{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T19:11:19Z","timestamp":1764961879417,"version":"3.46.0"},"reference-count":52,"publisher":"MDPI AG","issue":"23","license":[{"start":{"date-parts":[[2025,12,2]],"date-time":"2025-12-02T00:00:00Z","timestamp":1764633600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006141","name":"Military University of Technology","doi-asserted-by":"publisher","award":["UGB 056\/2025"],"award-info":[{"award-number":["UGB 056\/2025"]}],"id":[{"id":"10.13039\/501100006141","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The use of unmanned aerial vehicles (UAVs) as mobile sensor platforms has grown significantly in recent years, including applications where drones emulate radar targets or serve as dynamic measurement systems. This paper presents a novel approach to time-synchronized UAV trajectory planning for radar environment simulation. The proposed method considers a UAV equipped with a software-defined radio (SDR) capable of reproducing the radar signature of a simulated airborne object, e.g., a high-maneuverability or high-speed aerial platform. The UAV must follow a spatial trajectory that replicates the viewing geometry\u2014specifically, the observation angles\u2014of the reference target as seen from a ground-based radar. The problem is formulated within a factor graph framework, enabling joint optimization of the UAV trajectory, observation geometry, and temporal synchronization constraints. While factor graphs have been extensively used in robotics and sensor fusion, their application to trajectory planning under temporal and sensing constraints remains largely unexplored. The proposed approach enables unified optimization over space and time, ensuring that the UAV reproduces the target motion as perceived by the radar, both geometrically and with appropriate signal timing.<\/jats:p>","DOI":"10.3390\/s25237326","type":"journal-article","created":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T18:42:02Z","timestamp":1764960122000},"page":"7326","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Factor Graph-Based Time-Synchronized Trajectory Planning for UAVs in Ground Radar Environment Simulation"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7299-9579","authenticated-orcid":false,"given":"Pawe\u0142","family":"S\u0142owak","sequence":"first","affiliation":[{"name":"Institute of Radioelectronics, Faculty of Electronics, Military University of Technology, 00-908 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5190-1138","authenticated-orcid":false,"given":"Pawe\u0142","family":"Kaczmarek","sequence":"additional","affiliation":[{"name":"Institute of Radioelectronics, Faculty of Electronics, Military University of Technology, 00-908 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-2721-9960","authenticated-orcid":false,"given":"Adrian","family":"Kapski","sequence":"additional","affiliation":[{"name":"Institute of Radioelectronics, Faculty of Electronics, Military University of Technology, 00-908 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3873-2894","authenticated-orcid":false,"given":"Piotr","family":"Kaniewski","sequence":"additional","affiliation":[{"name":"Institute of Radioelectronics, Faculty of Electronics, Military University of Technology, 00-908 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Taylor, C., and Gross, J. (2024). Factor graphs for navigation applications: A tutorial. Navigation, 71.","DOI":"10.33012\/navi.653"},{"key":"ref_2","first-page":"141","article-title":"Factor Graphs: Exploiting Structure in Robotics. Annu. Rev. Control Robot","volume":"4","author":"Dellaert","year":"2021","journal-title":"Auton. Syst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1177\/0278364911430419","article-title":"iSAM2: Incremental Smoothing and Mapping Using the Bayes Tree","volume":"31","author":"Kaess","year":"2012","journal-title":"Int. J. Robot. Res."