Bofeng Li, Zhetao Zhang, Weikai Miao · Navigation: science and technology · 2025
This chapter offers an in-depth examination of integer ambiguity resolution in Global Navigation Satellite System (GNSS) positioning, a key process for achieving high-precision solutions. It begins by presenting the mixed-integer GNSS model, which serves as the foundational framework for all integer ambiguity resolution methods. The chapter then explores various strategies for integer estimation, addressing how unmodeled errors affect ambiguity resolution. Following this, the chapter outlines methods for evaluating and validating integer solutions. It also highlights the practical benefits of partial ambiguity resolution techniques, particularly in real-world GNSS applications. In response to the growing adoption of multi-frequency and multi-GNSS systems, the chapter further discusses methods for resolving multi-frequency ambiguities, providing valuable theoretical insights for GNSS users.
Cited by 0Open access
P. J. G. Teunissen · 2017
Cited by 29
Shengfeng Gu, Lizhe Fang, Weiping Jiang · Satellite Navigation · 2024
Abstract Among all the ambiguity resolution techniques, the Full Ambiguity Resolution (FAR), Partial Ambiguity Resolution (PAR) and Best Integer Equivariant (BIE) estimator are widely used. Although the researches have been done on the different classes of ambiguity resolution, we still hope to find the relationships among these specific algorithms. In this work, we unify the PAR and FAR algorithms under a whole framework of BIE by applying multiple integer candidates. A concise estimation formula of the variance of Gaussian BIE estimator based on the variance of float solution and the probability distribution of the candidates is first derived. Then, we propose an algorithm named Multiple Integer Candidates Ambiguity Resolution (MICAR) to discover as many ambiguities in the BIE as possible that can be estimated more precisely by PAR (FAR) algorithm instead of BIE. In the experiments, we utilize the simulated data of GPS (Global Positioning System) + BDS (BeiDou Navigation Satellite System) + Galileo (Galileo navigation satellite system) to contrast the effects of MICAR and single candidate estimator, i.e., FAR. By taking the threshold of 5 cm at 95% confidence level as an example, MICAR accelerates the convergence process by about 3.0 min. When the positioning sequence converges, MICAR reduces the root mean square of the positioning error by 9.8% in horizontal directions and 3.5% in vertical direction, which is attributed to more fixed NL.
Cited by 12Open access
A. Lannes, J.‐L. Prieur · Astronomische Nachrichten · 2014
Abstract Recent theoretical developments in astronomical aperture synthesis have revealed the existence of integer‐ambiguity prob‐lems. Those problems, which appear in the self‐calibration procedures of radio imaging, have been shown to be similar to the nearest‐lattice point (NLP) problems encountered in high‐precision geodetic positioning and in global navigation satellite systems. In this paper we analyse the theoretical aspects of the matter and propose new methods for solving those NLP problems. The related optimization aspects concern both the preconditioning stage, and the discrete‐search stage in which the integer ambiguities are finally fixed. Our algorithms, which are described in an explicit manner, can easily be implemented. They lead to substantial gains in the processing time of both stages. Their efficiency was shown via intensive numerical tests. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Cited by 1Open access
Orviská, Terézia · Brno University of Technology Digital Library (Brno University of Technology) · 2023
Cieľom tejto diplomovej práce je porovnanie presnosti a rýchlosti konvergencie metódy Precise Point Positioning (PPP) a PPP s celočíselným riešením ambiguit a posúdiť využiteľnosť metódy PPP v súvislosti s odhadom magnitúd zemetrasení. V práci sú popísané teoretické základy GNSS metódy PPP a dôležité systematické chyby a spôsoby ich eliminácie. Následne je predstavené riešenie praktickej časti. Dáta sú spracovávané z obdobia roka 2020 na 14 vybratých permanentných staniciach IGS. Výpočtu PPP predchádzalo kvalitatívne hodnotenie dát. Vo výsledku vykazuje metóda PPP s celočíselným riešením ambiguit s nekombinovanými observáciami najkonzistentnejšie a najpresnejšie výsledky s najrýchlejšou konvergenciou. Pre výpočet magnitúd boli vybraté 3 zemetrasenia na Aljaške z obdobia 2020-2021 a boli skúmané 4 kombinácie metód, pričom najlepšie výsledky poskytla znovu metóda PPP s celočíselným riešením ambiguit s nekombinovanými observáciami.
