Algorithm for Obtaining Cartesian Debiased Estimates of Target Coordinates from Range and Direction Measurements for an Arbitrary Radar Angular Orientation and Offset

Introduction. In order to increase the accuracy of estimating the coordinates of a target, it is necessary to compensate for systematic errors in the nonlinear transformation of coordinates from a spherical system to a Cartesian one. In this paper, we consider the generalization of the well-known coordinate transformation algorithm with compensation for systematic errors when the radar is positioned and oriented arbitrarily in the global Cartesian coordinate system. By applying the proposed algorithm, the coordinate vector and error correlation matrix can be obtained in the global Cartesian coordinate system in the presence of range measurements and angular positions in the local spherical coordinate system associated with the radar position. The proposed algorithm is discussed with respect to trajectory filtering. As a result of the compensation of systematic errors and the calculation of the correlation matrix of the coordinate vector, the accuracy of trajectory tracking is enhanced. Aim. To improve the accuracy of trajectory tracking when using the Kalman filter with converted measurements by means of the mathematical expressions obtained to estimate the coordinates of a target and the error correlation matrix in the global Cartesian coordinate system. Materials and methods. The problem was solved using the methods of mathematical statistics, statistical estimation theory, and computer simulation. Results. Mathematical expressions for calculating coordinates and the corresponding correlation matrix within the global Cartesian coordinate system were derived. Furthermore, comparative graphs illustrating trajectory tracking errors, associated with the use of various methodologies for constructing a Kalman filter based on transformed   measurements, were created. Conclusion. The use of explicit expressions for coordinate transformation, accompanied by compensation for systematic errors, illustrates the potential for a substantial enhancement in accuracy when the errors of primary measurements increase. This improvement can be achieved when applying both direct and straightforward coordinate recalculation methods and a Kalman filter for transformed measurements within the global Cartesian coordinate system.

1. Ryzhova D. A., Zavalin A. A., Karakotov S. D., Rud S. V. Peculiarities of State Control to Water Objects with Using Of Radar Systems. IOP Conf. Ser.: Earth and Environmental Science. 2021, vol. 1096, art. no. 012030. doi: 10.1088/1755-1315/1096/1/012030

2. Patole M., Torlak M., Wang D., Ali M. Automotive Radars: A Review of Signal Processing Techniques. IEEE Signal Processing Magazine. 2017, vol. 34, no. 2, pp. 22–35. doi: 10.1109/MSP.2016.2628914

3. Can C., Peng W., Bi T., Zhu H. Research on Au tomotive Safety Based on Multi-Sensor Information Fusion Technology. Asia-Pacific Conf. on Software Engineering, Social Network Analysis and Intelligent Computing, New Delhi, India, 10–12 Jan. 2024. IEEE, 2024, pp. 750–754. doi: 10.1109/SSAIC61213.2024.00152

4. Correas-Serrano A., Gonzalez-Huici M., Simoni R., Bredderman T., Warsitz E., Müller T. Performance Analysis and Design of a Distributed Radar Network for Automotive Application. 23rd Intern. Radar Symp. (IRS), Gdansk, Po land, 12–14 Sept. 2022. IEEE, 2022, pp. 30–35. doi: 10.23919/IRS54158.2022.9904987

5. Pirkani A., Kumar D., Cherniakov M., Gashinova M. Distributed Automotive Radar Multi-Modal Sensing. 20 th European Radar Conf. (EuRAD), Berlin, Germany, 20–22 Sept. 2023. IEEE, 2023, pp. 319–322. doi: 10.23919/EuRAD58043.2023.10289419

6. Chang G., Fengdeng G., Junkun Y., Tianyi J., Hongwei L. Fusion detection for networked radar aided by Doppler information. J. of Radars. 2023, vol. 12, no. 3, pp. 500–515. doi: 10.12000/JR22220

7. Kuzin A. A., Kuznetsov S. E., Miakinkov A. V., Fadeev R. S., Shabalin S. A. Experimental Study of Coherent Collaborative Processing Method in Distributed Automotive Radar. J. of the Russian Universities. Radioelectronics. 2025, vol. 28, no. 1, pp. 35–50. (In Russ.) doi: 10.32603/1993-8985-2025-28-1-35-50

8. Artyukhin I. V. High-resolution 2D-DoA Se quential Algorithm of Azimuth and Elevation Estimation in Automotive Distributed System of Coherent MIMO Radars. Russian Technological J. 2024, vol. 12, no. 3, pp. 65–77. (In Russ.) doi: 10.32362/2500-316X-2024-12-3-65-77

