Clock Error Research Articles

Cooperative current estimation based multi-AUVs localization for deep ocean applications

In deep ocean scenario where drift of Autonomous Underwater Vehicle(AUV) localization system can be fast since no bottom tracking data can be obtained by Doppler Velocity Logger(DVL), missions for AUV are either hardly to be carried out or depended on expensive inertial instruments. This paper presents a multi-AUVs cooperative ocean current estimation method by formulating factor graphs and solving nonlinear optimization problems which can be used for typical applications such as formation maneuvering and cooperative target searching. Considering that only intermittent information is broadcasted by acoustic modems, observability of the system in each case is analyzed with designed simple formation switching strategies. With the help of water tracking velocity acquired by Acoustic Doppler Current Profiler(ADCP) and estimation of ocean current, accuracy of AUV localization system can be improved to an acceptable level for formation control and reliable position of the target can be obtained even with accumulated clock error. At last, field data based simulations have been carried out to demonstrate the feasibility of the proposed methodology.

Gaussian Bounds of Sample Distributions for Integrity Analysis

We present a method to determine Gaussian overbounding distributions used in integrity analysis. This method overcomes the limitations of previous techniques by combining them, through the determination of an intermediate symmetric and unimodal overbounding distribution. This method is the basis of a MATLAB toolset that computes strict Gaussian overbounding distributions for any sample distribution. We apply the method to the determination of the overbounding distribution of the GPS clock and ephemeris errors for the advanced receiver autonomous integrity monitoring.

Evaluation of Tropospheric & Clock Errors for Precision GPS Positioning & Navigation

Positional inaccuracies in GPS are caused by severalerrors such as Ionospheric, Tropospheric, Satellite Clock, Receiver Clock etc., Instantaneous correction of these error aids in precise navigation. In the present work Original Hopfield model is considered for the tropospheric correction. The instantaneous tropospheric correction results in more precise position using GPS. The decreasing order of components on basis of effect are Ionospheric delay, Tropospheric delay, Clock error, satellite bias error, Receiver error, multipath error, Ephymeris error, random errors etc. It is a time taken process to calculate the individual error separately.so in this paper we only concentrated on simulation and analysis the tropospheric delay, clock error, ephemeris error. We used Modified Hopfield model to analysis Tropospheric delay, receiver instrumental bias for analysis Clock error in between we eliminate Ephymeris error, after obtained results are compared with and without time correction in original Hopfield model.

A Hypothetical Analysis on GPS Evolution, Error Sources, Accuracy Measures and Positioning Services

People have invented different solutions to find out the location of a person or an object on the earth. In the beginning, mariners used angular measurements to calculate their location for celestial bodies such as the sun and stars. In the earlier 1920s, implemented a more sophisticated radio navigation scheme depends on radio waves that give access to the navigators to determine the direction of the transmitters. Soon after, the advancement of satellites made it achievable to transmit with high accuracy, line of sight radio navigation signals created a new era in the technology of navigation. To find the position of an object, a 2-dimensional navigation system called transit was used in olden days. This method was to set the basic building blocks to introduce the global positioning system. This new GPS concept provided outstanding innovative methods to find the position of an object to the public, scientists, military, etc. The predominantly GPS known as Global Positioning System is a range-based positioning system that yields an obscure object's 3D position on top of the earth. This system collects estimates or range data from realized emanating sources to determine the position of obscure objects and the system is also called as a positioning system based on satellites. The precision of the object location frequently relies on the error of the satellite clock, atmospheric delays, multipath, positioning of the satellite, noise and multiple parameters of the receiver. Mostly, none of the above parameters have consistent behavior worldwide and should be checked to provide an exact solution. This paper mainly focuses on the study of GPS evolution, error sources, popular accuracy measures and positioning services of GPS.

