Improved Ambiguity Resolution Research Articles

Multi-sensor Attitude Estimation using Quaternion Constrained GNSS Ambiguity Resolution and Dynamics-Based Observation Synchronization

Recently, high accuracy and low-cost navigation hardware is becoming increasingly available that can be efficiently used for the control of autonomous vehicles. We present a sensor fusion method providing tightly coupled integration of pseudorange, carrier phase, and Doppler satellite measurements taken at multiple vehicle-mounted GNSS antennas with onboard inertial sensor observations. The key of accurate GNSS position and orientation estimation is the successful integer ambiguity resolution. We propose a method that uses the quaternion states as constraints to improve ambiguity resolution and to increase the accuracy of the GNSS based attitude determination. Generally, the low-cost hardware neither allows a hardware-level time synchronization between the GNSS receivers due to a lack of a common external oscillator nor provides the clock steering function available in geodetic GNSS receivers. The lack of observation synchronization causes several degrees of error in attitude estimation. To eliminate this effect, a dynamics-based solution is presented that synchronizes the observations by taking the dynamics of the moving platform into account. Compared to common external oscillator based sensor setups, our solution allows to increase both the number of rover receivers on the platform and the baselines between them easily, thus it opens up new possibilities in the attitude determination of large vehicles. We validate our approach against a tactical grade inertial navigation system. The results show that our approach using low-cost sensors provides the ambiguity success rate of 100% for the moving baselines, and the positioning and attitude error reached the centimeter and half a degree level, respectively.

Improving ambiguity resolution with common-antenna-based dual-board receiver for low-cost real-time kinematic positioning

Ambiguity resolution is of critical importance to the carrier phase-based real-time kinematic (RTK) positioning method. Improving the accuracy of float ambiguities is beneficial for achieving ambiguity resolution. However, the large measurement noise from low-cost receivers will worsen the estimation accuracy of float ambiguities, which affects the ambiguity resolution performance. In this contribution, to reduce the influence of large measurement noise on ambiguity resolution for low-cost receivers, an improved RTK method for ambiguity resolution is proposed to enhance the accuracy of float ambiguities by equipping the rover receiver with common-antenna-based dual global navigation satellite system (GNSS) boards instead of only one GNSS board. First, the dual-board design can increase the measurement redundancy of the same frequency to suppress the measurement noise. Second, because the common-antenna design can form a moving zero-baseline between the dual GNSS boards, the ambiguities between them can be easily fixed. Known fixed ambiguities can be used as constraints to strengthen the positioning model. Simulation and real-world static and kinematic experiments were conducted to test the proposed method. The results demonstrate that the proposed method can improve the accuracy of float ambiguities by increasing the redundancy of the measurements and introducing the constraints of the ambiguities, and the improved accuracy is about 20%. Compared with the traditional single-board RTK method, better ambiguity resolution performance can be achieved by taking advantage of the proposed common-antenna-based RTK method.

An Improved Ambiguity Resolution Algorithm for Smartphone RTK Positioning.

Ambiguity resolution based on smartphone GNSS measurements can enable various potential applications that currently remain difficult due to ambiguity biases, especially under kinematic conditions. This study proposes an improved ambiguity resolution algorithm, which uses the search-and-shrink procedure coupled with the methods of the multi-epoch double-differenced residual test and the ambiguity majority tests for candidate vectors and ambiguities. By performing a static experiment with Xiaomi Mi 8, the AR efficiency of the proposed method is evaluated. Furthermore, a kinematic test with Google Pixel 5 verifies the effectiveness of the proposed method with improved positioning performance. In conclusion, centimeter-level smartphone positioning accuracy is achieved in both experiments, which is greatly improved compared with the float and traditional AR solutions.

