Integer Ambiguity Research Articles

Differential positioning based on the orthogonal transformation algorithm with GNSS multi-system

Combining global navigation satellite systems (GNSSs) will significantly increase the number of visible satellites and, thus, will improve the geometry of observed satellites, resulting in improved positioning reliability and accuracy. We focus on GNSS multi-system differential positioning based on a single-system orthogonal transformation algorithm. The orthogonal transformation algorithm using single-difference measurements is proposed to avoid the high correlation between measurements and the unnecessary prominence to the reference satellite in double-difference positioning. In addition, the algorithm uses a more straightforward recursive least squares method to avoid the effect of uncertainties of the Kalman filter. We discuss the model differences between combined system positioning and single-system positioning and verify that the combining observations of different systems should start to be used after clock biases have been reduced, respectively. Moreover, as to rising and setting of satellites in multi-system differential positioning, we propose to use matrix transform to separate the setting satellites of combined systems at an epoch. This can avoid the correlation of initial integer ambiguity vectors of different systems. The experimental results show that the proposed method can handle the change of satellites automatically and combine multiple systems for reliable and accuracy differential positioning. The method especially outperforms the basic single-system orthogonal transformation positioning and traditional multi-system double-difference positioning in a complex environment.

Real Time Precise Relative Positioning with Moving Multiple Reference Receivers.

The stationary reference receiver with precisely known coordinates is difficult to establish in some special real-time relative positioning applications. To improve the relative position estimation accuracy and the reliability simultaneously for the RTK without a precisely known reference receiver, multiple receivers mounted on a moving platform are used as the base station. A code and phase measurement fusion model is proposed to reduce the communication burden and generate measurements at any virtual position where it is inconvenient to install the GPS receiver. To keep the integer property of the ambiguity of fused phase measurements, the RTK method with the moving reference receivers is proposed by implementing the integer ambiguity transformation and error absorption strategy based on the known geometry of multiple receivers. Static and kinematic experiments were carried out to evaluate the performance of the proposed relative positioning method. When compared with the single-receiver solution, static results have shown that the proposed method can improve position accuracy by 15.9% and 15.7% for the horizontal and the vertical component, respectively. The kinematic results have shown that the proposed method can achieve position accuracy enhancement by 26.9% for the vertical component.

Estimating the orbit error of BeiDou GEO satellites to improve the performance of multi-GNSS PPP ambiguity resolution

The Chinese BeiDou Navigation Satellite System (BDS) contains five geostationary earth orbit (GEO) satellites, which maintain almost stationary with respect to the earth. The accuracy of GEO orbit is very poor, which has negatively influenced multi-GNSS precise point positioning (PPP) integer ambiguity resolution (IAR). To overcome this problem, we estimate GEO orbit error corrections together with the narrow-lane fractional cycle bias (FCB), which is used to refine the orbit. We also estimate systematic bias in current GLONASS and BDS precise satellite clock products. Then, this solution is validated with combined GPS, GLONASS, and BDS PPP-IAR. A 7-day dataset of 45 stations was used in the experiment. It is demonstrated that, after considering GEO satellite orbit errors, the narrow-lane FCB estimates have comparable quality for each system. For kinematic PPP with an observation time of 6 min, only 14.7% of cases could be fixed by GPS alone, whereas the percentage decreased to 7.5% for GPS + GLONASS + BDS without considering GEO orbit errors. In contrast, if we consider that error, the percentage for GPS + GLONASS + BDS improved substantially to 96.9%.

Real-Time Phase Bias Estimation for BeiDou Satellites Based on Consideration of Orbit Errors

