Fractional Cycle Bias Research Articles

Parallel resolution of large GNSS networks using carrier ranges

Owing to the increasing numbers of navigation satellites, available frequencies, and tracking stations, the computational efficiency of large GNSS networks has become a critical issue, especially when undifferenced (UD) pseudorange and carrier phase observations are used. By converting UD carrier phases into carrier ranges before implementing the integrated processing of a whole network, the UD calculation time can be greatly decreased. However, because a common multicore hardware platform is not fully utilized to optimize the algorithm, the processing of a whole network is still time consuming. In this paper, this problem is overcome by applying the multicore parallel computing technology, namely the Task Parallel Library (TPL). To improve the processing efficiency of large GNSS networks using UD carrier ranges and based on the task parallelism and data parallelism offered through TPL, the wide-lane (WL) and narrow-lane (NL) fractional cycle biases (FCBs), and carrier ranges are computed in parallel, and the integrated processing of a whole network is performed by establishing the normal equation in parallel and adopting parallel computing for Cholesky decomposition. The proposed strategy is validated using GPS, BDS and Galileo observation data recorded from January 6, 2019, to January 26, 2019 by the IGS network with a global distribution of approximately 500 stations. The average positioning accuracies in the east, north and vertical components are 2.54, 2.34 and 4.41 mm, respectively, and the average SDBS ambiguity fixing rate is 93.18%. The average time consumptions for step 1, 2 and 3 are reduced from 151.45, 248.80 and 44.35 min to 32.20, 52.71 and 14.25 min, respectively, with acceleration ratios of 4.70, 4.72 and 3.11 times, respectively, on a six-core platform.

GPS/BDS-2/Galileo Precise Point Positioning Ambiguity Resolution Based on the Uncombined Model

In this study, an uncombined precise point positioning (PPP) model was established and was used for estimating fractional cycle bias (FCB) products and for achieving ambiguity resolution (AR), using GPS, BDS-2, and Galileo raw observations. The uncombined PPP model is flexible and efficient for positioning services and generating FCB. The FCBs for GPS, BDS-2, and Galileo were estimated using the uncombined PPP model with observations from the Multi-GNSS Experiment (MGEX) stations. The root mean squares (RMSs) of the float ambiguity a posteriori residuals associated with all of the three GNSS constellations, i.e., GPS, BDS-2, and Galileo, are less than 0.1 cycles for both narrow-lane (NL) and wide-lane (WL) combinations. The standard deviation (STD) of the WL combination FCB series is 0.015, 0.013, and 0.006 cycles for GPS, BDS-2, and Galileo, respectively, and the counterpart for the NL combination FCB series is 0.030 and 0.0184 cycles for GPS and Galileo, respectively. For the BDS-2 NL combination FCB series, the STD of the inclined geosynchronous orbit (IGSO) satellites is 0.0156 cycles, while the value for the medium Earth orbit (MEO) satellites is 0.073 cycles. The AR solutions produced by the uncombined multi-GNSS PPP model were evaluated from the positioning biases and the success fixing rate of ambiguity. The experimental results demonstrate that the growth of the amount of available satellites significantly improves the PPP performance. The three-dimensional (3D) positioning accuracies associated with the PPP ambiguity-fixed solutions for the respective only-GPS, GPS/BDS-2, GPS/Galileo, and GPS/BDS-2/Galileo models are 1.34, 1.19, 1.21, and 1.14 cm, respectively, and more than a 30% improvement is achieved when compared to the results related to the ambiguity-float solutions. Additionally, the convergence time based on the GPS/BDS-2/Galileo observations is only 7.5 min for the ambiguity-fixed solutions, and the results exhibit a 53% improvement in comparison to the ambiguity-float solutions. The values of convergence time based on the only-GPS observations are estimated as 22 and 10.5 min for the ambiguity-float and ambiguity-fixed solutions, respectively. Lastly, the success fixing rate of ambiguity is also dramatically raised for the multi-GNSS PPP AR. For example, the percentage is approximately 99% for the GPS/BDS-2/Galileo solution over a 10 min processing period. In addition, the inter-system bias (ISB) between GPS, BDS-2, and Galileo, which is carefully considered in the uncombined multi-GNSS PPP method, is modeled as a white noise process. The differences of the ISB series between BDS-2 and Galileo indicate that the clock datum bias of the satellite clock offset estimation accounts for the variation of the ISB series.

