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The long convergence time required to achieve high-precision position solutions with integer ambiguity resolution-enabled precise point positioning (PPP-RTK) is driven by the presence of ionospheric delays. When precise real-time ionospheric information is available and properly applied, it can strengthen the underlying model and substantially reduce the time required to achieve centimeter-level accuracy. In this study, we present and analyze the real-time PPP-RTK user performance using ionospheric corrections from multi-scale regional networks during a day with medium ionospheric disturbance. It is the goal of this contribution to measure the impact the network dimension has on the ambiguity-resolved user position through the predicted ionospheric corrections. The user-specific undifferenced ionospheric corrections are computed at the network side, along with the satellite phase biases needed for single-receiver ambiguity resolution, using the best linear unbiased predictor. Such corrections necessitate the parameterization of an estimable user receiver code bias, on which emphasis is given in this study. To this end, we process GPS dual-frequency data from four four-station evenly distributed CORS networks in the United States with varying station spacings in order to evaluate if and to what extent the ionospheric corrections from multi-scale networks can improve the user convergence times. Based on a large number of samples, our experimental results showed that sub-10 cm horizontal accuracy can be achieved almost instantaneously in the ionosphere-weighted partially-ambiguity-fixed kinematic PPP-RTK solutions based on corrections from a network with 68 km spacing. Most of the solutions (90%) were shown to require less than 6.0 min, compared to the ionosphere-float PPP solutions that needed 68.5 min. In case of sparser networks with 115, 174 and 237 km spacing, 50% of the horizontal positioning errors are shown to become less than one decimeter after 1.5, 4.0 and 7.0 min, respectively, while 90% of them require 10.5, 16.5 and 20.0 min. We also numerically demonstrated that the user’s convergence times bear a linear relationship with the network density and get shorter as the density increases, for both full and partial ambiguity resolution.

A synthesis of two prevailing global navigation satellite system positioning technologies, namely the precise point positioning and the network-based real-time kinematic, results in the emergence of the PPP-RTK, enabling single-receiver users to achieve high positioning accuracy with reasonable timeliness through integer ambiguity resolution. The realization of PPP-RTK needs to accomplish two sequential tasks. The first task is to determine a class of corrections including, among others, the satellite phase biases (SPBs) at the network level. With these corrections, the second task, then, is to solve for the ambiguity-fixed, absolute position at the user level. In this contribution, we revisit three variants (geometry-free, geometry-fixed and geometry-plus-satellite-clock-fixed) of the undifferenced and uncombined PPP-RTK network model and then point out their implications for practical use. We also carry out a case study using multi-day, dual-frequency global positioning system data from the crustal movement observation network of China stations, aiming to figure out what are the most appropriate linear combinations of the SPBs to be transmitted to the users from the viewpoint of decorrelation, and to assess the static and kinematic positioning performance.

We present practical attacks on IAR-CTR and IAR-CFB, AE modes proposed by Hwang and Gope to offer both privacy and authenticity in a single pass. Then we use the idea of iCOFB, which is originally defined with a tweakable random function, to fix the IAR-CTR mode using the tweakable block cipher, and propose the TIAR-CTR mode, which is a one-pass AEAD mode with provable security. However, the same fixed method does not work for the IAR-CFB mode.

Precise point positioning (PPP) has been a competitive global navigation satellite system (GNSS) technique for time and frequency transfer. However, the classical PPP is usually based on the ionosphere-free combination of dual-frequency observations, which has limited flexibility in the multi-frequency scenario. More importantly, the unknown integer ambiguities are not restored to the integer nature, making the advantage of high-precision carrier phase observations underutilized. In this contribution, using the undifferenced and uncombined (UDUC) observations, we derive the time and frequency transfer model suitable for multi-constellation and multi-frequency scenarios. Notably, in short- and medium-baseline time and frequency transfer, the ionosphere-fixed and ionosphere-weighted UDUC models are derived, respectively, by making full use of the single-differenced (SD) ionospheric constraints. The proposed model can be applied to short-, medium- and long-baseline time and frequency transfer. The ambiguities are solved in a double-differenced (DD) form and can thus be restored to integers. To verify the feasibility of the model, GPS data from several time laboratories were collected, and the performance of the time and frequency transfer were analyzed with different baseline lengths. The results showed that the ionosphere-fixed and ionosphere-weighted UDUC models with integer ambiguity resolution could improve the frequency stability by 25–60% and 9–30% at an averaging time of several tens of seconds to 1 day for short- and medium-baseline, respectively. Concerning the long-baseline, the UDUC model is 10–25% more stable than PPP for averaging time below a few thousands second and over 1 day.

