Integer Ambiguity Resolution Research Articles

ON THE REALISTIC STOCHASTIC MODEL OF GPS OBSERVABLES: IMPLEMENTATION AND PERFORMANCE

Abstract. High-precision GPS positioning requires a realistic stochastic model of observables. A realistic GPS stochastic model of observables should take into account different variances for different observation types, correlations among different observables, the satellite elevation dependence of observables precision, and the temporal correlation of observables. Least-squares variance component estimation (LS-VCE) is applied to GPS observables using the geometry-based observation model (GBOM). To model the satellite elevation dependent of GPS observables precision, an exponential model depending on the elevation angles of the satellites are also employed. Temporal correlation of the GPS observables is modelled by using a first-order autoregressive noise model. An important step in the high-precision GPS positioning is double difference integer ambiguity resolution (IAR). The fraction or percentage of success among a number of integer ambiguity fixing is called the success rate. A realistic estimation of the GNSS observables covariance matrix plays an important role in the IAR. We consider the ambiguity resolution success rate for two cases, namely a nominal and a realistic stochastic model of the GPS observables using two GPS data sets collected by the Trimble R8 receiver. The results confirm that applying a more realistic stochastic model can significantly improve the IAR success rate on individual frequencies, either on L1 or on L2. An improvement of 20% was achieved to the empirical success rate results. The results also indicate that introducing the realistic stochastic model leads to a larger standard deviation for the baseline components by a factor of about 2.6 on the data sets considered.

Assessment of the Contribution of BeiDou GEO, IGSO, and MEO Satellites to PPP in Asia-Pacific Region.

In contrast to the US Global Positioning System (GPS), the Russian Global Navigation Satellite System (GLONASS) and the European Galileo, the developing Chinese BeiDou satellite navigation system (BDS) consists of not only Medium Earth Orbit (MEO), but also Geostationary Orbit (GEO) as well as Inclined Geosynchronous Orbit (IGSO) satellites. In this study, the Precise Point Positioning (PPP) and PPP with Integer Ambiguity Resolution (IAR) are obtained. The contributions of these three different types of BDS satellites to PPP in Asia–Pacific region are assessed using data from selected 20 sites over more than four weeks. By using various PPP cases with different satellite combinations, in general, the largest contribution of BDS IGSO among the three kinds of BDS satellites to the reduction of convergence time and the improvement of positioning accuracy, particularly in the east direction, is identified. These PPP cases include static BDS only solutions and static/kinematic ambiguity-float and -fixed PPP with the combination of GPS and BDS. The statistical results demonstrate that the inclusion of BDS GEO and MEO satellites can improve the observation condition and result in better PPP performance as well. When combined with GPS, the contribution of BDS to the reduction of convergence time is, however, not as significant as that of GLONASS. As far as the positioning accuracy is concerned, GLONASS improves the accuracy in vertical component more than BDS does, whereas similar improvement in horizontal component can be achieved by inclusion of BDS IGSO and MEO as GLONASS.

FAST INTEGER AMBIGUITY RESOLUTION IN GPS KINEMATIC POSITIONING USING LEFT NULL SPACE AND MULTI-TIME (INVERSE) PAIRED CHOLESKY DECORRELATION

Aiming at the problems that huge amount of computation in ambiguity resolution with multiple epochs and high-order matrix inversion occurred in the GPS kinematic relative positioning, a modified algorithm for fast integer ambiguity resolution is proposed. Firstly, Singular Value Decomposition (SVD) is applied to construct the left null space matrix in order to eliminate the baselines components, which is able to separate ambiguity parameters from the position parameters efficiently. Kalman filter is applied only to estimate the ambiguity parameters so that the real-time ambiguity float solution is obtained. Then, sorting and multi-time (inverse) paired Cholesky decomposition are adopted for decorrelation of ambiguity. After diagonal elements preprocessing and diagonal elements sorting according to the results of Cholesky decomposition, the efficiency of decomposition and decorrelation is improved. Lastly, the integer search algorithm implemented in LAMBDA method is used for searching the integer ambiguity. To verify the validity and efficacy of the proposed algorithm, static and kinematic tests are carried out. Experimental results show that this algorithm has good performance of decorrelation and precision of float solution, with computation speed also increased effectively. The final positioning accuracy result with static baseline error less than 1 cm and kinematic error less than 2 cm, which indicates that it can be used for fast kinematic positioning of high precision carrier.