Correction of the fractional cycle bias (FCB) in the undifferenced ambiguity allows precise point positioning (PPP) integer ambiguity resolution (IAR) to be achieved, which can improve positioning accuracy significantly. In addition, in real-time PPP-IAR, integration of the BeiDou Navigation Satellite System (BDS) can provide a significant reduction in the initial fixing time of global positioning system (GPS)-only PPP-IAR. However, the FCB quality can be considerably affected by the low precision of the BDS orbit, which then severely hampers the GPS + BDS PPP-IAR performance. Therefore, a real-time FCB estimation strategy that takes the BDS satellite orbit error into consideration was developed in this study. The slant orbit error can be absorbed by the ionosphere-free (IF) ambiguity, which can then be recovered by fixing all IF ambiguities from all the tracking stations. The estimated orbit error is then used to refine the orbit, which is broadcast along with the FCBs to enable PPP ambiguity resolution. To evaluate the proposed strategy, an experiment using 60 tracking stations covering the China region is performed in a simulated real-time mode. The a posteriori residuals of both the wide- and narrow-lane ambiguities are checked to validate the efficiency of the proposed FCB strategy. The results show that when the proposed strategy is applied, the effect of the BDS orbit error on narrow-lane FCB estimation is eliminated and more than 94% of the narrow-lane residuals are within 0.1 cycles for both the GPS and the BDS. The fixing percentage within 20 min is 46.3% for the GPS-only solution but is only 4.8% when using GPS + BDS with the traditional method. However, when the proposed strategy is used, the fixing percentage for GPS + BDS improves significantly to 91.7%.

Estimating satellite phase fractional cycle biases based on Kalman filter

Phase fractional cycle biases (FCBs) originating from satellites and receivers destroy the integer nature of PPP carrier phase ambiguities. To achieve integer ambiguity resolution of PPP, FCBs of satellites are required. In former work, least squares methods are commonly adopted to isolate FCBs from a network of reference stations. However, it can be extremely time consuming concerning the large number of observations from hundreds of stations and thousands of epochs. In addition, iterations are required to deal with the one-cycle inconsistency among FCB measurements. We propose to estimate the FCB based on a Kalman filter. The large number of observations are handled epoch by epoch, which significantly reduces the dimension of the involved matrix and accelerates the computation. In addition, it is also suitable for real-time applications. As for the one-cycle inconsistency, a pre-elimination method is developed to avoid iterations and posterior adjustments. A globally distributed network consisting of about 200 IGS stations is selected to determine the GPS satellite FCBs. Observations recorded from DoY 52 to 61 in 2016 are processed to verify the proposed approach. The RMS of wide lane (WL) posterior residuals is 0.09 cycles while that of the narrow lane (NL) is about 0.05 cycles, which indicates a good internal accuracy. The estimated WL FCBs also have a good consistency with existing WL FCB products (e.g., CNES-GRG, WHU-SGG). The RMS of differences with respect to GRG and SGG products are 0.03 and 0.05 cycles. For satellite NL FCB estimates, 97.9% of the differences with respect to SGG products are within ± 0.1 cycles. The RMS of the difference is 0.05 cycles. These results prove the efficiency of the proposed approach.

An Improved PSO Algorithm and Its Application in GNSS Ambiguity Resolution

In this study, we proposed a novel method for global navigation satellite system (GNSS) ambiguity resolution (AR). The proposed method utilizes an improved particle swarm optimization (IPSO) algorithm to obtain the GNSS integer ambiguity with the double differenced (DD) float resolution and its corresponding covariance matrix. First, we introduced population maturity to the standard PSO (SPSO) algorithm for the adaptive adjustment of inertia weight. Next, to improve the global convergence and robustness of the SPSO algorithm, we adopted population classification and constructed a Gauss mutation for the particle evolution process of the optimal population. Then, we applied the IPSO algorithm in the field of GNSS AR, called IPSO–AR. Finally, we evaluated the performance of the IPSO–AR algorithm under different DD ambiguity float resolutions with various dimensions and precisions. Numerical results showed that compared with the SPSO–AR algorithm, the IPSO–AR algorithm has a superior correct rate, but low efficiency. Under the appropriate parameter settings, the efficiency of the IPSO–AR algorithm is mainly dependent on the dimensions of DD ambiguity, whereas the correct rate of the IPSO–AR algorithm is mainly dependent on the precision of DD ambiguity. The proposed IPSO–AR algorithm has potential applications under the conditions of few visible satellites or constrained baseline length.

GNSS Integer Ambiguity Resolution Methods Applied by Kalman Filter — The Review and Comparison

In this paper, we investigate several methods to resolve integer ambiguities in Global Navigation Satellite Systems (GNSS) carrier phase positioning applied by Kalman filter. Especially, we focus on integration methods of Least squares AMBiguity Decorrelation Adjustment (LAMBDA) [1, 2] method and Kalman filter. They are compared from theoretical and experimental points of view, and their characteristics and performance are investigated. We conduct experiments using real receiver data, and evaluate the integer ambiguity resolution performance. Based on the results, we show a guideline to resolve ambiguities with Kalman filter.

