Carrier Phase Integer Ambiguity Resolution Research Articles

Enhancing Real-Time Kinematic Relative Positioning for Unmanned Aerial Vehicles

Real-time kinematic (RTK) positioning of the global navigation satellite systems (GNSS) is used to provide centimeter-level positioning accuracy. There are several ways to implement RTK but a Kalman filter-based RTK is preferred because of its superior capability to resolve GNSS carrier phase integer ambiguities. However, the positioning performance of the Kalman filter-based RTK is often compromised by various factors when it comes to determining a precise relative position vector between moving unmanned aerial vehicles (UAVs) equipped with low-cost GNSS receivers and antennas, where the locations of both GNSS antennas are not accurately known and change over time. Some of the critical factors that lead to a high rate of incorrect resolutions of carrier phase integer ambiguities are measurement time differences between GNSS receivers, frequent cycle slips with high noise in code and carrier phase measurements, and an improper Kalman filter gain due to a newly risen satellite. In this paper, effective methods to deal with those factors to achieve a seamless Kalman filter-based RTK performance in moving UAVs are presented. Using our extensive 45 flight tests data sets, conducted over a duration of 3 to 12 min, the RTK positioning results showed that the root-mean-square position error (RMSE) decreased by up to 95.13%, with an average of 65.31%, and that the percentage of epochs that passed the ratio test, which is the most common method for validating double differenced carrier phase integer ambiguity resolution, increased by up to 130%, with an average of 23.54%.

GPS PPP-AR frequency transfer and its application for comparing atomic fountain primary frequency standards between NRC and PTB

We report on the first characterization of the frequency transfer performance of the GPS precise point positioning with carrier-phase integer ambiguity resolution (PPP-AR) implemented by the Natural Resources Canada (NRCan) online service CSRS-PPP. We show that continuous PPP-AR links of multiple days can be formed by carefully fixing the day boundary phase discontinuity. The stability of the links can reach and (Allan deviation) at 1 d and 7 d averaging times, respectively, and mid to low 10−17 at 20 d–30 d. We compared the atomic fountain primary frequency standards (PFS) between NRC in Ottawa, Canada and PTB in Braunschweig, Germany for 135 d via GPS PPP-AR and precise point positioning links. The NRC and PTB PFS agree within a few parts in 1016, which is well below the combined systematic uncertainty. It is expected that with uncertainty at 10−17, the PPP-AR links will become a powerful tool for remote comparisons of atomic optical clocks, essential for the future redefinition of the SI second.

Best integer equivariant estimation: performance analysis using real data collected by low-cost, single- and dual-frequency, multi-GNSS receivers for short- to long-baseline RTK positioning

The key to precise global navigation satellite system (GNSS) positioning is carrier phase integer ambiguity resolution with a high success rate. On the other hand when the success rate is too low, the user will normally prefer the float solution. The alternative can be to use the best integer equivariant (BIE) estimator, since it is optimal in the minimum mean squared error (MMSE) sense. Low-cost receiver real-time kinematic precise positioning has become possible through the many signals that can be obtained by combining several GNSSs, such as BDS, Galileo, QZSS and GPS. In this contribution, we will use both simulations and such low-cost multi-GNSS data to compare the performance of the BIE and integer least squares (ILS) estimator, based on full ambiguity resolution. The GNSS data are evaluated in Dunedin, New Zealand, with a short- (670 m) and long-baseline (112.9 km) where the relative atmospheric delays can be neglected and need to be estimated, respectively. We compare the BIE and ILS results by using both single-frequency and dual-frequency (DF) low-cost and survey-grade receivers and antennas. We demonstrate, for the first time, the distributional properties of BIE positioning, where it will be shown that a ‘star-like’ pattern reveals itself once the model gets stronger and the ILS success rate increases. It will further be shown that the DF low-cost receivers give a very good positioning performance, but still not yet competitive to the survey-grade counterparts for the long-baseline. We will also demonstrate that the positioning performance of the BIE estimator will always equal or be better than that of the float solutions. It will finally be shown that BIE will always be better in the MMSE sense than the ILS solution when the success rate is at low to medium levels, whereas for high success rates we get a similar performance to ILS.