Ambiguity of Residual Constraint-Based Precise Point Positioning with Partial Ambiguity Resolution under No Real-Time Network Corrections Using Real Global Positioning System (GPS) Data.

Although precise point positioning (PPP) is a well-established and promising technique with the use of precise satellite orbit and clock products, it costs a long convergence time to reach a centimeter-level positioning accuracy. The PPP with ambiguity resolution (PPP-AR) technique can improve convergence performance by resolving ambiguities after separating the fractional cycle bias (FCB). Now the FCB estimation is mainly realized by the regional or global operating reference station network. However, it does not work well in the areas where network resources are scarce. The contribution of this paper is to realize an ambiguity residual constraint-based PPP with partial ambiguity resolution (PPP-PARC) under no real-time network corrections to speed up the convergence, especially when the performance of the float solution is poor. More specifically, the update strategy of FCB estimation in a stand-alone receiver is proposed to realize the PPP-PAR. Thereafter, the solving process of FCB in a stand-alone receiver is summarized. Meanwhile, the influencing factors of the ambiguity success rate in the PPP-PAR without network corrections are analyzed. Meanwhile, the ambiguity residual constraint is added to adapt the particularity of the partial ambiguity-fixing without network corrections. Moreover, the positioning experiments with raw observation data at the Global Positioning System (GPS) globally distributed reference stations are conducted to determine the ambiguity residual threshold for post-processing and real-time scenarios. Finally, the positioning performance was verified by 22 GPS reference stations. The results show that convergence time is reduced by 15.8% and 26.4% in post-processing and real-time scenarios, respectively, when the float solution is unstable, compared with PPP using a float solution. However, if the float solution is stable, the PPP-PARC method has performance similar to the float solution. The method shows the significance of the PPP-PARC for future PPP applications in areas where network resource is deficient.

A method for undifferenced and uncombined PPP ambiguity resolution based on IF FCB

The integer characteristics of the undifferenced ambiguity can be recovered after the correction of fractional cycle biases (FCB), which can significantly shorten the initialization time of precise point positioning (PPP). In the former work, the ambiguity resolution model of the client end needs to be consistent with the FCB products server. However, the client may face the problem that inconsistent FCB products are provided and cannot be used to fix the ambiguity. We propose a new method for client of undifferenced and uncombined PPP ambiguity resolution based on FCB product of ionosphere-free combination PPP model. The hardware delay in server of ionosphere-free combination FCB (IF FCB) and undifferenced and uncombined FCB (L1L2 FCB) are deduced in detail. Constructing the narrow-lane ambiguity by using the raw ambiguity and the wide-lane ambiguity, the FCB product of the ionosphere-free combination PPP model is used to fix the ambiguity step by step. A globally distributed network consisting of about 118 MGEX stations is selected as the server end. Two types of FCB products, i.e., IF FCB and L1L2 FCB, are generated. 50 additional MGEX stations in global are selected as client end for performance assessment. The experimental results show that the new method for client of UU PPP model based on the IF FCB products has a comparable performance with the traditional method based on L1L2 FCB products, which improves the utilization of FCB products.

Research on empirical correction models of GPS Block IIF and BDS satellite inter-frequency clock bias

Triple-frequency observations will introduce an inter-frequency clock bias (IFCB) between the new frequency and the original dual-frequency observations. It has been verified that satellite IFCB can reach dozens of centimeters and several centimeters for GPS Block IIF satellite and BDS satellite, respectively. The existence of satellite IFCB will significantly affect undifferenced triple-frequency data processing. Based on 4-year data collected from 80 globally distributed stations, the long-term characteristics of IFCB coefficients obtained by using harmonic analysis have been studied. The results demonstrate that the coefficients of IFCB periodic model cannot be well fitted only by using sun elevation angle. Also, coefficients have obvious periodic characteristics and their periods differ among different satellites. Thus, a new linear-plus-periodic model is proposed to fit the long-term coefficients. Then, IFCB empirical correction models for 12 GPS Block IIF satellites and BDS GEO and IGSO satellites are built. In order to validate the correction model, IFCB standard deviation (STD), triple-frequency precise point positioning (PPP) and undifferenced extra-wide-lane (EWL) ambiguity resolution are employed. The results based on more than 4-year observations show that, with correction model applied, the average IFCB STD decreases by about 65.5% and 45.5% for GPS and BDS satellites, respectively. Compared to triple-frequency PPP without IFCB correction, triple-frequency PPP results with IFCB correction show that Up, North and East components accuracy are improved by 12.3%, 16.0% and 13.2%, respectively. Besides, IFCB correction will greatly improve the consistence of EWL fractional cycle bias among different stations and improve the success rate of EWL ambiguity resolution.

