Carrier Phase Ambiguity Resolution Research Articles

Partial ambiguity resolution with integrity risk constraint for high-performance Global Navigation Satellite System navigation

Carrier-phase ambiguity resolution of Global Navigation Satellite System in applications that require both high accuracy and high integrity is challenging. This paper proposes an efficient partial ambiguity resolution method with integrity risk constrain for high-performance navigation. First, the Global Navigation Satellite System observation model and the integer ambiguity resolution procedure, especially the partial ambiguity resolution using least squares ambiguity decorrelation adjustment, are described. Then an integrity risk constraint method and an improved integrity risk constraint method for ambiguity resolution are presented. Based on these methods, a hybrid strategy is further derived. Last, the simulation and analysis for the integrity risk constraint method versus the hybrid method are performed. Simulation results show that the first method is conservative and thus unnecessarily limits the navigation availability, while the hybrid method presents a tight upper-bound, so that it increases the integrity and accuracy of high-performance navigation significantly.

Zero-correlation transformation and threshold for efficient GNSS carrier phase ambiguity resolution

The key point of accurate and precise applications of Global Navigation Satellite Systems lies in knowing how to efficiently obtain correct integer ambiguity. One of the methods in solving the ambiguity resolution problem is applying the ambiguity searching technique coupled with an ambiguity decorrelation technique. Traditionally, an integer-valued limitation of the transformation matrix ensures that the integer characteristic of candidates exists after the inverse transformation, but this also makes the decorrelation imperfect. In this research, the float transformation matrix will be considered. To ensure both the integer characteristic and perfect decorrelation can be reached, the float transformation is used indirectly. To solve the ambiguity resolution problem, the problem is transformed by integer and float transformation matrices. The objective of integer transformation is reducing the number of candidates. The target of float transformation is validating these reduced candidates. A zero correlation domain or a near complete diagonalization covariance matrix can be obtained via the float transformation. A space in this domain will be used as the threshold; hence the zero correlation domain is called the threshold domain. The number of ambiguity candidates based on integer transformation can be reduced once again through the proposed method. The experiments in this paper prove that the method can make the ambiguity resolution become more efficient without any drop in the accuracy.

Instantaneous BeiDou–GPS attitude determination: A performance analysis

The advent of modernized and new global navigation satellite systems (GNSS) has enhanced the availability of satellite based positioning, navigation, and timing (PNT) solutions. Specifically, it increases redundancy and yields operational back-up or independence in case of failure or unavailability of one system. Among existing GNSS, the Chinese BeiDou system (BDS) is being developed and will consist of geostationary (GEO) satellites, inclined geosynchronous orbit (IGSO) satellites, and medium-Earth-orbit (MEO) satellites. In this contribution, a BeiDou–GPS robustness analysis is carried out for instantaneous, unaided attitude determination.Precise attitude determination using multiple GNSS antennas mounted on a platform relies on the successful resolution of the integer carrier phase ambiguities. The constrained Least-squares AMBiguity Decorrelation Adjustment (C-LAMBDA) method has been developed for the quadratically constrained GNSS compass model that incorporates the known baseline length. In this contribution the method is used to analyse the attitude determination performance when using the GPS and BeiDou systems. The attitude determination performance is evaluated using GPS/BeiDou data sets from a real data campaign in Australia spanning several days. The study includes the performance analyses of both stand-alone and mixed constellation (GPS/BeiDou) attitude estimation under various satellite deprived environments. We demonstrate and quantify the improved availability and accuracy of attitude determination using the combined constellation.

