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.     

Particle filter-based estimation of inter-system phase bias for real-time integer ambiguity resolution

Although double-differenced (DD) observations between satellites from different systems can be used in multi-GNSS relative positioning, the inter-system DD ambiguities cannot be fixed to integer because of the existence of the inter-system bias (ISB). Obviously, they can also be fixed as integer along with intra-system DD ambiguities if the associated ISBs are well known. It is critical to fix such inter-system DD ambiguities especially when only a few satellites of each system are observed. In most of the existing approaches, the ISB is derived from the fractional part of the inter-system ambiguities after the intra-system DD ambiguities are successfully fixed. In this case, it usually needs observations over long times depending on the number of observed satellites from each system. We present a new method by means of particle filter to estimate ISBs in real time without any a priori information based on the fact that the accuracy of a given ISB value can be qualified by the related fixing RATIO. In this particle filter-based method, the ISB parameter is represented by a set of samples, i.e., particles, and the weight of each sample is determined by the designed likelihood function related to the corresponding RATIO, so that the true bias value can be estimated successfully. Experimental validations with the IGS multi-GNSS experiment data show that this method can be carried out epoch by epoch to provide precise ISB in real time. Although there are only one, two, or at most three Galileo satellites observed, the successfully fixing rate increases from 75.5% for GPS only to 81.2%. In the experiment with five GPS satellites and one Galileo satellites, the first successfully fixing time is reduced to half of that without fixing the inter-system DD ambiguities.     

Improving BDS integer ambiguity resolution using satellite-induced code bias correction for precise orbit determination

GPS Solutions - Successful resolution of integer ambiguity over long baselines is a key to improve the accuracy of precise orbit determination for global navigation satellite system satellites. The...     

Grid point search algorithm for fast integer ambiguity resolution

A Grid Point Search Algorithm (GRIPSA) for fast integer ambiguity resolution is presented. In the proposed algorithm, after the orthogonal transformation of the original ambiguity parameters, the confidence ellipsoid of the new parameters is represented by a rectangular polyhedron with its edges parallel to the corresponding axes. A cubic grid covering the whole polyhedron is then identified and transformed back to the original coordinate system. The integer values of the corresponding transformed grid points are obtained by rounding off the transformed values to their nearest integer values. These values are then tested as to whether they are located inside the polyhedron. Since the identification of the grid points in the transformed coordinate system greatly reduces the search region of the integer ambiguities, marked improvements are obtained in the computational effort. Received: 13 October 1997 / Accepted: 9 June 1998     

差分GPS载波相位整周模糊度快速解算方法

本文提出了一种整周模糊度的快速求解方法,将差分GPS的测量值分配到主要测量值集合和次要测量值集合中,用主要集合中的相位测量值限定简约搜索空间,而次要集合中的相位测量值用来验证候选集合。利用已知的基线长度的约束条件,对搜索空间进行了简约,提高了求解整周模糊度的速度,同时,通过Cholesky分解提高搜索效率。     

A comparison of three PPP integer ambiguity resolution methods

Precise point positioning (PPP) integer ambiguity resolution with a single receiver can be achieved using advanced satellite augmentation corrections. Several PPP integer ambiguity resolution methods have been developed, which include the decoupled clock model, the single-difference between-satellites model, and the integer phase clock model. Although similar positioning performances have been demonstrated, very few efforts have been made to explore the relationship between those methods. Our aim is to compare the three PPP integer ambiguity resolution methods for their equivalence. First, several assumptions made in previous publications are clarified. A comprehensive comparison is then conducted using three criteria: the integer property recovery, the system redundancy, and the necessary corrections through which the equivalence of these three PPP integer ambiguity resolution methods in the user solution is obtained.     