Combined precise orbit determination of GPS and GLONASS with ambiguity resolution

Precise orbit products of Global Navigation Satellite Systems (GNSS) are an essential precondition for precise positioning. Ambiguity resolution (AR) can enhance the orbit accuracy in precise orbit determination (POD). To improve the quality of orbits, we propose a method of combined POD for GPS and GLONASS with AR. Firstly, GLONASS wide-lane and narrow-lane fractional cycle biases (FCBs) are daily estimated. Then, by applying the estimated FCBs, GLONASS and GPS double-differenced wide-lane and narrow-lane ambiguities are successfully resolved, even for the baselines of up to several thousand kilometers. Finally, the ambiguity-resolved solutions are achieved by introducing the constraints of the resolved ambiguities into the real-valued solutions. To prove the contribution of the AR to GPS and GLONASS POD, a network including 141 sites is processed over 2018. The results show that the receiver types and firmware versions seriously affect the stability of the daily wide-lane FCBs. The fluctuation of the inter-system biases between two adjacent days is obviously larger than a half narrow-lane wavelength, causing an irregular change of the daily narrow-lane FCBs. After FCB calibration, the success rate of GLONASS can reach up to 90% over the whole year, which is at the same level compared with that of GPS. The improvements of GLONASS and GPS orbits after AR are confirmed by the orbit comparison with the International GNSS Service final products, the orbit misclosures at day boundaries and satellite laser ranging residuals. Due to some other issues, such as the GLONASS frequency-division multiple access and the high noise of observations, the improvement of GLONASS orbit is still less obvious than that of GPS orbit.

Multi-GNSS fractional cycle bias products generation for GNSS ambiguity-fixed PPP at Wuhan University

The School of Geodesy and Geomatics (SGG) at Wuhan University has been generating GPS fractional cycle bias (FCB) products for users to realize ambiguity-fixed precise point positioning (PPP) since 2015. Along with the development of multiple Global Navigation Satellite Systems (GNSS), there is an urgent need to provide multi-GNSS FCB products for the PPP ambiguity resolution (AR) with multi-constellation observations. This study focuses on the multi-GNSS FCB estimation, in which the FCB products of GPS, Galileo, BDS and QZSS are generated. We describe here the detailed estimation method and the significant improvements to the new service. The FCB quality, as well as the PPP AR performance, is evaluated. The mean standard deviations of wide-lane FCBs relative to CODE are 0.019, 0.005, 0.015 and 0.008 cycles, while those of narrow-lane are 0.021, 0.021, 0.057 and 0.010 cycles for GPS, Galileo, BDS and QZSS, respectively. The comparison with CNES GPS and Galileo FCBs indicates their good consistency with the corresponding FCBs. Compared with GPS-only PPP AR, the convergence time and time to first fix of the four-system PPP AR can be reduced by 27.3 and 29.4% in the static mode, respectively, while the corresponding improvements are 42.6 and 51.9% in the kinematic mode, respectively. These results demonstrate that our SGG FCB service can provide high-precision and reliable four-system FCB corrections for worldwide users to conduct ambiguity-fixed PPP processing.

Enhanced orbit determination for formation-flying satellites through integrated single- and double-difference GPS ambiguity resolution