3D-RTK CAPABILITY OF SINGLE GNSS RECEIVERS

Abstract. Small, aerial objects are now being utilised in many areas of civil object capture and monitoring. As a rule, the standard application of a simple GPS receiver with code solutions serves the 3D-positioning of the trajectories or recording positions. Without GPS correction information, these can be calculated at an accuracy of 10–20 metres. Corrected code solutions (DGPS) generally lie in the metre range. A precise 3D-positioning of the UAV (unmanned aerial vehicle) trajectories in the centimetre range provides significant improvements. In addition, the recording time of each sensor can be synchronized with the exact time stamp of the GNSS low-cost system. In recent years, increasing works on positioning from L1 GPS raw data have been published. Along with this, the carrier phase measurements with the established evaluation algorithms are analysed in the post processing method to centimetre-exact positions or to high-precision 3D trajectories [e.g. Schwieger and Gläser, 2005 or Korth and Hofmann 20011]. The use of reference information from local reference stations or a reference network serves the purpose of carrier phase ambiguity resolution. Furthermore, there are many activities worldwide in the area of PPP techniques (Precise Point Positioning). However, dual frequency receivers are primarily used in this instance. Moreover, very long initialisation times must be scheduled for this. A research project on the subject of low-cost RTK GNSS was developed for real-time applications at the Beuth Hochschule für Technik Berlin University of Applied Sciences [Stempfhuber 2012]. The overall system developed for the purpose of real-time applications with centimetre accuracy is modularly constructed and can be used for various applications (http://prof.beuthhochschule.de/stempfhuber/seite-publikation/). With hardware costing a few hundred Euro and a total weight of 500–800 g (including the battery), this system is ideally suited for UAV applications. In addition, the GNSS data processed with the RTK method can be provided in standardised NMEA format. Through the reduced shadowing effects of the aerial objects, GNSS external factors such as multipath cause few problems. With L1 carrier phase analysis, the baseline computation must nevertheless remain limited at the range of a few kilometres. With distances of more than 5 kilometres between the reference station and the rover station position, mistakes arise in the decimetre area. The overall modular system consists of a low-cost, single-frequency receiver (e.g. uBlox LEA4T or 6T receiver), a L1 antenna (e.g. the Trimble Bullet III), a developed data logger including an integrated WLAN communication module for storage and securing of the raw data as well as a power supply. Optimisation of the L1 antenna has shown that, in this instance, many problems relating to signal reception can be reduced. A calibration of the choke-ring adaptors for various antenna calibration facilities results in good and homogeneous antenna parameters. In this situation, the real-time algorithm from the Open Source project RTKLib [Takasu, 2010] generally runs on a small computer at the reference station. In this case, the data transfer from the L1 receiver to the PC is realisable through a serial cable. The rover station can transfer the raw data to the computing algorithm over a WLAN network or through a data radio. Of course, this computational algorithm can also be adapted to an integrated computing module for L1 carrier phase resolutions. The average time to first fix (TTFF) amounts to a few minutes depending on the satellite constellation. Different test series in movement simulators and in moving objects have shown that a stable, fixed solution is achieved with a normal satellite constellation. A test series with a Microdrones quadrocopter could also be conducted. In comparison of the RTK positions with a geodetic dual frequency receiver, differences are in millimetre ranges. In addition, reference systems (based on total stations) are present for the precise examination of the kinematically captured positioning [Eisenbeiss et al. 2009].

Fast GNSS ambiguity resolution by ant colony optimisation

Global Navigation Satellite System (GNSS) carrier phase ambiguity resolution (AR) is the key technique to high precision positioning and navigation. Ant colony optimisation (ACO) as a stochastic meta-heuristic method solves combinatorial optimisation problems by construsting solutions iteratively using a colony of ants guided by pheromone trails and heuristic information. This paper seeks to explore the effectiveness of ACO to deal with the AR problem and closest lattice point problem. The performance of this new method is evaluated considering several simulated examples with different dimensions. The results show that the proposed algorithm can compete efficiently with other promising approaches to the problem and provide integer optimal solutions in often simulated scenarios. We hope that this paper provides a starting point for researches in applying ACO algorithm and other stochastic methods in the AR problem and other GNSS problems due to the simplicities involved in algebraic manipulation.

Inertial Navigation Position and Orientation Estimation with Occasional Galileo Satellite Position Fixes and Stereo Camera Measurements

Abstract In this paper an adaptive unscented Kalman filter based mixing filter is used to integrate kinematic satellite aided inertial navigation system with vision based measurements of five representative points on a runway in a modern receiver that incorporates carrier phase smoothing and ambiguity resolution. Using high resolution multiple stereo camera based measurements of five points on the runway, in addition to a set of typical pseudo-range estimates that can be obtained from a satellite navigation system such GPS or GNSS equipped with a carrier phase receiver, the feasibility of generating high precision estimates of the typical outputs from an inertial navigation system is demonstrated. The methodology may be developed as a stand-alone system or employed in conjunction with a traditional strapped down inertial navigation systems for purposes of initial alignment. Moreover the feasibility of employing adaptive mixing was explored as it facilitates the possibility of using the system for developing a vision based automatic landing controller.