Cited by 0
André Lannes · Journal of the Optical Society of America A · 2001
Phase calibration is the key operation of phase closure imaging. In the case of nonredundant arrays, the related problem amounts to finding the node of a Z lattice closest to the end of a vector, the components of which are the differences between the closure phases of the data and those of the model. The aim of the paper is to show that this integer ambiguity problem can be solved in a very efficient manner. Its potential instabilities can also be well identified. The corresponding approach, which can be extended to redundant arrays, revisits and completes the analysis presented in a recent paper entitled "Phase calibration on interferometric graphs" [J. Opt. Soc. Am. A 16, 443 (1999)]. The procedures of self-calibration must be modified accordingly. The spinoffs of this approach also concern the integer ambiguity problems encountered in the global positioning system.
Cited by 16
Shahram Jazaeri, Alireza Amiri-Simkooei, Mohammad Ali Sharifi · Survey Review · 2013
In this contribution, modified versions of Agrell, Eriksson, Vardy, Zeger (AEVZ) algorithms for integer ambiguity resolution of GNSS phase observations are presented. This modification removes many redundant mathematical operations of the AEVZ algorithm based on a recursive function. We also introduce G-based and H-based versions of AEVZ and its modification and compare the modified versions with the AEVZ method. Numerical results reveal that the H-based version of the modified algorithm performs the best. Therefore, it accelerates solving the high-dimensional weighted integer least squares problem. This results in faster integer ambiguity resolution.
Cited by 3
Ming Wei, K. P. Schwarz · International Association of Geodesy symposia · 1996
Cited by 1
Longjiang Tang, Jungang Wang, Liangwei Nie et al. · 2023
Along with other space geodetics, the Global Navigation Satellite Systems (GNSS) technique has a profound impact on the realization of terrestrial reference frames and the provision of geodetic and geophysical parameters such as Earth Rotation Parameters (ERP) and geocenter (GCC) coordinates. In GNSS data processing ambiguity resolution (AR) is critical to achieve high-precision estimates. Due to the uncalibrated phase delay (UPD), currently most International GNSS Service (IGS) Analysis Centers (ACs) adopt the double-differenced (DD) ambiguity resolution strategy, which eliminates UPD by between-satellite and between-station differencing. Undifferenced (UD) ambiguity resolution, which recovers the integer characteristics of the UD ambiguities by directly estimating the UPD, was originally developed for Precise Point Positioning and nowadays is also applied to network processing, for example for Precise Orbit Determination (POD). In this study, we conducted a comparative study on the UD and DD ambiguity resolution for GNSS POD data processing. We demonstrated that the UD strategy significantly improves the solution compared to the DD solutions. The GPS orbit precision is improved by 20%, 12%, and 5% in the along, cross, and radial component, respectively. Moreover, the ERP are significantly improved, and the agreement with the IGS final products is improved by 28% and 23% for the offsets of x-pole and y-pole, respectively, and by 23%, and 17% for the corresponding rates. Additionally, the repeatability of station coordinates and GCC coordinates is also slightly improved. More important, we confirmed that the two strategies are theoretically equivalent and the outperformance of UD strategy could be explained by its strong robustness to wrong-fixings and its capability to fix the most UD ambiguities, while wrong-fixings and missing fixings are hardly avoidable in the DD strategy due to the requirement of forming DD-ambiguities.
Cited by 0Open access
Chansik Park, Ilsun Kim · 2002
The attitude of a vehicle can be precisely determined using GPS (Global Positioning System) carrier phase measurements from more than two antennas attached to the vehicle with properly applied efficient integer ambiguity resolution technique. Many methods utilizing the known baseline length as a constraint have been proposed to resolve integer ambiguity in real-time manner. Three-dimensional search space is reduced to two-dimensional search space with this constraint. But there exist still strong requirements for the fast integer ambiguity resolution, which uses single epoch measurement of longer baseline. In this paper, a new constraint from the geometry of multiple baselines is derived. With this new constraint, two-dimensional search space is further reduced to one-dimensional search space.
Cited by 10
Xiaoyong Yu, Hongru Wang · 2024
Hyperbolic Tangent Iterative Optimizer (HTIO) algorithm is introduced for ambiguity resolution in double-difference positioning, with reduced complexity and higher success-rate. The DO (De-rounding operation) algorithm is also proposed to address inconsistencies in the geometric properties of search spaces for various ambiguity elements. Applying the HTIO algorithm to ambiguity vectors processed by the DO algorithm significantly enhances the success rate of ambiguity resolution and reduces fixing time. The HTIO algorithm was compared with LAMBDA algorithms and other 6 swarm intelligence optimization algorithms through ambiguity resolution simulation experiment. Comparatively, the HTIO algorithm outperforms the LAMBDA algorithm with a 5.54% increase in search success-rate and a 91.04% reduction in time, and surpasses the IBOA algorithm with a 4.39% increase in success-rate and an 82.14% time reduction.
