Clock Bias Research Articles

BDS-3/GNSS multi-frequency precise point positioning ambiguity resolution using observable-specific signal bias

With the completion of the BeiDou Global Navigation Satellite System (BDS-3), more available frequencies have brought opportunities for precise point positioning ambiguity resolution (PPP-AR). This research theoretically deduced the multi-frequency PPP-AR model and the rule of observable-specific signal bias (OSBs) transformation. Then 34 International GNSS Service stations were selected to investigate the multi-frequency BDS-3 and multi-GNSS PPP-AR. The results suggested that the OSBs products provided by Centre National D'Etudes Spatiales (CNES) have absorbed the impact of inter-frequency clock bias and could be directly added to the original observations to restore the integer characteristics of the multi-frequency ambiguity. Meanwhile, the antenna phase center correction should keep consistent when using the CNES OSBs to restore the Melbourne-Wübbena ambiguity. The BDS-3 quad-frequency ambiguity-fixed convergence time was 39.0 and 25.1 min for kinematic and static PPP, which was shortened by nearly 15.0 and 3.4 min than the dual-frequency ambiguity-fixed solution. The quad-system multi-frequency PPP-AR showed the optimal state, which was 0.66, 0.76 and 2.66 cm for kinematic mode while 0.31, 0.31 and 1.02 cm for static mode in east, north, and up directions, respectively.

Clock and Power-Induced Bias Correction for UWB Time-of-Flight Measurements

Ultra-Wide Band (UWB) communication systems can be used to design low cost, power efficient and precise navigation systems for mobile robots, by measuring the Time of Flight (ToF) of messages traveling between on-board UWB transceivers to infer their locations. Theoretically, decimeter level positioning accuracy or better should be achievable, at least in benign propagation environments where Line-of-Sight (LoS) between the transceivers can be maintained. Yet, in practice, even in such favorable conditions, one often observes significant systematic errors (bias) in the ToF measurements, depending for example on the hardware configuration and relative poses between robots. This letter proposes a ToF error model that includes a standard transceiver clock offset term and an additional term that varies with the received signal power (RxP). We show experimentally that, after fine correction of the clock offset term using clock skew measurements available on modern UWB hardware, much of the remaining pose dependent error in LoS measurements can be captured by the (appropriately defined) RxP-dependent term. This leads us to propose a simple bias compensation scheme that only requires on-board measurements (clock skew and RxP) to remove most of the observed bias in LoS ToF measurements and reliably achieve cm-level ranging accuracy. Because the calibrated ToF bias model does not depend on any extrinsic information such as receiver distances or poses, it can be applied before any additional error correction scheme that requires more information about the robots and their environment.

Understanding the characteristic of GLONASS inter-frequency clock bias using both FDMA and CDMA signals

Understanding the characteristic of GLONASS inter-frequency clock bias using both FDMA and CDMA signals

Analysis of BDS-3 Onboard Clocks Based on GFZ Precise Clock Products

The characteristics and performance of satellite clocks are important to the positioning, navigation, and timing (PNT) services of Global Navigation Satellite System (GNSS) users. Although China’s BeiDou-3 Navigation Satellite System (BDS-3) has been fully operational for more than one year, there is still a lack of comprehensive research on the onboard clocks of the entire BDS-3 constellation. In this study, the precise clock products of GeoForschungsZentrum (GFZ) from day-of-year (DOY) 1, 2021 to DOY 300, 2021 were used to analyze the characteristics and performance of BDS-3 onboard clocks from the following aspects: clock bias, frequency, drift rate, fitting residuals, periodicity, and frequency stability. Compared with BDS-2, the clock quality of BDS-3 satellites has been greatly improved, but there are still jumps in the clock offsets and frequency series of BDS-3 clocks. The drift rate of BDS-3 clocks varies within the range between −2×10−18 and 2×10−18 s/s2. The daily model fitting residuals of passive hydrogen masers (PHM) on BDS-3 medium Earth orbit (MEO), inclined geosynchronous orbit (IGSO), and geostationary (GEO) satellites are 0.15, 0.28, and 0.46 ns, respectively. The overlapping Allan deviation (OADEV) of BDS-3 MEO clocks is 4.0 × 10−14 s/s at a time interval of 1000 s. The PHMs on BDS-3 MEO satellites exhibit fewer periodic signals than those of Rb clocks. In addition, the precise clock offsets of the BDS-3 PHMs carried on the MEO, IGSO, and GEO satellites show different periodicities, which are similar to those of the corresponding types of BDS-2 satellites.