},{"key":"ref_4","first-page":"1","article-title":"Motion Planning as Probabilistic Inference Using Gaussian Processes and Factor Graphs","volume":"12","author":"Dong","year":"2016","journal-title":"Proc. RSS"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Bhardwaj, M., Boots, B., and Mukadam, M. (August, January 31). Differentiable Gaussian Process Motion Planning. Proceedings of the 2020 IEEE International Conference on Robotics and Automation (ICRA), Paris, France.","DOI":"10.1109\/ICRA40945.2020.9197260"},{"key":"ref_6","unstructured":"Xie, M., Escontrela, A., and Dellaert, F. (2020). A Factor-Graph Approach for Optimization Problems with Dynamics Constraints. arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"100614","DOI":"10.1016\/j.cosrev.2023.100614","article-title":"Deep Learning for Unmanned Aerial Vehicles Detection: A Review","volume":"51","author":"Alenezi","year":"2024","journal-title":"Comput. Sci. Rev."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"107709","DOI":"10.1016\/j.neunet.2025.107709","article-title":"BRSR-OpGAN: Blind radar signal restoration using operational generative adversarial network","volume":"190","author":"Zahid","year":"2025","journal-title":"Neural Netw."},{"key":"ref_9","first-page":"105499","article-title":"Design and processing of anti-IFM receiver WOFDM-MCFC low intercept radar signal","volume":"168 Pt A","author":"Ma","year":"2025","journal-title":"Digit. Signal Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"109246","DOI":"10.1016\/j.sigpro.2023.109246","article-title":"Compound Radar Jamming Recognition Based on Signal Source Separation","volume":"214","author":"Zhou","year":"2024","journal-title":"Signal Process"},{"key":"ref_11","unstructured":"(2025, October 25). Radar Simulator Market Size, Share & Industry Analysis, By End Use Industry (Military and Commercial), By Commercial (Aviation, Automobiles, and Others), By Application (Designing Testing and Operator Training), By System (Hardware and Software), and Regional Forecast, 2024\u20132032, Last Updated 4 August 2025. Available online: https:\/\/www.fortunebusinessinsights.com\/radar-simulator-market-108213."},{"key":"ref_12","first-page":"83","article-title":"Advanced Signal Processing and Modeling Techniques for Automotive Radar: Challenges and Innovations in ADAS Applications","volume":"144","author":"Biswas","year":"2025","journal-title":"CMES Comput. Model. Eng. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"859","DOI":"10.1177\/00375497231152651","article-title":"Ad Hoc HLA Simulation Model Derived from a Model-Based Traffic Scenario","volume":"99","author":"Reiher","year":"2023","journal-title":"Simulation"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"257","DOI":"10.3390\/vehicles3020016","article-title":"Radar Target Simulation for Vehicle-in-the-Loop Testing","volume":"3","author":"Diewald","year":"2021","journal-title":"Vehicles"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"117786","DOI":"10.1016\/j.measurement.2025.117786","article-title":"Development and implementation of an FPGA-based embedded real-time digital simulator","volume":"254","author":"Sarac","year":"2025","journal-title":"Measurement"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"105331","DOI":"10.1016\/j.mejo.2021.105331","article-title":"Radio frequency analog-to-digital converters: Systems and circuits review","volume":"119","author":"Li","year":"2022","journal-title":"Microelectron. J."},{"key":"ref_17","first-page":"118453","article-title":"Latency mitigation in digital radar target simulation","volume":"257 Part B","author":"Sobotka","year":"2025","journal-title":"Measurement"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1013","DOI":"10.1016\/j.procs.2023.08.082","article-title":"DRFM-Based Jamming Signal Recognition Method Guided by Target Detection","volume":"221","author":"Chen","year":"2023","journal-title":"Procedia Comput. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"130528","DOI":"10.1016\/j.optcom.2024.130528","article-title":"Photonics-based wideband multi-format tunable radar jamming signal generator","volume":"562","author":"Shan","year":"2024","journal-title":"Opt. Commun."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"102019","DOI":"10.1016\/j.softx.2024.102019","article-title":"SARrawSim: Synthetic Aperture Radar Raw Data Simulator","volume":"29","author":"Sahad","year":"2025","journal-title":"SoftwareX"},{"key":"ref_21","first-page":"1363","article-title":"HQNN-SFOP: Hybrid Quantum Neural Networks with Signal Feature Overlay Projection for Drone Detection Using Radar Return Signals\u2014A Simulation","volume":"81","author":"Wang","year":"2024","journal-title":"Comput. Mater. Contin."