MATHEMATICAL FOUNDATIONS OF GPS AND RTK POSITIONING

164 “Al-Farg‘oniy avlodlari” elektron ilmiy jurnali ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil "Descendants of Al-Farghani" electronic scientific journal. ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 year Электронный научный журнал "Потомки АльФаргани" ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год https://al-fargoniy.uz/ MATHEMATICAL FOUNDATIONS OF GPS AND RTK POSITIONING Khonturaev Sardorbek, PhD student of the Tashkent State Technical University Email: sardorisroil[email protected] Muhammadjonova Shukronakhon, Student of the Fergana State Technical University Abstract: Drones equipped with Global Navigation Satellite Systems have transformed modern geospatial mapping, precision agriculture, and surveying. Standard Global Positioning System technology provides meter-level accuracy, but its precision is often insufficient for cartographic applications. Real-Time Kinematic techniques enhance GPS accuracy by applying carrier-phase corrections from reference stations, achieving centimeter-level positioning. This paper examines the integration of drones with GPS and RTK, reviews mathematical formulas underlying location determination, and highlights applications in geospatial cartography. Keywords: GNSS, GPS, RTK, UAV, drone, real-time, position, 3d surface Introduction. Geospatial positioning technologies form the backbone of modern mapping and surveying [1]. Traditional methods of ground-based measurement are labor-intensive and limited in spatial coverage. The advent of drones (UAVs) combined with satellitebased positioning systems has revolutionized data collection by offering rapid, high-resolution, and automated geospatial mapping [2]. While the Global Positioning System (GPS) is widely used, its typical error margin of 5–10 meters are insufficient for many scientific and engineering tasks. [3] To overcome this limitation, Real-Time Kinematic (RTK) positioning was developed, offering centimeterlevel accuracy by utilizing correction data from a reference station [4]. Drone platforms serve as ideal carriers for GNSS/RTK sensors due to their ability to cover difficult terrain and capture high-resolution data in real time [5]. Moreover, as Khalilov et al. [6] emphasize, advanced computational techniques such as neural networks and self-learning algorithms may further enhance positional accuracy through adaptive error correction and signal processing. This paper explores how GPS and RTK determine location, which mathematical formulas are used in coordinate computation, and how these methods are applied in drone-based geospatial cartography. Methods. The Global Positioning System (GPS) determines a receiver’s location using trilateration from satellite signals [7]. Each satellite transmits its position (𝑥𝑖, 𝑦𝑖, 𝑧𝑖) and a precise timestamp. The receiver calculates the travel time of signals to estimate distances called pseudoranges. The pseudorange equation is [8]: 𝜌𝑖=√(𝑥−𝑥𝑖)2+(𝑦−𝑦𝑖)2+(𝑧−𝑧𝑖)2+𝑐∗∆𝑡 Where: 𝜌 = measured pseudo range from satellite i (x, y, z) = receiver coordinates (unknowns to be solved) (𝑥𝑖, 𝑦𝑖, 𝑧𝑖) = known satellite coordinates c = speed of light Δt = receiver clock bias With at least four satellites, the receiver solves four nonlinear equations to determine (x, y, z, Δt) [3]. RTK enhances GPS by incorporating carrierphase measurements [9]. Instead of only pseudo ranges, RTK measures the phase of the carrier wave [4]: 165 “Al-Farg‘oniy avlodlari” elektron ilmiy jurnali ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil "Descendants of Al-Farghani" electronic scientific journal. ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 year Электронный научный журнал "Потомки АльФаргани" ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год https://al-fargoniy.uz/ 𝜑𝑖= 1 𝜆(√(𝑥−𝑥𝑖)2+(𝑦−𝑦𝑖)2+(𝑧−𝑧𝑖)2+𝑐∗∆𝑡− 𝑁𝑖∗𝜆) Where: 𝜑𝑖 = measured carrier phase (in cycles) 𝜆 = carrier wavelength 𝑁𝑖 = integer ambiguity (unknown number of whole wavelengths) A base station at a known location provides corrections by comparing its measured values to true distances, eliminating common-mode errors such as ionospheric delays [10]. The rover (drone) applies these corrections in real time to achieve centimeter accuracy. Drones equipped with RTK-GNSS modules act as rovers, integrating positioning data with onboard sensors (e.g., cameras, LiDAR, IMU) [11]. As Khonturaev notes, this integration enhances geospatial cartography by reducing survey time and improving accuracy in inaccessible terrain [2]. Uljaev, Ubaydullaev, and Khonturaev [5] detail drone-based coordinate determination technologies, showing that UAVs combined with RTK can replace traditional ground surveying methods while maintaining high precision. Khalilov et al. [6] suggest that advanced neural network methods can optimize weight coefficients for error reduction, providing future pathways for autonomous calibration of drone navigation systems. Results. The