Improving ambiguity resolution success rate in the joint solution of GNSS-based attitude determination and relative positioning with multivariate constraints

Usually, the global navigation satellite system (GNSS)-based attitude determination and the GNSS-based relative positioning are treated as separate problems. In this contribution, the two problems are resolved as a combined solution of a joint model with multivariate constraints, in which the known geometry of the antenna array on the rover platform is used as constraints, thereby improving the parameter estimation in not only the attitude determination but also the relative positioning. The objective function of the joint model is decomposed and then reduced so that only the terms concerning the constraints are retained, thus transforming the objective function into the exact form of that of the GNSS-based multivariate constrained attitude determination model. Then the multivariate constrained least-squares ambiguity decorrelation adjustment method together with the efficient shrinkage or expansion with bounds strategies, which are shown to work well in the attitude determination, can be extended to the joint model. In experimental validation, the joint model is tested with both simulated and real GPS/BDS datasets in terms of ambiguity resolution success rate and the precision of solutions. The results confirm that the joint model ensures the highest ambiguity resolution success rate among the existing unaided GNSS-based relative positioning methods, and the improvement is more obvious when the observation model is weaker, for example, when the number of available satellites is fewer or when the precision of observations is poorer, thus showing an overall best performance in both reliability and precision aspects.

Estimating multi-frequency satellite phase biases of BeiDou using maximal decorrelated linear ambiguity combinations

Improving ambiguity resolution (AR) in multi-frequency undifferenced and uncombined precise point positioning (PPP) benefits from accurate uncalibrated phase delays (UPD), which are often estimated from linear combinations of float ambiguities. The traditional linear ambiguity combinations for estimating these UPDs in case of triple-frequency observations are typically extra-wide-lane, wide-lane, and L1 combinations. We proposed the method for estimating UPDs from triple-frequency ambiguities using maximal decorrelated linear ambiguity combinations obtained by the least-squares ambiguity decorrelation adjustment Z-transformation. To validate the quality and availability of estimating UPDs for the BeiDou navigation satellite system, based on maximal decorrelated linear ambiguity combinations, tests using observations from stations of the Crustal Movement Observation Network of China and the Asia-Pacific Reference Frame project are performed using undifferenced and uncombined PPP-AR. The results show the internal precision of combined satellite UPDs estimated from the maximal decorrelated linear ambiguity combinations is better than that estimated from traditional combinations in terms of temporal stability and RMS of posteriori residuals. Furthermore, the statistical results also demonstrated that triple-frequency PPP-AR using the improved UPDs reduces the average convergence time by 8.9 and 12.3% in horizontal and vertical directions, and also improves the positioning accuracy for 3?h of observations by 11.1, 9.1 and 8.3% in the east, north and up directions, respectively, compared with triple-frequency PPP-AR using the UPDs derived from the traditional combinations.

Angle-of-Arrival Estimation in Formation Flying Satellites: Concept and Demonstration

One of the challenges of future space missions is the deployment of instruments distributed across multiple satellites that fly in an autonomous formation. This type of mission requires that the spacecraft communicate using inter-satellite links (ISL) and collaborate to fulfil the role of a large-scale, complex scientific instrument. In this context, precise information regarding the relative positioning of the satellites in terms of distance and orientation is essential. In radio frequency (RF) based techniques, obtaining the relative orientation between the satellites starts with estimating the angle-of-arrival (AoA) of the radio signal. This paper proposes an AoA estimation technique that can be used in Orthogonal Frequency-Division Multiplexing (OFDM) communications. Our approach consists in obtaining the AoA using the ISL, specifically the preamble whose primary roles are data link synchronization and channel estimation. Our technique allows a more accurate measurement of the AoA due to an improved ambiguity resolution of phase measurements. We give theoretical results regarding the accuracy of the AoA estimator and we validate them by tests on a real-time functioning testbed consisting in multiple National Instruments Universal Software Radio Peripheral platforms. We show that the carrier frequency offset introduces only a negligible bias in the AoA estimation and we discuss the effect of multipath propagation.

Review of triple-frequency GNSS: ambiguity resolution, benefits and challenges

Triple-frequency GNSS has been intensively studying in the past decades, especially with the open service of China’s BeiDou system. In this review, we will address the ambiguity resolution, benefits gained from additional frequency signals compared to the dual-frequency GNSS signals, as well as analyse the challenges of triple-frequency GNSS for future development. We first review and compare the three carrier ambiguity resolution models of geometry-based, geometry-free, geometry-ionosphere-free (GIF). The benefits gained from triple-frequency GNSS are then comprehensively examined with respect to dual-frequency case, including the improved ambiguity resolution, extra-widelane based RTK, the augmented RTK service, the shortened PPP convergence, the improved availability and reliability. In addition, some challenges are discussed from both theoretical and practical aspects to open eyes for future research.

