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Global navigation satellite systems (GNSSs) have been widely used to provide real-time and high-precision attitude information for land vehicles, ships and aircraft over the past few decades. With the joint use of emerging multi-GNSS common-clock receivers and the single-differenced (SD) model, the accuracy of pitch and roll can be significantly improved to the same level as that of yaw. However, the prerequisite is that the frequency-dependent phase line biases (LBs) in multiple GNSS systems and frequencies are accurately and rapidly estimated. In this contribution, we intend to solve this problem by using a multi-dimensional particle filter (PF)-based approach. We first investigate the relationship between the ratio value and the multi-dimensional phase LBs. Results have revealed that the ratio value can be used to judge the quality of multi-dimensional phase LBs and represent the likelihood function of the observations. We then present the procedure of multi-dimensional PF-based phase LBs estimation for SD ambiguity resolution and attitude determination. An improved strategy is also proposed to reduce the computation time. Finally, we take the two-dimensional case as a representative example to evaluate the performance of the proposed method in aspects of the convergence and accuracy of phase LB estimates, the attitude determination accuracy, and the computation time. Experimental results from two static datasets have demonstrated that the two-dimensional phase LBs basically rapid converge within 20 epochs. Moreover, compared with the double-differenced method, the proposed multi-dimensional PF-based SD method could provide comparable yaw accuracy and much better pitch accuracy. The pitch accuracy is improved to the same level as yaw by approximately 42.9%–50.0%. With regard to the computation time, it is found that with the proposed modification strategy, the single-epoch computation times are significantly reduced by approximately 90.7%–93.5%, and they are mostly within 0.05 s for most of the epochs on a personal computer.

Five BDS-3 satellites were launched successfully in 2015 and 2016, including two new-generation BDS-3 IGSO satellites I1-S and I2-S, and three new-generation BDS-3 Medium Earth Orbit (MEO) satellites M1-S, M2-S and M3-S. The BDS-3 satellites are able to transmit several new navigation frequencies including B1C at 1575.42 MHz, B2a at 1176.45 MHz, and B2b at 1207.14 MHz, in addition to B1 (1561.098 MHz) and B3 (1268.520 MHz) frequencies transmitted by BDS-2 satellites. Since the satellite-induced and elevation-dependent code biases were observed for the 14 older BDS-2 satellites (C01-C14), a comprehensive analysis and characterization of the code observations for the BDS-3 satellites were conducted based on observation data collected at 14 stations including 13 of the International GNSS Monitoring and Assessment System (IGMAS) and one station named SGG0 operated at the School of Geodesy and Geomatics (SGG) of Wuhan University. The Multipath (MP) combination, Melbourne-Wubbena (MW) combination and wide-lane uncalibrated phase delays (UPDs) are calculated to verify the existence and magnitude of the satellite-induced code bias for the BDS-3 satellites. The elevation-dependent polynomial models of MP combination for BDS-3 satellites were also established in order to analyze the characteristic of satellite-induced bias observed at different receiver types or different frequencies. For comparison, the MP combination of a newly launched BDS-2 satellite I6 was also calculated. The results indicate that the satellite-induced code bias is negligible for the BDS-3 satellites while newly launched BDS-2 satellite I6 has similar elevation-dependent code bias as the 14 older BDS-2 satellites.

