Ambiguity Resolution Success Rate Research Articles

A Triple Checked Partial Ambiguity Resolution for GPS/BDS RTK Positioning.

Reliable and accurate carrier phase ambiguity resolution is the key to high-precision Global Navigation Satellite System (GNSS) positioning and application. With the fast development of modern GNSS, the increased number of satellites and ambiguities makes it hard to fix all ambiguities completely and correctly. The partial ambiguity fixing technique, which selects a suitable subset of high-dimensional ambiguities to fix, is beneficial for improving the fixed success rate and reliability of ambiguity resolution. In this contribution, the bootstrapping success rate, bounded fixed-failure ratio test, and the new defined baseline precision defect are used for the selection of the ambiguity subset. Then a model and data dual-driven partial ambiguity resolution method is proposed with the above three checks imposed on it, which is named the Triple Checked Partial Ambiguity Resolution (TC-PAR). The comprehensive performance of TC-PAR compared to the full-fixed LAMBDA method is also analyzed based on several criteria including the fixed rate, the fixed success rate and correct fixed rate of ambiguity as well as the precision defect and RMS of the baseline solution. The results show that TC-PAR could significantly improve the fixed success rate of ambiguity, and it has a comparable baseline precision to the LAMBDA method, both of which are at centimeter level after ambiguities are fixed.

Extended long-short ambiguity resolution in multi-antenna GNSS-over-fiber systems for enhanced attitude determination

An extended long-short ambiguity resolution (AR) approach to multi-antenna GNSS-over-fiber systems is proposed and experimentally demonstrated, providing enhanced attitude determination. GNSS signals are remotely acquired from M extended antennas and then transferred to a central node. Through virtual baseline processing, multiple baselines with intermediate length are constructed. Consequently, although the adjacent baseline ratio is limited, the overall aperture is expanded. The carrier phase measurements of equal length virtual baselines are averaged by following the parity detection process for reducing noise, leading to refined baseline estimation. Also, the extended long-short AR method is a single-epoch and search-free one, enabling both improved accuracy and real-time implementations. Both the AR success rate and accuracy of baseline estimation are remarkably enhanced, thus resulting in enhanced attitude determination. Full 3-D attitude determination is also possible with the proposed angle-type or cross-type planar array. In the experiments, the ambiguity is successfully resolved for more than 1400 epochs. The standard deviation of the baseline vector is reduced to almost one-third in both horizontal and vertical directions, without specific delay monitoring hardware.

Performance Analysis of Positioning Solution Using Low-Cost Single-Frequency U-Blox Receiver Based on Baseline Length Constraint.

With the rapid development of the satellite navigation industry, low-cost and high-precision Global Navigation Satellite System (GNSS) positioning has recently become a research hotspot. The traditional application of GNSS may be further extended thanks to the low cost of measuring instruments, but effective methods are also desperately needed due to the low quality of the data obtained using these instruments. Thus, in this paper, we propose the analysis and evaluation of the ambiguity fixed-rate and positioning accuracy of single-frequency Global Positioning System (GPS) and BeiDou Navigation Satellite System (BDS) data, collected from a low-cost u-blox receiver, based on the Constrained LAMBDA (CLAMBDA) method with a baseline length constraint, instead of the classical LAMBDA method. Three sets of experiments in different observation environments, including two sets of static short-baseline experiments and a set of dynamic vehicle experiments, are adopted in this paper. The experiment results show that, compared to classical LAMBDA method, the CLAMBDA method can significantly improve the success rate of the GNSS ambiguity resolution. When the ambiguity is fixed correctly, the baseline solution accuracy reaches 0.5 and 1 cm in a static scenario, and 1 and 2 cm on a dynamic platform.