Enabling Plug-and-Play and Crowdsourcing SLAM in Wireless Communication Systems

Simultaneous localization and mapping (SLAM) during communication is emerging. This technology promises to provide information on propagation environments and transceivers’ location, thus creating several new services and applications for the Internet of Things and environment-aware communication. Using crowdsourcing data collected by multiple agents appears to be much potential for enhancing SLAM performance. However, the measurement uncertainties in practice and biased estimations from multiple agents may result in serious errors. This study develops a robust SLAM method with measurement plug-and-play and crowdsourcing mechanisms to address the above problems. First, we divide measurements into different categories according to their unknown biases and realize a measurement plug-and-play mechanism by extending the classic belief propagation (BP)-based SLAM method. The proposed mechanism can obtain the time-varying agent location, radio features, and corresponding measurement biases (such as clock bias, orientation bias, and received signal strength model parameters), with high accuracy and robustness in challenging scenarios without any prior information on anchors and agents. Next, we establish a probabilistic crowdsourcing-based SLAM mechanism, in which multiple agents cooperate to construct and refine the radio map in a decentralized manner. Our study presents the first BP-based crowdsourcing that resolves the “double count” and “data reliability” problems through the flexible application of probabilistic data association methods. Numerical results reveal that the crowdsourcing mechanism can further improve the accuracy of the mapping result, which, in turn, ensures the decimeter-level localization accuracy of each agent in a challenging propagation environment.

Development, Verification, and Analysis of a Small Satellite Thrust Determination Filter

A current challenge for the Advanced Propulsion Experiment (APEX) CubeSat, under development by Missouri University of Science and Technology, is the accurate determination of thrust for its experimental multimodal propulsion system main payload. A consider batch least-squares filter is proposed that processes downlinked dual-frequency GPS pseudorange data to estimate the average thrust magnitude of a maneuver in conjunction with the initial dynamic states, static coefficients of drag and solar radiation pressure, and GPS receiver clock biases from each measurement epoch. Two Monte Carlo analyses were performed to test the statistical consistency of the filter and the robustness under the influence of unmodeled dynamics. The filter was found to perform nominally with matching truth and filter dynamics models and acceptably when the truth model used to synthesize measurements was a higher fidelity than the filter model. Filter performance was assessed using flight data from NASA’s Gravity Recovery and Climate Experiment Follow-On mission. A trade study was conducted to identify acceptable orbital regimes for the APEX mission. Thrust estimates using this method were shown to meet APEX mission requirements for a range of acceptable orbits.

The benefit of Galileo E6 signals and their application in the real-time instantaneous decimeter-level precise point positioning with ambiguity resolution

With the development of commercial services in the European Union, the advanced designs of Galileo E6 signals can be freely offered to Global Navigation Satellite System (GNSS) users. However, only a few studies have analyzed the benefit of Galileo E6 signals, and their application in the real-time instantaneous precise point positioning (PPP) with ambiguity resolution (AR) has not been investigated. In this study, based on the triple-frequency ionosphere-free combination, the positioning performance of instantaneous decimeter-level PPP AR was assessed with real-time orbit, clock and phase biases provided by the Centre National d’Études Spatiales Navigation Team (CNES/NAV). Experimenting on the GPS/Galileo observations from the 42 globally distributed stations, the wide-lane (WL) ambiguity residuals within ± 0.25 cycle for GPS L1/L2/L5, Galileo E1/E5a/E5b and E1/E5a/E6 frequency combinations were 92.54%, 90.72% and 96.48% with single-epoch data, respectively. When the WL ambiguities are fixed instantaneously, owing to the low combination noise of Galileo E1/E5a/E6, the positioning performance of instantaneous PPP AR with GPS/Galileo (E1/E5a/E6 used) outperforms that of GPS/Galileo (E1/E5a/E5b used). If the position dilution of precision is <1.5, the Root Mean Square (RMS) of GPS/Galileo (E1/E5a/E6 used) for east, north and up components are 0.13 m, 0.13 m and 0.57 m, respectively, while those of GPS/Galileo (E1/E5a/E5b used) are 0.17 m, 0.19 m and 0.62 m, respectively. Moreover, the WL ambiguity fixing rate of GPS/Galileo (E1/E5a/E6 used) is higher than that of GPS/Galileo (E1/E5a/E5b used), which shows an improvement of approximately 9% in the relatively low satellite elevation angles (20°< elevation angle < 40°). The results indicate that the Galileo E6 signal can bring a significant improvement in real-time instantaneous decimeter-level PPP AR.