Robust Vehicular Localization and Map Matching in Urban Environments Through IMU, GNSS, and Cellular Signals

A framework for ground vehicle localization that uses cellular signals of opportunity (SOPs), a digital map, an inertial measurement unit (IMU), and a Global Navigation Satellite System (GNSS) receiver is developed. This framework aims to enable localization in an urban environment where GNSS signals could be unusable or unreliable. The proposed framework employs an extended Kalman filter (EKF) to fuse pseudorange observables extracted from cellular SOPs, IMU measurements, and GNSS-derived position estimates (when available). The EKF is coupled with a map-matching approach. The framework assumes the positions of the cellular towers to be known, and it estimates the vehicle's states (position, velocity, orientation, and IMU biases) along with the difference between the vehicle-mounted receiver clock error states (bias and drift) and each cellular SOP clock error state. The proposed framework is evaluated experimentally on a ground vehicle navigating in a deep urban area with a limited sky view. Results show a position root-mean-square error (RMSE) of 2.8 m across a 1,380-m trajectory when GNSS signals are available and an RMSE of 3.12 m across the same trajectory when GNSS signals are unavailable for 330 m. Moreover, compared to localization with a loosely coupled GNSS?IMU integrated system, a 22% reduction in the localization error is obtained whenever GNSS signals are available, and an 81% reduction in the localization error is obtained whenever GNSS signals are unavailable.

Ground Vehicle Navigation in GNSS-Challenged Environments Using Signals of Opportunity and a Closed-Loop Map-Matching Approach

A ground vehicle navigation approach in a global navigation satellite system (GNSS)-challenged environments is developed, which uses signals of opportunity (SOPs) in a closed-loop map-matching fashion. The proposed navigation approach employs a particle filter that estimates the ground vehicle’s state by fusing pseudoranges drawn from ambient SOP transmitters with road data stored in commercial maps. The problem considered assumes the ground vehicle to have a priori knowledge about its initial states as well as the position of SOPs. The proposed closed-loop approach estimates the vehicle’s states for subsequent time as it navigates without the GNSS signals. In this approach, a particle filter is employed to continuously estimate the vehicle’s position and velocity along with the clock error states of the vehicle-mounted receiver and SOP transmitters. The simulation and experimental results with cellular long-term evolution (LTE) SOPs are presented, evaluating the efficacy and accuracy of the proposed framework in different driving environments. The experimental results demonstrate a position root-mean-squared error (RMSE) of: 1.6 m over a 825-m trajectory in an urban environment with five cellular LTE SOPs, 3.9 m over a 1.5-km trajectory in a suburban environment with two cellular LTE SOPs, and 3.6 m over a 345-m trajectory in a challenging urban environment with two cellular LTE SOPs. It is demonstrated that incorporating the proposed map-matching algorithm reduced the position RMSE by 74.88%, 58.15%, and 46.18% in these three environments, respectively, from the RMSE obtained by an LTE-only navigation solution.

Along-strike variation in slab geometry at the southern Mariana subduction zone revealed by seismicity through ocean bottom seismic experiments

SUMMARYWe have conducted the first passive Ocean Bottom Seismograph (OBS) experiment near the Challenger Deep at the southernmost Mariana subduction zone by deploying and recovering an array of 6 broad-band OBSs during December 2016–June 2017. The obtained passive-source seismic records provide the first-ever near-field seismic observations in the southernmost Mariana subduction zone. We first correct clock errors of the OBS recordings based on both teleseismic waveforms and ambient noise cross-correlation. We then perform matched filter earthquake detection using 53 template events in the catalogue of the US Geological Survey and find >7000 local earthquakes during the 6-month OBS deployment period. Results of the two independent approaches show that the maximum clock drifting was ∼2 s on one instrument (OBS PA01), while the rest of OBS waveforms had negligible time drifting. After timing correction, we locate the detected earthquakes using a newly refined local velocity model that was derived from a companion active source experiment in the same region. In total, 2004 earthquakes are located with relatively high resolution. Furthermore, we calibrate the magnitudes of the detected earthquakes by measuring the relative amplitudes to their nearest relocated templates on all OBSs and acquire a high-resolution local earthquake catalogue. The magnitudes of earthquakes in our new catalogue range from 1.1 to 5.6. The earthquakes span over the Southwest Mariana rift, the megathrust interface, forearc and outer-rise regions. While most earthquakes are shallow, depths of the slab earthquakes increase from ∼100 to ∼240 km from west to east towards Guam. We also delineate the subducting interface from seismicity distribution and find an increasing trend in dip angles from west to east. The observed along-strike variation in slab dip angles and its downdip extents provide new constraints on geodynamic processes of the southernmost Mariana subduction zone.