A $2q$-Order Difference-Set Approach to Eliminate Phase Ambiguity of a Single-Frequency Signal

The resolution of the integer phase ambiguity among a group of spatially separated sensors is considered, where the smallest inter-sensor spacing exceeds half wavelength ( $\lambda /2$ ) of a single-frequency signal. The sufficient condition that the integer phase ambiguities can be unambiguously determined is derived and verified numerically. The traditional receiver of a triplet is identified as a particular example of this general framework. An efficient strategy for averaging redundant virtual observations is outlined. The performance in terms of the success rate of ambiguity resolution is significantly improved due to the averaging process.

Estimating inter-system biases for tightly combined Galileo/BDS/GPS RTK

The overlapping carrier frequencies L1/E1, L5/E5a and B2/E5b from GPS/Galileo/BDS allow inter-system double-differencing of observations, which shows a clear advantage over differencing of the observations of each constellation independently. However, the inter-system biases destroy the integer nature of the inter-system double-differencing ambiguities. Two methods of direct rounding and parameter estimation are used to determine the ISB value. By analyzing data collected from Curtin University from 2015 to 2018, the phase and code inter-system bias (ISB) are related to the receiver type, firmware version and the selected overlapping frequency. Upgrade of receiver firmware version results in changes of ISB values. For example, the upgrade of Javad firmware in Dec, 15, 2017 causes the difference of 0.5 cycles ISB between BDS GEO and non-GEO satellites. By comparing the three dynamic models which include white noise process, random walk process, and random constant in the parameter estimation method, the ISB determined by the random constant model is consistent with the value obtained by the direct rounding method. After the calibration of ISBs, the performances of tightly combined positioning are assessed. The success rate of ambiguity resolution and accuracy of positioning for the tight combination (TC) are significantly improved in comparison with that for the loose combination (LC) over short baselines. For L5/E5a, on which only few satellites can be observed, the maximum increase in success rates of ambiguity resolution can reach 31.7%, i.e., from 54.9% of LC to larger than 86.6% of TC, and the positioning accuracies can even be increased by 0.13 m, i.e., from 0.208 of LC to 0.074 m of TC in East direction for the mix-receiver TRIMBLE NETR9-SEPT POLARX4 in 2018.

Calibration of BeiDou Triple-Frequency Receiver-Related Pseudorange Biases and Their Application in BDS Precise Positioning and Ambiguity Resolution.

Global Navigation Satellite System pseudorange biases are of great importance for precise positioning, timing and ionospheric modeling. The existence of BeiDou Navigation Satellite System (BDS) receiver-related pseudorange biases will lead to the loss of precision in the BDS satellite clock, differential code bias estimation, and other precise applications, especially when inhomogeneous receivers are used. In order to improve the performance of BDS precise applications, two ionosphere-free and geometry-free combinations and ionosphere-free pseudorange residuals are proposed to calibrate the raw receiver-related pseudorange biases of BDS on each frequency. Then, the BDS triple-frequency receiver-related pseudorange biases of seven different manufacturers and twelve receiver models are calibrated. Finally, the effects of receiver-related pseudorange bias are analyzed by BDS single-frequency single point positioning (SPP), single- and dual-frequency precise point positioning (PPP), wide-lane uncalibrated phase delay (UPD) estimation, and ambiguity resolution, respectively. The results show that the BDS SPP performance can be significantly improved by correcting the receiver-related pseudorange biases and the accuracy improvement is about 20% on average. Moreover, the accuracy of single- and dual-frequency PPP is improved mainly due to a faster convergence when the receiver-related pseudorange biases are corrected. On the other hand, the consistency of wide-lane UPD among different stations is improved significantly and the standard deviation of wide-lane UPD residuals is decreased from 0.195 to 0.061 cycles. The average success rate of wide-lane ambiguity resolution is improved about 42.10%.