Real-Time BDS-3 Clock Estimation with a Multi-Frequency Uncombined Model including New B1C/B2a Signals

The global system of BDS (BeiDou Navigation Satellite System), i.e., BDS-3, is characterized with a multi-frequency signal broadcasting capability, which was demonstrated as beneficial for GNSS (Global Navigation Satellite System) data processing. However, research on real-time BDS-3 clock estimation with multi-frequency signals is quite limited, especially for the new B1C and B2a signals. In this study, we developed models for BDS-3 multi-frequency real-time data processing, including the uncombined model for clock estimation and the GFIF (Geometry-Free Ionosphere-Free) combined model for IFCB (Inter-Frequency Clock Bias) determination. Based on the models, simulated real-time numerical experiments with about 80 global IGS (International GNSS Service) network stations are conducted for validation and analysis. The results indicate that: (1) the uncombined model with multi-frequency signals can achieve comparable accuracy with the traditional dual-frequency IF model in terms of clock estimation, and the double-differenced clock STDs (Standard Deviations) are generally less than 0.05 ns with post-processed clocks as a reference; (2) unlike the B1C and B1I/B3I signals, the satellite IFCBs generated from multi-frequency clock estimation show apparent temporal variations for B2a and B1I/B3I signals, further investigation with GFIF models confirm the variations mainly result from the errors of receiver antenna corrections. Therefore, we addressed the feasibility of the uncombined model and the importance of accurate antenna information in the multi-frequency data processing.

GNSS observable-specific phase biases for all-frequency PPP ambiguity resolution

An unwritten rule to resolve GNSS ambiguities in precise point positioning (PPP-AR) is that users should follow faithfully the frequency choices and observable combinations mandated by satellite clock and phase bias providers. Switching to other frequencies of measurements requires that the satellite clocks be converted, albeit in a roundabout way, to agree with the new frequencies of code biases. Satellite phase biases, on the other hand, are prescribed conventionally as wide-lane and narrow-lane combinations, which prevents users from resolving other phase combinations in the case of multi-frequency observables. We therefore develop an approach to compute observable-specific phase biases (phase OSBs) in concert with the legacy, but ambiguity-fixed, satellite clocks to enable PPP-AR over any frequency choices and observable combinations at the user end, i.e., all-frequency PPP-AR. In particular, the phase OSBs on the baseline frequencies (e.g., L1/L2 for GPS and E1/E5a for Galileo) are estimated by decoupling the code OSBs pre-aligned with the satellite clocks; then satellite clocks are re-estimated by holding pre-resolved undifferenced ambiguities and phase OSBs on the baseline frequencies; finally, all third-frequency phase OSBs are determined by introducing the ambiguity-fixed satellite clocks above. We used a global network of multi-frequency GPS/Galileo data over a month to verify this approach. In dual-frequency PPP-AR using GPS L1/L2, L1/L5, Galileo E1/E5a, E1/E5b, E1/E5 and E1/E6 signals, over 95% of wide-lane and narrow-lane ambiguity residuals were within ±0.25 and ±0.15 cycles, respectively, after the code and phase OSB corrections on raw GNSS measurements. As a result, the ambiguity fixing rates reached around 95% in all PPP-AR tests, though it was only the satellite clocks aligned with the GPS L1/L2 and Galileo E1/E5a pseudorange that were applied throughout. We stress that the key to computing such phase OSBs for all-frequency PPP-AR is that the code OSBs have the same bias datum as that of the satellite clocks.