Retrieving geophysical signals from GPS in the La Plata River region

Over the last few years, efforts to model short-term deformations of the Earth’s crust have multiplied. Sudden water level rise can cause sporadic, but significant, motions in the solid Earth’s surface. In this work, we address the problem of retrieving reliable estimates of the vertical displacement of a Global Positioning System (GPS) station located very close to the eastern shore of the La Plata River, during a strong storm surge event. Capturing sub-daily GPS displacements demands an elaborate processing strategy because several highly correlated parameters must be estimated simultaneously. We present a successful strategy that reduces the number of unknowns that have to be estimated simultaneously, by using an empirical model that describes the elastic response of the Earth’s crust to the hydrological load variations. We incorporate this model into the observation equations so that, instead of estimating the station position, we estimate every epoch a single parameter of the empirical model, i.e., the empirical elastic parameter EEP, that is assumed to be a constant of the Earth’s crust in the region of the GPS station. We verify that the estimated parameter agrees well with the value calculated from the CRUST 1.0-A model of the Earth’s crust. The GPS receiver was tied to an external cesium clock, which allowed us to process the data according to two different strategies: (a) estimating the receiver clock error (Δt) as an epoch-wise free parameter, which is equivalent to ignoring the presence of the external clock, and (b) conditioning the variability of that estimate with a small a priori variance compatible with the external clock’s variability. We find that, without having an external atomic clock, the estimation of all the parameters, i.e., the zenith tropospheric delay, Δt, and the EEP, worsens when the GPS station is affected by sub-daily vertical displacements.

Evaluation of the GPS Precise Orbit and Clock Corrections from MADOCA Real-Time Products.

The Japanese Quasi-Zenith Satellite System (QZSS) is a regional navigation satellite system covering the entire Asia-Oceania region. Except for the standard satellite navigation signals, QZSS satellites also broadcast L6E augmentation signals with real time GNSS precise orbit every 30 s and clock messages every 1 s, which is very important and necessary for Real-Time precise point positioning (RTPPP) applications. In this paper, the MADOCA real-time services derived from L6E augmentation signals were evaluated for both accuracy and availability compared with IGS final products. To avoid the datum difference of GPS orbit between MADOCA real-time and IGS final products, the 7-parameters Helmert transformation was firstly used in this paper, and then the orbit was evaluated on radial, along, and cross-track directions. On the clock evaluation, the mean satellites clock errors were taken as reference clock error, and then the standard deviation (STD) was calculated for each satellite. Furthermore, the signal in space range errors (SISRE) were also summarized to evaluate the ranging-measurement accuracy. Seven-day evaluation results show that satellite orbit, clock errors, and the final SISRE errors range as being 1.8–3.9 cm, 0.04–0.15 ns (1.2–4.5 cm), and 5–10 cm, respectively. For the one-year long-term evaluation, daily SISRE errors in 2018 show consistent performance with that of seven days. Furthermore, the open source software RTKLIB was used to evaluate the kinematic PPP performance based on the MADOCA real-time products, and it shows that the daily positioning accuracy of the 20 globally distributed IGS stations can reach 4.9, 4.2, 11.7, and 12.1 cm in the east, north, up, and 3D directions, respectively. Hence, it is concluded that the current MADOCA real-time ephemeris products can provide orbit and clock products with SISRE on centimeters level with high interval, which could meet the demands of the RTPPP solution and serve real-time users who can access the MADOCA real-time products via L6E signal or internet.

Comparison of orientational error of an optically pumped quantum sensor in on-board equipment of Galileo and GPS satellite systems

A comparison of the hypothetical orientational shift of the rubidium frequency standards on gas cell used in the on-board equipment of the Galileo and GPS satellites navigation systems is presented. The calculated dependences of the orientational error of the rubidium atomic clock as a function of the time of movement of the Galileo and GPS satellites in orbit are given. The calculated values of the correlation coefficients of the dependences of the orientational shift of the atomic clocks and the experimental data of the temporal dependence of the orientational error of the satellites are presented. There is a significant excess of the correlation coefficients in the GPS satellite system in comparison with the Galileo system. This may be due to the difference in the contribution of the orientational error of atomic clocks to the total error of the satellite ephemeris.