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"28375","DOI":"10.1109\/JSTARS.2025.3626140","article-title":"Generation of Accurate, High-Resolution Digital Elevation Models from Ultra-Wideband, Drone-Borne, Repeat-Pass Interferometric SAR","volume":"18","author":"Kim","year":"2025","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"J\u0119drzejewski, K., Suwi\u0144ski, K., and W\u00f3jtowicz, P. (May, January 21). Simulator of Passive Radar for Detecting Space Objects in Low-Earth Orbit. Proceedings of the 2025 26th International Radar Symposium (IRS), Hamburg, Germany.","DOI":"10.23919\/IRS64527.2025.11046106"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.cja.2021.09.001","article-title":"Graph Based Navigation and Positioning for Control System Design: A Review","volume":"35","author":"Wu","year":"2022","journal-title":"Chin. J. Aeronaut."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"28315","DOI":"10.1109\/ACCESS.2025.3534993","article-title":"Factor Graphs in Optimization-Based Robotic Control\u2014A Tutorial and Review","volume":"13","author":"Abdelkarim","year":"2025","journal-title":"IEEE Access"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1319","DOI":"10.1177\/0278364918790369","article-title":"Continuous-Time Gaussian Process Motion Planning via Probabilistic Inference","volume":"37","author":"Mukadam","year":"2018","journal-title":"Int. J. Robot. Res."},{"key":"ref_27","unstructured":"Suzuki, T. (2021, January 20\u201324). Global Optimization of Position and Velocity by Factor Graph Optimization. Proceedings of the 34th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2021), St. Louis, MI, USA."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Fan, G., Wang, Q., Yang, G., and Liu, P. (2024). RFG-TVIU: Robust Factor Graph for Tightly Coupled Vision\/IMU\/UWB Integration. Front. Neurorobot., 18.","DOI":"10.3389\/fnbot.2024.1343644"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"15501477211064758","DOI":"10.1177\/15501477211064758","article-title":"A Hybrid Cooperative Navigation Method for UAV Swarm Based on Factor Graph and Kalman Filter","volume":"18","author":"Chen","year":"2022","journal-title":"Int. J. Distrib. Sens. Netw."},{"key":"ref_30","first-page":"1234","article-title":"Continuous-Time Radar-\/LiDAR-Inertial Odometry Using Gaussian Process Motion Priors","volume":"40","author":"Burnett","year":"2024","journal-title":"IEEE Trans. Robot."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Michalczyk, T., Scherer, J., and Biber, P. (2024, January 14\u201318). Tightly-Coupled Factor-Graph Radar\u2013Inertial Odometry. Proceedings of the 2024 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Abu Dhabi, United Arab Emirates.","DOI":"10.1109\/IROS58592.2024.10801945"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Melebari, A., Nergis, P., Eskandari, S., Costa, P.R., and Moghaddam, M. (2024). Absolute Calibration of a UAV-Mounted Ultra-Wideband SDRadar in the Near-Field. Remote Sens., 16.","DOI":"10.3390\/rs16020231"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TRO.2016.2597321","article-title":"On-Manifold Preintegration for Real-Time Visual-Inertial Odometry","volume":"33","author":"Forster","year":"2017","journal-title":"IEEE Trans. Robot."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.24425\/mms.2025.154681","article-title":"Testing Positioning System for Ground Penetrating Radar Using Multi-Degree-of-Freedom Industrial Robot","volume":"32","author":"Kraszewski","year":"2025","journal-title":"Metrol. Meas. Syst."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Mir, I., Gul, F., Mir, S., Khan, M.A., Saeed, N., Abualigah, L., Abuhaija, B., and Gandomi, A.H. (2022). A Survey of Trajectory Planning Techniques for Autonomous Systems. Electronics, 11.","DOI":"10.3390\/electronics11182801"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1874","DOI":"10.1109\/TRO.2021.3075644","article-title":"ORB-SLAM3: An Accurate Open-Source Library for Visual, Visual\u2013Inertial, and Multi-Map SLAM","volume":"37","author":"Campos","year":"2021","journal-title":"IEEE Trans. Robot."