Snow Water Equivalent of Dry Snow Derived From GNSS Carrier Phases

Snow water equivalent (SWE) is a key variable for various hydrological applications. It is defined as the depth of water that would result upon complete melting of a mass of snow. However, until now, continuous measurements of the SWE are either scarce, expensive, labor-intense, or lack temporal or spatial resolution especially in mountainous and remote regions. We derive the SWE for dry-snow conditions using carrier phase measurements from the Global Navigation Satellite System (GNSS) receivers. Two static GNSS receivers are used, whereby one antenna is placed below the snow and the other antenna is placed above the snow. The carrier phase measurements of both receivers are combined in double differences (DDs) to eliminate clock offsets and phase biases and to mitigate atmospheric errors. Each DD carrier phase measurement depends on the relative position between both antennas, an integer ambiguity due to the periodic nature of the carrier phase signal, and the SWE projected into the direction of incidence. The relative positions of the antennas are determined under snow-free conditions with millimeter accuracy using real-time kinematic positioning. Subsequently, the SWE and carrier phase integer ambiguities are jointly estimated with an integer least-squares estimator. We tested our method at an Alpine test site in Switzerland during the dry-snow season 2015–2016. The SWE derived solely by the GNSS shows very high correlation with conventionally measured snow pillow (root mean square error: 11 mm) and manual snow pit data. This method can be applied to dense low-cost GNSS receiver networks to improve the spatial and temporal information on snow.

A dynamic single epoch ambiguity resolution algorithm with straight trajectory constraints

The key problem of dynamic single epoch GPS positioning is the rapid decomposition of ambiguity. In some special dynamic positioning, the coordinates of the GPS receiver placed on the rover station often have the constraint condition, if this useful information is taken into account, it will help to determine the ambiguity accurately. For this reason, the straight motion trajectory of GPS receiver is used as the constraint condition for single epoch ambiguity resolution, in this algorithm, the wide lane ambiguities are calculated by using the MW combination, and the corresponding alternative combinations of wide lane ambiguity are established, which are determined according to the minimum residual sum of squares criterion firstly, it is used to lay the foundation for the subsequent L1 and L2 frequency ambiguity solution. And then the straight trajectory constraint condition and the phase observation equation are used to solve the integer ambiguity N1. When the constraint information of the trajectory is unknown, the space linear equation is established as the constraint condition according to the coordinate solution by wide lane relative positioning to solve the unknown problem. By designing the corresponding experiments, the results of this algorithm are proved to be feasible by using different experimental schemes, such as without constraint condition, the constraint condition with known prior information, introducing dynamic constraint condition and so on, to solve the ambiguity, and to compare and analyze the effect of linear constraint condition on single epoch ambiguity resolution.

Multipath mitigation method based on Gaussian mixture model in RF relative measurement

Radio Frequency (RF) technology represents a high-precision relative navigation solution that has significant potential for application to earth-orbiting satellites. In precision applications, multipath errors dominate the total error because observables, which are used to estimate carrier-phase integer ambiguity, are not always subject to a Gaussian distribution when dual-frequency ambiguity estimation methods are used in the presence of multipath. As it has been shown that ranging observables obey a Gaussian mixture distribution, this study proposes improvements to the accuracy of estimation based on multipath mitigation founded on the Gaussian mixture model. To this end, such a model is created for integer ambiguity resolution in the presence of multipath, using which the theoretical error in dual-frequency ambiguity estimation is derived. Expectation Maximization (EM), which aids dual-frequency ambiguity estimation, is subsequently proposed to reduce the effect of multipath errors. Finally, two experimental scenarios are implemented to test the performance of the proposed method. The results show that EM-aided dual-frequency ambiguity estimation reduces the range error to approximately 20% in comparison with simple dual-frequency ambiguity estimation. Therefore the proposed technique is effective for multipath mitigation in RF relative measurement.