An Improved Ambiguity Resolution of Three Carriers in Precise Point Positioning

Precise point positioning (PPP) is of great importance in fields requiring coordinates with high accuracy, such as geophysics, meteorology, and geodesy. Compared with relative positioning restricted by the length of baseline, PPP using a single receiver is more flexible. Carrier phase observations with much higher precision than pseudorange measurements are used in the positioning process of PPP. However, accurate estimation of integer ambiguity associated with carrier phase observations is a prerequisite. Then the integrity of the phase observable can be completed as a range measurement with high precision. With triple-frequency signals of global navigation satellite system (GNSS) available, combinations of original observations with longer wavelengths and lower noises are preferable. In this paper, we propose an improved ambiguity resolution (AR) method based on the traditional bootstrapping method to make it suitable for PPP and increase the precision of AR and positioning. Virtual ambiguities of extra-wide-lane, wide-lane, and modified narrow-lane combinations are resolved first for their relatively better characteristics. Furthermore, extra pseudoranges are included to decrease the ionospheric delay which cannot be ignored as it is in relative positioning. Then, original ambiguities on the three frequencies can be recovered from the combined ambiguities using three linear equations which are independent to each other. Finally, the performance of the improved method is tested with real GPS navigation data on three frequencies which is provided by the international GNSS service. Compared with the traditional bootstrapping method, it shows that the ambiguity residuals of the improved method have decreased which indicates more accurate estimation. The precision of the positioning results has increased accordingly.

Initial assessment of the COMPASS/BeiDou-3: new-generation navigation signals

The successful launch of five new-generation experimental satellites of the China’s BeiDou Navigation Satellite System, namely BeiDou I1-S, I2-S, M1-S, M2-S, and M3-S, marks a significant step in expanding BeiDou into a navigation system with global coverage. In addition to B1I (1561.098 MHz) and B3I (1269.520 MHz) signals, the new-generation BeiDou-3 experimental satellites are also capable of transmitting several new navigation signals in space, namely B1C at 1575.42 MHz, B2a at 1176.45 MHz, and B2b at 1207.14 MHz. For the first time, we present an initial characterization and performance assessment for these new-generation BeiDou-3 satellites and their signals. The L1/L2/L5 signals from GPS Block IIF satellites, E1/E5a/E5b signals from Galileo satellites, and B1I/B2I/B3I signals from BeiDou-2 satellites are also evaluated for comparison. The characteristics of the B1C, B1I, B2a, B2b, and B3I signals are evaluated in terms of observed carrier-to-noise density ratio, pseudorange multipath and noise, triple-frequency carrier-phase ionosphere-free and geometry-free combination, and double-differenced carrier-phase and code residuals. The results demonstrate that the observational quality of the new-generation BeiDou-3 signals is comparable to that of GPS L1/L2/L5 and Galileo E1/E5a/E5b signals. However, the analysis of code multipath shows that the elevation-dependent code biases, which have been previously identified to exist in the code observations of the BeiDou-2 satellites, seem to be not obvious for all the available signals of the new-generation BeiDou-3 satellites. This will significantly benefit precise applications that resolve wide-lane ambiguity based on Hatch–Melbourne–Wubbena linear combinations and other applications such as single-frequency precise point positioning (PPP) based on the ionosphere-free code–carrier combinations. Furthermore, with regard to the triple-frequency carrier-phase ionosphere-free and geometry-free combination, it is found that different from the BeiDou-2 and GPS Block IIF satellites, no apparent bias variations could be observed in all the new-generation BeiDou-3 experimental satellites, which shows a good consistency of the new-generation BeiDou-3 signals. The absence of such triple-frequency biases simplifies the potential processing of multi-frequency PPP using observations from the new-generation BeiDou-3 satellites. Finally, the precise relative positioning results indicate that the additional observations from the new-generation BeiDou-3 satellites can improve ambiguity resolution performance with respect to BeiDou-2 only positioning, which indicates that observations from the new-generation BeiDou-3 satellites can contribute to precise relative positioning.