The formation-flying technique is a fundamental concept for earth observing satellite missions, which usually require both absolute and relative orbit accuracies. Their precise orbit determinations are usually exclusively performed based on spaceborne GNSS data, where integer ambiguity resolution (IAR) plays a crucial role in achieving the best orbit accuracy. However, it is found that single-receiver IAR by resolving the single-difference (SD) ambiguities between GNSS satellites at each individual formation-flying satellite cannot achieve the same relative orbit accuracy that is attained by double-difference (DD) IAR between formation-flying satellites. To unravel this problem, 1 year of GPS data collected by the Gravity Recovery and Climate Experiment (GRACE) mission are used and four types of orbits are derived for comparison: (1) orbits where no ambiguities are fixed; (2) orbits where SD IAR is performed for both satellites; (3) orbits where only DD ambiguities between the twin GRACE satellites are resolved; (4) and orbits where SD IAR is carried out on only GRACE A while DD IAR is further accomplished between the twin satellites, namely the integrated SD IAR and DD IAR solutions. They are then evaluated through comparison to the reduced-dynamic orbit generated at the Jet Propulsion Laboratory, residual analysis of satellite laser ranging (SLR), and K-Band ranging (KBR) measurements. As expected, the integrated SD IAR and DD IAR solutions can achieve the highest absolute and relative orbit accuracies simultaneously. Specifically, SLR residuals in case of the integrated IAR are reduced by at least 25% for the kinematic orbit, when compared to the case of DD IAR. KBR residuals in case of the integrated IAR are reduced by 35 and 16% for the dynamic and kinematic orbit, respectively, when compared to those of SD IAR. Importantly, we find that errors in GPS clocks and/or narrow-lane fractional-cycle biases are in part responsible for the deteriorated relative accuracy of SD IAR achieved orbits. Therefore, we suggest that the integrated SD IAR and DD IAR scheme should be implemented for the best orbit solutions of formation-flying missions.

Modeling and Assessment of GPS/Galileo/BDS Precise Point Positioning with Ambiguity Resolution

Multi-frequency and multi-GNSS integration is currently becoming an important trend in the development of satellite navigation and positioning technology. In this paper, GPS/Galileo/BeiDou (BDS) precise point positioning (PPP) with ambiguity resolution (AR) are discussed in detail. The mathematical model of triple-system PPP AR and the principle of fractional cycle bias (FCB) estimation are firstly described. With the data of 160 stations in Multi-GNSS Experiment (MGEX) from day of year (DOY) 321-350, 2018, the FCBs of the three systems are estimated and the experimental results show that the range of most GPS wide-lane (WL) FCB is within 0.1 cycles during one month, while that of Galileo WL FCB is 0.05 cycles. For BDS FCB, the classification estimation method is used to estimate the BDS FCB and divide it into GEO and non-GEO (IGSO and MEO) FCB. The variation range of BDS GEO WL FCB can reach 0.5 cycles, while BDS non-GEO WL FCB does not exceed 0.1 cycles within a month. However, the accuracies of GPS, Galileo, and BDS non-GEO narrow-lane (NL) FCB are basically the same. In addition, the number of visible satellites and Position Dilution of Precision (PDOP) values of different combined systems are analyzed and evaluated in this paper. It shows that the triple-system combination can significantly increase the number of observable satellites, optimize the spatial distribution structure of satellites, and is significantly superior to the dual-system and single-system. Finally, the positioning characteristics of single-, dual-, and triple-systems are analyzed. The results of the single station positioning experiment show that the accuracy and convergence speed of the fixed solutions for each system are better than those of the corresponding float solutions. The average root mean squares (RMSs) of the float and the fixed solution in the east and north direction for GPS/Galileo/BDS combined system are the smallest, being 0.92 cm, 0.52 cm and 0.50 cm, 0.46 cm respectively, while the accuracy of the GPS in the up direction is the highest, which is 1.44 cm and 1.27 cm, respectively. Therefore, the combined system can accelerate the convergence speed and greatly enhance the stability of the positioning results.

Analysis of BDS Fractional Cycle Biases and PPP Ambiguity Resolution.

It is difficult to enable traditional precise point positioning (PPP) with ambiguity resolution (AR) due to fractional cycle biases (FCBs), which limit the accuracy and reliability of positioning results. The BeiDou Navigation Satellite System (BDS) has been providing continuous positioning, navigation, and timing (PNT) services in the global region since the end of 2018. The BDS constellation includes geostationary earth orbit (GEO), inclined geostationary orbit (IGSO), and medium earth orbit (MEO) satellites. However, its hybrid constellation structure and the satellite-side multipath effect have hindered the BDS PPP AR for two main reasons: (1) some receivers have half-cycle biases between GEO and non-GEO satellites, which result in the inconsistency of hardware delays for each satellite type; (2) the correction model for elevation-dependent satellite-side multipath effect is only applicable to IGSO and MEO, while in the case of GEO the effect cannot be effectively weakened or eliminated. To rectify these problems, a suitable strategy is proposed for estimating BDS FCBs, whereby the GEO FCBs and non-GEO FCBs are estimated independently. Results show that the FCBs estimated by the new strategy for GEO and non-GEO are more stable compared to the traditional strategy. The GEO wide-lane (WL) FCBs fluctuate less than 0.3 cycle in one month, except for C05, while the variation of non-GEO WL FCBs is about 0.1 cycle. In addition, compared to the traditional strategy, the fractions of GEO WL ambiguities after the removal of FCBs estimated by the new strategy can be improved noticeably from 53.5% to 78.5%, and from 71.8% to 92.3% for <0.15 cycle and <0.25 cycle respectively, which could be comparable with non-GEO. Simultaneously, the improvement of GEO narrow-lane (NL) ambiguities is from 28.9% to 40.2%, and from 40.4% to 53.3% for <0.10 cycle and <0.15 cycle respectively, are less noticeable. This is mainly due to the low precision IGS products for GEO. After PPP AR, the mean convergence time is shorted from 56.0 min to 43.6 min, and from 71.6 min to 62.7 min for static PPP and kinematic PPP, respectively.