Attitude, Angle of Attack and Angle of Sideslip Determination Using Carrier Phase Integer Ambiguity Estimation

The Global Navigation Satellite System (GNSS) positioning solution may be computed to a cm level of error, if the signal carrier phase is properly considered during the calculations. With such a precision, even attitude angles can be precisely estimated if more than one antennae is available. However, using the carrier phase information depends on solving the carrier phase integer ambiguity problem which is intrinsic to this approach. Since only a fraction of the carrier phase is observed by the GNSS receivers, the solution for the integer ambiguity is usually obtained by means of search algorithms, based on least squares. This contribution investigates the performance of the integer ambiguity solution for the attitude determination problem and also proposes a novel application: angle of attack and angle of sideslip estimation for specific aeronautical application. This novel application benefits aeronautical industry by introducing a new alternative for measuring reliable airdata parameters, namely angle of attack and angle of sideslip, by using a nonsusceptible to icing conditions data. The results presented here are based on the constrained LAMBDA method. Dynamic real data are used for evaluation of the proposed novel application.

Improvements on the particle-filter-based GLONASS phase inter-frequency bias estimation approach

The carrier phase inter-frequency bias (IFB) of GLONASS between receivers of different types is usually not zero. This bias must be estimated and removed in data processing so that the integer double difference (DD) ambiguities can be fixed successfully. Recently, the particle filter approach has been proposed to estimate the IFB rate in real time. In this approach, the IFB rate samples are first generated and used to correct the phase IFB in the GLONASS observations. Then, the weights of the rate samples are updated with a function related to RATIO which is for ambiguity acceptance testing in integer ambiguity resolution. Afterwards, the IFB rate is estimated according to the weighted particles. This approach can estimate IFB accurately with short convergence time and without prior information. However, when the system noise is set too low, the estimated results are unstable due to the serious problem of particle diversity-loss, even though the system model is accurate. Additionally, the computational burden is dependent on the number of particles, which has to be optimized for the computation at hand. Therefore, this study proposes two improvements for the IFB estimation in regard to the above two aspects. The first improvement is to solve the noise setting problem by employing a regularized particle filter (RPF). The second improvement optimizes the number of particles in the resampling step according to the standard deviation (STD) of the weighted particles via a controlling function. The two improvements result in significantly better performances. The regularization method allows for the system noise to be set as zero without disturbing the estimates, and consequently, more precise estimates can be achieved. In addition, the approach using the controlling function for adapting the number of particles has comparable performance in precision but the computation load is largely reduced.

A novel ambiguity search algorithm for high accuracy differential GNSS relative positioning

It is well understood that the key to high accuracy differential global navigation satellite system (DGNSS) relative positioning is the resolution of the integer ambiguities within the carrier phase measurements and the continuous tracking of them. The resolution process is usually converted into an integer least square optimization problem, e.g., among them is the most notable Least-squares Ambiguity Decorrelation Adjustment (LAMBDA) algorithm. This paper adds additional statistical constraints to the existing LAMBDA approach to improve the performance of ambiguity resolution process when the LAMBDA ratio is below a certain predefined threshold. In order to fix the integer ambiguity as soon as possible, a novel ambiguity search algorithm is proposed, which is like the threshold based algorithm but explicitly exploits the correlation structure of the double difference covariance model and the measurement accuracy difference. To do that, the relationship between global navigation satellite system (GNSS) pseudorange measurement accuracy and the resolution of the integer ambiguity of carrier phase measurements is analyzed based on the GNSS distance measurement equations. Analytic statistical models of the single and double differences of the distance measurements are presented. Based on the analysis, it is found that the conventional choice of the highest elevation satellite as the reference satellite may not be a superior selection in single epoch algorithms. In fact, code phase measurements are used for ambiguity resolution, and the covariance of baseline vector is independent with reference satellite selection when the least square problem is optimally weighted. The novel ambiguity search algorithm is presented to fix the ambiguity as soon as possible. Simulation and field test data validate the analysis and the ambiguity search algorithm.

Estimation of satellite position, clock and phase bias corrections

Precise point positioning with integer ambiguity resolution requires precise knowledge of satellite position, clock and phase bias corrections. In this paper, a method for the estimation of these parameters with a global network of reference stations is presented. The method processes uncombined and undifferenced measurements of an arbitrary number of frequencies such that the obtained satellite position, clock and bias corrections can be used for any type of differenced and/or combined measurements. We perform a clustering of reference stations. The clustering enables a common satellite visibility within each cluster and an efficient fixing of the double difference ambiguities within each cluster. Additionally, the double difference ambiguities between the reference stations of different clusters are fixed. We use an integer decorrelation for ambiguity fixing in dense global networks. The performance of the proposed method is analysed with both simulated Galileo measurements on E1 and E5a and real GPS measurements of the IGS network. We defined 16 clusters and obtained satellite position, clock and phase bias corrections with a precision of better than 2 cm.