Improved ambiguity resolution for URTK with dynamic atmosphere constraints

Raw observation processing method with prior knowledge of ionospheric delay could strengthen the ambiguity resolution (AR), but it does not make full use of the relatively longer wavelength of wide-lane (WL) observation. Furthermore, the accuracy of calculated atmospheric delays from the regional augmentation information has quite different in quality, while the atmospheric constraint used in the current methods is usually set to an empirical value. A proper constraint, which matches the accuracy of calculated atmospheric delays, can most effectively compensate the residual systematic biases caused by large inter-station distances. Therefore, the standard deviation of the residual atmospheric parameters should be fine-tuned. This paper presents an atmosphere-constrained AR method for undifferenced network RTK (URTK) rover, whose ambiguities are sequentially fixed according to their wavelengths. Furthermore, this research systematically analyzes the residual atmospheric error and finds that it mainly varies along the positional relationship between the rover and the chosen reference stations. More importantly, its ionospheric part of certain location will also be cyclically influenced every day. Therefore, the standard deviation of residual ionospheric error can be modeled by a daily repeated cosine or other functions with the help of data one day before, and applied by rovers as pseudo-observation. With the data collected at 29 stations from a continuously operating reference station network in Guangdong Province (GDCORS) in China, the efficiency of the proposed approach is confirmed by improving the success and error rates of AR for 10-20 % compared to that of the WL-L1-IF one, as well as making much better positioning accuracy.

Improving Ambiguity Resolution for Medium Baselines Using Combined GPS and BDS Dual/Triple-Frequency Observations.

The regional constellation of the BeiDou navigation satellite system (BDS) has been providing continuous positioning, navigation and timing services since 27 December 2012, covering China and the surrounding area. Real-time kinematic (RTK) positioning with combined BDS and GPS observations is feasible. Besides, all satellites of BDS can transmit triple-frequency signals. Using the advantages of multi-pseudorange and carrier observations from multi-systems and multi-frequencies is expected to be of much benefit for ambiguity resolution (AR). We propose an integrated AR strategy for medium baselines by using the combined GPS and BDS dual/triple-frequency observations. In the method, firstly the extra-wide-lane (EWL) ambiguities of triple-frequency system, i.e., BDS, are determined first. Then the dual-frequency WL ambiguities of BDS and GPS were resolved with the geometry-based model by using the BDS ambiguity-fixed EWL observations. After that, basic (i.e., L1/L2 or B1/B2) ambiguities of BDS and GPS are estimated together with the so-called ionosphere-constrained model, where the ambiguity-fixed WL observations are added to enhance the model strength. During both of the WL and basic AR, a partial ambiguity fixing (PAF) strategy is adopted to weaken the negative influence of new-rising or low-elevation satellites. Experiments were conducted and presented, in which the GPS/BDS dual/triple-frequency data were collected in Nanjing and Zhengzhou of China, with the baseline distance varying from about 28.6 to 51.9 km. The results indicate that, compared to the single triple-frequency BDS system, the combined system can significantly enhance the AR model strength, and thus improve AR performance for medium baselines with a 75.7% reduction of initialization time on average. Besides, more accurate and stable positioning results can also be derived by using the combined GPS/BDS system.