FCB estimation with three different PPP models: equivalence analysis and experiment tests

Precise and reliable fractional cycle bias (FCB) products are critical for ambiguity resolution of precise point positioning (PPP). Three PPP models are usually adopted to generate the FCB products, i.e., the traditional ionospheric-free combined PPP (IF-PPP) model, the uncombined and unconstrained PPP model (UU-PPP) as well as the ionospheric-constrained PPP (IC-PPP) model. Considering that different observation models and ionospheric delay constraints are used, the applicability and interoperability of the obtained FCB products are assessed. We presented the equivalent conversion formulas of different FCB products obtained with different PPP models, which are then converted and compared. The root mean square (RMS) of the wide-lane (WL) FCB differences for IF-UU, IF-IC and UU-IC is 0.021, 0.024 and 0.010 cycles, while the RMS of the narrow-lane FCB differences is 0.028, 0.018 and 0.021 cycles. The precision of the WL FCBs based on the uncombined PPP models is higher than that based on the IF-PPP model since the new WL ambiguities derived from the uncombined ambiguities are free of the pseudorange noise. The equivalence of the FCB products estimated from the three different PPP models is confirmed in theory and by experiment results. To further evaluate the performance of the three PPP models, the positioning accuracy, the convergence time and the ambiguity fixing success rate are calculated using IGS data. The maximum positioning difference is less than 0.9 mm among the three PPP models. Compared to the float solutions of the IF-PPP, UU-PPP, IC-PPP with GIM model or with re-injected ionospheric delay corrections, the convergence time is shortened by 38.5%, 46.2%, 50.0% and 87.8% and the positioning accuracy improved by 27.1%, 38.9%, 41.1% and 25.7%, respectively. The ambiguity fixing success rate of the uncombined PPP models is slightly higher than that of the combined model, and fast ambiguity-fixed solution can proceed with high-precise ionospheric delay corrections.

Parallel computation of regional CORS network corrections based on ionospheric-free PPP

Global navigation satellite system real-time processing requires low latency, high timeliness, and high computational efficiency. A typical application is providing corrections using data from a regional Continuously Operating Reference Station (CORS) network. Usually the wide-lane and narrow-lane fractional cycle biases (FCBs) are determined at the server and broadcast to users to fix undifferenced ambiguity. Also, a tropospheric model is established at the server and broadcast to users to obtain accurate and reliable a priori zenith total delays for precise point positioning (PPP) using the ionospheric-free (IF) observation combination. Currently, serial methods are typically applied, i.e., all reference stations are involved in estimating the wide-lane and narrow-lane FCBs and establishing a regional tropospheric delay model. To improve the efficiency and shorten the latency, we develop a parallel computation method for regional CORS network corrections based on IF PPP by adopting a multicore parallel computing technology task parallel library, wherein parallel computations involving the FCBs, tropospheric delays, and tropospheric model are successively performed based on data parallelism, in which the same operation is performed concurrently on elements in an array, and task parallelism, which refers to one or more independent tasks running concurrently. Data covering four seasons from the Hong Kong and southwestern America CORS networks are utilized in the experiment. The single differenced FCBs between satellites are determined within each full pass, and a tropospheric model with an internal accuracy better than 1.4 cm and an external accuracy better than 1.6 cm is derived at the server. With the parallel implementation, the speedup ratios of FCB estimation and tropospheric modeling are 1.79, 3.15, 5.59, and 9.69 times higher for dual-core, quad-core, octa-core, and hexadeca-core platforms, respectively, than for a single-core platform.