The Unified Form of Code Biases and Positioning Performance Analysis in Global Positioning System (GPS)/BeiDou Navigation Satellite System (BDS) Precise Point Positioning Using Real Triple-Frequency Data

Multi- system and multi-frequency are two key factors that determine the performance of precise point positioning. Both multi-frequency and multi-system lead to new biases, which are not solved systematically. This paper concentrates on mathematical models of biases, influences of these biases, and positioning performance analysis of different observation models. The biases comprise the inter-frequency clock bias in multi-frequency and the inter-system clock bias in multi-system. The former is the residual differential code biases (DCBs) from receiver clock and satellite clock and usually occurs at the third frequency, the latter is the deviation of the receiver clock errors in different systems. Unified mathematical models of the biases are presented by analyzing the general formula of observation equations. The influences of these biases are validated by experiments with corresponding observation models. Subsequently, the experiments, which are based on the data at five globally distributed stations in Multi-Global Navigation Satellite System (GNSS) Experiment (MGEX) on day of year 100, 2018, assess positioning performance of different observation models with combination of frequencies (dual-frequency or triple- frequency) and systems (BeiDou Navigation Satellite System (BDS) or Global Positioning System (GPS)). The results show that the performances of triple-frequency models are almost as the same level as the dual-frequency models. They provide scientific support for the triple-frequency ambiguity-fixed solution which has a better convergence characteristic than dual-frequency ambiguity-fixed solution. Furthermore, the biases are expressed as an unified form that gives an important and valuable reference for future research on multi-frequency and multi-system precise point positioning.

Indoor non-line-of-sight and multipath detection using deep learning approach

Global navigation satellite systems (GNSSs) provide sufficient position accuracy outdoors but can perform poorly in indoor situations. Therefore, various techniques have been presented to meet the indoor requirements, among which the pseudolite system (PLS) is widely studied. A PLS has several error sources including clock error, thermal noise and path-dependent error; among them path-dependent errors, e.g., non-line-of-sight (NLOS, i.e., the direct ray is blocked, only reflected rays received) and multipath (i.e., the direct ray is received together with the reflected rays), are the most intractable to deal with and always cause large range errors. The indoor furniture and flat surfaces of the walls and ceilings make NLOS and multipath quite severe. Detecting NLOS and multipath is a classification problem, which can be well tackled by deep learning approaches. Deep learning algorithms are functions of nonlinear regression using thousands of parameters that distributed in many hidden layers whose values are determined by training phase. We present an NLOS and multipath detecting network (NMDN) that consists of five convolution layers and two fully connected layers; we feed NMDN with the outputs of a receiver’s tracking loop; subsequently, it tells us the detection results. The training and testing data sets are generated by a GNSS software receiver using intermediate frequency signal collected from an indoor PLS. The presented method is compared with two support vector machines, which are the traditional methods for classification, and shows an improvement of up to 45% in overall classification accuracy. The impacts of indoor NLOS and multipath are also analyzed with double difference observables. The results show NLOS causes more serious range error than multipath. The proposed method is suitable for detecting NLOS and multipath when the receiver operating environment is not stable.

WASP-4b Arrived Early for the TESS Mission

Abstract The Transiting Exoplanet Survey Satellite (TESS) recently observed 18 transits of the hot Jupiter WASP-4b. The sequence of transits occurred 81.6 ± 11.7 s earlier than had been predicted, based on data stretching back to 2007. This is unlikely to be the result of a clock error, because TESS observations of other hot Jupiters (WASP-6b, 18b, and 46b) are compatible with a constant period, ruling out an 81.6 s offset at the 6.4σ level. The 1.3 day orbital period of WASP-4b appears to be decreasing at a rate of ms per year. The apparent period change might be caused by tidal orbital decay or apsidal precession, although both interpretations have shortcomings. The gravitational influence of a third body is another possibility, though at present there is minimal evidence for such a body. Further observations are needed to confirm and understand the timing variation.