GNSS triple-frequency geometry-free and ionosphere-free track-to-track ambiguities

During the last few years, more and more GNSS satellites have become available sending signals on three or even more frequencies. Examples are the GPS Block IIF and the Galileo In-Orbit-Validation (IOV) satellites. Various investigations have been performed to make use of the increasing number of frequencies to find a compromise between eliminating different error sources and minimizing the noise level, including the investigations in the triple-frequency geometry-free (GF) and ionosphere-free (IF) linear combinations, which eliminate all the geometry-related errors and the first-order term of the ionospheric delays.In contrast to the double-difference GF and IF ambiguity resolution, the resolution of the so-called track-to-track GF and IF ambiguities between two tracks of a satellite observed by the same station only requires one receiver and one satellite. Most of the remaining errors like receiver and satellite delays (electronics, cables, etc.) are eliminated, if they are not changing rapidly in time, and the noise level is reduced theoretically by a factor of square root of two compared to double-differences. This paper presents first results concerning track-to-track ambiguity resolution using triple-frequency GF and IF linear combinations based on data from the Multi-GNSS Experiment (MGEX) from April 29 to May 9, 2012 and from December 23 to December 29, 2012. This includes triple-frequency phase and code observations with different combinations of receiver tracking modes. The results show that it is possible to resolve the combined track-to-track ambiguities of the best two triple-frequency GF and IF linear combinations for the Galileo frequency triplet E1, E5b and E5a with more than 99.6% of the fractional ambiguities for the best linear combination being located within ±0.03 cycles and more than 98.8% of the fractional ambiguities for the second best linear combination within ±0.2 cycles, while the fractional parts of the ambiguities for the GPS frequency triplet L1, L2 and L5 are more disturbed by errors as e.g. the uncalibrated Phase Center Offsets (PCOs) and Phase Center Variations (PCVs), that have not been considered. The best two GF and IF linear combinations between tracks are helpful to detect problems in data and receivers. Furthermore, resolving the track-to-track ambiguities is helpful to connect the single-receiver ambiguities on the normal equation level and to improve ambiguity resolution.

Combined GPS and BeiDou Instantaneous RTK Positioning

The Chinese BeiDou-2/COMPASS Navigation Satellite System has attained regional operational status and is expected to reach the same level of popularity as GPS once it has reached its full constellation. This contribution considers combined BeiDou+GPS relative code positioning and single- and multiple-frequency real-time kinematic (RTK) positioning. A combined system increases the redundancy for solving the unknown GNSS parameters and thus allows for more precise position estimates, improved reliability and robustness against failure of any of the systems. The performance is evaluated by ambiguity success rates and by comparing the estimated positions to very precise benchmark coordinates. We make use of the LAMBDA method for integer ambiguity resolution, in combination with the Fixed Failure-rate Ratio Test to validate the resolved ambiguities. The combined model will be shown to allow for improved ambiguity resolution performance and positioning robustness and accuracy over the BeiDou- and GPS-only solutions. Copyright © 2014 Institute of Navigation.

GPS/Galileo long baseline computation: method and performance analyses

Modernized GPS and Galileo will provide triple-frequency signals for civil use, generating a high interest to examine the improvement of positioning performance using the triple-frequency signals from both constellations over baselines up to hundreds or thousands of kilometers. This study adopts a generalized GPS/Galileo long-range approach to process the mutually compatible GPS and Galileo triple-frequency measurements for high-precision long baseline determination. The generalized approach has the flexibility to deal with GPS and Galileo constellations separately or jointly, and also the capability to handle dual or triple-frequency measurements. We compared the generalized long-range approach with the Bernese v5.0 software on two test baselines located in East Asia and obtained highly compatible computational results. Further, in order to assess possible improvement of GPS/Galileo long baseline determination compared with the current dual-frequency (L1/L2) GPS, we simulated GPS and Galileo measurements of the test baselines. It is shown that the current level of accuracy of daily baseline solutions can be improved by using the additional Galileo constellation. Both the additional constellation and the triple-frequency measurements can improve ambiguity resolution performance, but single-constellation triple-frequency ambiguity resolution is more resistant to the influences of code noise and multipath than dual-constellation dual-frequency ambiguity resolution. Therefore, in environments where large code noise or multipath is present, the use of triple-frequency measurements is the main factor for improving ambiguity resolution performance.