On the estimability of parameters in undifferenced, uncombined GNSS network and PPP-RTK user models by means of $$\mathcal {S}$$ S -system theory

The concept of integer ambiguity resolution-enabled Precise Point Positioning (PPP-RTK) relies on appropriate network information for the parameters that are common between the single-receiver user that applies and the network that provides this information. Most of the current methods for PPP-RTK are based on forming the ionosphere-free combination using dual-frequency Global Navigation Satellite System (GNSS) observations. These methods are therefore restrictive in the light of the development of new multi-frequency GNSS constellations, as well as from the point of view that the PPP-RTK user requires ionospheric corrections to obtain integer ambiguity resolution results based on short observation time spans. The method for PPP-RTK that is presented in this article does not have above limitations as it is based on the undifferenced, uncombined GNSS observation equations, thereby keeping all parameters in the model. Working with the undifferenced observation equations implies that the models are rank-deficient; not all parameters are unbiasedly estimable, but only combinations of them. By application of $$\mathcal {S}$$ -system theory the model is made of full rank by constraining a minimum set of parameters, or S-basis. The choice of this S-basis determines the estimability and the interpretation of the parameters that are transmitted to the PPP-RTK users. As this choice is not unique, one has to be very careful when comparing network solutions in different $$\mathcal {S}$$ -systems; in that case the S-transformation, which is provided by the $$\mathcal {S}$$ -system method, should be used to make the comparison. Knowing the estimability and interpretation of the parameters estimated by the network is shown to be crucial for a correct interpretation of the estimable PPP-RTK user parameters, among others the essential ambiguity parameters, which have the integer property which is clearly following from the interpretation of satellite phase biases from the network. The flexibility of the $$\mathcal {S}$$ -system method is furthermore demonstrated by the fact that all models in this article are derived in multi-epoch mode, allowing to incorporate dynamic model constraints on all or subsets of parameters.

Robust and precise baseline determination of distributed spacecraft in LEO

Recent experience with prominent formation flying missions in Low Earth Orbit (LEO), such as GRACE and TanDEM-X, has shown the feasibility of precise relative navigation at millimeter and sub-millimeter levels using GPS carrier phase measurements with fixed integer ambiguities. However, the robustness and availability of the solutions provided by current algorithms may be highly dependent on the mission profile. The main challenges faced in the LEO scenario are the resulting short continuous carrier phase tracking arcs along with the observed rapidly changing ionospheric conditions, which in the particular situation of long baselines increase the difficulty of correct integer ambiguity resolution. To reduce the impact of these factors, the present study proposes a strategy based on a reduced-dynamics filtering of dual-frequency GPS measurements for precise baseline determination along with a dedicated scheme for integer ambiguity resolution, consisting of a hybrid sequential/batch algorithm based on the maximum a posteriori and integer aperture estimators. The algorithms have been tested using flight data from the GRACE, TanDEM-X and Swarm missions in order to assess their robustness to different formation and baseline configurations. Results with the GRACE mission show an average 0.7mm consistency with the K/Ka-band ranging measurements over a period of more than two years in a baseline configuration of 220km. Results with TanDEM-X data show an average of 3.8mm consistency of kinematic and reduced-dynamic solutions in the along-track component over a period of 40days in baseline configurations of 500m and 75km. Data from Swarm A and Swarm C spacecraft are largely affected by atmospheric scintillation and contain half cycle ambiguities. The results obtained under such conditions show an overall consistency between kinematic and reduced-dynamic solutions of 1.7cm in the along-track component over a period of 30days in a variable baseline of approximately 60–175km. An analysis of one orbital period excluding a region where errors due to atmospheric scintillation occur, shows a consistency between kinematic and reduced-dynamic solutions of 3mm in the along-track direction.

