## 5 Conclusions  
Accurate measurement of the track axis coordinates is fundamental for evaluating existing geometric layouts on railway lines. The authors' mobile GNSS measurement method--augmented by an IMU--enables rapid reconstruction of track geometry within the national spatial reference framework. However, to maintain effectiveness under high GNSS uncertainty, a multi-receiver configuration is essential.  
This article describes an algorithm used to assess the geometric configuration of a multi-receiver GNSS antenna array, which is designed to reduce coordinate measurement uncertainty. The proposed method has proven highly effective at limiting uncertainty, thereby enhancing the reliability of the measurement system--even under challenging terrain conditions.  
Overall, the application of the proposed algorithm markedly increased the length of the surveyed railway section, yielding results within the specified accuracy threshold--particularly for stringent requirements on the order of 1 cm.  
In summary, the algorithm proposed for monitoring the geometric configuration of a multi-receiver antenna array effectively reduces measurement uncertainty and enhances track axis determination-- even in environments with dense urban infrastructure, forest cover, deep cuttings, and other, similarly challenging conditions.  
Figure 16: Selected results of measurement—case IV.  
Within the InnoSatTrack project, a range of algorithms was developed to accurately map the geometric layout of measured railway lines. The next phase of the authors' comprehensive work involves extending these methods by integrating the identified geometric layout with a track database (covering both lines and stations) along with corresponding kilometrage data. Track-selective localization is crucial, not only for effective and fully functional railway infrastructure inventory but also for modern railway traffic control systems.  
In summary, since the project's completion and product delivery, InnoSatTrack has been gradually implemented nationwide by PKP PLK S.A.  
Conceptualization: [PERSON]. ([PERSON]), W.K., A.W., P.C. and P.S.D.; Data curation: [PERSON]. ([PERSON]) and P.S.D.; Formal analysis: [PERSON]. ([PERSON]); Funding acquisition: W.K., A.W. and P.C.; Investigation: [PERSON]. ([PERSON]), W.K., A.W., P.C., P.S.D., R.L., K.K., S.J., M.M., [PERSON] ([PERSON]) and S.G.; Methodology: [PERSON]. ([PERSON]), W.K., A.W. and P.S.D.; Software: [PERSON]. ([PERSON]); Supervision: W.K. and A.W. Project administration: A.W.; Resources: A.W. and P.S.D.; Validation: [PERSON]. ([PERSON]) and [PERSON]; Visualization: [PERSON]. ([PERSON]), [PERSON] and P.S.D.; Writing--original draft: [PERSON]. ([PERSON]) and [PERSON]; Writing--review and editing: [PERSON] and P.S.D. All authors have read and agreed to the published version of the manuscript.  
This research was co-financed by the National Centre for Research and Development under the Operational Programme Smart Growth (POIR) and by PKP Polskie Linie Kolejowe S.A. (Project No. POIR.04.01.01-00-0017/17-00). The project \"Developing an Innovative Method to Determine the Precise Rail Vehicle Trajectory\" ran from 28 June 2018 to 31 May 2021 as part of the joint BRIK initiative (\"Research and Development in Railway Infrastructure\"). BRIK was co-managed by the National Centre for Research and Development and PKP Polskie Linie Kolejowe S.A., under the acronym InnoSatTrack.  
All data in the form of spatial coordinates, as well as information regarding the location of the railway line, have been provided to the infrastructure manager. These data have not been made publicly available as open datasets.  
The authors thank Pomeranian Mechanical and Track Enterprise (PPMT S.A.) for providing the measurement draisine with platform wagon used in data acquisition. They also gratefully acknowledge the Railway Line Department in Gdynia, PKP PLK, for granting access to the railway infrastructure as part of the InnoSatTrack project.  
The authors declare no conflicts of interest.