9. Greitans K. M. A study on Automated Railway Level Crossing Control System Using FMCW Radar for Accident Prevention. Intern. Conf. on Appl. Electronics, Pilsen, Czech Republic, 06–07 Sept. 2023. IEEE, 2023, pp. 1–6. doi: 10.1109/AE58099.2023.10274197

10. Ryndyk A. G., Myakinkov A. V., Balashova D. M., Burov V. N., Shabalin S. A., Mikhaylov A. D. Railway Safety Radar System with Use of FSR. 21st Intern. Radar Symp., Berlin, Germany, 21–22 June 2021. IEEE, 2021, pp. 1–13. doi: 10.23919/IRS51887.2021.9466229

11. Liu B., Yan X., Li Q., Huang Sh. An Improved Method for Traffic Control Relying on Close-Loop Control Theory. 2nd Intern. Asia Conf. on Informatics in Control, Automation and Robotics, Wuhan, China, 06–07 March 2010. IEEE, 2010, pp. 48–50. doi: 10.1109/CAR.2010.5456724

12. Suchomski P. Explicit Expressions for Debiased Statistics of 3D Converted Measurements. IEEE Trans. on Aerospace and Electronic Systems. 1999, vol. 35, no. 1, pp. 368–370. doi: 10.1109/7.745708

13. Spitzmiller J. N., Adhami R. R. Tracking with Spherical-Estimate-Conditioned Debiased Converted Measurements. IEEE Radar Conf., Arlington, USA, 10–14 May 2010. IEEE, 2010, pp. 134–139. doi: 10.1109/RADAR.2010.5494637

14. Bordonaro S. V., Luginbuhl T. E., Walsh M. J. A Generalized Converted Measurement Kalman Filter. NUWC-NPT Techn. Rep. 2023, vol. 12, p. 472. doi: 10.48550/arXiv.2502.08375

15. Wang K., Li X., Wu P. Modified Unbiased Converted Measurement Kalman Filter with Outlier Detection. 42nd Chinese Control Conf., Tianjin, China, 24–26 July 2023. IEEE, 2023, pp. 3660–3665. doi: 10.23919/CCC58697.2023.10240538

16. Bordonaro S. V., Willett P., Bar-Shalom Y. Un biased Tracking with Converted Measurements. IEEE Radar Conf., Atlanta, USA, 07–11 May 2012. IEEE, 2012, pp. 0741–0745. doi: 10.1109/RADAR.2012.6212236

17. Wang H., Huo L., Zhang J. Target Tracking Algorithm Based on Dynamic Template and Kalman Fil ter. IEEE 3 rd Intern. Conf. on Communication Software and Networks, Xi'an, China, 27–29 May 2011. IEEE, 2011, pp. 330–333. doi: 10.1109/ICCSN.2011.6013726

18. Konovalov A. A. Osnovy traektornoi obrabotki radiolokatsionnoi informatsii [Fundamentals of Trajectory Processing of Radar Information]. SPb., Izd-vo SPbGETU "LETI", 2013, 164 p. (In Russ.)

19. Subramaniam M., Tharmarasa R., McDonald M., Kirubarajan T. Passive Tracking with Sensors of Opportunity Using Passive Coherent Location. Proc. SPIE6969, Signal and Data Processing of Small Targets. Orlando, Florida, 16 April 2008, vol. 6969, p. 69691F. doi: 10.1117/12.779598

20. Mallick M., Xiaoqing T., Liu J. Evaluation of Measurement Converted KF, EKF, UKF, CKF, and PF in GMTI Filtering. 10th Intern. Conf. on Control, Au tomation and Information Sciences, Xi'an, China, 14– 17 Oct. 2021. IEEE, 2021, pp. 21–27. doi: 10.1109/ICCAIS52680.2021.9624559

21. Palais B., Palais R., Rodi S. A Disorienting Look at Euler’s Theorem on the Axis of a Rotation. The American Mathematical Monthly. 2009, vol. 116, no. 10, pp. 892–909. doi: 10.4169/000298909X477014

22. Draper N. R., Smith H. Applied Regression Analysis. John Wiley & Sons, Inc, 1998, 736 p. doi: 10.1002/9781118625590

23. Burov V. N., Myakinkov A. V. Targets Tracking Algorithm Which Are Maneuvering Along Curvilinear Trajectories in Forward-Scattering Radar. J. of the Russian Universities. Radioelectronics. 2011, no. 6, pp. 88–94.

24. Burov V. N. Multi-Model Trajectory Tracking of Targets in a Bistatic Lumen Radar System. Design and Technology of Electronic Devices. 2012, no. 2, pp. 2–4. (In Russ.)