## 5 Conclusions  
A dual/triple-frequency multi-constellation PPP-RTK method with atmospheric augmentation is proposed, to shorten convergence time and improve positioning accuracy, especially for kinematic positioning. The performance of PPP-Float, PPP-AR, PPP-RTK, and short-baseline RTK was assessed in a set of experiments conducted in Wuhan. The positioning results indicate that dual-frequency PPP-RTK achieved a performance similar to short-baseline RTK, demonstrating that PPP-RTK has the potential for lane-level navigation.  
For static positioning experiments, the accuracies of ionospheric and tropospheric interpolation products were within 2-4 cm and better than 2 cm, respectively. Both multi-system dual-frequency PPP-AR and PPP-RTK achieved centimeter-level positioning accuracy, with horizontal RMS of 1.43 cm and 1.39 cm and vertical RMS of 2.97 cm and 2.62 cm, respectively. Furthermore, the convergence time of PPP-RTK was reduced from 166 s in PPP-AR to just 1 s, achieving instantaneous ambiguity resolution in open areas. The addition of atmospheric products provided a more significant improvement in terms of accuracy and convergence time for BDS only.  
For vehicle kinematic experiments, the positioning accuracy of dual-frequency PPP-RTK in the north, east, and up directions was 2.6 cm, 2.6 cm, and 7.6 cm, respectively, and the success rate of ambiguity resolution was 97.96%. Moreover, PPP-RTK demonstrated excellent positioning availability of over 99%, and it was able to achieve fast ambiguity resolution within several seconds after outage and blocking. Compared with PPP-Float and PPP-AR, the improvements in terms of positioning accuracy were 75.9%, 75.9%, and 75.7%, and 77.4%, 77.6%, and 74.4% in the three directions, respectively. In addition, the performances of triple-frequency PPP-AR and PPP-RTK were also evaluated. The results showed that compared with the dual-frequency designs, the positioning accuracy of PPP-AR underwent an 18.9% improvement, while the improvement for PPP-RTK was not significant. Triple-frequency PPP-RTK can further enhance the robustness and accuracy of positioning. In summary, PPP-RTK also had better performance and shorter convergence time in terms of kinematic positioning.  
Conceptualization, Z.L.; methodology, Z.-FS. and D.-W.G.; validation, Z.-Y.Q. and S.-Y.X.; data curation, S.-Y.X. and X.-T.L.; writing--original draft preparation, Z.-FS. and D.-W.G.; writing--review and editing, S.-Y.X. and X.-T.L.; project administration, Z.L.; funding acquisition, Z.L. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The raw data supporting the conclusions of this article will be made available by the authors on request.  
Author [PERSON] was employed by the company Kepler Satellite Orbit Science & Technology (Wuhan) Co., Ltd. The remaining authors declare that the research was conducted  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Solution** & **Horizontal Accuracy** & **Vertical Accuracy** \\ \hline \multirow{2}{*}{Dual-frequency} & \textless{}5 cm & 92.83\% & \textless{}10 cm & 96.46\% \\  & \textless{}10 cm & 99.69\% & \textless{}20 cm & 99.61\% \\ \multirow{2}{*}{TRIP-frequency} & \textless{}5 cm & 96.36\% & \textless{}10 cm & 98.83\% \\  & \textless{}10 cm & 99.89\% & \textless{}20 cm & 99.93\% \\ \hline \hline \end{tabular}
\end{table}
Table 5: Dual-frequency and triple-frequency PPP-RTK positioning availability.  
in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.