저비용 GNSS 수신기를 이용한 반송파 위상 시각간 차분 측정치 기반의 정밀 상대위치 결정 기법

본 논문에서는 저비용 단일 GNSS(Global Navigation Satellite System) 수신기를 이용하여 정밀한 위치를 추정하는 방법으로 반송파 위상 TD(Time Differenced) 측정치를 이용한 상대위치 결정기법을 제안하고 성능을 분석하였다. 제안하는 상대위치 결정기법은 반송파 위상 측정치를 활용하지만 시각 간 차분을 사용하므로 하나의 GNSS 수신기로 동작하며, 미지정수 결정 문제를 고려하지 않아도 된다. 또한 차분 간격이 짧은 시간일 경우 공간적 공통오차 및 위성시계 오차를 효율적으로 제거할 수 있다. 제안하는 위치결정 알고리즘은 오차해석을 통해 코드측정치를 이용한 절대위치 결정기법보다 성능이 우수함을 증명하고 실차 실험으로 구현된 알고리즘 및 오차해석 결과를 검증하였다. 그 결과, 위치추정 성능은 약 10분 동안 3m이내에 수렴하여 코드기반의 절대위치기법에 비하여 약 4배 이상 향상된 정확도 및 정밀도를 확인할 수 있었다. In this paper, a precise relative positioning with TD(time differenced) carrier phase measurements from a low-cost GNSS(Global Navigation Satellite System) receiver is proposed and analysed. The proposed method is using carrier phase measurement from a single GNSS receiver that reference receiver is not required and stand alone positioning is possible. TD operation removes the troublesome integer ambiguity resolution problem, and if the time interval is short, other error, such as, ionospheric, tropospheric delay and ephemeris error are effectively eliminated. The error analysis of the proposed method shows that a precise and positioning with carrier phase is possible. The implemented system is evaluated using a real car experiments. The results show that the horizontal positioning error was less than 3m during 10 minutes experiments, which is 4 times more precise than the results of normal code based absolute positioning.

Partial Ambiguity Resolution for Ground and Space-Based Applications in a GPS+Galileo scenario: A simulation study

Integer Ambiguity Resolution (IAR) is the key to fast and precise GNSS positioning. The proper diagnostic metric for successful IAR is provided by the ambiguity success rate being the probability of correct integer estimation. In this contribution we analyse the performance of different GPS+Galileo models in terms of number of epochs needed to reach a pre-determined success rate, for various ground and space-based applications. The simulation-based controlled model environment enables us to gain insight into the factors contributing to the ambiguity resolution strength of the different GPS+Galileo models. Different scenarios of modernized GPS+Galileo are studied, encompassing the long baseline ground case as well as the medium dynamics case (airplane) and the space-based Low Earth Orbiter (LEO) case. In our analyses of these models the capabilities of partial ambiguity resolution (PAR) are demonstrated and compared to the limitations of full ambiguity resolution (FAR). The results show that PAR is generally a more efficient way than FAR to reduce the time needed to achieve centimetre-level positioning precision. For long single baselines, PAR can achieve time reductions of fifty percent to achieve such precision levels, while for multiple baselines it even becomes more effective, reaching reductions up to eighty percent for four station networks. For a LEO, the rapidly changing observation geometry does not even allow FAR, while PAR is then still possible for both dual- and triple-frequency scenarios. With the triple-frequency GPS+Galileo model the availability of precise positioning improves by fifteen percent with respect to the dual-frequency scenario.

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

GLONASS could hardly reach the positioning performance of GPS, especially for fast and real-time precise positioning. One of the reasons is the phase inter-frequency bias (IFB) at the receiver end prevents its integer ambiguity resolution. A number of studies were carried out to achieve the integer ambiguity resolution for GLONASS. Based on some of the revealed IFB characteristics, for instance IFB is a linear function of the received carrier frequency and L1 and L2 have the same IFB in unit of length, most of recent methods recommend estimating the IFB rate together with ambiguities. However, since the two sets of parameters are highly correlated, as demonstrated in previous studies, observations over several hours up to 1 day are needed even with simultaneous GPS observations to obtain a reasonable solution. Obviously, these approaches cannot be applied for real-time positioning. Actually, it can be demonstrated that GLONASS ambiguity resolution should also be available even for a single epoch if the IFB rate is precisely known. In addition, the closer the IFB rate value is to its true value, the larger the fixing RATIO will be. Based on this fact, in this paper, a new approach is developed to estimate the IFB rate by means of particle filtering with the likelihood function derived from RATIO. This approach is evaluated with several sets of experimental data. For both static and kinematic cases, the results show that IFB rates could be estimated precisely just with GLONASS data of a few epochs depending on the baseline length. The time cost with a normal PC can be controlled around 1 s and can be further reduced. With the estimated IFB rate, integer ambiguity resolution is available immediately and as a consequence, the positioning accuracy is improved significantly to the level of GPS fixed solution. Thus the new approach enables real-time precise applications of GLONASS.