Integer Ambiguity Resolution by Mixture Kalman Filter for Improved GNSS Precision

Accurate carrier-phase integer ambiguity resolution is fundamental to high-precision global navigation satellite systems (GNSSs). Real-time GNSSs typically resolve the ambiguities by a combination of recursive estimators and integer least-squares solvers, which need to be reset when satellites are added or cycle slip occurs. In this article, we propose a mixture Kalman filter solution to integer ambiguity resolution. By marginalizing out the set of ambiguities and exploiting a likelihood proposal for generating the ambiguities, we can bound the possible values to a tight and dense set of integers. Thus, we extract the state and integer estimates from a mixture Kalman filter. The proposed approach yields an integrated method to detect cycle slip and initialize new satellites. Numerical analysis and experimental results indicate that the proposed method achieves reliable position estimates, repeatedly finds the correct integers in cases when other methods may fail, and is more robust to cycle slip.

Stochastic modeling of between-receiver single-differenced ionospheric delays and its application to medium baseline RTK positioning

Global navigation satellite system (GNSS) carrier phase integer ambiguity resolution is one of the key issues and challenges for precise relative positioning. For baselines greater than 10 km, integer ambiguity resolution becomes difficult because the ionospheric delay effects on the single-differenced (SD) observations are significant. One way to deal with this difficulty is to weight the ionospheric delays, instead of treating them in a deterministic way, giving rise to the so-called ionosphere-weighted model. With this model relative positioning technology is enabled to work with much larger inter-station distances than its current status. Here we seek to gain insight into how to reasonably weight the SD ionospheric delays, normally introduced in the ionosphere-weighted model as pseudo observables, which consists of two sequential steps. The first step seeks to construct realistic stochastic models for the GNSS observables with the ionosphere-fixed model where the SD ionospheric delays are assumed to be absent. With the results so obtained in the second step we estimate the variance of the SD ionospheric delays with the ionosphere-float model, in which the SD ionospheric delays are assumed to be fully unknown. Numerical tests based on GPS data from the National Geodetic Survey CORS network demonstrate a clear improvement in medium baseline real time kinematic (RTK) positioning performance, as compared to the ionosphere-weighted model that employs the unrealistic stochastic models for the pseudo SD ionospheric observables.

The Potency of the Modernized GNSS Signals for Real-Time Kinematic Positioning

GNSS positioning has become popular in the past decade as an efficient method of precise and real-time positioning. It is relatively low cost and ease-of-use. Up to now, several parameters were defined to characterize the performance of real-time positioning: availability, precision, accuracy. This article evaluates the performance of signal linear combinations for real-time positioning, both for static as well as the kinematic positioning. This article starts with the investigation of linear combinations (LC) rising from the carrier frequencies of the GNSS systems. Some Linear Combination shows potential benefits in carrier phase integer ambiguity resolution, particularly utilizing the Galileo and Beidou signal phase carrier. For each system, a set of combinations was studied, analyzed, and then selected during the development of GNSS positioning method utilizing the Least-squares Ambiguity Decorrelation Adjustment (LAMBDA). Special signal selection can affect the estimated position and its standard deviation. To further analyze, the results obtained from data processing are compared with respect to baselines and signals. The ambiguity fixing rate is correlated with the baseline length and the method as well as the signals that were used. The analysis of the measurement noise level was first conducted to set a baseline for the real-time GNSS positioning application. According to the results and to assess the data quality and positioning performance of GNSS in respect with GPS (Global Positioning System), an experimental test has established using MGEX data. This research investigates the satellite visibilities, multipath, Signal to Noise Ratio (SNR), and positioning performance. It is shown that in every epoch, at least 8 satellites are visible. The SNR’s are up to 60 dBHz, the code multipath residuals varies within ~1 m, while the phase residuals varies by about ~2 cm. hence the modernized GNSS signals have potencies to improves the RTK positioning.