Enhanced Precise Point Positioning for GNSS Users

This paper summarizes the main results obtained during the development of an Enhanced Precise Point Positioning (EPPP) Global Navigation Satellite Systems multifrequency user algorithm. The main innovations include the application of precise ionospheric corrections to facilitate the resolution of undifferenced carrier phase ambiguities, ambiguity validation, and integrity monitoring. The performance of the EPPP algorithm in terms of accuracy, convergence time, and integrity is demonstrated with actual GPS and simulated Galileo data. This can be achieved with very limited bandwidth requirements for EPPP users (less than 300 b/s for dual-frequency GPS data).

Instantaneous Global Navigation Satellite System (GNSS)-Based Attitude Determination for Maritime Applications

Global Navigation Satellite System (GNSS) is a valuable technology for a large number of maritime applications. Other than providing the absolute positioning service, it aids many demanding applications, such as precise docking, formation of surface craft, autonomous vehicles, sinkage monitoring, etc. GNSS carrier-phase-based algorithms provide high-precision positioning solutions, but an integer number of cycles inherent to the observed signal have to be resolved. A newly developed GNSS carrier-phase ambiguity resolution method is tested. The new method solves for the unknown number of integer cycles by exploiting the known placement of the GNSS antennas aboard the vessel. The a priori information on the antennas baseline separation is employed as a hard constraint. A simplified (linearized) version of the method, suitable for large vessels, is also analyzed. The new method was tested against the most challenging scenario when processing GNSS data: single-frequency, single-epoch, unaided ambiguity resolution. Through different tests, the high performance of the new method is demonstrated: high fixing rate, large robustness, and short time-to-fix after initialization, cycle slips, and/or loss of locks. Considerations about the wide spectra of maritime applications are given, and a specific experiment is carried out to demonstrate the capabilities of the method for navigation in shallow waters.

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.

Troposphere modeling for precise GPS rapid static positioning in mountainous areas

In global navigation satellite system precise positioning, double differencing of the observations is the common approach that allows for significant reduction of correlated atmospheric effects. However, with growing distance between the receivers, tropospheric errors decorrelate causing large residual errors affecting the carrier phase ambiguity resolution and positioning quality. This is especially true in the case of height differences between the receivers. In addition, the accuracy achieved by using standard atmosphere models is usually unsatisfactory when the tropospheric conditions at the receiver locations are significantly different from the standard atmosphere. This paper presents an evaluation of three different approaches to troposphere modeling: (a) neglecting the troposphere, (b) using a standard atmosphere model, and (c) estimating tropospheric delays at the reference station network and providing interpolated tropospheric corrections to the user. All these solutions were repeated with various constraints imposed on the tropospheric delays in the least-squares adjustment. The quality of each solution was evaluated by analyzing the residual height errors calculated by comparing the estimated results to the reference coordinates. Several permanent GPS stations of the EUPOS (European Position Determination System) active geodetic network located in the Carpathian Mountains were selected as a test reference network. The distances between the reference stations ranged from 64 to 122 km. KRAW station served as a simulated user receiver located inside the reference network. The user receiver ellipsoidal height is 267 m and the reference station heights range from 277 to 647 m. The results show that regardless of station height differences, it is recommended to model the tropospheric delays at the reference stations and interpolate them to the user receiver location. The most noticeable influence of the residual (unmodeled) tropospheric errors is observed in the station height component. In many cases, mismodeling of the troposphere disrupts ambiguity resolution and, therefore, prevents the user from obtaining an accurate position.

チャネル間ハードウェアバイアス較正によるRTK-GPS/GLONASS測位の性能評価

The purpose of this paper is to improve GLONASS positioning performance in RTK-GPS/GLONASS between different types of receivers. Tables of inter-channel code biases in various receivers were obtained with a zero baseline test in order to examine the magnitude of inter-channel code bias between the receivers. We showed the improvement of fix rate and positioning accuracy of RTK-GPS/GLONASS between different types of receivers by using table of the calibrated GLONASS biases and partial ambiguity fixing solution of GPS and GLONASS for the aid of GPS carrier-phase ambiguity resolution. Furthermore, we showed that the bias correction method suggested in this paper is applicable to RTK-GPS/GLONASS with multi-epoch measurements as well as with single-epoch measurements.