Instantaneous and controllable integer ambiguity resolution: review and an alternative approach

In the high-precision application of Global Navigation Satellite System (GNSS), integer ambiguity resolution is the key step to realize precise positioning and attitude determination. As the necessary part of quality control, integer aperture (IA) ambiguity resolution provides the theoretical and practical foundation for ambiguity validation. It is mainly realized by acceptance testing. Due to the constraint of correlation between ambiguities, it is impossible to realize the controlling of failure rate according to analytical formula. Hence, the fixed failure rate approach is implemented by Monte Carlo sampling. However, due to the characteristics of Monte Carlo sampling and look-up table, we have to face the problem of a large amount of time consumption if sufficient GNSS scenarios are included in the creation of look-up table. This restricts the fixed failure rate approach to be a post process approach if a look-up table is not available. Furthermore, if not enough GNSS scenarios are considered, the table may only be valid for a specific scenario or application. Besides this, the method of creating look-up table or look-up function still needs to be designed for each specific acceptance test. To overcome these problems in determination of critical values, this contribution will propose an instantaneous and CONtrollable (iCON) IA ambiguity resolution approach for the first time. The iCON approach has the following advantages: (a) critical value of acceptance test is independently determined based on the required failure rate and GNSS model without resorting to external information such as look-up table; (b) it can be realized instantaneously for most of IA estimators which have analytical probability formulas. The stronger GNSS model, the less time consumption; (c) it provides a new viewpoint to improve the research about IA estimation. To verify these conclusions, multi-frequency and multi-GNSS simulation experiments are implemented. Those results show that IA estimators based on iCON approach can realize controllable ambiguity resolution. Besides this, compared with ratio test IA based on look-up table, difference test IA and IA least square based on the iCON approach most of times have higher success rates and better controllability to failure rates.

Performance analysis on carrier phase-based tightly-coupled GPS/BDS/INS integration in GNSS degraded and denied environments.

The integration of Global Navigation Satellite Systems (GNSS) carrier phases with Inertial Navigation System (INS) measurements is essential to provide accurate and continuous position, velocity and attitude information, however it is necessary to fix ambiguities rapidly and reliably to obtain high accuracy navigation solutions. In this paper, we present the notion of combining the Global Positioning System (GPS), the BeiDou Navigation Satellite System (BDS) and low-cost micro-electro-mechanical sensors (MEMS) inertial systems for reliable navigation. An adaptive multipath factor-based tightly-coupled (TC) GPS/BDS/INS integration algorithm is presented and the overall performance of the integrated system is illustrated. A twenty seven states TC GPS/BDS/INS model is adopted with an extended Kalman filter (EKF), which is carried out by directly fusing ambiguity fixed double-difference (DD) carrier phase measurements with the INS predicted pseudoranges to estimate the error states. The INS-aided integer ambiguity resolution (AR) strategy is developed by using a dynamic model, a two-step estimation procedure is applied with adaptively estimated covariance matrix to further improve the AR performance. A field vehicular test was carried out to demonstrate the positioning performance of the combined system. The results show the TC GPS/BDS/INS system significantly improves the single-epoch AR reliability as compared to that of GPS/BDS-only or single satellite navigation system integrated strategy, especially for high cut-off elevations. The AR performance is also significantly improved for the combined system with adaptive covariance matrix in the presence of low elevation multipath related to the GNSS-only case. A total of fifteen simulated outage tests also show that the time to relock of the GPS/BDS signals is shortened, which improves the system availability. The results also indicate that TC integration system achieves a few centimeters accuracy in positioning based on the comparison analysis and covariance analysis, even in harsh environments (e.g., in urban canyons), thus we can see the advantage of positioning at high cut-off elevations that the combined GPS/BDS brings.