The paper addresses the problem of attitude determination using Global Navigation Satellite System (GNSS) measurements from multiple antennas mounted on a navigation platform. To achieve attitude determination by GNSS with typical accuracy down to tenths of a degree for one-meter baselines, GNSS phase measurements are employed. A key challenge with phase measurements is the presence of unknown integer ambiguities. Consequently, the attitude determination problem traditionally reduces to a nonlinear, non-convex optimization problem with integer constraints. No closed-form solution to this problem is known, and its real-time calculation is computationally intensive. Given an a priori initial attitude approximation, we propose a new algorithm for attitude tracking based on the reduction of the nonlinear orthogonality-constrained attitude estimation problem to a linear integer least squares problem, for which numerical methods are well known and computationally much less demanding. Additionally, a simple a priori model for GNSS measurement error variance is introduced, grounded on the geometry of satellite signal propagation through vacuum and the Earth’s atmosphere, providing a clear physical interpretation. Applying the algorithm to a real dataset collected from a quasi-static multi-antenna, multi-GNSS system with sub-meter baselines, we obtain promising results.

Single-station PPP-RTK is a special case of PPP-RTK in that corrections are computed, instead of a network, by only one single GNSS receiver. The present contribution aims to develop a framework to generate multi-epoch, single-station corrections, thereby providing PPP-RTK users the capability to time-predict corrections that are subject to time delay or latency. By presenting analytical expressions of the user ambiguity variance matrix, we address how the ambiguity resolution performance is driven by the correction latency and therefore by the uncertainty involved in the time-prediction of single-station PPP-RTK corrections. Supported by numerical results, our analytical study shows that the number of satellites and number of frequencies work in tandem to enable one to increase the correction latency, yet ensuring successful single-receiver ambiguity resolution.

PPP-RTK extends the precise point positioning (PPP) concept by incorporating the idea of integer ambiguity resolution underlying the real-time kinematic (RTK) technique, making rapid initialization and high accuracy attainable with a standalone receiver. While PPP-RTK has been well achieved by using global navigation satellite system code division multiple access observables, GLONASS PPP-RTK is nonetheless challenging due to the nature of frequency division multiple access (FDMA) observables. In this work, we present a GLONASS PPP-RTK concept that takes advantage of the integer-estimable FDMA (IE-FDMA) model recently proposed in Teunissen (in GPS Solut 23(4):1–19, 2019. https://doi.org/10.1007/s10291-019-0889-0 ) to guarantee rigorous integer ambiguity resolution and simultaneously takes care of the presence of the inter-frequency biases (IFBs) in homogeneous and heterogeneous network configurations. When conducting GLONASS PPP-RTK based on a network of homogeneous receivers, code and phase observation equations are used to construct the IE-FDMA model, in which the IFBs are implicitly eliminated through reparameterization. For a network consisting of heterogeneous receivers, we exclude the code observables and develop a phase-only IE-FDMA model instead, thereby circumventing the adverse effects of IFBs. For verification purposes, we collect a set of five-day global positioning system (GPS) and GLONASS data from two regional networks: one equipped with homogeneous receivers and another with heterogeneous receivers. The results show that the GLONASS-specific network corrections, including satellite clocks, satellite phase biases, and ionospheric delays estimated by the two networks, are as precise as those of their GPS-specific counterparts. Via satellite clock and phase bias corrections, we succeed in fixing both GPS and GLONASS ambiguities, shortening the convergence time to 5 (12) min, compared to 11 (18) min of ambiguity-float positioning in the case of a homogeneous (heterogeneous) network with a data sampling rate of 30 s. For ambiguity-fixed positioning, the convergence time defined in this work also indicates the time to first fix since the positioning error converges to the centimeter level once successful integer ambiguity resolution is achieved. Adding ionospheric corrections further speeds up the initialization in the two networks, with the convergence time being reduced to 0.5 (3) min. Compared with GPS-only positioning, the integration of GPS and GLONASS yields an improvement of 8–34% in accuracy and leads to a reduction of 25–50% in convergence.