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"112926","DOI":"10.1016\/j.measurement.2023.112926","article-title":"A Multi-Sensor Fusion Positioning Approach for Indoor Mobile Robot Using Factor Graph","volume":"216","author":"Zhang","year":"2023","journal-title":"Measurement"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"9115","DOI":"10.1109\/LRA.2022.3189824","article-title":"Continuous-Time Factor Graph Optimization for Trajectory Smoothness of GNSS\/INS Navigation in Temporarily GNSS-Denied Environments","volume":"7","author":"Zhang","year":"2022","journal-title":"IEEE Robot. Autom. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Xia, T., and Chen, H. (2024). A Survey of Autonomous Vehicle Behaviors: Trajectory Planning Algorithms, Sensed Collision Risks, and User Expectations. Sensors, 24.","DOI":"10.3390\/s24154808"},{"key":"ref_40","first-page":"3463","article-title":"Linear vs. Nonlinear Model Predictive Control for Trajectory Tracking of Micro Aerial Vehicles","volume":"50","author":"Kamel","year":"2017","journal-title":"IFAC-Pap."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Tordesillas, J., Lopez, B.T., and How, J.P. (2019, January 3\u20138). FASTER: Fast and Safe Trajectory Planner for Flights in Unknown Environments. Proceedings of the IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Macau, China.","DOI":"10.1109\/IROS40897.2019.8968021"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1177\/0278364914554813","article-title":"Keyframe-Based Visual\u2013Inertial Odometry Using Nonlinear Optimization","volume":"34","author":"Leutenegger","year":"2015","journal-title":"Int. J. Robot. Res."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Barfoot, T.D. (2017). State Estimation for Robotics, Cambridge University Press.","DOI":"10.1017\/9781316671528"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1080\/13658810801949850","article-title":"Finding Shortest Paths on Real Road Networks: The Case for A*","volume":"23","author":"Zeng","year":"2009","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_45","first-page":"20","article-title":"A Comparison of RRT, RRT*, and RRT*-Smart Path Planning Algorithms","volume":"16","author":"Noreen","year":"2016","journal-title":"Int. J. Comput. Sci. Netw. Secur."},{"key":"ref_46","unstructured":"Silver, D., Lever, G., Heess, N., Degris, T., Wierstra, D., and Riedmiller, M. (2014, January 21\u201326). Deterministic Policy Gradient Algorithms. Proceedings of the 31st International Conference on Machine Learning (ICML) 2014, Beijing, China."},{"key":"ref_47","unstructured":"Nocedal, J., and Wright, S.J. (2006). Numerical Optimization, Springer. [2nd ed.]."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s10589-024-00589-1","article-title":"The Levenberg\u2013Marquardt Method: An Overview of Modern Convergence Theories and More","volume":"89","author":"Fischer","year":"2024","journal-title":"Comput. Optim. Appl."},{"key":"ref_49","unstructured":"Blackman, S.S., and Popoli, R. (1999). Design and Analysis of Modern Tracking Systems, Artech House."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Rahman, M., Sakar, N., and Lutui, R. (2025). A Survey on Multi-UAV Path Planning: Classification, Algorithms, Open Research Problems, and Future Directions. Drones, 9.","DOI":"10.3390\/drones9040263"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"S\u0142owak, P., Kraszewski, T., and Kaniewski, P. (2025). Handheld Ground-Penetrating Radar Antenna Position Estimation Using Factor Graphs. Sensors, 25.","DOI":"10.3390\/s25113275"},{"key":"ref_52","first-page":"213","article-title":"Factor Graph-Based Trajectory Estimation for Radar-Based Missile Tracking","volume":"101","year":"2025","journal-title":"Przegl\u0105d Elektrotechniczny"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/25\/23\/7326\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T18:45:40Z","timestamp":1764960340000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/25\/23\/7326"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,2]]},"references-count":52,"journal-issue":{"issue":"23","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["s25237326"],"URL":"https:\/\/doi.org\/10.3390\/s25237326","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,2]]}}}