1 × 10−16 frequency transfer by GPS PPP with integer ambiguity resolution

For many years, the time community has been using the precise point positioning (PPP) technique which uses GPS phase and code observations to compute time and frequency links. However, progress in atomic clocks implies that the performance of PPP frequency comparisons is a limiting factor in comparing the best frequency standards. We show that a PPP technique where the integer nature of phase ambiguities is preserved consitutes significant improvement of the classical use of floating ambiguities. We demonstrate that this integer-PPP technique allows frequency comparisons with 1 × 10−16 accuracy in a few days and can be readily operated with existing products.

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.

Role of stochastic model on GPS integer ambiguity resolution success rate

An important step in the high-precision GPS positioning is double-difference integer ambiguity resolution (IAR). The fraction or percentage of success among a number of integer ambiguity fixing is called the success rate. We investigate the ambiguity resolution success rate for the GPS observations for two cases, namely a nominal and a realistic stochastic model of the GPS observables. In principle, one would expect to have higher reliability on IAR success rates if a realistic GPS observables stochastic model is employed. The GPS geometry-based observation model is employed in which a more realistic stochastic model of GPS observables is determined using the least-squares variance component estimation. Two short and one GPS long baseline datasets and one simulated dataset are employed to evaluate the efficacy of the proposed algorithm. The results confirm that a more realistic stochastic model can significantly improve the IAR success rate on individual frequencies, either on L1 or on L2. An improvement of 25 % was achieved to the empirical success rate results. The results are of interest for many applications in which single-frequency observations can be used. This includes applications like attitude determination using single frequency single epoch of GPS observations.

Research on Fast Integer Ambiguity Resolution Using Pheromone-based Particle Swarm Optimization Algorithm

Research on Fast Integer Ambiguity Resolution Using Pheromone-based Particle Swarm Optimization Algorithm

Timing group delay and differential code bias corrections for BeiDou positioning

This article first clearly figures out the relationship between parameters of timing group delay (TGD) and differential code bias (DCB) for BDS, and demonstrates the equivalence of TGD and DCB correction models combining theory with practice. The TGD/DCB correction models have been extended to various occasions for BDS positioning, and such models have been evaluated by real triple-frequency datasets. To test the effectiveness of broadcast TGDs in the navigation message and DCBs provided by the Multi-GNSS Experiment (MGEX), both standard point positioning (SPP) and precise point positioning (PPP) tests are carried out for BDS signals with different schemes. Furthermore, the influence of differential code biases on BDS positioning estimates such as coordinates, receiver clock biases, tropospheric delays and carrier phase ambiguities is investigated comprehensively. Comparative analysis show that the unmodeled differential code biases degrade the performance of BDS SPP by a factor of two or more, whereas the estimates of PPP are subject to varying degrees of influences. For SPP, the accuracy of dual-frequency combinations is slightly worse than that of single-frequency, and they are much more sensitive to the differential code biases, particularly for the B2B3 combination. For PPP, the uncorrected differential code biases are mostly absorbed into the receiver clock bias and carrier phase ambiguities and thus resulting in a much longer convergence time. Even though the influence of the differential code biases could be mitigated over time and comparable positioning accuracy could be achieved after convergence, it is suggested to properly handle with the differential code biases since it is vital for PPP convergence and integer ambiguity resolution.

Efficient high-rate satellite clock estimation for PPP ambiguity resolution using carrier-ranges.

In order to catch up the short-term clock variation of GNSS satellites, clock corrections must be estimated and updated at a high-rate for Precise Point Positioning (PPP). This estimation is already very time-consuming for the GPS constellation only as a great number of ambiguities need to be simultaneously estimated. However, on the one hand better estimates are expected by including more stations, and on the other hand satellites from different GNSS systems must be processed integratively for a reliable multi-GNSS positioning service. To alleviate the heavy computational burden, epoch-differenced observations are always employed where ambiguities are eliminated. As the epoch-differenced method can only derive temporal clock changes which have to be aligned to the absolute clocks but always in a rather complicated way, in this paper, an efficient method for high-rate clock estimation is proposed using the concept of “carrier-range” realized by means of PPP with integer ambiguity resolution. Processing procedures for both post- and real-time processing are developed, respectively. The experimental validation shows that the computation time could be reduced to about one sixth of that of the existing methods for post-processing and less than 1 s for processing a single epoch of a network with about 200 stations in real-time mode after all ambiguities are fixed. This confirms that the proposed processing strategy will enable the high-rate clock estimation for future multi-GNSS networks in post-processing and possibly also in real-time mode.