Fast integer least-squares estimation for GNSS high-dimensional ambiguity resolution using lattice theory

GNSS ambiguity resolution is the key issue in the high-precision relative geodetic positioning and navigation applications. It is a problem of integer programming plus integer quality evaluation. Different integer search estimation methods have been proposed for the integer solution of ambiguity resolution. Slow rate of convergence is the main obstacle to the existing methods where tens of ambiguities are involved. Herein, integer search estimation for the GNSS ambiguity resolution based on the lattice theory is proposed. It is mathematically shown that the closest lattice point problem is the same as the integer least-squares (ILS) estimation problem and that the lattice reduction speeds up searching process. We have implemented three integer search strategies: Agrell, Eriksson, Vardy, Zeger (AEVZ), modification of Schnorr–Euchner enumeration (M-SE) and modification of Viterbo-Boutros enumeration (M-VB). The methods have been numerically implemented in several simulated examples under different scenarios and over 100 independent runs. The decorrelation process (or unimodular transformations) has been first used to transform the original ILS problem to a new one in all simulations. We have then applied different search algorithms to the transformed ILS problem. The numerical simulations have shown that AEVZ, M-SE, and M-VB are about 320, 120 and 50 times faster than LAMBDA, respectively, for a search space of dimension 40. This number could change to about 350, 160 and 60 for dimension 45. The AEVZ is shown to be faster than MLAMBDA by a factor of 5. Similar conclusions could be made using the application of the proposed algorithms to the real GPS data.     

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.     

Improved method to estimate undifferenced satellite fractional cycle biases using network observations to support PPP ambiguity resolution

GPS Solutions - Fixing ambiguities is beneficial to improve accuracy and convergence time of precise point positioning (PPP). In recent years, several methods have been proposed to estimate...     

Robustness of GNSS integer ambiguity resolution in the presence of atmospheric biases

Both the underlying model strength and biases are two crucial factors for successful integer GNSS ambiguity resolution (AR) in real applications. In some cases, the biases can be adequately parameterized and an unbiased model can be formulated. However, such parameterization will, as trade-off, reduce the model strength as compared to the model in which the biases are ignored. The AR performance with the biased model may therefore be better than with the unbiased model, if the biases are sufficiently small. This would allow for faster AR using the biased model, after which the unbiased model can be used to estimate the remaining unknown parameters. We assess the bias-affected AR performance in the presence of tropospheric and ionospheric biases and compare it with the unbiased case. As a result, the maximum allowable biases are identified for different situations where CORS, static and kinematic baseline models are considered with different model settings. Depending on the size of the maximum allowable bias, a user may decide to use the biased model for AR or to use the unbiased model both for AR and estimating the other unknown parameters.     

Network-based geometry-free three carrier ambiguity resolution and phase bias calibration

Continuously operating reference stations (CORS) are increasingly used to deliver real-time and near-real-time precise positioning services on a regional basis. A CORS network-based data processing system uses either or both of the two types of measurements: (1) ambiguity-resolved double-differenced (DD) phase measurements, and (2) phase bias calibrated zero-differenced (ZD) phase measurements. This paper describes generalized, network-based geometry-free models for three carrier ambiguity resolution (TCAR) and phase bias estimation with DD and ZD code and phase measurements. First, the geometry-free TCAR models are constructed with two Extra-Widelane (EWL)/Widelane (WL) virtual observables to allow for rapid ambiguity resolution (AR) for DD phase measurements without distance constraints. With an ambiguity-resolved WL phase measurement and the ionospheric estimate derived from the two EWL observables, an additional geometry-free equation is formed for the third virtual observable linearly independent of the previous two. AR with the third geometry-free model requires a longer period of observations for averaging than the first two, but is also distance-independent. A more general formulation of the geometry-free model for a baseline or network is also introduced, where all the DD ambiguities can be more rigorously resolved using the LAMBDA method. Second, the geometry-free models for calibration of three carrier phase biases of ZD phase measurements are similarly defined for selected virtual observables. A network adjustment procedure is then used to improve the ZD phase biases with known DD integer constraints. Numerical results from experiments with 24-h dual-frequency GPS data from three US CORS stations baseline lengths of 21, 56 and 74 km confirm the theoretical predictions concerning AR reliability of the network-based geometry-free algorithms.