Precise point positioning ambiguity resolution by integrating BDS-3e into BDS-2 and GPS

With the development of BDS-3, more BDS satellites are in orbits which can contribute to the reduction in the initial time of ambiguity fixing as well as the increase in the ambiguity fixing rate. We focus on the improvement in the BDS-based PPP AR as well as the GPS- and BDS-integrated PPP AR, using newly available BDS-3e satellites. To achieve this goal, the wide-lane (WL) and narrow-lane (NL) fractional cycle biases (FCBs) of B1I and B3I observations of BDS-2 IGSO and MEO satellites as well as of observations of BDS-3e satellites are generated using a network of globally distributed reference stations, while the BDS-2 GEO satellites are excluded from the FCBs estimation for their poor orbit accuracy due to the poor geometry. Both static and kinematic PPP AR solutions have been compared and analyzed in five combination strategies, including BDS-2 AR, BDS-2/3e AR, GPS AR, GPS/BDS-2 AR and GPS/BDS-2/3e AR. The experimental results illustrate that the inclusion of BDS-3e satellites is able to significantly improve the performance of the BDS-based PPP AR but only marginally improves the performance of the GPS- and BDS-integrated PPP AR. An average TTFF of 57.2 min (static) and 60.3 min (kinematic) and a fixing rate of 88.7% (static) and 87.3% (kinematic) have been achieved in the static and kinematic PPP AR for BDS-2/3e. The average time of TTFF is shortened to 15.3 min (static) and 16.4 min (kinematic) with a fixing rate of 96.9% (static) and 96.2% (kinematic) for GPS/BDS-2. The PPP AR of GPS/BDS-2/3e is found to perform the best among the five combination strategies of solutions, and an average TTFF of 13.1 min (static) and 14.3 min (kinematic) and a fixing rate of 97.0% (static) and 96.7% (kinematic) have been obtained.

The realization and evaluation of mixed GPS/BDS PPP ambiguity resolution

Aiming at shortening convergence time and improving positioning accuracy, multi-GNSS precise point positioning (PPP) ambiguity resolution (PPP-AR) has been an important issue in the past decade. In this paper, a mixed (or inter-system) GPS and BDS PPP-AR model with inter-system biases considered is proposed. Datasets from the IGS MGEX network are utilized in the study to evaluate the proposed model. As a critical correction in multi-GNSS PPP-AR, the inter-system bias (ISB) can be treated as a fixed constant or unknown estimate. The effects of various ISB processing methods on other key corrections for PPP-AR, such as fractional cycle bias (FCB) and inter-system phase bias (ISPB), are analyzed. Experimental results indicate that fixing or estimating ISB approaches will not affect GPS FCB estimations. However, various ISB dealing methods will have a significant influence on some BDS FCB and ISPB estimations at some stations because of the limited BDS tracking satellites over long periods of observation. Regardless of the presence of unstable FCB products on some BDS satellites, narrow-lane FCBs on other satellites are time-continuous, and their daily changes are within the range of 0.3 cycle. And in aspect of the time to first fix (TTFF), fixing ISB is superior to estimating it. The performance of the mixed GPS and BDS PPP-AR is evaluated. Experimental results indicate that compared with the intra-system PPP-AR, the mixed method has no superiority when ISB is estimated. While it has a slight improvement in TTFF, i.e., from 969.64 to 897.96 s, however, the total fixed rate decreases from 86.5 to 85.56% when ISB is fixed as a constant. In addition, the mixed PPP-AR shows significant improvement over the intra-system PPP-AR under circumstances with limited satellite visibility.

Comparison of convergence time and positioning accuracy among BDS, GPS and BDS/GPS precise point positioning with ambiguity resolution