FARSE scheme for single epoch GPS solution based on DUFCOM and DC algorithm and its performance analysis

Fast integer ambiguity resolution for single epoch observation is one of main issues of GPS precise positioning in real time surveying applications. An improved solution of dual frequency correlation method (DUFCOM) and direct calculation method (DC), named as fast ambiguity resolution for single epoch scheme (FARSE), is proposed in this paper. A software based on the proposed scheme for monitoring of construction cranes, Gsertcas, is developed. With the help of Gsertcas, the performance and suitability of FARSE are investigated through simulation experiments. The result of the experiment demonstrates that the success rate for ambiguity resolution is above 97% and the root mean square of the position solution with correct ambiguity resolution is better than 3·8 mm.

On reliable data-driven partial GNSS ambiguity resolution

In high-precision global navigation satellite system applications, it is often not possible to simultaneously meet the requirements for fast and reliable integer ambiguity resolution. For a given r...

Success rate improvement of single epoch integer least-squares estimator for the GNSS attitude/short baseline applications with common clock scheme

The sub-centimeter level relative positioning can be accomplished anytime and anywhere on Earth, provided that the integer ambiguities of the very precise Global Navigation Satellite System (GNSS) carrier-phase observables are correctly resolved. The Integer Least-Squares (ILS) estimator is known to be efficient and optimal for Integer Ambiguity Resolution (IAR) of the unconstrained GNSS model, and the stronger the model strength, the higher the success rate. In this contribution, we investigate the model strength of the common clock scheme, in which the two receivers of the short baseline feed a common clock and the cables have the same delay and many GNSS-based attitude/short baseline applications benefit from this approach. For this scheme, the clock biases between two receivers will be zero and can be omitted, and thus it has a stronger model strength and a higher success rate, compared with the non-common clock scheme. In this contribution, the well-known Ambiguity Dilution of Precision (ADOP), which captures the main characteristics of the ambiguity precision, is utilized to assess the model strengthening when the perfectly synchronized receivers are employed. The mathematical expression of the single epoch ADOP is deduced by the given design parameters, and thus the ADOP-based approximate ILS success rate can be easily computed with the specified measurement set-up which includes the number of available satellites and frequencies and the precision of observation. We focus on the success rate assessment of the common clock scheme for stand-alone, unaided, single-frequency, single epoch observation with only five visible satellites, since it is the most challenging case of GNSS relative positioning and in this case the performance of the employed ambiguity resolution method can be investigated effectively. Thus, one can predict whether or not the required performance will be obtained for the worst case.

Array-based satellite phase bias sensing: theory and GPS/BeiDou/QZSS results

Single-receiver integer ambiguity resolution (IAR) is a measurement concept that makes use of network-derived non-integer satellite phase biases (SPBs), among other corrections, to recover and resolve the integer ambiguities of the carrier-phase data of a single GNSS receiver. If it is realized, the very precise integer ambiguity-resolved carrier-phase data would then contribute to the estimation of the receiver’s position, thus making (near) real-time precise point positioning feasible. Proper definition and determination of the SPBs take a leading part in developing the idea of single-receiver IAR. In this contribution, the concept of array-based between-satellite single-differenced (SD) SPB determination is introduced, which is aimed to reduce the code-dominated precision of the SD-SPB corrections. The underlying model is realized by giving the role of the local reference network to an array of antennas, mounted on rigid platforms, that are separated by short distances so that the same ionospheric delay is assumed to be experienced by all the antennas. To that end, a closed-form expression of the array-aided SD-SPB corrections is presented, thereby proposing a simple strategy to compute the SD-SPBs. After resolving double-differenced ambiguities of the array’s data, the variance of the SD-SPB corrections is shown to be reduced by a factor equal to the number of antennas. This improvement in precision is also affirmed by numerical results of the three GNSSs GPS, BeiDou and QZSS. Experimental results demonstrate that the integer-recovered ambiguities converge to integers faster, upon increasing the number of antennas aiding the SD-SPB corrections.