Impact of GPS antenna phase center models on zenith wet delay and tropospheric gradients

Today Global Navigation Satellite Systems (GNSS) tropospheric products, such as zenith total delays (ZTD) and zenith wet delays (ZWD), are widely used as complementary data sets in numerical weather prediction models. In particular, the wet delays are treated as unknown parameters in GNSS processing and are estimated with other parameters such as station coordinates. In this study, we investigate the effects of Phase Center Correction (PCC) models on ZWD, integrated water vapor (IWV) and horizontal gradients derived from Global Positioning System (GPS) observations. Two solutions were generated using the GAMIT software over the European Reference Frame (EUREF) Permanent GNSS Network (EPN). The first (reference) solution was derived by applying the International GNSS Service (IGS) type-mean PCC models, while for the second solution PCC models from individual calibrations were used. The solutions were generated identically, except for the PCC model differences. The tropospheric products from the two solutions were then compared, with the assumption that common signals would be differenced out. The comparison of the two solutions clearly shows a bias in all tropospheric products, which can be attributed to PCC model deficiencies. Overall, mean biases of 1.8, 0.3, 0.14 and 0.19 mm are evident in ZWD, IWV, North–South and East–West gradients, respectively. Moreover, the differences between the two solutions show seasonal variations. For all antenna types, the ZWD and IWV differences are dominated by white plus power-law noise, with the latter characterizing the low-frequency spectrum. On the other hand, the horizontal gradients exhibit a white plus first-order autoregressive noise characteristic with less than 1% white noise. The individual PCC model provides a better fit to an external independent model in terms of gradient estimates and also provides up to 3% more carrier phase integer ambiguity resolution.

Ionosphere influence on success rate of GPS ambiguity resolution in a satellite formation flying

Satellite formation flying is one of the most promising technologies for future space missions. The distribution of sensors and payloads among different satellites provides more redundancy, flexibility, improved communication coverage, among other advantages. One of the fundamental issues in spacecraft formation flying is precise position and velocity determination between satellites. For missions in low Earth orbits, GPS system can meet the precision requirement in relative positioning, since the satellite dynamics is modeled properly. The key for high accuracy GPS relative positioning is to resolve the ambiguities to their integer values. Ambiguities resolved successfully can improve the positioning accuracy to decimetre or even millimetre-level. So, integer carrier phase ambiguity resolution is often a prerequisite for high precision GPS positioning. The determination of relative position was made using an extended Kalman filter. The filter must take into account imperfections in dynamic modeling of perturbations affecting the orbital flight, and changes in solar activity that affects the GPS signal propagation, for mitigating these effects on relative positioning accuracy. Thus, this work aims to evaluate the impact of ionosphere variation, caused by changes in solar activity, in success rate of ambiguity resolution. Using the Ambiguity Dilution of Precision (ADOP) concept, the ambiguity success rate is analyzed and the expected precision of the ambiguity-fixed solution is calculated. Evaluations were performed using actual data from GRACE mission and analyzed for their performance in real scenarios. Analyses were conducted in different configurations of relative position and during different levels of solar activity. Results bring the impact of various disturbances and modeling of solar activity level on the success rate of ambiguity resolution.

Integer aperture ambiguity resolution based on difference test

Carrier-phase integer ambiguity resolution (IAR) is the key to highly precise, fast positioning and attitude determination with Global Navigation Satellite System (GNSS). It can be seen as the process of estimating the unknown cycle ambiguities of the carrier-phase observations as integers. Once the ambiguities are fixed, carrier phase data will act as the very precise range data. Integer aperture (IA) ambiguity resolution is the combination of acceptance testing and integer ambiguity resolution, which can realize better quality control of IAR. Difference test (DT) is one of the most popular acceptance tests. This contribution will give a detailed analysis about the following properties of IA ambiguity resolution based on DT: All these conclusions are verified based on the single and combination GNSS simulation experiments. The experiment results indicate the correctness of these conclusions. These properties demonstrate the special properties of DT-IA estimator, and also provide the research frame to investigate other IA estimators. They are helpful to realize better use of IA estimators in quality control and precise positioning in future.