To ensure the compatibility and interoperability with modernized GPS, Galileo satellites are capable of broadcasting navigation signals on carrier phase frequencies that overlap with GPS, i.e., GPS/Galileo L1-E1/L5-E5a. Moreover, the GPS/Galileo L2-E5b signals have different frequencies with wavelength differences smaller than 4.2 mm. Such overlapping and narrowly spaced signals between GPS and Galileo bring the opportunity to use the tightly combined double-differenced (DD) model for precise real-time kinematic (RTK) positioning, resulting in improved performance of ambiguity resolution and positioning with respect to the classical standard or loosely combined DD model. In this paper, we focus on the model and performance assessment of tightly combined GPS/Galileo L1-E1/L2-E5b/L5-E5a RTK for short and long baselines. We first investigate the tightly combined GPS/Galileo DD observational model for both short and long baselines with simultaneously considering the GPS/Galileo overlapping and non-overlapping frequencies. Particularly, we introduce a reparameterization approach to solve the rank deficiency that caused by the correlation between the DISB parameters and the DD ionospheric parameters for both overlapping and non-overlapping frequencies. Then we present performance assessment for the tightly combined GPS/Galileo RTK model with real-time estimation of the differential inter-system bias (DISB) parameters for short and long baselines in terms of ratio value, ambiguity dilution of precision (ADOP), ambiguity conditional number, decorrelation number, search count, empirical success rate, time-to-first-fix (TTFF), and positioning accuracy. Results from both static and kinematic experiments demonstrated that compared to the loosely combined model, the tightly combined model can deliver improved performance of ambiguity resolution and precise positioning with different satellite visibility. For the car-driven short baseline experiment with 10° elevation cut-off angle, the tightly combined model can not only significantly increase the ratio value by approximately 27.5% (from 16.0 to 20.4), but also reduce the ambiguity ADOP, the conditional number, and the search count in LAMBDA by approximately 22.2% (from 0.027 to 0.021 cycles), 14.9% (from 199.2 to 169.6), and 25.4% (from 150.1 to 112.0), respectively. Comparable decorrelation number, empirical success rate, and positioning accuracy are also obtained. For the car-driven long baseline experiment, it is also observed that the ambiguity resolution performance in terms of the ratio value, the decorrelation number, the condition number, and the search count are significantly improved by approximately 18.5% (from 2.7 to 3.2), 22.0% (from 0.186 to 0.227), 55.9% (from 937.6 to 413.7), and 10.3% (from 43.8 to 39.3), respectively. Moreover, comparable ADOP, empirical success rate, and positioning accuracy are obtained as well. Additionally, the TTFF can be reduced (from 54.1 to 51.8 epochs with 10° elevation cut-off angle) as well from the results of static experiments.

Rapid and reliable ambiguity resolution is the key to high-precision global navigation satellite system-based applications. To improve the efficiency and reliability of ambiguity resolution, one effective method is equipping the reference station with an antenna array of known geometry instead of a single antenna. In this contribution, the benefits of reference antenna array-aided real-time kinematic (RTK) positioning are investigated. The mathematical relations between the number of reference antennas and the float baseline and ambiguity solutions are explored, and the closed-form formula of ambiguity dilution of precision (ADOP) is further presented. It is demonstrated that the maximum decrease in ADOP or the improvement in precision of float solutions is approximately 29.29% with the increase in the number of reference antennas. Then, we analyze the impact of errors (noises or biases) on the float baseline and ambiguity solutions. Finally, the performance of the array-aided RTK is evaluated with raw BDS and GPS observations in terms of the ambiguity resolution success rate, false alarm, wrong detection alarm, and the time-to-first-fix (TTFF). It is demonstrated that the array-aided RTK can deliver improved ambiguity success rates with respect to the standard one-reference-antenna RTK, especially when only single-frequency, single-system observations from fewer satellites are available. Moreover, the array-aided RTK can provide much more robust ambiguity resolution in the presence of biases. And the performance of suppressing false alarm and wrong detection alarm is improved as well. Additionally, the TTFF with single-frequency observations is also significantly shortened. One order of magnitude improvements in TTFF are achieved for most of the cases from the standard one-reference-antenna solutions to the three-reference-antenna solutions. The results confirm that the array-aided RTK ensures more reliable and efficient ambiguity resolution.

Relative positioning with a moving base is required in many vehicle-to-vehicle applications. Due to unavoidable latency, conventional synchronous real-time kinematic (SRTK) positioning, which refers a fixed base, cannot fulfill the requirements of high-speed safety–critical situations since the movement of the base during the latency must be considered. We present an asynchronous RTK (ARTK)/time-differenced carrier phase (TDCP) integration method. We first introduce the ARTK method to directly obtain the asynchronous baseline between moving base and rover at their respective observation epochs. The position increments of the moving base during the latency are then calculated by the TDCP technique. Finally, the two positioning results are combined to achieve low-latency, high-rate, and high-precision relative positioning. With the proposed method, the size of transmitted data packets can be significantly reduced, which is beneficial for real-time implementation. It is also noted that the proposed method can be easily merged with global navigation satellite system/inertial navigation system (INS) integration to further improve the output rate of relative position. The performance is evaluated by a field test using two moving vehicles with one being the moving base and the other the rover. Results show that the baseline error is less than 1 cm compared to reliably post-processed SRTK results when latencies of raw observations remain below 1 s. Even when the latency is 15 s, centimeter-level accuracy can still be guaranteed. With ARTK/TDCP/INS integration, decimeter-to-centimeter-level higher-rate (> 100 Hz) relative positioning results can be obtained.