accuracy of positioning systems varies significantly depending on the method used. Standalone GPS typically provides an accuracy of about 5–10 meters, which is sufficient for everyday navigation in cars, smartphones, and logistics [7]. However, this accuracy is strongly influenced by factors such as ionospheric and tropospheric delays, multipath interference caused by signal reflections from terrain or buildings, and the geometric configuration of visible satellites (Dilution of Precision) [3]. While inexpensive and globally available, standalone GPS cannot be used in tasks requiring high precision such as cadastral mapping or construction surveying. To improve on this, Differential GPS (DGPS) was developed, which reduces common-mode errors by using correction data from a stationary reference receiver located at a precisely known position. By broadcasting the correction information to mobile users, DGPS can improve accuracy to about 0.5–3 meters. This makes DGPS suitable for applications such as agricultural field mapping, marine navigation, and general geospatial monitoring. However, even this method does not achieve the level of precision needed for geodetic surveying or engineering projects. The highest precision in real-time is obtained with Real-Time Kinematic (RTK) GPS, which achieves 1-3-centimeter accuracy under favorable conditions [4]. RTK uses not only pseud orange measurements but also carrier-phase observations, combined with correction data from a base station [10]. This makes it possible to eliminate most errors, including atmospheric delays and satellite clock biases. The technology is widely applied in geospatial cartography [2], precision agriculture, construction, and the navigation of drones and autonomous vehicles [5]. The main limitation is the requirement for continuous, reliable communication between the rover (e.g., drone) and the base station or reference network. Another advanced method worth mentioning is Precise Point Positioning (PPP), which does not require a base station but instead uses precise satellite orbits and clock corrections provided by global services [12]. PPP can achieve accuracies of about 5–20 centimeters after a convergence time of 10–30 minutes. This approach is useful for geodetic applications in remote areas where local reference stations are not available. Table 1. The comparison of positioning methods. Technology Accuracy Applications Standalone GPS 5–10 m Smartphones, car navigation, logistics [7] DGPS 0.5–3 m Agriculture, marine navigation, mapping [8] 166 “Al-Farg‘oniy avlodlari” elektron ilmiy jurnali ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil "Descendants of Al-Farghani" electronic scientific journal. ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 year Электронный научный журнал "Потомки АльФаргани" ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год https://al-fargoniy.uz/ Technology Accuracy Applications RTK GPS 1–3 cm Surveying, drones, precision cartography [4] PPP 5–20 cm Geodesy, remote monitoring, scientific use [12] Unmanned Aerial Vehicles (UAVs) equipped with Real-Time Kinematic (RTK) systems have transformed cartography by providing high-precision spatial data for diverse applications. Their ability to generate centimeter-level accurate geospatial datasets allows for detailed analysis in topography, agriculture, disaster management, and urban planning. Topographic Mapping. UAV-RTK systems produce high-resolution Digital Elevation Models (DEMs) by capturing overlapping images and applying photogrammetric processing. The elevation Z at a point (x, y) can be computed from stereo imagery using the collinearity equations: 𝑥−𝑥0 =−𝑓𝜏11(𝑋−𝑋𝑠)+𝜏12(𝑌−𝑌𝑠)+𝜏13(𝑍−𝑍𝑠) 𝜏31(𝑋−𝑋𝑠)+𝜏32(𝑌−𝑌𝑠)+𝜏33(𝑍−𝑍𝑠) 𝑦−𝑦0 =−𝑓𝜏21(𝑋−𝑋𝑠)+𝜏22(𝑌−𝑌𝑠)+𝜏23(𝑍−𝑍𝑠) 𝜏31(𝑋−𝑋𝑠)+𝜏32(𝑌−𝑌𝑠)+𝜏33(𝑍−𝑍𝑠) where (𝑋𝑠, 𝑌𝑠, 𝑍𝑠) is the camera position, 𝜏𝑖𝑗 are elements of the rotation matrix, and 𝑓 is the focal length. DEMs generated can be visualized as 3D surfaces or contour maps, providing essential information for slope analysis, watershed delineation, and flood modeling. A typical graph is a contour map of elevation vs. geographic coordinates (x, y). Agriculture. Precision agriculture leverages UAVs to produce vegetation indices such as the Normalized Difference Vegetation Index (NDVI): 𝑁𝐷𝑉𝐼 =𝑁𝐼𝑅−𝑅𝑒𝑑 𝑁𝐼𝑅+𝑅𝑒𝑑 where NIR and Red are the near-infrared and red reflectance values of crops. NDVI maps enable farmers to identify stressed areas and generate prescription maps for variable-rate fertilization. Graphs of NDVI values over field coordinates can highlight spatial heterogeneity, allowing targeted interventions to increase yield and reduce chemical usage. Disaster Response. UAVs provide rapid situational awareness by generating ortho mosaics and 3D reconstructions of affected regions. The volume of debris or floodwater can be estimated using: 𝑉 =∑(𝑍𝑖−𝑍𝑟𝑒𝑓)∗𝐴 𝑛 𝑖=1 where 𝑍𝑖 is the measured elevation, 𝑍𝑟𝑒𝑓 is a reference surface, and A is