Combined BeiDou-2 and BeiDou-3 instantaneous RTK positioning: stochastic modeling and positioning performance assessment

ABSTRACT The global BeiDou navigation satellite system (BDS-3) has been in operation recently and many related applications have been already compatible with BDS-3 satellites. Most applications still use the empirical stochastic models for combined regional BeiDou navigation satellite system (BDS-2) and BDS-3 positioning. However, BDS-3 has different satellite design, and the empirical stochastic models may not describe the observation precisions and the correlation characteristics of BDS-3 signals well and finally degrade the positioning performance. Hence, in this contribution, the variance-covariance component estimation is applied to calculate the precisions of observations and their spatiotemporal correlations as well as the cross-correlations for BDS-2 and BDS-3 multi-frequency signals. With the retrieved stochastic model, the performance of integer ambiguity resolution and positioning are then numerically demonstrated on short baseline. The results show that the observation precisions are in general elevation-dependent for both BDS-2 and BDS-3 except for BDS-2 geostationary earth orbit (GEO) satellites. For both BDS-2 and BDS-3, the spatial correlations are adequately small to be ignored in practical data processing. For the receivers we used, the cross-correlations are found significantly amongst the BDS-2 multi-frequency phase signals. Moreover, the temporal correlations differ between different frequencies and observations. The combined BDS-2 and BDS-3 can improve the ambiguity resolution success rate and positioning accuracy.

Differential Inter-System Biases Estimation and Initial Assessment of Instantaneous Tightly Combined RTK with BDS-3, GPS, and Galileo

In this contribution, we assess, for the first time, the tightly combined real-time kinematic (RTK) with GPS, Galileo, and BDS-3 operational satellites using observations from their overlapping L1-E1-B1C/L5-E5a-B2a frequencies. First, the characteristics of B1C/B2a signals from BDS-3 operational satellites is evaluated compared to GPS/Galileo L1-E1/L5-E5a signals in terms of observed carrier-to-noise density ratio, pseudorange multipath and noise, as well as double-differenced carrier phase and code residuals using data collected with scientific geodetic iGMAS and commercial M300Pro receivers. It’s demonstrated that the observational quality of B1C/B2a signals from BDS-3 operational satellites is comparable to that of GPS/Galileo L1-E1/L5-E5a signals. Then, we investigate the size and stability of phase and code differential inter-system bias (ISB) between BDS-3/GPS/Galileo B1C-L1-E1/B2a-L5-E5a signals using short baseline data collected with both identical and different receiver types. It is verified that the BDS-3/GPS/Galileo ISBs are indeed close to zero when identical type of receivers are used at both ends of a baseline. Moreover, they are generally present and stable in the time domain for baselines with different receiver types, which can be easily calibrated and corrected in advance. Finally, we present initial assessment of single-epoch tightly combined BDS-3/GPS/Galileo RTK with single-frequency and dual-frequency observations using a formal and empirical analysis, consisting of ambiguity dilution of precision (ADOP), ratio values, the empirical ambiguity resolution success rate, and the positioning accuracy. Experimental results demonstrate that the tightly combined model can deliver much lower ADOP and higher ratio values with respect to the classical loosely combined model whether for GPS/BDS-3 or GPS/Galileo/BDS-3 solutions. The positioning accuracy and the empirical ambiguity resolution success rate are remarkably improved as well, which could reach up to approximately 10%∼60% under poor observational conditions.

Data-Driven Generalized Integer Aperture Bootstrapping for High-Integrity Positioning

A new method is developed for integer ambiguity resolution in carrier-phase differential global navigation satellite system positioning. The method is novel in that it is simultaneously data-driven, generalized to include partial ambiguity resolution, and amenable to a full analytical characterization of the prior probabilities of correctly and incorrectly resolved ambiguities. The technique is termed generalized integer aperture bootstrapping (GIAB). A full development of the GIAB is provided herein, including sizing its integer aperture to usually produce a higher prior probability of full ambiguity resolution than comparable existing methods. In Monte-Carlo simulations, the GIAB is shown to provide nearly optimal ambiguity resolution success rates of full ambiguity resolution for relevant integrity requirements under strong models while enabling partial ambiguity resolution.