An alternative approach to calculate the posterior probability of GNSS integer ambiguity resolution

When precise positioning is carried out via GNSS carrier phases, it is important to make use of the property that every ambiguity should be an integer. With the known float solution, any integer vector, which has the same degree of freedom as the ambiguity vector, is the ambiguity vector in probability. For both integer aperture estimation and integer equivariant estimation, it is of great significance to know the posterior probabilities. However, to calculate the posterior probability, we have to face the thorny problem that the equation involves an infinite number of integer vectors. In this paper, using the float solution of ambiguity and its variance matrix, a new approach to rapidly and accurately calculate the posterior probability is proposed. The proposed approach consists of four steps. First, the ambiguity vector is transformed via decorrelation. Second, the range of the adopted integer of every component is directly obtained via formulas, and a finite number of integer vectors are obtained via combination. Third, using the integer vectors, the principal value of posterior probability and the correction factor are worked out. Finally, the posterior probability of every integer vector and its error upper bound can be obtained. In the paper, the detailed process to calculate the posterior probability and the derivations of the formulas are presented. The theory and numerical examples indicate that the proposed approach has the advantages of small amount of computations, high calculation accuracy and strong adaptability.     

Modeling and quality control for reliable precise point positioning integer ambiguity resolution with GNSS modernization

Recent research has demonstrated that the undifferenced integer ambiguities can be recovered using products from a network solution. The standard dual-frequency PPP integer ambiguity resolution consists of two aspects: Hatch-Melbourne-Wübbena wide-lane (WL) and ionosphere-free narrow-lane (NL) integer ambiguity resolution. A major issue affecting the performance of dual-frequency PPP applications is the time it takes to fix these two types of integer ambiguities, especially if the WL integer ambiguity resolution suffers from the noisy pseudorange measurements and strong multipath effects. With modernized Global Navigation Satellite Systems, triple-frequency measurements will be available to global users and an extra WL (EWL) model with very long wavelength can be formulated. Then, the easily resolved EWL integer ambiguities can be used to construct linear combinations to accelerate the PPP WL integer ambiguity resolution. Therefore, we propose a new reliable procedure for the modeling and quality control of triple-frequency PPP WL and NL integer ambiguity resolution. First, we analyze a WL integer ambiguity resolution model based on triple-frequency measurements. Then, an optimal pseudorange linear combination which is ionosphere-free and has minimum measurement noise is developed and used as constraint in the WL and the NL integer ambiguity resolution. Based on simulations, we have investigated the inefficiency of dual-frequency WL integer ambiguity resolution and the performance of EWL integer ambiguity resolution. Using almanacs of GPS, Galileo and BeiDou, the performances of the proposed triple-frequency WL and NL models have been evaluated in terms of success rate. Comparing with dual-frequency PPP, numerical results indicate that the proposed triple-frequency models can outperform the dual-frequency PPP WL and NL integer ambiguity resolution. With 1 s sampling rate, generally, only several minutes of data are required for reliable triple-frequency PPP WL and NL integer ambiguity resolution. Under benign observation situations and good geometries, the integer ambiguity can be reliably resolved even within 10 s.     

Triple-frequency carrier ambiguity resolution for Beidou navigation satellite system

The Chinese Beidou system, also known as Compass, has entered its trial operational stage and can already provide services for triple-frequency users. Using triple-frequency signals is expected to be of great benefit for ambiguity resolution. Based on error characteristic analysis of the Beidou frequencies, we introduce the procedure of selecting the best combinations of triple-frequency signals. The geometry-based model and geometry-free model of triple-frequency signals are presented. Three triple-frequency carrier ambiguity resolution (TCAR) methods are described, which include the cascading rounding method, the stepwise AR method and the modified stepwise AR method. In order to evaluate the performance of these methods, observations from baselines of various lengths were collected using Beidou triple-frequency receivers and were processed epoch-by-epoch using the three methods. The same observation data were also processed in a dual-frequency mode for comparison. The results show that, compared to the dual-frequency based solution, the single epoch ambiguity resolution success rate with triple frequency improved nearly 30 % for the short baselines (<20 km) and 100 % for the mid-length baselines (20–50 km) using the proposed modified stepwise AR method.