the area of each grid cell. Graphs such as heatmaps of elevation differences or time-series of affected area coverage can assist emergency planners in prioritizing interventions. Urban Planning. In urban environments, UAVderived point clouds generate accurate 3D city models. Surface models can be used to calculate building heights H and shadow lengths L for solar exposure studies: 𝐿 =𝐻∗tan⁡(∅) where ∅ is the solar elevation angle. Graphical outputs can include 3D renderings, shadow simulation plots, and density maps for infrastructure planning. Additionally, integration with Geographic Information Systems (GIS) allows overlaying multiple datasets for multi-criteria analysis. Integration of UAV Data. A comprehensive graph can be a multi-layered map showing elevation, NDVI, and urban infrastructure simultaneously. Mathematical interpolation methods like Kriging or Inverse Distance Weighting (IDW) can be applied to generate continuous surfaces from discrete UAV points: 𝑍(𝑥0)=∑𝑍(𝑥𝑖) 𝑑(𝑥0,𝑥𝑖)𝑝 𝑛 𝑖=1 ∑1 𝑑(𝑥0,𝑥𝑖)𝑝 𝑛 𝑖=1 where 𝑑(𝑥0,𝑥𝑖) is the distance between grid point 𝑥0 and measurement 𝑥𝑖 and p is a weighting exponent. Overall, UAV-RTK systems provide unparalleled spatial precision and versatility, enabling quantitative analysis and visualization that support 167 “Al-Farg‘oniy avlodlari” elektron ilmiy jurnali ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil "Descendants of Al-Farghani" electronic scientific journal. ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 year Электронный научный журнал "Потомки АльФаргани" ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год https://al-fargoniy.uz/ topography, agriculture, disaster management, and urban planning. The integration of mathematical models, graphs, and GIS tools maximizes the utility of UAV-derived cartographic data. Discussion. The integration of drones with GPS and RTK enables centimeter-level geospatial mapping, which is critical for modern cartography, construction, and environmental monitoring [13]. While GPS provides global coverage, its accuracy is limited by atmospheric delays, satellite geometry, and clock biases [3]. RTK overcomes these limitations but requires reliable communication between base and rover [4]. Beyond accuracy, the combination of drones and RTK offers significant efficiency advantages [11]. Traditional surveying methods often require extensive manual labor and time, whereas drones can quickly capture high-resolution data over large and difficult-toaccess areas. This allows for near real-time monitoring of construction sites, agricultural fields, or environmental changes such as coastal erosion or deforestation. However, the effectiveness of RTK is influenced by several operational factors. Signal obstruction caused by dense vegetation, urban canyons, or terrain can degrade positioning accuracy [10]. Multipath errors, where GPS signals reflect off surfaces before reaching the receiver, can also introduce discrepancies. To mitigate these issues, many systems integrate additional sensors such as inertial measurement units (IMUs), barometers, or visual odometry, providing redundancy and improving reliability in challenging environments [14]. The data obtained through RTK-enabled drones is not only accurate but also highly versatile [11]. It can be used to generate digital elevation models (DEMs), orthomosaic maps, and 3D models that support decision-making across multiple industries. For instance, in precision agriculture, centimeter-level mapping allows for optimized irrigation planning, crop monitoring, and yield estimation. In construction, RTK drones can monitor site progress, detect deviations from design plans, and enhance safety by reducing the need for manual inspections in hazardous areas. Finally, as drone technology and RTK solutions continue to evolve, integration with cloud computing, AI-driven analytics, and autonomous flight systems promises further enhancements in mapping speed, data accuracy, and operational scalability [6]. These advancements indicate a future where drones with RTK may become a standard tool for geospatial intelligence, environmental monitoring, and infrastructure management [12]. Conclusion. The practical implications of this enhanced accuracy, as detailed in the results and applications sections, are profound and span a diverse range of industries. In topographic mapping, the ability to generate high-resolution Digital Elevation Models and Digital Terrain Models without the intensive labor of placing numerous ground control points has revolutionized surveying. The mathematical rigor of photogrammetry, governed by the collinearity equations, is now underpinned by precise geotagging, resulting in reliable 3D models and contour maps. In precision agriculture, the combination of RTK positioning with spectral analysis, such as the calculation of the Normalized Difference Vegetation Index, allows for the creation of accurate prescription maps. This facilitates