Improving reliability and efficiency of RTK ambiguity resolution with reference antenna array: BDS + GPS analysis and test

Rapid and reliable ambiguity resolution is the key to high-precision global navigation satellite system-based applications. To improve the efficiency and reliability of ambiguity resolution, one effective method is equipping the reference station with an antenna array of known geometry instead of a single antenna. In this contribution, the benefits of reference antenna array-aided real-time kinematic (RTK) positioning are investigated. The mathematical relations between the number of reference antennas and the float baseline and ambiguity solutions are explored, and the closed-form formula of ambiguity dilution of precision (ADOP) is further presented. It is demonstrated that the maximum decrease in ADOP or the improvement in precision of float solutions is approximately 29.29% with the increase in the number of reference antennas. Then, we analyze the impact of errors (noises or biases) on the float baseline and ambiguity solutions. Finally, the performance of the array-aided RTK is evaluated with raw BDS and GPS observations in terms of the ambiguity resolution success rate, false alarm, wrong detection alarm, and the time-to-first-fix (TTFF). It is demonstrated that the array-aided RTK can deliver improved ambiguity success rates with respect to the standard one-reference-antenna RTK, especially when only single-frequency, single-system observations from fewer satellites are available. Moreover, the array-aided RTK can provide much more robust ambiguity resolution in the presence of biases. And the performance of suppressing false alarm and wrong detection alarm is improved as well. Additionally, the TTFF with single-frequency observations is also significantly shortened. One order of magnitude improvements in TTFF are achieved for most of the cases from the standard one-reference-antenna solutions to the three-reference-antenna solutions. The results confirm that the array-aided RTK ensures more reliable and efficient ambiguity resolution.

Galileo Augmenting GPS Single-Frequency Single-Epoch Precise Positioning with Baseline Constrain for Bridge Dynamic Monitoring

The Single-frequency Single-epoch double-differenced baseline resolution technique of Global Positioning System (GPS) provides a good opportunity for monitoring the displacement or deflection behavior of bridges under different loading conditions in real-time. However, for single GPS, a high success rate baseline solution is difficult to achieve due to the lack of sufficient visible satellites and the low accuracy of float solutions. Galileo Satellite Navigation System (Galileo) has 14 medium earth orbit satellites (as of May 2018) that can be used to supplement GPS. The frequency bands of Galileo overlap with that of GPS on E1/L1 and E5a/L5, which is conducive to the combination of observations in integration positioning. Accordingly, Galileo augmenting GPS is an effective and necessary approach to improve the positioning availability and reliability. Moreover, using the baseline length constraint can improve the accuracy of float solutions, narrow the search space, and finally increase the success rate of ambiguity resolution and positioning. The single-frequency single-epoch double-differenced GPS/Galileo mathematical model with baseline length constraint is deduced in this paper. Two sets of GNSS real bridge data were used for further analysis on the improvement of GPS/Galileo with baseline length constraint when compared to single GPS. Finally, a Fast Fourier Transformation (FFT) algorithm was adopted for precisely detecting the local dominant frequencies of XB, YB, and ZB direction of the two stations.

An Improved Method of Ambiguity Resolution in GNSS Positioning

High precision navigation with positioning accuracy of decimeter or even centimeter is becoming more and more important in many fields. Accurate carrier phase observations are the prerequisites for this requirement. Because they have much lower measurement noises than the code observations which are primarily used nowadays. However, the existence of integer ambiguity when counting cycles of carrier phases prohibits the straightforward application of the measurements. Once the integer ambiguity has been resolved, the application of carrier phase measurement is almost equivalent to pseudorange however with much higher precision. Therefore ambiguity resolution is key for fast and high precision positioning. The success rate of ambiguity resolution is affected by the ionospheric delay and observation noise. In previous methods, ionospheric errors are simply ignored. To resolve ambiguity reliably and quickly, we propose a modified method for precise point positioning. We analyze the characteristics of the dual-frequency combinations of original observations and the ones with longer wavelengths and lower noises are preferable. The wide-lane and sub-wide-lane combinations with higher success rates are chosen for ambiguity resolution. Extra pseudoranges are included to eliminate the ionospheric delay which hampers ambiguity resolution seriously. After the combined ambiguities have been resolved, the original ambiguities of each frequency can be calculated from the linearly independent equations. Based on real Global positioning system (GPS) navigation data, performances of the modified method are tested and compared with that of the traditional method. The results show that the modified method is less affected by ionospheric delay and can obtain more accurate ambiguity resolution.