Short-term satellite clock bias forecast based on complementary ensemble empirical mode decomposition and quadratic polynomial

The nonlinear and nonstationary characteristics of satellite clock bias (SCB) have a harmful effect on the accuracy and stability of SCB forecast. To eliminate the influence of nonlinearity and non-stationarity, a hybrid forecast model was constructed that combines complementary ensemble empirical mode decomposition (CEEMD) and quadratic polynomial (QP), called CEEMD-QP. First, the SCB sequence is decomposed into several intrinsic mode function (IMF) components and one residual term by CEEMD. Second, permutation entropy (PE) and the correlation coefficient are used to quantitatively determine the IMF component with more noise and weak correlation with the original SCB signal. Finally, the SCB is reconstructed with the IMFs and residual components, and QP model is used to fit and forecast the clock bias. We adapt the observation part of ultra-rapid precise SCB data of GPS provided by IGS to forecast experiments. The results show that the CEEMD-QP method has obvious advantages of forecast accuracy and stability in short-term forecast.

Real-Time Precise Orbit Determination for LEO between Kinematic and Reduced-Dynamic with Ambiguity Resolution

The real-time integer-ambiguity resolution of the carrier-phase observation is one of the most effective approaches to enhance the accuracy of real-time precise point positioning (PPP), kinematic precise orbit determination (KPOD), and reduced-dynamic precise orbit determination (RPOD) for low earth orbit (LEO) satellites. In this study, the integer phase clock (IPC) and wide-lane satellite bias (WSB) products from CNES (Centre National d’Etudes Spatiales) are used to fix ambiguity in real time. Meanwhile, the three models of real-time PPP, KPOD, and RPOD are applied to validate the contribution of ambiguity resolution. Experimental results show that (1) the average positioning accuracy of IGS stations for ambiguity-fixed solutions is improved from about 7.14 to 5.91 cm, with an improvement of around 17% compared to the real-time float PPP solutions, with enhancement in the east-west direction particularly significant, with an improvement of about 29%; (2) the average accuracy of the estimated LEO orbit with ambiguity-fixed solutions in the real-time KPOD and RPOD mode is improved by about 16% and 10%, respectively, with respect to the corresponding mode with the ambiguity-float solutions; (3) the performance of real-time LEO RPOD is better than that of the corresponding KPOD, regardless of fixed- or float-ambiguity solutions. Moreover, the average ambiguity-fixed ratio can reach more than 90% in real-time PPP, KPOD, and RPOD.

GNSS satellite inter-frequency clock bias estimation and correction based on IGS clock datum: a unified model and result validation using BDS-2 and BDS-3 multi-frequency data

GNSS satellite inter-frequency clock bias estimation and correction based on IGS clock datum: a unified model and result validation using BDS-2 and BDS-3 multi-frequency data

Urban positioning: 3D mapping‐aided GNSS using dual‐frequency pseudorange measurements from smartphones

<h3>Abstract</h3> A smartphone with a highly sensitive antenna receiving numerous unhealthy measurements suffers from non-line-of-sight (NLOS) reception and multipath effects. 3D mapping-aided (3DMA) GNSS has been proven to be effective in urban environments. However, the multipath effect remains challenging for urban positioning. In nature, the new GNSS civilian L5-band signal with a shorter chip length shows a much better resistibility to multipath than the conventional L1-band signal. Therefore, this study integrated the multi-constellation L5-band measurements into 3DMA GNSS to improve the positioning performance in urban canyons, namely the L1-L5 3DMA GNSS. Furthermore, this study compares different approaches on the receiver clock biases estimation for 3DMA GNSS. Finally, the integration of different 3DMA GNSSs is presented. The experiments conducted using smartphone data show that the L1-L5 3DMA GNSS is available for a better position solution than the 3DMA GNSS with L1-band only, thereby achieving a positioning accuracy within 10 m on average.

Comparison Analysis on the Accuracy of Galileo PPP Using Different Frequency Combinations in Europe