targeted interventions, optimizing resource use and boosting crop yields. Furthermore, in construction and engineering, the centimeter-level accuracy ensures that as-built models can be reliably compared against design plans, enabling proactive project management and quality control. The capacity for rapid volumetric calculations and change detection in environmental monitoring and disaster response further underscores the transformative utility of drone-RTK systems. However, the deployment of these systems is not without its challenges. The operational efficacy of RTK is contingent upon a stable and reliable communication link between the base station and the rover, which can be compromised in remote areas or 168 “Al-Farg‘oniy avlodlari” elektron ilmiy jurnali ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil "Descendants of Al-Farghani" electronic scientific journal. ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 year Электронный научный журнал "Потомки АльФаргани" ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год https://al-fargoniy.uz/ terrain with obstructed lines of sight. Factors such as signal multipath, satellite geometry, and initialization times for integer ambiguity resolution remain practical considerations. To mitigate these limitations, the integration of RTK with supplementary sensors— Inertial Measurement Units for attitude and trajectory smoothing, barometers for altitude validation, and even visual odometry systems—creates a robust, multisensor positioning solution that maintains accuracy even during temporary signal outages. Looking forward, the evolution of drone-based positioning is poised to continue its rapid advancement. As highlighted by Khalilov et al. [6], the next frontier lies in the integration of artificial intelligence and neural networks with traditional GNSS/RTK systems. These self-learning algorithms promise to further enhance accuracy through adaptive error modeling, intelligent signal filtering in noisy environments, and autonomous system calibration. Furthermore, technologies like Precise Point Positioning and its hybrid variant, PPP-RTK, offer a compelling future direction by potentially delivering global, centimeterlevel accuracy without the absolute requirement for a local base station, thereby increasing operational flexibility, especially in remote or maritime environments. In conclusion, drones equipped with RTK have firmly established themselves as an indispensable tool in the modern geospatial toolkit. By translating complex mathematical principles into practical, high-precision applications, they have not only democratized access to accurate mapping but have also opened new horizons for scientific inquiry, industrial efficiency, and intelligent environmental management. The continued convergence of this technology with AI and cloud-based analytics promises a future where real-time, centimeter-accurate geospatial intelligence is seamlessly integrated into the fabric of smart infrastructure, sustainable agriculture, and proactive environmental stewardship. References 1. Colomina, I. &. (2014). Unmanned aerial systems for photogrammetry and remote sensing: A review. ISPRS. Journal of Photogrammetry and Remote Sensing. 2. Hofmann-Wellenhof, B. L. (2007). GNSS– Global Navigation Satellite Systems: GPS, GLONASS, Galileo, and more. Springer. 3. Kaplan, E. D. (2017). Understanding GPS/GNSS: Principles and applications. . Artech House. 4. Khalilov, D., Bozorova, S., Khonturaev, S., Khoitkulov, A., Sotvoldieva, D., & Toshmatov, S. (2024). Self-learning system and methods of selection of weight coefficients of neural network. In E3S Web of Conferences (Vol. 508, p. 04011). EDP Sciences. 5. Leick, A. R. (2015). GPS satellite surveying. John Wiley & Sons. 6. Misra, P. &. (2011). Global Positioning System: Signals, measurements, and performance. Ganga-Jamuna Press. 7. Siebert, S. &. (2014). Mobile 3D mapping for surveying earthwork projects using an Unmanned Aerial Vehicle (UAV) system. Automation in Construction. 8. Takasu, T. &. (2009). Development of the low-cost RTK-GPS receiver with an open-source program package RTKLIB. 9. Teunissen, P. J. (2017). Springer handbook of global navigation satellite systems. Springer. 10. Yigit, C. O. (2017). Experimental study on the precise positioning with GPS, GLONASS and GPS/GLONASS combined static and kinematic RTK. Measurement. 11. Zarchan, P. (2019). Fundamentals of GPS. Progress in Astronautics and Aeronautics. 12. Zhang, J. &. (2015). Visual-lidar odometry and mapping: Low-drift, robust, and fast. IEEE International Conference on Robotics and Automation. 13. Улжаев, Э., Убайдуллаев, У., & Хонтураев, С. (2025). ТЕХНОЛОГИИ ОПРЕДЕЛЕНИЯ КООРДИНАТ С ПОМОЩЬЮ ДРОНОВ. Techscience. uz-Texnika fanlarining dolzarb masalalari, 3(5), 25-29. 14. Хонтураев, С. (2025). ПРИМЕНЕНИЕ ДРОНОВ В СОВРЕМЕННОЙ ГЕОПРОСТРАНСТВЕННОЙ КАРТОГРАФИИ. Techscience. uz-Texnika fanlarining dolzarb masalalari, 3(4), 29-32.