Quality assessment of the affine-constrained GNSS attitude model

By mounting two or more GNSS antennas on one platform, the attitude of the platform can be determined. Rapid, reliable integer ambiguity resolution is the key to the viability of GNSS-based attitude determination. To reduce the complexity of the search space and retain the high ambiguity resolution success rates, Teunissen developed the affine-constrained multivariate attitude model. The quality measures of the affine-constrained GNSS attitude model are addressed. When there are enough satellites in common-view visibility, the affine constraints will probably provide a marginal contribution to the quality measures. However, the strength of the model validation can be improved significantly by equipping the array with more antennas in case that there are only five common-view satellites available. Although the phase-slip minimal detectable bias measures in a roving-platform case will be worse than those in a stationary-platform case, the code-spike minimal detectable bias measures are practically independent of whether the platform is moving or stationary. The main contribution of the affine constraints is that one can now ‘trade’ satellites for receiver/antennas.

Influencing Factors of GNSS Differential Inter-System Bias and Performance Assessment of Tightly Combined GPS, Galileo, and QZSS Relative Positioning for Short Baseline

This paper first investigates the influencing factors of between-receiver Differential Inter-System Bias (DISB) between overlapping frequencies of the Global Positioning System (GPS), Galileo and the Quasi-Zenith Satellite System (QZSS). It was found that the receiver reboot and the type of observations may have an impact on DISBs. The impact of receiver firmware upgrades and the activation of anti-multipath filters are also investigated and some new results are presented. Then a performance evaluation is presented of tightly combined relative positioning for a short baseline with GPS/Galileo/QZSS L1-E1-L1/L5-E5a-L5 observations with the current constellations, in which the recently launched Galileo and QZSS satellites will also be included. It is demonstrated that when DISBs are a priori calibrated and corrected, the tightly combined model can deliver a much higher empirical ambiguity resolution success rate and positioning accuracy with respect to the classical loosely combined model, especially under environments where the observed satellites for each system are limited and only single-frequency observations are available. The ambiguity dilution of precision, bootstrapping success rate, and ratio values are analysed to illustrate the benefits of the tightly combined model as well.

Improved pitch-constrained ambiguity function method for integer ambiguity resolution in BDS/MIMU-integrated attitude determination

The initialization is a critical step in the BeiDou Navigation Satellite System and microelectromechanical inertial measurement unit (MIMU)-integrated attitude determination. One of the primary tasks in the initializing process is integer ambiguity resolution. The rapidity and reliability of integer ambiguity resolution play a particularly important role in kinematic attitude determination. However, the common ambiguity function method (AFM) is time consuming and unreliable due to searching over the entire yaw and pitch range. In view of the fact that MIMU can provide initial pitch by self-alignment with the three-axis accelerometers, we present a pitch-constrained AFM (PCAFM) for single-epoch and single-frequency integer ambiguity resolution. Although only pitch information is available, both pitch and yaw search spaces can be constrained in PCAFM. The pitch search space is constrained by the initial pitch from MIMU, and then the yaw search candidates are reduced by the constrained pitch range with the mathematical relationship between yaw and pitch in the DD carrier-phase observation equation. Experimental results demonstrate that the yaw and pitch search candidates of PCAFM are greatly decreased by 67.55% and 97.51%, respectively. Meanwhile, the success rate of integer ambiguity resolution is improved compared with the AFM.