Relative positioning of formation-flying spacecraft using single-receiver GPS carrier phase ambiguity fixing

In recent years, differential carrier phase-based relative positioning (or “baseline determination”) with precision at the millimeter and submillimeter levels has been demonstrated for the GRACE, TanDEM-X and Swarm missions in offline processing. Specific features of such missions have included the use of spacecraft of similar shapes placed in almost identical orbits as well as the use of consistent geodetic-class GPS receivers. These elements have proven to be advantageous for the computation of baseline solutions with such precision levels. Particularly, they have allowed to fully leverage the use of differential GPS techniques, including the estimation and use of carrier phase integer ambiguities. Similarly, the aforementioned spacecraft and orbit characteristics have made it possible to tightly constrain the relative dynamics of formations in the generation of reduced-dynamic solutions. Other than the former examples, prospective formation-flying mission proposals, such as SAOCOM-CS and PICOSAR, may comprise spacecraft with very different characteristics, including dissimilar GPS/GNSS receivers. Such cases may no longer provide favorable conditions for relative orbit determination strategies. As an alternative, absolute orbit solutions may be computed individually for each spacecraft and used for the generation of precise baseline products. This study aims at the assessment of the potential of single-receiver ambiguity fixing for the generation of precise baseline solutions. Results using flight data from the GRACE, TanDEM-X and Swarm missions exhibit baseline accuracy better than 5 mm (3D RMS) for a one-month test period in June 2016. As such, the presented solutions may be considered for prospective formation-flying remote sensing missions with baseline precision requirements at the subcentimeter level. Likewise, the method is considered of particular interest for future multi-spacecraft formations and swarms that require efficient determination of a large number of individual baselines.

Detection and elimination method for deception jamming based on an antenna array

A global positioning system is an important way of locating an aircraft, while deception jamming can affect the positioning accuracy of such navigation. Considering this, a detection and elimination method for deception jamming is proposed based on a specially designed array for the aircraft. The jamming is detected by comparing the double-difference observation of the carrier phases of two different signals to a certain threshold derived according to the measurement errors of the receiver. To estimate the jamming direction with high accuracy, meanwhile considering the feasibility of airborne installation, a novel configurated array combining medium-length baseline with short baseline is designed, and a fast method to solve the integer ambiguity is discussed. After jamming detection, the nulling of the array beam is pointed to the jamming source through the orthogonal vector weighting to suppress jamming. The validity of the method is verified by computer simulations.

GNSS Spoofing Detection Technique Using Fraction Parts of Double-difference Carrier Phases

With electronic warfare increasingly complicated and military confrontations increasingly intense, the potential security threat to satellite navigation has become a difficult issue to deal with. Traditional satellite navigation anti-interference technology generally refers to jamming, but less consideration has been given to spoofing. It should be noted that the potential risk induced by spoofing interference is worse than that caused by jamming as the loss of positioning integrity may not be immediately obvious. This paper introduces a spoofing detection method based on a two-antenna structure using fraction parts of double-difference carrier phase observables. If all spoofing signals are transmitted by one single antenna, a spoofing detection hypothesis test can be carried out through the normalisation of double-difference carrier phase observables, without need to consider the integer ambiguity problem, measure the baseline vector and estimate the real directions of signals. The detection scheme adopts an M of N algorithm (if M or more of the test values exceed the threshold, the algorithm declares the presence of a spoofing signal), integrating carrier phase measurements of all the available satellites. Finally, the proposed method is verified by real experiments. This spoofing detection method can easily be applied to GNSS anti-spoofing receivers without changing their architecture and has simple and effective characteristics.

