Kalman or Kalman-related filtering methods are routinely applied in precise point positioning (PPP). However, in robot simultaneous localization and mapping (SLAM) systems, factor graph optimization (FGO) has proven advantages over filtering methods in recent years, e.g. reducing the linearization error and support of plug-and-play features for multiple sensor fusion. Therefore, it would be interesting to apply the FGO to PPP. In addition, it will also facilitate the tight integration of PPP with Visual/LiDAR SLAM. In this work, PPP is solved under the FGO framework. A factor graph for PPP has been constructed. Results from 268 IGS-MGEX stations show that the FGO method can achieve a similar performance to that of Kalman filtering. First, the positioning accuracy in the convergence period can be improved for PPP based on FGO because it optimizes the entire state variables based on all the available observations. For applications that do not require real-time processing, the observation after the current state, e.g. future observations, can also be used to enhance the current state estimation. Second, the accuracy of static PPP is almost the same for the two methods with millimeter-accuracy for horizontal directions and centimeter-accuracy for vertical directions. Third, the kinematic PPP for both methods can achieve centimeter-level accuracy in horizontal directions and decimeter-level accuracy in vertical directions. Although the performance is comparable, it is noted that the computational efficiency of the FGO method is still a problem. For each epoch, the average elapsed time for Kalman filtering is 132 microseconds, while that of FGO method is 9664 microseconds. The elapsed time of the FGO method can be further improved if the fix-window optimization technique is applied, which will be investigated in the future.

Factor graph optimization has been widely used for state estimation in robotic SLAM community. Extensive algorithms have been proposed for camera/LiDAR/INS based SLAM. However, GNSS positioning based on factor graph optimization is limited, which prevents the introduction of high precise GNSS to robotic SLAM community. The current implementations are focused on pseudorange or RTK based positioning. PPP with ambiguity resolution (AR) is the state-of-the-art positioning technique for the past decade. Therefore, the PPP AR based on factor graph optimization is proposed, in which the pseudorange and carrier phases factors are constructed from the error equations of raw observations, while the ambiguity resolution factor is built from the ambiguity resolution. Results from 80 MGEX stations show that the average accuracy of static PPP is improved from 1.25, 0.61 and 2.29 cm to 0.81, 0.5 and 2.1 cm, corresponding to improvements of 35.1%, 18.7% and 8.7% in east, north, and up directions, respectively. As for kinematic PPP, the average accuracy is improved from 2.62, 2.21 and 5.8 cm to 1.64, 1.74 and 5.37 cm, corresponding to improvements of 37.5%, 21.6% and 7.4% in east, north, and up directions, respectively. The kinematic PPP was also verified with real-world data collected from a moving vehicle. After the first ambiguity fixing, the accuracy of PPP is improved from 3.7, 2.1 and 10.1 cm to 1.6, 2.0 and 9.0 cm for east, north and up component, respectively, corresponding to improvements of 32%, 5% and 11%. The above results confirm the efficiency of the proposed algorithm.

The European Galileo High Accuracy Service (HAS) started to provide freely and openly accessible real-time precise satellite orbit, clock and code bias products to global users on January 24, 2023. Combined with the already running BeiDou PPP-B2b service, the launch of a variety of satellite-based PPP services provided more choices to users. However, different satellite-based PPP services provide services for different GNSS systems, which hamper users to make full use of multi-GNSS systems. Therefore, the combination of different satellite-based products can further improve the availability of corrections, usage of multi-GNSS observation data and positioning performance. This paper proposes to combine HAS and PPP-B2b products by the Helmert coordinate transformation method. To validate the algorithm, HAS and PPP-B2b products of day of year (DOY) 308–317 in 2023 were collected in Zhengzhou, China. First, they are evaluated in terms of correction availability, orbit and clock quality. Then the HAS and PPP-B2b products are combined by the Helmert coordinate transformation method. Two combination strategies are proposed. The first strategy is integrating BDS satellites of PPP-B2b products into HAS products (denoted as C_H), while the second strategy is integrating Galileo satellites of HAS products into PPP-B2b products (denoted as C_B). Finally, the combined strategies are evaluated with static and kinematic data. Based on the static data of 18 Multi-GNSS Experiment (MGEX) stations in China and its surrounding areas, the results show that, when separately using HAS and PPP-B2b products for PPP, the average accuracy in horizontal and vertical directions are (2.4, 2.7 cm) and (2.4, 2.0 cm), respectively. The average accuracy of C_H strategy is 2.1 and 1.7 cm, which was improved by 31.3% compared with separately using the products. Similarly, the average accuracy of C_B strategy is 2.1 and 1.9 cm, corresponding to improvements of 29.6%. When comparing the two combined strategies, it is noted that the C_B strategy converges faster. Based on the data from vehicle platform, the results show that the horizontal and vertical accuracy of the C_B strategy is 8.6 and 15.7 cm respectively. The accuracy improvement of C_B is better than that of C_H strategy, and the average accuracy is 68.4% better than that of separately using the products. The above results show that the two combined strategies can improve positioning accuracy. In addition, the improvements in accuracy and convergence speed of C_B strategy are more significant. Users are advised to use C_B strategy for the combination of HAS and PPP-B2b products, which will greatly expand the application of HAS and PPP-B2b services.

With the modernization of Global Navigation Satellite System (GNSS), triple- or multi-frequency signals have become available from more and more GNSS satellites. The additional signals are expected to enhance the performance of precise point positioning (PPP) with ambiguity resolution (AR). To deal with the additional signals, we propose a unified modeling strategy for multi-frequency PPP AR based on raw uncombined observations. Based on the unified model, the fractional cycle biases (FCBs) generated from multi-frequency observations can be flexibly used, such as for dual- or triple- frequency PPP AR. Its efficiency is verified with Galileo and BeiDou triple-frequency observations collected from globally distributed MGEX stations. The estimated FCB are assessed with respect to residual distributions and standard deviations. The obtained results indicate good consistency between the input float ambiguities and the generated FCBs. To assess the performance of the triple-frequency PPP AR, 11 days of MGEX data are processed in three-hour sessions. The positional biases in the ambiguity-fixed solutions are significantly reduced compared with the float solutions. The improvements are 49.2%, 38.3%, and 29.6%, respectively, in east/north/up components for positioning with BDS, while the corresponding improvements are 60.0%, 29.0%, and 21.1% for positioning with Galileo. These results confirm the efficiency of the proposed approach, and that the triple-frequency PPP AR can bring an obvious benefit to the ambiguity-float PPP solution.