On-The-Fly Ambiguity Resolution Based on Double-Differential Square Observation.

Global navigation systems provide worldwide positioning, navigation and navigation services. However, in some challenging environments, especially when the satellite is blocked, the performance of GNSS is seriously degraded or even unavailable. Ground based positioning systems, including pseudolites and Locata, have shown their potentials in centimeter-level positioning accuracy using carrier phase measurements. Ambiguity resolution (AR) is a key issue for such high precision positioning. Current methods for the ground based systems need code measurements for initialization and/or approximating linearization. If the code measurements show relatively large errors, current methods might suffer from convergence difficulties in ground based positioning. In this paper, the concept of double-differential square observation (DDS) is proposed, and an on-the-fly ambiguity resolution (OTF-AR) method is developed for ground based navigation systems using two-way measurements. An important advantage of the proposed method is that only the carrier phase measurements are used, and code measurements are not necessary. The clock error is canceled out by two-way measurements between the rover and the base stations. The squared observations are then differenced between different rover positions and different base stations, and a linear model is then obtained. The floating integer values are easy to compute via this model, and there is no need to do approximate linearization. In this procedure, the rover’s approximate coordinates are also directly obtained from the carrier measurements, therefore code measurements are not necessary. As an OTF-AR method, the proposed method relies on geometric changes caused by the rover’s motion. As shown by the simulations, the geometric diversity of observations is the key factor for the AR success rate. Moreover, the fine floating solutions given by our method also have a fairly good accuracy, which is valuable when fixed solutions are not reliable. A real experiment is conducted to validate the proposed method. The results show that the fixed solution could achieve centimeter-level accuracy.

A Low-Cost INS-Integratable GNSS Ultra-Short Baseline Attitude Determination System.

Traditional attitude determination using global navigation satellite system (GNSS) carrier phases is mostly applied on geodetic-grade receivers with sufficient baseline length. However, for some special applications such as mobile communication base station smart antenna attitude determination, only low-cost receivers with ultra-short baselines can be employed, and the environments are more challenging. When solving the ambiguity resolution (AR) problem with low-cost receivers, it is hard for the traditional methods in ambiguity domain to estimate float ambiguities accurately due to the large code pseudorange noises; thus, such systems fail to determine the correct ambiguities. Aiming at improving the AR success rate for ultra-short baselines attitude determination with low-cost receivers, we provide an objective function named Mean Square Residual (MSR) based on the geometrical relationship among the position spherical search space, the fractional carrier phases, and the possible ambiguities. The method can be calculated without code pseudoranges, and thus, can provide a higher AR success rate when using low-cost receivers. The corresponding analysis and acceptance test method are discussed in this contribution, and further, as an extension for more complicated urban dynamic applications, a GNSS/Inertial Navigation System (INS) integrated system is introduced. Several experiments have been conducted to verify performance.

A new strategy of stochastic modeling aiming at BDS hybrid constellation in precise relative positioning

A realistic stochastic model is the prerequisite in global navigation satellite system (GNSS) positioning applications. Considering that Beidou satellite navigation system (BDS) consists of geostationary earth orbit (GEO), inclined geosynchronous satellite orbit (IGSO) and medium earth orbit (MEO) satellites operating in different orbits with different heights, only one model being used in stochastic modeling is probably inadequate, which is the usual case for Global Positioning System (GPS). In this contribution, a new strategy of stochastic modeling for hybrid constellation BDS precise relative positioning is proposed, namely multiple elevation-dependent models (MEDM). In MEDM, parameters of three elevation-dependent models for GEO, IGSO, MEO satellites are estimated independently. After that, variances of phase and code observations for different orbits satellites can be estimated, then the stochastic model specific for BDS can be constructed in real time. The experimental results reveal that, when MEDM is compared with the sole elevation-dependent model and the carrier-to-noise power density ratios (C/N0)-dependent model, the maximum improvements in ambiguity resolution success rates (SR) in 5, 6, 7, 8 available satellites cases are about 0.2%, 13%, 3%, 1% for B1 frequency, and 1%, 6%, 2.5%, 0.2% for B2 frequency, respectively. The overall improvements in baseline solutions by using MEDM are in millimeter or submillimeter level. Hence, MEDM is more suitable for hybrid constellation of BDS in precise relative positioning applications.