Precise point positioning (PPP) usually takes about 30 min to obtain centimetre-level accuracy, which greatly limits its application. To address the drawbacks of convergence speed and positioning accuracy, we develop a PPP model with integrated GPS and BDS observations. Based on the method, stations with global coverage are selected to estimate the fractional cycle bias (FCB) of GPS and BDS. The short-term and long-term time series of wide-lane (WL) FCB, and the single day change of narrow-lane (NL) FCB are analysed. It is found that the range of GPS and BDS non-GEO (IGSO and MEO) WL FCB is stable at up to a 30-day-time frame. At times frame of up to 60 days, the stability is reduced a lot. Whether for short-term or long-term, the changes in the BDS GEO WL FCB are large. Moreover, BDS FCB sometimes undergoes a sudden jump. Besides, 17 and 10 stations were used respectively to investigate the convergence speed and positioning errors with six strategies: BDS ambiguity-float PPP (Bfloat), GPS ambiguity-float PPP (Gfloat), BDS/GPS ambiguity-float PPP (BGfloat), BDS ambiguity-fixed PPP (Bfix), GPS ambiguity-fixed (Gfix), and BDS/GPS ambiguity-fixed (BGfix). The average convergence speed of the ambiguity-fixed solution is greatly improved compared with the ambiguity-float solution. In terms of the average convergence time, the Bfloat is the longest and the BGfix is the shortest among these six strategies. Whether for ambiguity-float PPP or ambiguity-fixed PPP, the convergence reduction time in three directions for the combined system is the largest compared with the single BDS. The average RMS value of the Bfix in three directions (easting (E), northing (N), and up (U)) are 2.0 cm, 1.5 cm, and 5.9 cm respectively, while those of the Gfix are 0.8 cm, 0.5 cm, and 1.7 cm. Compared with single system, the BDS/GPS combined ambiguity-fixed system (BGfix) has the fastest convergence speed and the highest accuracy, with average RMS as 0.7 cm, 0.5 cm, and 1.9 cm for the E, N, U components, respectively.

A Unified Model for Multi-Frequency PPP Ambiguity Resolution and Test Results with Galileo and BeiDou Triple-Frequency Observations

With the modernization of Global Navigation Satellite System (GNSS), triple- or multi-frequency signals have become available from more and more GNSS satellites. The additional signals are expected to enhance the performance of precise point positioning (PPP) with ambiguity resolution (AR). To deal with the additional signals, we propose a unified modeling strategy for multi-frequency PPP AR based on raw uncombined observations. Based on the unified model, the fractional cycle biases (FCBs) generated from multi-frequency observations can be flexibly used, such as for dual- or triple- frequency PPP AR. Its efficiency is verified with Galileo and BeiDou triple-frequency observations collected from globally distributed MGEX stations. The estimated FCB are assessed with respect to residual distributions and standard deviations. The obtained results indicate good consistency between the input float ambiguities and the generated FCBs. To assess the performance of the triple-frequency PPP AR, 11 days of MGEX data are processed in three-hour sessions. The positional biases in the ambiguity-fixed solutions are significantly reduced compared with the float solutions. The improvements are 49.2%, 38.3%, and 29.6%, respectively, in east/north/up components for positioning with BDS, while the corresponding improvements are 60.0%, 29.0%, and 21.1% for positioning with Galileo. These results confirm the efficiency of the proposed approach, and that the triple-frequency PPP AR can bring an obvious benefit to the ambiguity-float PPP solution.

Estimating and assessing Galileo satellite fractional cycle bias for PPP ambiguity resolution

Due to the rapid deployment of the Galileo constellation, Galileo is now able to contribute to GNSS precise point positioning (PPP) ambiguity resolution (AR) with 17 operational satellites as of De...

Toward global instantaneous decimeter-level positioning using tightly coupled multi-constellation and multi-frequency GNSS

Autonomous driving represents one of the emerging applications that require both high-precision positions and highly critical timeliness to reach stringent safety standards. We develop a method to potentially achieve global instantaneous decimeter-level positioning by virtue of tightly coupled multi-GNSS triple-frequency observations contributing to precise point positioning (PPP). Inter-system phase biases (ISPBs) for two wide-lane observables are first computed for each station to form inter-GNSS resolvable ambiguities, and then correspondingly two wide-lane fractional-cycle biases (FCBs) are computed for each satellite to recover the integer property of single-station ambiguities. With both ISPB and FCB products, we can accomplish tightly coupled multi-GNSS PPP wide-lane ambiguity resolution (PPP-WAR) using only a single epoch of triple-frequency observations on a global scale. To verify this method, we used 1 month of GPS/BeiDou/Galileo/QZSS data from 107 globally distributed stations and 1 h of such multi-GNSS data collected on a vehicle moving in an urban area. We found that both ISPB and FCB products could be estimated every 24 h with high precisions of around or below 0.1 cycles; 83–98% of their day-to-day variations fell within 0.1 cycles, facilitating their precise predictions for real-time applications. Using these corrections, we achieved both instantly and reliably ambiguity-fixed solutions at 91.2% of all epochs at the 107 stations on average; the resultant single-epoch positions reached a mean accuracy of 0.22 m, 0.18 m and 0.63 m for the east, north and up components, respectively, in case of abundant triple-frequency observations from over 15 satellites. Similarly, for the vehicle-borne test, we obtained instantaneous PPP-WAR solutions at 99.31% of all epochs and achieved a positioning accuracy of 0.29, 0.35 and 0.77 m for the east, north and up components, respectively, which improved significantly the identification of road lanes as opposed to other single-epoch solutions. Finally, we expect that the prospect of instantaneous PPP-WAR in aiding driverless vehicles can be more promising if, whenever possible, integrated with inertial sensors and/or smoothed through multi-epoch data.