Ephemeris Errors and the Gravitational-wave Signal: Harmonic Mode Coupling in Pulsar Timing Array Searches

Abstract Any unambiguous detection of a stochastic gravitational wave background (GWB) by a pulsar timing array will rest on the measurement of a characteristic angular correlation between pulsars. The ability to measure this correlation will depend on the geometry of the array. However, spatially correlated sources of noise, such as errors in the planetary ephemeris or clock errors, can produce false-positive correlations. The severity of this contamination will also depend on the geometry of the array. This paper quantifies these geometric effects with a spherical harmonic analysis of the pulsar timing residuals. At least nine well-spaced pulsars are needed to simultaneously measure a GWB and separate it from ephemeris and clock errors. Uniform distributions of pulsars can eliminate the contamination for arrays with large numbers of pulsars, but pulsars following the galactic distribution of known millisecond pulsars will always be affected. We quantitatively demonstrate the need for arrays to include many pulsars and for the pulsars to be distributed as uniformly as possible. Finally, we suggest a technique to cleanly separate the effect of ephemeris and clock errors from the gravitational wave signal.

Airborne Pseudolite Distributed Positioning based on Real-time GNSS PPP

Airborne-Pseudolite (A-PL) systems have been proposed to augment Global Navigation Satellite Systems (GNSSs) in difficult areas where GNSS-only navigation cannot be guaranteed due to signal blockages, signal jamming, etc. One of the challenges in realising such a system is to determine the coordinates of the A-PLs to a high accuracy. The GNSS Precise Point Positioning (PPP) technique is a possible alternative to differential GNSS techniques such as those that generate Real-Time Kinematic (RTK) solutions. To enhance the A-PL positioning performance in GNSS challenged areas, it is assumed that inter-PL range measurements are also used in addition to GNSS measurements. When processing these new measurements, cross-correlations among A-PL estimated states introduced during measurement updates need to be accounted for so as to obtain consistent estimated states. In this paper, a distributed algorithm based on a Split Covariance Intersection Filter (SCIF) is proposed. Three commonly used means of implementing the SCIF algorithm are analysed. Another challenge is that real-time GNSS PPP relies on the use of precise satellite orbit and clock information. One problem is that these real-time orbit and satellite clock error corrections may not be always available, especially for moving A-PLs in challenging environments when signal outages occur. To maintain A-PL positioning accuracy using GNSS PPP, it is necessary to predict these error corrections during outages. Different prediction models for orbit and clock error corrections are discussed. A test was conducted to evaluate the A-PL positioning based on GNSS PPP and inter-PL ranges, as well as the performance of error prediction modelling. It was found that GNSS PPP combined with inter-PL ranges could achieve better converged positioning accuracy and a reduction in convergence time of GNSS PPP. However, the performance of GNSS PPP with inter-PL ranges was degraded by observing A-PLs with limited positioning accuracy. Although the performance improvement achieved by the SCIF-based distributed algorithms was smaller than that by the centralised algorithm, greater robustness in dealing with deteriorated observed A-PLs' trajectory data was demonstrated by the distributed algorithms. In addition, short-term prediction models were analysed, and their performance was shown to reduce the effect of error correction outages on A-PL positioning accuracy.

X-Ray Pulsar-Based Navigation Considering Spacecraft Orbital Motion and Systematic Biases

The accuracy of X-ray pulsar-based navigation is greatly affected by the Doppler effect caused by the spacecraft orbital motion and the systematic biases introduced by the pulsar directional error, spacecraft-borne clock error, etc. In this paper, an innovative navigation method simultaneously employing the pulse phase (PP), the difference of two neighbor PPs (DPP) and the Doppler frequency (DF) of X-ray pulsars as measurements is proposed to solve this problem. With the aid of the spacecraft orbital dynamics, a single pair of PP and DF relative to the spacecraft’s state estimation error can be estimated by using the joint probability density function of the arrival photon timestamps as the likelihood function. The systematic biases involved to the PP is proved to be nearly invariant over two adjacent navigation periods and the major part of it is eliminated in the DPP; therefore, the DPP is also exploited as additional navigation measurement to weaken the impact of systematic biases on navigation accuracy. Results of photon-level simulations show that the navigation accuracy of the proposed method is remarkably better than that of the method only using PP, the method using both PP and DF and the method using both PP and DPP for Earth orbit.