Precise line-of-sight vector estimation based on an inter-satellite radio frequency system

This paper proposes a precise line-of-sight (LOS) vector estimation using an inter-satellite radio frequency system. GNSS-like technology is inherited such that the ranging signals are locally generated inside the formation. However, the approach differs from the standard GNSS model usage in that the LOS vector to be of a unit length is fully explored as a priori constraint for the carrier phase integer ambiguity resolution. The constraint is lumped to the mapping process from the real-valued ambiguities to the integers by what is called validation or subset ambiguity bounding. These two approaches have the same rules of regarding the constraint as a gateway to accept or reject the ambiguity candidates, but differ by using “all-ambiguity-set” and “subset-ambiguity”. Both show remarkable improvement with up to 80% lower integer fixing failure rates than without treating the constraint. Validation provides a slightly better performance than the subset ambiguity bounding in terms of the integer fixing failure rates and the computational efficiency. The predefined tolerance regions that are critical for these two methods are analytically determined as function of the carrier noise. The paper also introduces a LOS dependent ambiguity dilution of precision (ADOPLOS) measure that can serve as a metric to characterize the expectation of being able to successfully resolve the ambiguities. The region of ADOPLOS lower than 0.21 is empirically summarized as the safe region where the integer fixing failure rates are less than 1%. A closed form of the ADOPLOS is derived which is able to capture the impact of the various factors. Antenna baseline geometries and multiple frequencies in the form of an ultra-BOC signal structure are demonstrated as the most important influencing factors. With multiple properly arrayed antennas and using ultra-BOC structure, instantaneous ambiguity resolution can be achieved and the LOS accuracy can reach millimeter level.

Reliable Integer Ambiguity Resolution: Multi-Frequency Code Carrier Linear Combinations and Statistical A Priori Knowledge of Attitude

This paper provides two methods to improve the reliability of carrier phase integer ambiguity resolution: The first one is a group of multi-frequency linear combinations that include both code and carrier phase measurements, and allow an arbitrary scaling of the geometry, an arbitrary scaling of the ionospheric delay, and any preferred wavelength. The maximization of the ambiguity discrimination results in combinations with a wavelength of several meters and a noise level of a few centimeters. These combinations could be beneficial for both Real-Time Kinematics (RTK) and Precise Point Positioning (PPP). The second method incorporates some statistical a priori knowledge of attitude into the actual fixing. The a priori knowledge includes the length and direction of the baseline between two receivers and is given either as a uniform or Gaussian distribution. It enables a substantial reduction of the search space volume but also ensures a large robustness over errors in the a priori information. Both methods improve the accuracy of the float solution, which motivates a simple rounding for ambiguity fixing. A method is described, which enables an efficient computation of its success rate with a few integral transformations. Copyright © 2012 Institute of Navigation.

A New Approach for GNSS Ambiguity Decorrelation

Integer carrier phase ambiguity resolution is the key to fast and high-precision Global navigation satellite system(GNSS) positioning and application. LAMBDA method is one of the best methods for fixing integer ambiguity. The principle of LAMBDA is discussed. For incompleteness of Cholesky decomposition and complexity of Integer Gauss transformation, a new approach for GNSS ambiguity decorrelation is proposed based on symmetric pivoting strategy and united inverse integer strategy. The new algorithm applies symmetric pivoting strategy to ambiguity covariance matrix while doing Cholesky decomposition, then finds the inverse and integer matrix of ‘L’. This method not only uses Cholesky decomposition to improve efficiency, but also avoids complicated Integer Gauss transformations. The feasibility and advantage of the method are verified using randomly simulation covariance matrix.