A new single-epoch method for resolving GNSS ambiguity with segmented search in short baseline

Correct ambiguity resolution (AR) is the key for high-precision positioning. In order to reliably resolve ambiguity, an improved single-epoch segmented ambiguity algorithm is proposed in this paper. First, the double-difference wide-lane ambiguities resolved by the Melbourne-Wubbena (MW) method are divided into two categories based on signal-to-noise ratio (SNR). One is the primary ambiguity group that can be resolved easily, and the other is the secondary ambiguity group that is difficult to resolve. Second, the primary ambiguity group is divided into different levels in a (4,1,1…1) pattern based on SNR, and the different levels’ ambiguities are resolved successively, and the method being that the integer ambiguity is selected from two candidate values built by rounding up and down. Third, the resolved primary ambiguity group is used as a constraint to solve the secondary ambiguity group. Finally, the wide-lane observation equation and the basic observation equation are combined to solve the ambiguity of the L1 carrier phase, and the resolved ambiguity can be obtained by directly rounding. The candidate values of the primary ambiguity group is limited to two cycles through this step, and the (4,1,1…1) pattern can reduce ambiguity search times. Since an appropriate selection enables the correct resolution of primary wide-lane ambiguities, the selection of the threshold for SNR is the key to group wide-lane ambiguity, and we use the data from Novatel receiver as an example to discuss how to confirm the SNR threshold in this paper. Static short baseline data and kinematic short baseline that were less than 20 km were collected to validate the proposed algorithm. These results were compared with the commercial software IE8.60. These experiments show that the successful rate reaches 100%, 99.78% and 99.42%, respectively, for the new single-epoch segmented ambiguity method in the three data sets, and the accuracy is similar to that of IE8.60 for the combined double difference.

Multi-GNSS PPP-RTK: From Large- to Small-Scale Networks

Precise point positioning (PPP) and its integer ambiguity resolution-enabled variant, PPP-RTK (real-time kinematic), can benefit enormously from the integration of multiple global navigation satellite systems (GNSS). In such a multi-GNSS landscape, the positioning convergence time is expected to be reduced considerably as compared to the one obtained by a single-GNSS setup. It is therefore the goal of the present contribution to provide numerical insights into the role taken by the multi-GNSS integration in delivering fast and high-precision positioning solutions (sub-decimeter and centimeter levels) using PPP-RTK. To that end, we employ the Curtin PPP-RTK platform and process data-sets of GPS, BeiDou Navigation Satellite System (BDS) and Galileo in stand-alone and combined forms. The data-sets are collected by various receiver types, ranging from high-end multi-frequency geodetic receivers to low-cost single-frequency mass-market receivers. The corresponding stations form a large-scale (Australia-wide) network as well as a small-scale network with inter-station distances less than 30 km. In case of the Australia-wide GPS-only ambiguity-float setup, 90% of the horizontal positioning errors (kinematic mode) are shown to become less than five centimeters after 103 min. The stated required time is reduced to 66 min for the corresponding GPS + BDS + Galieo setup. The time is further reduced to 15 min by applying single-receiver ambiguity resolution. The outcomes are supported by the positioning results of the small-scale network.

Satellite-Clock Modeling in Single-Frequency PPP-RTK Processing

The real-time kinematic precise point positioning (PPP-RTK) technique enables integer ambiguity resolution by providing singlereceiver users with information on the satellite phase biases next to the standard PPP corrections. Using undifferenced and uncombined observations, rank deficiencies existing in the design matrix need to be eliminated to formestimable parameters. In this contribution, the estimability of the parameters was studied in single-frequency ionosphere-weighted scenario, given a dynamic satellite-clock model in the network Kalman filter. In case of latency of the network corrections, the estimable satellite clocks, satellite phase biases, and ionospheric delays need to be predicted over short time spans. With and without satellite-clock models incorporated in the network Kalman filter, different approaches were used to predict the network corrections. This contribution shows how the predicted network corrections responded to the presence and absence of satellite-clock models. These differences in the predicted network corrections were also reflected in the user positioning results. Using three different 1-Hz global positioning system (GPS) single-frequency data sets, two user stations in one small-scale network were used to compute the positioning results, applying predicted network corrections. The latency of the network products ranges from 3 to 10 s. It was observed that applying strong satellite-clock constraints in the network Kalman filter (i.e., with the process noise of 1 or 0.5mm per square root of second) reduced the root-mean squares (RMS) of the user positioning results to centimeters in the horizontal directions and decimeters in the vertical direction for latencies larger than 6 s, compared to the cases without a satellite-clock model.