Cited by 0
Kirill Starikov, С. Н. Петров · 2024
With the help of modern navigation satellite systems it is possible to determine the coordinates of points with millimeter accuracy due to the precision measurement of the carrier phase of the navigation radio signal. However, in the course of achieving this accuracy, a nontrivial problem of resolving the integer phase ambiguity arises. The most difficult part is to determine the integer number of wavelengths between the phase centers of the transmitting antenna of the navigation satellite and the receiving antenna. An exact solution to this problem using the integer least squares method, although it exists in theory, is not applicable in practice due to the prohibitively high computational complexity. In this regard, a number of authors have developed various suboptimal methods for resolving integer phase ambiguity, which allow solving a problem with varying degrees of uncertainty. In this case, the problem can be considered satisfactorily solved for relative coordinate determinations from GPS (global positioning system) and GLONASS (global navigation satellite system) observations, as well as for absolute coordinate determinations from GPS. As for the absolute determinations of coordinates from GLONASS observations, the situation is complicated by the fact that GLONASS, unlike GPS, uses frequency-division multiple access. In other words, if all GPS navigation spacecraft broadcast navigation radio signals on a single frequency, then GLONASS satellites broadcast on different frequencies. This paper generalizes one of the modern and effective methods for resolving integer ambiguity for GPS observations, MLAMBDA, to the GLONASS system. The ultimate goal of the work is an independent algorithm for absolute determination of coordinates with millimeter accuracy using the GLONASS system. In the course of the work, the MLAMBDA formalism was generalized to navigation systems with frequency division of access, and the first results of independent coordinate determinations from GLONASS observations in absolute mode were obtained.
Cited by 0
Junbo Shi · 2012
Cited by 0Open access
Yiming Chen, Sheng Zhao, Dongfang Zheng et al. · 2014
This article presents a Contemplative Realtime (CRT) framework to resolve a vector of carrier phase integer ambiguities existing in Real-Time Kinematic GPS aided inertial navigation systems (RTK GPS/INS). Within this CRT framework, a Maximum-a-Posteriori (MAP) estimation method is derived to represent the RTK GPS/INS problem, and solved by Nonlinear Mixed Integer Least Square (NMILS) approach. This approach allows the utilization of multiple epochs of GPS data with INS defined motion constraints over a time interval. The NMILS approach over time intervals allows hypothesis testing on each interval to contemplate alternative fault detection hypotheses. The objectives are to enhance reliability and accuracy. Implementation results are included that demonstrate the performance of the proposed method achieving centimeter position accuracy.
Cited by 4
Mamoun F. Abdel–Hafez, Y. J. LEE, Walton R. Williamson et al. · NAVIGATION Journal of the Institute of Navigation · 2003
ABSTRACT: An efficient method for obtaining the admissible integer ambiguity hypotheses within a probabilistic volume is combined with an integer hypothesis testing method to reduce the convergence time to the GPS carrier-phase integers with high probability. The baseline coordinates, float ambiguities, and their covariance matrices are first estimated using a Kalman filter. To reduce the hypothesis set, the covariance matrix decorrelation method, which is the major contribution of the least-squares ambiguity decorrelation adjustment (LAMBDA) method, is then applied. The reduced admissible hypotheses are tested by the multiple-hypothesis Wald sequential probability test (MHWSPT) to select, with a given probability, the most likely hypothesis. This method considerably reduces the computational time needed to solve the integer ambiguity resolution problem at each epoch.
Cited by 6
Yanze Wu, Xianwen Yu, Jiafu Wang · Journal of Applied Geodesy · 2022
Abstract Ensuring that ambiguity cycles are correctly fixed as integers is a critical prerequisite for ensuring the reliability of GNSS high-precision carrier positioning results. As a result, it is both theoretically and practically important to investigate the performance of the ambiguity validation test by selecting an appropriate threshold. To begin, two statistics are proposed in this paper to quantitatively describe the performance of the validation test, namely the true negative rate and the false positive rate, which are based on the percentage of Type I errors (discarded-truth) in the total number of failed tests and the percentage of Type II errors (false positive) in the total number of passed tests. Following that, this paper employs the false positive rate and the true negative rate as primary and secondary criteria for evaluating the performance of the R-ratio test, respectively, and develops simulation experiments to evaluate the performance of different thresholds under different ambiguity dimensions and data accuracy, and finally provides decisions for test threshold selection: (1) For ambiguities with 4 to 9 dimensions, a reference table for the selection of thresholds is given. (2) For ambiguities of 10 dimensions or more, the threshold value should be no less than 2.0 (where data with a mean value of more than 3.7 for the main diagonal elements of the variance matrix should not be fixed).