High-Precision Kinematic Satellite and Doppler Aided Inertial Navigation System

In this paper an adaptive unscented Kalman filter based mixing filter is used to develop a high-precision kinematic satellite aided inertial navigation system with a modern receiver that incorporates carrier phase smoothing and ambiguity resolution. Using carrier phase measurements with multiple antennas, in addition to a set of typical pseudo-range estimates that can be obtained from a satellite navigation system such as GPS or GLONASS, the feasibility of generating high precision estimates of the typical outputs from an inertial navigation system is demonstrated. The methodology may be developed as a stand-alone system or employed in conjunction with a traditional strapped down inertial navigation system for purposes of initial alignment. Moreover the feasibility of employing adaptive mixing facilitates the possibility of using the system in an interoperable fashion with satellite navigation measurements.

Computed success rates of various carrier phase integer estimation solutions and their comparison with statistical success rates

The success rate of carrier phase ambiguity resolution (AR) is the probability that the ambiguities are successfully fixed to their correct integer values. In existing works, an exact success rate formula for integer bootstrapping estimator has been used as a sharp lower bound for the integer least squares (ILS) success rate. Rigorous computation of success rate for the more general ILS solutions has been considered difficult, because of complexity of the ILS ambiguity pull-in region and computational load of the integration of the multivariate probability density function. Contributions of this work are twofold. First, the pull-in region mathematically expressed as the vertices of a polyhedron is represented by a multi-dimensional grid, at which the cumulative probability can be integrated with the multivariate normal cumulative density function (mvncdf) available in Matlab. The bivariate case is studied where the pull-region is usually defined as a hexagon and the probability is easily obtained using mvncdf at all the grid points within the convex polygon. Second, the paper compares the computed integer rounding and integer bootstrapping success rates, lower and upper bounds of the ILS success rates to the actual ILS AR success rates obtained from a 24 h GPS data set for a 21 km baseline. The results demonstrate that the upper bound probability of the ILS AR probability given in the existing literatures agrees with the actual ILS success rate well, although the success rate computed with integer bootstrapping method is a quite sharp approximation to the actual ILS success rate. The results also show that variations or uncertainty of the unit–weight variance estimates from epoch to epoch will affect the computed success rates from different methods significantly, thus deserving more attentions in order to obtain useful success probability predictions.

A New Method For DGPS Ambiguity Resolution

This paper deals with the problem of determining the baseline vector between two GPS receivers in single frequency (L1) using the basic principles of interferometric Differential GPS, therefore using the interferometric relations between the two receivers and the satellites visible to both receivers. As a preliminary step, ambiguity identification was solved using the results provided by the Kalman filter; these results were optimized by evaluating the Dilution Of Precision indexes for satellites in view of the receivers. Results achieved by applying this first procedure to data collected are discussed. To increase the accuracy of the results, a new, computationally fast algorithm for carrier phase ambiguity resolution on data collected from static and dynamics acquisitions was developed, implemented and tested. The new algorithm permitted an increase of accuracy of about two orders of magnitude with respect to results given by filtered Double Difference in the resolution of baseline vector.

Fast GNSS ambiguity resolution as an ill-posed problem

A linear observational equation system for real-time GNSS carrier phase ambiguity resolution (AR) is often severely ill-posed in the case of poor satellite geometry. An ill-posed system may result in unreliable or unsuccessful AR if no care is taken to mitigate this situation. In this paper, the GNSS AR model as an ill-posed problem is solved by regularizing its baseline and ambiguity parameters, respectively, with the threefold contributions: (i) The regularization parameter is reliably determined in context of minimizing mean square error of regularized solution where the covariance matrix of initial values of unknowns is used as an approximate smoothness term instead of the quadratic matrix of the true values of unknowns; (ii) The different models for computing initial values of unknowns are systematically discussed in order to address the potential schemes in real world applications; (iii) The superior performance of the regularized AR are demonstrated through the numerically random simulations as well as the real GPS experiments. The results show that the proposed regularization strategies can effectively mitigate the model’s ill-condition and improve the success AR probability of the observational system with a severely ill-posed problem.

