an important alternative to real-time kinematic (RTK) positioning for deformation monitoring. Global Navigation Satellite System (GNSS) multi-frequency and multi-system integration can benefit PPP dynamic deformation monitoring due to increased measurement redundancy and facilitated ambiguity resolution (AR). This study investigates the performance of GNSS multi-frequency uncombined (UC) PPP-AR with GPS triple-frequency, BDS-3 triple-frequency, and Galileo five-frequency signals for monitoring high-rate vibrations based on 20-Hz observations from a shaking table. The GNSS multi-frequency UC fractional cycle bias (FCB) estimation method and PPP-AR method are developed. The results of vibration tests indicate that the displacement monitoring accuracy of GPS/BDS-3/Galileo multi-frequency PPP-AR solution is 5.3–8.4 mm. BDS-3 dual-frequency PPP-AR, BDS-3 triple-frequency PPP-AR, and GPS/BDS-3/Galileo multi-frequency PPP-AR solutions improve the accuracy by 8%, 17%, and 46% over the BDS-3 dual-frequency float solutions, respectively. GNSS multi-frequency UC PPP-AR can accurately identify the vibration frequency and amplitude, but the PPP-derived displacement is nosier than that of RTK.

BDS-3/BDS-2 FCB estimation considering different influencing factors and precise point positioning with ambiguity resolution

Since the full system deployment of BDS-3 in July 2020, it has provided stable and reliable positioning, navigation and timing (PNT) services for users around the world. It is an urgent task to explore the potential performance of precise point positioning (PPP) based on the full constellation of BDS-2/BDS-3. This research presented the global satellite visibility of BDS-2/BDS-3, analyzed the stability and accuracy of BDS-2 and BDS-3 rubidium clocks and hydrogen clocks, jointly estimated the BDS-2 and BDS-3 fractional cycle biases (FCB) and evaluated the PPP performance with ambiguity resolution (AR). Due to the fact that International GNSS Service (IGS) precise products have a great impact on narrow-lane (NL) FCB estimation, the FCB obtained by using the precise products from CODE (Center for Orbit Determination in Europe) and GBM (Deutsches GeoForschungsZentrum) in two different analysis centers and the FCB obtained by using GBM precise products with different calculation strategies are analyzed. In addition, the experimental data from Multi-GNSS Experiment (MGEX), EUREF (EPN) and Germany (GPN) observation networks are used to evaluate the influence of orbital errors on BDS FCB estimation. For wide-lane (WL) FCB, MGEX WL FCB has the highest accuracy, followed by EPN and GPN. On the contrary, for NL FCB, MGEX NL FCB has the worst accuracy, followed by EPN, and GPN is the best. Eventually, numerous experiment results showed that compared with the static float solution, the PPP fixed solutions in East (E), North (N) and Up (U) directions are improved by 32.9%, 27.7%, 13.8% for BDS-3 and 10.0%, 7.5%, 3.8% for BDS-2/BDS-3. Compared with the kinematic float solution, the PPP fixed solutions in E, N and U directions are improved by 27.9%, 23.4%, 16.6% for BDS-3 and 15.9%, 8.9%, 5.3% for BDS-2/BDS-3.

Best Integer Equivariant Estimation With Quality Control in GNSS RTK for Canyon Environments

Global Navigation Satellite System (GNSS) has been widely used for navigation and positioning in many areas, from professional applications to the mass market. However, in the real-time kinematic (RTK) positioning, the ambiguities usually cannot be fixed in natural and urban canyon environments due to the high occlusion, strong reflection, and frequent maneuvering. In this paper, the best integer equivariant (BIE) estimation with quality control is proposed. Specifically, the quality control issues based on the observation and state domains are added to the BIE estimator. First, an improved multipath processing approach is used to mitigate the effects of multipath and diffraction, and a modified detection, identification, and adaptation procedure is adopted for processing the outliers and non-line-of-sight reception. For state-domain quality control, a two-step selection and segment estimation are used to refine the ambiguity candidates. Finally, the BIE estimator is obtained. To validate the effectiveness of the proposed method, two experiments, including natural and urban canyon environments, are conducted. Compared with the float, fixed, and traditional BIE solutions, the BIE solution with quality control has the best positioning performance, including precision and reliability. Moreover, the problem of ambiguity resolution is also alleviated to a great extent. Specifically, in real-time monitoring, the real-time single-epoch positioning accuracy in east, north, and up directions are approximately 0.023, 0.014, and 0.044 m, respectively. For the low-cost urban vehicle data, the proposed method performs best regarding availability, precision, and reliability.

The enhanced search procedure in a coordinate domain for precise satellite positioning

Recently, there has been an increase in the availability of GNSS satellites for observation, which poses new challenges in the computational process of precise satellite positioning. Ambiguity resolution is a crucial step in achieving accurate relative positioning. In this regard, an improved version of a searching procedure in a three-dimensional coordinate domain has been proposed and tested. The method starts with a float solution, similar to classic methods like the LAMBDA. However, the subsequent step, the search procedure, is conducted in the coordinate space instead of the ambiguity space. This shift to a three-dimensional coordinate space leads to reduced computation time. A computational experiment was performed to validate the proposed method, which demonstrated a significant reduction in processing time. The advantages of this method are particularly prominent when dealing with a large number of satellites.

Assessment of the double-parameter iterative Tikhonov regularization for single-epoch measurement model-based precise GNSS positioning

The double-parameter iterative Tikhonov regularization is assessed for strengthening the single-epoch model-based precise GNSS positioning within the framework of cognitive meanings. The simultaneous iterative Tikhonov regularization of the least-squares (LS) estimators of the parameters of interest in the single-epoch GNSS model is analyzed to enhance their accuracy properties. Regularization parameters (RP) are collected in the regularization operator, which can play a standardizing role in the regularization principle. Thus, the double-parameter approach can consider the heteroscedasticity of the LS estimators of interest in the regularization principle. This approach employs the quality-based mean squared error (mse) matrix trace minimization criterion to select optimal double-RP values simultaneously. Then, the unconstrained LS estimation is iteratively regularized, obtaining regularized float solutions. The numerical tests indicate that the double-parameter approach successfully strengthens the single-epoch GNSS measurement models due to considering the heteroscedasticity of the LS estimators of interest in the regularization principle. The regularized variance–covariance (vc) matrix describes float solutions of improved precision properties at the cost of losing the LS estimator’s unbiasedness. The accuracy is thus superior in the sense of mse. Therefore, the regularized LS estimator is well-scaled but with a biased localization. It also provides a more peaked probability density function (PDF) of real-valued ambiguities, obtaining a lower failure rate (FR) of integer least-squares (ILS) ambiguity resolution (AR). In this way, the regularized ILS estimator performs well in the ambiguity domain with the presence of regularized bias.

An Improved Ambiguity Resolution Algorithm for Smartphone RTK Positioning.

Ambiguity resolution based on smartphone GNSS measurements can enable various potential applications that currently remain difficult due to ambiguity biases, especially under kinematic conditions. This study proposes an improved ambiguity resolution algorithm, which uses the search-and-shrink procedure coupled with the methods of the multi-epoch double-differenced residual test and the ambiguity majority tests for candidate vectors and ambiguities. By performing a static experiment with Xiaomi Mi 8, the AR efficiency of the proposed method is evaluated. Furthermore, a kinematic test with Google Pixel 5 verifies the effectiveness of the proposed method with improved positioning performance. In conclusion, centimeter-level smartphone positioning accuracy is achieved in both experiments, which is greatly improved compared with the float and traditional AR solutions.