Estimating and assessing Galileo satellite fractional cycle bias for PPP ambiguity resolution

Due to the rapid deployment of the Galileo constellation, Galileo is now able to contribute to GNSS precise point positioning (PPP) ambiguity resolution (AR) with 17 operational satellites as of December 2017. We estimate the satellite fractional cycle bias (FCB) based on globally distributed MGEX stations and assess the Galileo FCB quality by a comparison with that of GPS and BDS. Results of 60 days indicate that the quality of Galileo wide-lane (WL) FCB is better than GPS and BDS in terms of data usage rate, residual distribution, as well as standard deviation of daily estimates. The RMS of Galileo WL FCB residuals is 0.071 cycles, while that of GPS and BDS are 0.089 and 0.117 cycles, respectively. The standard deviation of Galileo daily WL FCB is 0.010 cycles, while that of GPS and BDS is 0.018 and 0.043 cycles. We attribute the better quality of Galileo WL FCB to its signal modulation, AltBOC, which significantly compresses the multipath effect for pseudorange measurement. Within the Galileo constellation, the performance of In-Orbit Validation (IOV) satellites WL FCB is worse than that of Full Operational Capability (FOC) satellites as a result of a reduction in the power of the transmitted signal. The performance of the two highly eccentric satellites is comparable to other FOC satellites. The overall quality of Galileo narrow-lane (NL) FCB is slightly worse than that of GPS but better than that of BDS. The RMS of Galileo NL FCB residuals is 0.062 cycles, while that for GPS and BDS is 0.050 and 0.086 cycles respectively. In addition, the NL FCB quality of FOC, IOV (except E19), as well as the two eccentric satellites, shows no significant difference in terms of data usage rates and residuals. Galileo PPP AR solutions are conducted at 20 MGEX stations with 3-h sessions for 10 days. The positional biases of AR solutions are 0.7, 0.6, and 2.1 cm for east, north and up components respectively, while those for float solutions are 2.1, 1.1, and 2.7 cm, corresponding to the improvements of 67, 45, and 22%, respectively. These results demonstrate that, currently, Galileo FCB can be estimated with accuracy comparable with GPS and BDS, and the Galileo observations can bring an obvious benefit to ambiguity-fixed PPP.

Correction of fractional cycle bias of pseudolite system for user integer ambiguity resolution

Synchronized pseudolite systems allow single-point positioning (SPP) using carrier-phase measurements with centimeter-level accuracy. Locata, developed by the Locata Corporation, emerged from the development of synchronized pseudolite technology. Numerous studies have claimed to achieve Locata SPP with centimeter-level accuracy; however, the ambiguities of Locata observations have been typically estimated as floating-point values. Similar to precise point positioning of global navigation satellite systems, pseudolite SPP that is based on ambiguity fixing can improve the positioning accuracy and shorten the convergence time, compared to the float solution. However, integer ambiguity fixing may be prevented due to the clock offset among the pseudolite transmitters. To recover the integer nature of single-difference ambiguity, a fractional cycle bias (FCB) correction method is proposed; this method is based on two-way time synchronization. The FCB correction method employed to estimate and correct the bias by a pseudolite network itself has two steps: two-way FCB measurement and half-cycle bias estimation. A ground-based testbed is constructed to test and verify the proposed method. It is shown that the FCB between the pseudolite transmitters have a stability better than 15 ps even considering the influence of variable multipath delay; the bias can be estimated with high precision and reliability. Statistical results support the theoretical findings that half-cycle bias may be caused by two-way clock offset measuring via carrier phase measurement. A two-dimensional positioning experiment is performed to evaluate the performance of the proposed FCB correction method. The single-difference ambiguities are fixed to integer values using the LAMBDA method with the fixed failure-rate ratio test. The results indicated that the positioning accuracy is at the sub-centimeters to centimeter-level compared with the real-valued solutions for all surveyed points.