Three Carrier Ambiguity Resolutions: Generalised Problems, Models and Solutions

In this paper, the problems of three carrier phase ambiguity resolution (TCAR) and position estimation (PE) are generalized as real time GNSS data processing problems for a continuously observing network on large scale. In order to describe these problems, a general linear equation system is presented to uniform various geometry-free, geometry-based and geometry-constrained TCAR models, along with state transition questions between observation times. With this general formulation, generalized TCAR solutions are given to cover different real time GNSS data processing scenarios, and various simplified integer solutions, such as geometry-free rounding and geometry-based LAMBDA solutions with single and multiple-epoch measurements. In fact, various ambiguity resolution (AR) solutions differ in the floating ambiguity estimation and integer ambiguity search processes, but their theoretical equivalence remains under the same observational systems models and statistical assumptions. TCAR performance benefits as outlined from the data analyses in some recent literatures are reviewed, showing profound implications for the future GNSS development from both technology and application perspectives.

Global Navigation Satellite System Ambiguity Resolution with Constraints from Normal Equations

Carrier phase ambiguity resolution is the key to precise positioning with a global navigation satellite system (GNSS); therefore, quite a few ambiguity resolution methods have been developed in the past two decades. In this paper, a new ambiguity searching algorithm by treating part of normal equations as constraints is developed. The process starts with the truncation of the terms with respect to the small eigenvalues from the normal equations of a least-squares estimation problem. The remaining normal equations are employed as the constraint equations for the efficient searching of integer ambiguities. In the case of short single baseline rapid GNSS positioning with double differenced phase measurements, there are only three real parameters of the position to be estimated. Therefore three terms of the normal equations should be truncated off due to the fact that there is a large difference between the last three eigenvalues of the normal matrix of the float solution and the others, and then the remainin...

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.

Fixed point theorems of GPS carrier phase ambiguity resolution and their application to massive network processing: Ambizap

Precise point positioning (PPP) has become popular for Global Positioning System (GPS) geodetic network analysis because for n stations, PPP has O(n) processing time, yet solutions closely approximate those of O(n3) full network analysis. Subsequent carrier phase ambiguity resolution (AR) further improves PPP precision and accuracy; however, full‐network bootstrapping AR algorithms are O(n4), limiting single network solutions to n < 100. In this contribution, fixed point theorems of AR are derived and then used to develop “Ambizap,” an O(n) algorithm designed to give results that closely approximate full network AR. Ambizap has been tested to n ≈ 2800 and proves to be O(n) in this range, adding only ∼50% to PPP processing time. Tests show that a 98‐station network is resolved on a 3‐GHz CPU in 7 min, versus 22 h using O(n4) AR methods. Ambizap features a novel network adjustment filter, producing solutions that precisely match O(n4) full network analysis. The resulting coordinates agree to ≪1 mm with current AR methods, much smaller than the ∼3‐mm RMS precision of PPP alone. A 2000‐station global network can be ambiguity resolved in ∼2.5 h. Together with PPP, Ambizap enables rapid, multiple reanalysis of large networks (e.g., ∼1000‐station EarthScope Plate Boundary Observatory) and facilitates the addition of extra stations to an existing network solution without need to reprocess all data. To meet future needs, PPP plus Ambizap is designed to handle ∼10,000 stations per day on a 3‐GHz dual‐CPU desktop PC.

Investigation of Instantaneous Carrier Phase Ambiguity Resolution with the GPS/Galileo Combination using the General Ambiguity Search Criterion

The success of high-precision Global Navigation Satellite Systems (GNSS) kinematic positioning depends partly on the ability to resolve the integer phase ambiguities. In this paper, we propose a new algorithm for instantaneous kinematic ambiguity resolution, for present and modernised GPS and for GALILEO. This approach - the GEneral Criterion Cascading Ambiguity Resolution (GECCAR) - selects the integer set of ambiguities using the General Ambiguity Search Criterion (GASC). Simulation runs have shown that single-epoch L1/E1 frequency ambiguity resolution was possible 99% of the considered epochs, when the three frequencies from both systems were used together. This new approach shows an improvement in the selection of the correct set of ambiguities when compared with the selection made by the Integer Least Squares Criterion (ILSC). We conclude that the GECCAR approach is a very promising algorithm for instantaneous ambiguity resolution.