The affine constrained GNSS attitude model and its multivariate integer least-squares solution

A new global navigation satellite system (GNSS) carrier-phase attitude model and its solution are introduced in this contribution. This affine-constrained GNSS attitude model has the advantage that it avoids the computational complexity of the orthonormality-constrained GNSS attitude model, while it still has a significantly improved ambiguity resolution performance over its unconstrained counterpart. The functional and stochastic model is formulated in multivariate form, for one-, two- and three-dimensional antenna arrays, tracking GNSS signals on an arbitrary number of frequencies with two or more antennas. The stochastic model includes the capability to capture variations in the antenna-quality within the array. The multivariate integer least-squares solution of the model parameters is given and the model’s ambiguity success-rate is analysed by means of the ambiguity dilution of precision (ADOP). A general closed-form expression for the affine-constrained ADOP is derived, thus providing an easy-to-use and insightful rule-of-thumb for the ambiguity resolution capabilities of the affine constrained GNSS attitude model.

Multiplatform Instantaneous GNSS Ambiguity Resolution for Triple‐ and Quadruple‐Antenna Configurations with Constraints

Traditionally the relative positioning and attitude determination problem are treated as independent. In this contribution we will investigate the possibilities of using multiantenna (i.e., triple and quadruple) data, not only for attitude determination but also for relative positioning. The methods developed are rigorous and have the additional advantage that they improve ambiguity resolution on the unconstrained baseline(s) and the overall success rate of ambiguity resolution between a number of antennas.

Improving Ambiguity Resolution Rate with an Accurate Ionospheric Differential Correction

Ambiguity resolution is essential for precise range determination. As it is difficult to process, a good ionospheric model is essential to get unambiguous results or to reduce time to solve the ambiguities. In this paper, a developed model to determine the differential ionospheric error to sub-centimetre accuracy is described. As a function of elevation angle and TEC, the model is applicable at any location and only requires a single frequency receiver provided the TEC over the reference station is known. It has been evaluated using real GPS measurements at spaced stations in Glasgow (UK) and Stirling (UK), where the results showed good correlation. It was found that the variance ratio and reference variance of the ambiguity resolution rate and the quality of the differential positioning solution are improved. Significant improvements of more than 50% have also been found by correcting the differential ionospheric delay in the measurements for the estimated positions.

Vanishing points and three-dimensional lines from omni-directional video

This paper describes a system for structure from motion using vanishing points and three-dimensional lines extracted from omni-directional video sequences. To track lines, we use a novel dynamic programming approach to improve ambiguity resolution, and we use delayed states to aid in the initialization of landmarks. By reobserving vanishing points we get direct measurements of the robot's three-dimensional attitude that are independent of its position. Using vanishing points simplifies the representation since parallel lines share the same direction states. We show the performance of the system in various indoor and outdoor environments and include comparisons with independent two-dimensional reference maps for each experiment .

A performance comparison of single and dual frequency GPS ambiguity resolution strategies

The impact of observation selection, observation combination and model parameterization on GPS carrier phase ambiguity resolution and position accuracy under operational conditions is investigated. The impact of an ionospheric bias for a generic linear combination of L1 and L2 measurements is assessed and the results are used to clearly outline the desirable characteristics for improving ambiguity resolution versus positioning accuracy performance. Ambiguity resolution performance and position accuracy are shown for widelane (WL), L1-only, and ionospheric-free (IF) combinations. Several techniques for dealing with the ionospheric bias are also presented and compared, including stochastic ionospheric modelling. Multiple carrier phase combination solutions estimated in the same filter are also compared. The concept of an optimal processing strategy—in terms of both reliable ambiguity resolution and high accuracy positions—is presented. In total, eight strategies, which vary in observables and parameters, are tested on several datasets ranging from 13 km to 43 km.

Improving ambiguity resolution by applying ionosphere corrections from a permanent GPS array

Fast and high precision relative GPS positioning over distances up to 100 km is mainly limited by errors in the GPS signals due to propagation through the Earth’s ionosphere. With permanent GPS arrays, which are present in many countries nowadays, it becomes possible to correct a user’s GPS measurements to a certain extent for these ionospheric delays. A way to do so is to interpolate the ionospheric delays which have been estimated from the network of permanent stations. When these ‘interpolated corrections’ are applied to the user’s data, the ionospheric delays may be reduced, which may lead to an improved ambiguity resolution for his (long) baseline.