Integer least-squares theory for the GNSS compass

Global navigation satellite system (GNSS) carrier phase integer ambiguity resolution is the key to highprecision positioning and attitude determination. In this contribution, we develop new integer least-squares (ILS) theory for the GNSS compass model, together with efficient integer search strategies. It extends current unconstrained ILS theory to the nonlinearly constrained case, an extension that is particularly suited for precise attitude determination. As opposed to current practice, our method does proper justice to the a priori given information. The nonlinear baseline constraint is fully integrated into the ambiguity objective function, thereby receiving a proper weighting in its minimization and providing guidance for the integer search. Different search strategies are developed to compute exact and approximate solutions of the nonlinear constrained ILS problem. Their applicability depends on the strength of the GNSS model and on the length of the baseline. Two of the presented search strategies, a global and a local one, are based on the use of an ellipsoidal search space. This has the advantage that standard methods can be applied. The global ellipsoidal search strategy is applicable to GNSS models of sufficient strength, while the local ellipsoidal search strategy is applicable tomodels for which the baseline lengths are not too small. We also develop search strategies for the most challenging case, namely when the curvature of the non-ellipsoidal ambiguity search space needs to be taken into account. Two such strategies are presented, an approximate one and a rigorous, somewhat more complex, one. The approximate one is applicable when the fixed baseline variance matrix is close to diagonal. Both methods make use of a search and shrink strategy. The rigorous solution is efficiently obtained by means of a search and shrink strategy that uses nonquadratic, but easy-to-evaluate, bounding functions of the ambiguity objective function. The theory presented is generally valid and it is not restricted to any particular GNSS or combination of GNSSs. Its general applicability also applies to the measurement scenarios (e.g. single-epoch vs. multiepoch, or single-frequency vs. multi-frequency). In particular it is applicable to the most challenging case of unaided, single frequency, single epoch GNSS attitude determination. The success rate performance of the different methods is also illustrated.

The GNSS ambiguity ratio-test revisited: a better way of using it

Integer carrier phase ambiguity resolution is the key to fast and high-precision global navigation satellite system (GNSS) positioning and application. Apart from integer estimation, also acceptance tests are part of the ambiguity resolution process. A popular acceptance test is the so-called ratio-test. In this contribution we study the properties and the underlying concepts of the ratio-test. We discuss some misconceptions of the ratio-test and in particular show that the ratio-test is not a test for testing the correctness of the integer least-squares solution. We also show that the common usage of the ratio-test with a fixed critical value has shortcomings. Instead, the fixed failure rate approach is recommended. This approach is part of the more general theory of integer aperture estimation, and enables that the failure rate does not exceed a user-defined value. Results of the fixed failure-rate ratio-test and its improved performance are illustrated with a number of examples.

Efficiency of carrier-phase integer ambiguity resolution for precise GPS positioning in noisy environments

Precise GPS positioning relies on tracking the carrier-phase. The fractional part of carrier-phase can be measured directly using a standard phase-locked loop, but the integer part is ambiguous and the ambiguity must be resolved based on sequential carrier-phase measurements to ensure the required positioning precision. In the presence of large phase-measurement noise, as can be expected in a jamming environment for example, the amount of data required to resolve the integer ambiguity can be large, which requires a long time for any generic integer parameter estimation algorithm to converge. A key question of interest in significant applications of GPS where fast and accurate positioning is desired is then how the convergence time depends on the noise amplitude. Here we address this question by investigating integer least-sqaures estimation algorithms. Our theoretical derivation and numerical experiments indicate that the convergence time increases linearly with the noise variance, suggesting a less stringent requirement for the convergence time than intuitively expected, even in a jamming environment where the phase noise amplitude is large. This finding can be useful for practical design of GPS-based systems in a jamming environment, for which the ambiguity resolution time for precise positioning may be critical.