Triple-Frequency Code-Phase Combination Determination: A Comparison with the Hatch-Melbourne-Wübbena Combination Using BDS Signals

Considering the influence of the ionosphere, troposphere, and other systematic errors on double-differenced ambiguity resolution (AR), we present an optimal triple-frequency code-phase combination determination method driven by both the model and the real data. The new method makes full use of triple-frequency code measurements (especially the low-noise of the code on the B3 signal) to minimize the total noise level and achieve the largest AR success rate (model-driven) under different ionosphere residual situations (data-driven), thus speeding up the AR by directly rounding. With the triple-frequency Beidou Navigation Satellite System (BDS) data collected at five stations from a continuously-operating reference station network in Guangdong Province of China, different testing scenarios are defined (a medium baseline, whose distance is between 20 km and 50 km; a medium-long baseline, whose distance is between 50 km and 100 km; and a long baseline, whose distance is larger than 100 km). The efficiency of the optimal code-phase combination on the AR success rate was compared with that of the geometry-free and ionosphere-free (GIF) combination and the Hatch-Melbourne-Wübbena (HMW) combination. Results show that the optimal combinations can always achieve better results than the HMW combination with B2 and B3 signals, especially when the satellite elevation angle is larger than 45°. For the wide-lane AR which aims to obtain decimeter-level kinematic positioning service, the standard deviation (STD) of ambiguity residuals for the suboptimal combination are only about 0.2 cycles, and the AR success rate by directly rounding can be up to 99%. Compared with the HMW combinations using B1 and B2 signals and using B1 and B3 signals, the suboptimal combination achieves the best results in all baselines, with an overall improvement of about 40% and 20%, respectively. Additionally, the STD difference between the optimal and the GIF code-phase combinations decreases as the baseline length increases. This indicates that the GIF combination is more suitable for long baselines. The proposed optimal code-phase combination determination method can be applied to other multi-frequency global navigation satellite systems, such as new-generation BDS, Galileo, and modernized GPS.

A Comparison Between Hard- and Soft-Constrained RTK Positioning With Multiple Reference Antennas

Real-time kinematic (RTK) ambiguity resolution can be improved by engaging multiple reference antennas. The baseline information between the antennas can be taken as true values and used as a hard-constraint; or it can be regarded as weighted, a priori measurements, and used as a soft-constraint. In this contribution, a comparison between the two constraints is made to explicate their difference in RTK ambiguity resolution for some specific applications, in which the coordinates of the reference antennas cannot be accurately calibrated in advance. The functional and stochastic models of the hard and soft-constrained RTK positioning are given. The float ambiguity precision of the unconstrained, the hard and the soft-constrained cases is compared. The closed-formulae of the ambiguity dilution of precision (ADOP) for the unconstrained and the hard-constrained models are derived, and they are proven to be the upper and lower bound of the soft-constrained ADOP, respectively. A simulated and a real data show that, with the hard or soft-constraints, the single-frequency ambiguity resolution success rate, as well as the performance in time-to-first-fix (TTFF), is improved significantly. When there is a bias smaller than 6 cm in the a priori baseline vectors between the reference antennas, the two constraints have a comparable performance in ambiguity success rate. Also, a sharp increase in the hard-constrained TTFF is observed at the same time. When the bias gets larger, the hard-constrained ambiguity success rates are decreased drastically. In contrast, the soft-constraint maintains a relatively good performance, showing a much greater tolerance for bias.