Stochastic Observability and Uncertainty Characterization in Simultaneous Receiver and Transmitter Localization

The stochastic observability of simultaneous receiver and transmitter localization is studied. A mobile vehicle-mounted receiver is assumed to draw pseudorange measurements from multiple unknown radio frequency transmitters and to fuse these measurements through an extended Kalman filter (EKF) to simultaneously localize the receiver and transmitters together with estimating the receiver's and transmitters’ clock errors. The receiver is assumed to have perfect a priori knowledge of its initial states, while the transmitters’ states are unknown. It is shown that the receiver's and transmitters’ clock biases are stochastically unobservable and that their estimation error variances will diverge. A lower bound on the divergence rate of the estimation error variances of the receiver's and transmitters’ clock biases is derived and demonstrated numerically. Simulation and experimental results are presented for an unmanned aerial vehicle navigating without GPS signals, using pseudoranges made on unknown terrestrial transmitters. It is demonstrated that despite the receiver's and transmitters’ clock biases being stochastically unobservable, the EKF produces bounded localization errors.

Integrated navigation for Mars final approach based on Doppler radar and X-ray pulsars with atomic clock error

Final approach phase has been recognized as one of the key phases in Mars landing progress. The performance of the navigation system in this phase determines the positioning accuracy of the atmospheric entry point, which further influences the actualization of Mars high-precision pinpoint landing. Regarding this phase, conventional navigation strategies rely on the Earth-based measurement whose time delay is serious due to the long distance between the spacecraft and the ground telemetering station. Therefore, an X-ray pulsars and Doppler radar integrated navigation algorithm is proposed in the current study to ensure the real-time measurement information acquisition and to enhance the reliability of the navigation system. The non-Gaussian bias error of the atomic clock is also considered in the measurement model to improve the accuracy of estimation results. Meanwhile, the augmented state rank Kalman filter (ARKF) is introduced to match the new nonlinear navigation model involving non-Gaussian noises. Numerical simulation results show that the proposed navigation algorithm can make full use of the navigation information provided by X-ray pulsars and Doppler radar, and can effectively reduce the influence of atomic clock errors and guarantee a better performance than traditional methods.

A new test of gravitational redshift using Galileo satellites: The GREAT experiment

Abstract We present the result of the analysis of the GREAT (Galileo gravitational Redshift test with Eccentric sATellites) experiment. An elliptic orbit induces a periodic modulation of the fractional frequency difference between a ground clock and the satellite clock, partly due to the gravitational redshift, while the good stability of Galileo clocks allows one to test this periodic modulation to a high level of accuracy. GSAT0201 and GSAT0202, with their large eccentricity and on-board H-maser clocks, are perfect candidates to perform this test. Satellite laser ranging data allows us to partly decorrelate the orbit perturbations from the clock errors. By analyzing several years of Galileo tracking data, we have been able to improve the Gravity probe A test (1976) of the gravitational redshift by a factor of 5.6, providing, to our knowledge, the first reported improvement since more than 40 years.

Lane-Level Localization and Mapping in GNSS-Challenged Environments by Fusing Lidar Data and Cellular Pseudoranges

A method for achieving lane-level localization in global navigation satellite system (GNSS)-challenged environments is presented. The proposed method uses the pseudoranges drawn from unknown ambient cellular towers as an exclusive aiding source for a vehicle-mounted light detection and ranging (lidar) sensor. The following scenario is considered. A vehicle aiding its lidar with GNSS signals enters an environment where these signals become unusable. The vehicle is equipped with a receiver capable of producing pseudoranges to unknown cellular towers in its environment. These pseudoranges are fused through an extended Kalman filter to aid the lidar odometry, while estimating the vehicle's own state (3-D position and orientation) simultaneously with the position of the cellular towers and the difference between the receiver's and cellular towers’ clock error states (bias and drift). The proposed method is computationally efficient and is demonstrated to achieve lane-level accuracy in different environments. Simulation and experimental results with the proposed method are presented illustrating a close match between the vehicle's true trajectory and estimated using the cellular-aided lidar odometry over a 1 km trajectory. The proposed method yielded a 68% reduction in the 2-D position root mean-squared error (RMSE) over lidar odometry-only.