New Global Navigation Satellite System Ambiguity Resolution Method Compared to Existing Approaches

Integer carrier phase ambiguity resolution is the process of resolving unknown cycle ambiguities of doubledifferenced carrier phase data as integers, and it is a prerequisite for rapid and high-precision global navigation satellite system positioning and navigation. Besides integer estimation, integer ambiguity resolution also involves validation of the integer estimates. In this contribution a new ambiguity resolution method is presented, based on the class of integer aperture estimators, which for the first time reveals an overall approach to the combined problem of integer estimation and validation. Furthermore, it is shown how the different discrimination tests that are currently in use in practice can be cast into the framework of the new approach.

On InSAR Ambiguity Resolution For Deformation Monitoring

On InSAR Ambiguity Resolution For Deformation Monitoring

High accuracy of GPS/Pseudolite/INS integration: Carrier phase measurement processing issues

This paper addresses solutions to the challenges of carrier phase integer ambiguity resolution and cycle slip detection/identification, for maintaining high accuracy of an integrated GPS/Pseudolite/INS system. Such a hybrid positioning and navigation system is an augmentation of standard GPS/INS systems in localized areas. To achieve the goal of high accuracy, the carrier phase measurements with correctly estimated integer ambiguities must be utilized to update the system integration filter’s states. The occurrence of a cycle slip that is undetected is, however, can significantly degrade the filter’s performance. This contribution presents an effective approach to increase the reliability and speed of integer ambiguity resolution through using pseudolite and INS measurements, with special emphasis on reducing the ambiguity search space. In addition, an algorithm which can effectively detect and correct the cycle slips is described as well. The algorithm utilizes additional position information provided by the INS, and applies a statistical technique known as the cumulative-sum (CUSUM) test that is very sensitive to abrupt changes of mean values. Results of simulation studies and field tests indicate that the algorithms are performed pretty well, so that the accuracy and performance of the integrated system can be maintained, even if cycle slips exist in the raw GPS measurements.

An integer ambiguity resolution procedure for GPS/pseudolite/INS integration

In order to achieve a precise positioning solution from GPS, the carrier-phase measurements with correctly resolved integer ambiguities must be used. Based on the integration of GPS with pseudolites and Inertial Navigation Systems (INS), this paper proposes an effective procedure for single-frequency carrier-phase integer ambiguity resolution. With the inclusion of pseudolites and INS measurements, the proposed procedure can speed up the ambiguity resolution process and increase the reliability of the resolved ambiguities. In addition, a recently developed ambiguity validation test, and a stochastic modelling scheme (based on-line covariance matrix estimation) are adapted to enhance the quality of ambiguity resolution. The results of simulation studies and field experiments indicate that the proposed procedure indeed improves the performance of single-frequency ambiguity resolution in terms of both reliability and time-to-fix-ambiguity.

High accuracy of GPS/pseudolite/INS integration: Carrier phase measurement processing issues

This paper addresses solutions to the challenges of carrier phase integer ambiguity resolution and cycle slip detection/ identification, for maintaining high accuracy of an integrated GPS/Pseudolite/INS system. Such a hybrid positioning and navigat ion system is an augmentation of standard GPS/INS systems in localized areas. To achieve the goal of high accuracy, the carrier phase measurements with correctly estimated integer ambiguities must be utilized to update the system integration filter's states. The occurrence of a cycle slip that is undetected is, however, can significantly degrade the filter's performance. This contribution presents an effective approach to increase the reliability and speed of integer ambiguity resolution through using pseudolite and INS measurements, with special emphasis on reducing the ambiguity search space. In addition, an algorithm which can effectively detect and correct the cycle slips is described as well. The algorithm utilizes additional position information provided by the INS, and applies a statistical technique known as the cumulative-sum (CUSUM) test that is very sensitive to abrupt changes of mean values. Results of simulation studies and field tests indicate that the algorithms are performed pretty well, so that the accuracy and performance of the integrated system can be maintained, even if cycle slips exist in the raw GPS measurements.