This paper first investigates the influencing factors of between-receiver Differential Inter-System Bias (DISB) between overlapping frequencies of the Global Positioning System (GPS), Galileo and the Quasi-Zenith Satellite System (QZSS). It was found that the receiver reboot and the type of observations may have an impact on DISBs. The impact of receiver firmware upgrades and the activation of anti-multipath filters are also investigated and some new results are presented. Then a performance evaluation is presented of tightly combined relative positioning for a short baseline with GPS/Galileo/QZSS L1-E1-L1/L5-E5a-L5 observations with the current constellations, in which the recently launched Galileo and QZSS satellites will also be included. It is demonstrated that when DISBs are a priori calibrated and corrected, the tightly combined model can deliver a much higher empirical ambiguity resolution success rate and positioning accuracy with respect to the classical loosely combined model, especially under environments where the observed satellites for each system are limited and only single-frequency observations are available. The ambiguity dilution of precision, bootstrapping success rate, and ratio values are analysed to illustrate the benefits of the tightly combined model as well.

In precise relative positioning applications, an effective approach to improve the interoperability of GNSS systems is the tightly combining or inter-system double-differencing of observations from the common frequencies that are shared by different constellations. As the BeiDou satellites are currently transmitting a B2 signal at 1207.14 MHz that is identical to the Galileo E5b signal, the inter-system double-differenced observations can also be created between observations from both systems at that particular frequency. In this paper, we will focus on the instantaneous ambiguity resolution performance analysis of tightly combining BeiDou B2 and Galileo E5b observations. The size and stability of phase and code Differential Inter-System Biases (DISBs) between BeiDou B2 and Galileo E5b signals are first investigated, in which the new BeiDou and Galileo satellites launched recently will also be included. Then, first results of the Tightly Combined Model (TCM) with a priori corrected DISBs (TCM_C) are evaluated in comparison to the Loosely Combined Model (LCM) and tightly combined model with unknown DISBs (TCM_F) in an instantaneous approach. It is demonstrated that the instantaneous integer ambiguity resolution performance can be improved using the TCM_C with respect to LCM and TCM_F.

A prerequisite for a Global Positioning System (GPS) attitude determination is to calculate baselines between antennae with accuracy at the millimetre level simultaneously. However, in order to have a low cost attitude determination system, a set of Commercial-Off-The-Shelf (COTS) receivers with separate clocks are used. In this case, if the receiver clocks are not precisely synchronized, the baseline vector between antennae will be calculated from the GPS signals received at different times. This can be a significant error source for high-kinematic applications. In this paper, two equivalent and effective approaches are developed to compensate this significant bias for baseline estimation and attitude determination. Test results using real airborne GPS data demonstrate that the receiver time misalignment between the two receivers can result in a 5 cm baseline offset for an aircraft with a 50 m/s velocity; the corresponding attitude errors can reach about 0·50° in yaw and 0·10° in pitch respectively for the attitude determination system with a baseline length of 3·79 m. With the proposed methods, these errors can be effectively eliminated.

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.

In ground-based pseudolite positioning system, one of critical problems is to calibrate the antenna phase center (APC) of the pseudolites in the positioning system frame. In this paper, a convenient method is proposed to resolve the problem. At first, the pseudolite antenna acts as a GPS receiver antenna. Therefore, the APC in the antenna frame can be determined in field. In the positioning system frame, the reference point (ARP) and the attitude of antenna can be determined with conventional ground method. Finally, APC can be transformed to the positioning system frame. Positioning results in a calibrated system demonstrate that within a specific indoor experiment an RMS error of less than 1.0 cm was obtained for the horizontal coordinates and 1.1 cm for the vertical one.