Despite the benefits of integer ambiguity resolution (IAR) in precise point positioning (PPP), observation outages and harsh signal environments still impact float ambiguity estimation in kinematic surveying, consequently resulting in ambiguity-fixed failure. The inertial navigation system (INS) is an autonomous and spontaneous positioning one, which could provide continuous and superior positioning accuracy over short time. Thus, the INS attains more accurate position than code solution. Moreover, the tight integration of INS and PPP is capable of continuous operation where there are less than four satellites available. These advantages can improve float ambiguity estimation and assist in re-initializing the interrupted ambiguity and PPP solution. Based on the good quality of float ambiguity, the ambiguity dilution precision (ADOP) and the size of integer ambiguity search space are reduced, and then, the IAR-PPP is improved. In this work, the INS aiding effect on IAR-PPP was revealed by the sufficient theoretical analysis and performance assessment. A ring laser gyroscope-based navigation-grade IMU and a fiber optic gyroscope-based tactical-grade IMU were utilized to conduct experiments in an open-sky environment and urban area. The assessment adopted the following aspects of ADOP, bootstrapping success rate, time to fix and position errors. It is found that IAR-PPP with INS aiding achieves an enhanced performance during GPS outage when INS could deliver a superior accurate position. For the navigation- and tactical-grade IMU, the INS-aided ambiguity re-fixing performance can be classified as three levels: significant improvement for the outage duration less than 10 s, moderate improvement for the outage duration from 10 to 60 s and a little or zero improvement for the outage duration longer than 60 s. From the viewpoint of the INS-predicted position domain, an accuracy better than 0.1 m and 1.0 m is required for the significant and moderate improvement, while one can only achieve a little or zero improvement if the position error is larger than 1.0 m. Besides, we also performed the INS-aided IAR-PPP in real urban environment. For the urban environments, the span of clean data is often shorter than 30 min due to intermittent signal interruptions; thus, ambiguity re-fixing for PPP always fails. INS-aided information could bridge the data gaps and achieve fast ambiguity re-fixing. In summary, INS aiding information is capable of improving IAR-PPP performance significantly over a short GPS outage.

Extra-wide-lane real-time kinematic (ERTK) is a technique that makes full use of extra-wide-lane (EWL) observations to realize instantaneous precise positioning. Beyond the previous study by using triple-frequency signals, the hexa- and penta-frequency signals, referred to as hyper-frequency signals in this study, are currently available for Beidou-3 and Galileo systems, respectively, which will be definitely beneficial to ERTK. In this study, the advantages and performance of hyper-frequency ERTK (HERTK) are profoundly addressed. The mathematical model of generalized HERTK is deduced with canonical formulae to show how model parameters profit from additional signals and high-precision EWL/WL observations. Specifically, the optimal linear combinations of hyper-frequency signals are determined in terms of ionosphere-weighted and ionosphere-float models. The precision gains of both position and ambiguity parameters are numerically demonstrated for single- and multi-epoch, accompanied by a comprehensible explanation of the hyper-frequency enhancement mechanism. The performance of HERTK is evaluated with three long baselines from 248.4 to 511.0 km. The results show that the HERTK achieves instantaneous decimeter-level solutions without the need for complicated narrow-lane (NL) ambiguity resolution (AR). Furthermore, centimeter HERTK can be realized by only accumulating NL phase data over approximately 20 epochs, which essentially leverages the more precise between-epoch information to smooth the noisy solutions. Besides the smoothed positions, the precision of NL ambiguity is also significantly improved, thus enabling rapid and reliable NL AR for long baselines. Higher accuracy of 1–2 cm solutions is achieved within 10–30 epochs.

PPP-RTK corrections, aiding GNSS users to achieve single-receiver integer ambiguity-resolved parameter solutions, are often estimated in a recursive manner by a provider. Such recursive, multi-epoch, estimation of the corrections relies on a set of S -basis parameters that are chosen by the provider so as to make the underlying measurement setup solvable. As a consequence, the provider can only estimate estimable forms of the corrections rather than the original corrections themselves. It is the goal of the present contribution to address the consequence of the corrections’ dependency on the provider’s S -basis, showcasing the characteristics of their multi-epoch solutions, thereby identifying potential pitfalls which the PPP-RTK user should avoid when evaluating such solutions. To this end, we develop a simulation platform that allows one to have full control over the properties of PPP-RTK corrections and demonstrate various misleading temporal behaviors that exist when interpreting the multi-epoch solutions of their estimable forms. The roles of the correction latency and time correlation in the multi-epoch user positioning performance are quantified, while the deviation of the user-reported positioning precision description from its user-actual counterpart is measured under a misspecified user stochastic model.