The Galileo constellations are characterized by transmitting GNSS signals on multi-frequencies, which can benefit the robustness and accuracy of the solutions. However, the dual-frequency E1/E5a combinations are generally used for precise point positioning (PPP). In this paper, the performance of Galileo static and kinematic PPP using different dual- and multi-frequency combinations are assessed using observations from the European region. Overall, the accuracy of daily PPP achieved by the dual-frequency GPS, Galileo, and BDS is better than 5 mm in the horizontal direction and better than 10 mm in the vertical direction. Though the number of observed Galileo satellites is less than GPS, the horizontal accuracy can reach 1.6 mm/2.3 mm/5.7 mm on North/East/Up component, which is improved by 59.0% and 12.3% compared to the GPS in the north and up direction. Then, the accuracy of Galileo static PPP is analyzed using different dual- and multi-frequency combinations. Results indicate that the Galileo E1/E5b PPP can degrade the accuracy due to the inter-frequency clock biases between the E1/E5a and E1/E5b combinations. Best accuracy can be achieved for the triple- and four-frequency PPP, which is 4.8 mm in the up direction. The hourly accuracy for the static PPP can reach 5.6 mm/9.2 mm/12.6 mm in the north/east/up direction using the GPS/Galileo/GLONASS/BDS combinations. Finally, a positioning convergence ratio (PCR) indicator, which represents the accuracy of PPP over a period, is used to analyze the convergence time of kinematic PPP. Results indicated that the multi-frequency Galileo observations contribute minorly to the convergence of kinematic PPP. However, Galileo shows the best convergence performance for the single GNSS positioning, and the GPS/Galileo combined PPP achieved the best performance for the PPP using different GNSS combinations.

A method for precisely predicting satellite clock bias based on robust fitting of ARMA models

A method for precisely predicting satellite clock bias based on robust fitting of ARMA models

A cost‐efficient joint target location and clock bias estimation technique using multiple time step measurements in mobile bistatic sonar

Active localization of a stationary target is one key issue in applications of bistatic sonar based on bistatic range (BR) measurement. However, the complex underwater environment makes achieving the synchronization between the transmitted station and the received station difficult. For solving the synchronization problem, this study relies on mobile bistatic sonar to acquire measurement information at multiple time steps. After that, the target location and clock bias are jointly estimated by solving a non-linear least square problem. To this end, a recursive scheme is proposed to optimise the measurement cost and reduce the impact of the initial value to achieve accurate estimation by taking full advantage of the redundant measurement information. Orthogonal transformation is used to ensure computational efficiency and numerical reliability. The Cramer–Rao Lower Bound (CRLB) is also derived as a lower bound on the error in estimating the target position and clock bias. Simulation results show that the proposed method achieves the CRLB performance and optimises the measurement cost over the small error region under Gaussian noise.

A Detection and Weakening Method for GNSS Time-Synchronization Attacks

In human society, the normal operation of important infrastructures, such as the power system, communication system, and computer network depend on the precise time provided by stationary satellite timing receivers, which are vulnerable to a kind of spoofing interference called time-synchronization attack. The aim of the attacks is to change the time provided by receivers to other systems, resulting in system disorders. This paper presents a time-synchronization attack detection and weakening method to solve this problem. The proposed method can estimate the binomial model parameters of receiver clock bias and the clock bias attack values at all observation instants by solving an optimization problem. Once the existence of the attacks is detected, the estimated clock bias attack values are used to correct the clock bias, so as to weaken the time-synchronization attacks. According to the numerical simulation results and analysis, we find that the proposed method outperforms other algorithms mentioned in this paper. Because it can not only realize real-time detection, but also greatly weaken the impact of time-synchronization attacks, making the attacks almost invalid.

Multipath Mitigation via Synthetic Aperture Beamforming for Indoor and Deep Urban Navigation

A synthetic aperture navigation (SAN) system that exploits downlink cellular long-term evolution (LTE) signals and an inertial measurement unit (IMU) is developed. The system is suitable for multipath-rich environments, such as indoors and deep urban canyons. The proposed SAN system mitigates multipath via a spatial discriminator, which utilizes the motion of a single antenna element to synthesize a geometrically separated antenna array from time-separated snapshots, alleviating the need for a physical antenna array. Signals from the synthesized antenna array are used to beamform towards the line-of-sight (LOS) LTE direction, while suppressing multipath components. Different stages of the beamforming process are discussed, and the computational complexity of the proposed system is analyzed. To deal with the unknown clock biases of the LTE eNodeBs, two navigation frameworks are developed: (1) base/rover and (2) standalone rover. The proposed SAN system is validated experimentally, and the navigation solutions achieved from the following systems are compared: (i) IMU only, (ii) LTE only, (iii) feedforward LTE-SAN, (iv) feedback LTE-SAN, (v) LTE-IMU, and (vi) feedback LTE-SAN-IMU. In the performed experiment, a pedestrian-mounted receiver navigated indoors for 109 m in 50 seconds, while receiving LTE signals from 5 LTE eNodeBs. The proposed LTE-SAN-IMU system exhibited a two-dimensional position root mean-squared error (RMSE) of 1.44 m with a standard deviation of 1.85 m.