Cited by 1
P. Henkel and C. Günther · mediaTUM – the media and publications repository of the Technical University Munich (Technical University Munich) · 2012
Cited by 0Open access
Hongmin Zhang, Weisong Wen, Li‐Ta Hsu · Proceedings of the Satellite Division's International Technical Meeting (Online)/Proceedings of the Satellite Division's International Technical Meeting (CD-ROM) · 2023
Fixing ambiguity to a correct integer value is an essential part of high-accuracy positioning using the global navigation satellite system (GNSS) real-time kinematic (RTK). A lot of research has been done on this problem in the past decades. The least-squares ambiguity decorrelation adjustment (LAMBDA) is widely applied for estimating integer ambiguity. In general, the float ambiguity variables are resolved based on the least square estimation using the double-differenced code and carrier measurements. Then the integer ambiguity is resolved by the LAMBDA method which exploits the information content of the full ambiguity variance-covariance matrix, with statistical decorrelation as the objective in constructing the ambiguity transformation. One of the well-recognized challenges of the LAMBDA method is its reliance on the covariance matrix associated with float ambiguity. Moreover, the correlation between the float parameters and the integer ambiguities is not fully explored in the two-step optimization. Instead of resolving the integer ambiguity in a two-step-based manner, can we directly resolve the float state (position of the user) and integer ambiguities simultaneously in a tighter manner? To answer this question, this paper exploits to use of the mixed-integer program (MIP) to solve the GNSS-RTK positioning problem. In particular, the classical branch and bound algorithm of MIP is applied in this research. This algorithm uses a divide-and-conquer strategy to partition the solution space into subproblems and then optimizes individually over each subproblem, in order to find the best solution globally. At each subproblem in the branch and bound algorithm, the position parameter and integer ambiguity are estimated simultaneously to minimize the objective function. In this paper, epoch-by-epoch positioning experiment shows that the positioning performance can be improved a lot based on an MIP method when compared with the conventional method. Moreover, integer ambiguity can be obtained for all epochs. For some epochs, when LAMBDA cannot fix ambiguity successfully, the MIP method can get a reasonable integer value for ambiguity to improve the position accuracy. Compared with the conventional method, the newly proposed algorithm doesn’t directly rely heavily on the quality of float ambiguity estimation and the associated covariance which is indispensable for conventional LAMBDA algorithms. We can get integer ambiguity directly along with ambiguity-fixed position results in one step via the MIP method.
Cited by 0
Jakub Nosek, Pavel Václavovic, Lewen Zhao et al. · Digital Repository (National Repository of Grey Literature)
Precise Point Positioning (PPP) is a powerful method for GNSS data processing, when observations from only one receiver are utilized. Since any systematic error is not eliminated by between-stations differencing, introduced precise products and models are crucial for precise positioning. While precise satellite orbits and clocks corrections are essential for traditional float PPP, satellite phase biases are required for integer ambiguity resolution (PPP-AR). When considering also local ionospheric and tropospheric parameters, the integer ambiguity resolution is achievable very fast, which helps to reduce the initial convergence significantly. When exploiting such precise atmospheric corrections, the solution can be considered an alternative for the popular Real Time Kinematic (RTK) method. Therefore, the solution is usually abbreviated by PPP-RTK. We processed daily GNSS data from EUREF Permanent Network (EPN) using PPP, PPP-AR, and PPP-RTK. The tropospheric delay was not introduced as a product but estimated together with other parameters. The assessment of the convergence time has shown a significant benefit of the resolved integer ambiguities. The average time to reach 10 cm horizontal accuracy was 22, 11, and 7 min when processing in PPP, PPP-AR, and PPP-RTK mode, respectively. The first fixed epoch was obtained after 10 and 2 minutes in the case of PPP-AR and PPP-RTK, respectively. The accuracy assessment of Slant Total Electron Content (STEC) estimated at collocated stations showed their good consistency expressed by the average standard deviation of 0.08 TEC Units.
Cited by 0
E. K. Sutton · 2003
GPS attitude determination systems use carrier phase measurements at three or more antennas to determine platform orientation. Since the antenna separation is usually more than half of the GPS carrier wavelength, the measurements contain an integer cycle ambiguity that must be resolved before attitude can be calculated. This paper will present a new algorithm to instantaneously and reliably resolve the integer cycle ambiguities under conditions of low satellite visibility using at least four antennas in a non-planar array. The new algorithm makes it possible to establish and maintain an attitude solution with as few as two satellites if clock differences are calibrated or three satellites if they are not. Analysis of antenna array geometry and algorithm performance using simulated data will also be presented.
Cited by 3