Comparison and assessment of long-term performance of BDS-2/BDS-3 satellite atomic clocks

Recently, the performance of the new-generation BeiDou Navigation Satellite System (BDS) satellite clocks has attracted considerable attention. There are two types of BDS clocks developed in China—namely, the new rubidium (Rb-II) and passive hydrogen maser (PHM)—and these can affect positioning, navigation and timing (PNT) services. In this paper, to comprehensively evaluate and analyze the performance of BDS satellite atomic clocks, the physical characteristics are exhibited, including phase, frequency, frequency drift and noise. The physical characteristics are applied to detect the switching of satellite clocks. The frequency stability of atomic clocks is assessed by overlapping Allan deviation (ADEV) and overlapping Hadamard deviation (HDEV). The periodic characteristics of satellite clock bias data are studied through spectrum analysis. Based on 550 day precision satellite clock bias data provided by Wuhan University, China, experiments were performed, focusing on the performance of the Rb-II and PHM clocks installed on BDS-3. Some valuable conclusions are obtained: (a) the BDS-3 satellite clocks perform better than BDS-2. The root mean square value of the frequency drift of BDS-3 satellite clocks is enhanced by 41% compared to BDS-2. The noise of BDS-3 is 0.216 ns, which is 55% better than BDS-2. (b) The frequency stability of BDS-2 satellite clocks is worse than that of BDS-3. The mean daily stability of BDS-2 and BDS-3 satellite clocks by ADEV is 5.29 × 10−14 and 5.36 × 10−14, respectively, while that with HDEV is 2.35 × 10−14 and 0.70 × 10−14. For PHM clocks, the stability at 10 000 s with ADEV and HDEV is 2.29 × 10−14 and 2.25 × 10−14, respectively, while the daily stability is 0.77 × 10 −14 and 0.67 × 10−14, respectively. (c) BDS geostationary orbit (GEO) and inclined GEO satellite atomic clocks have multiple significant periodic terms while the medium earth orbit satellite clocks only own one or two, which are related to the corresponding satellite orbital periods.

Modeling and Performance Evaluation of Precise Positioning and Time-Frequency Transfer with Galileo Five-Frequency Observations

The present Global Navigation Satellite System (GNSS) can provide at least double-frequency observations, and especially the Galileo Navigation Satellite System (Galileo) can provide five-frequency observations for all constellation satellites. In this contribution, precision point positioning (PPP) models with Galileo E1, E5a, E5b, E5 and E6 frequency observations are established, including a dual-frequency (DF) ionospheric-free (IF) combination model, triple-frequency (TF) IF combination model, quad-frequency (QF) IF combination model, four five-frequency (FF) IF com-bination models and an FF uncombined (UC) model. The observation data of five stations for seven days are selected from the multi-GNSS experiment (MGEX) network, forming four time-frequency links ranging from 454.6 km to 5991.2 km. The positioning and time-frequency transfer performances of Galileo multi-frequency PPP are compared and evaluated using GBM (which denotes precise satellite orbit and clock bias products provided by Geo Forschung Zentrum (GFZ)), WUM (which denotes precise satellite orbit and clock bias products provided by Wuhan University (WHU)) and GRG (which denotes precise satellite orbit and clock bias products provided by the Centre National d’Etudes Spatiales (CNES)) precise products. The results show that the performances of the DF, TF, QF and FF PPP models are basically the same, the frequency stabilities of most links can reach sub10−16 level at 120,000 s, and the average three-dimensional (3D) root mean square (RMS) of position and average frequency stability (120,000 s) can reach 1.82 cm and 1.18 × 10−15, respectively. The differences of 3D RMS among all models are within 0.17 cm, and the differences in frequency stabilities (in 120,000 s) among all models are within 0.08 × 10−15. Using the GRG precise product, the solution performance is slightly better than that of the GBM or WUM precise product, the average 3D RMS values obtained using the WUM and GRG precise products are 1.85 cm and 1.77 cm, respectively, and the average frequency stabilities at 120,000 s can reach 1.13 × 10−15 and 1.06 × 10−15, respectively.