journal article·2025-12-30Open access

Non-Linear Method of Vehicle Velocity Determination Based on Tensor Product B-Spline Approximation with Probabilistic Weights for NHTSA Database of Compact Vehicle Class

Milos Poliak, Przemysław Kubiak, Mateusz Krukowski, Filip Turoboś, Marek Jaśkiewicz, Justyna Jaśkiewicz, Damian Frej

Applied Sciences· Volume 16· Issue 1

Abstract

This research article focuses on the method of vehicle crash velocity evaluation based on the tensor product approximation by B-splines. Weights associated with each observation are introduced in the least square method, which is based on probabilistic reasoning. The presented calculation algorithm is built based on the Compact vehicle class from the NHTSA database, which consists of 338 records of frontal crashes with stationary obstacles. The presented model is restricted to velocities in the range between 43 and 93.5 km/h. The relative error obtained for the presented calculation method is 5.2%. Hence, improvement has been observed in comparison with the linear model with the same probabilistic weights approach, for which relative error is equal to 5.47%.
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Research context

OpenAlex topics: Automotive and Human Injury Biomechanics · Transportation Safety and Impact Analysis · Cellular and Composite Structures

SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.

Affiliations

Department of Road and Urban Transport, University of Žilina, 010-26 Žilina, Slovakia

Division of Ecotechnics, Lodz University of Technology, 90-924 Łódź, Poland

Institute of Mathematics, Lodz University of Technology, 90-924 Łódź, Poland

Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, 44-100 Gliwice, Poland

Department of Automotive Engineering and Transport, Kielce University of Technology, Avenue Tysiaclecia Państwa Polskiego 7, 25-314 Kielce, Poland

Wydział Humanistyczny, Uniwersytet im. Jana Kochanowskiego w Kielcach (Jan Kochanowski University of Kielce), ul Żeromskiego 5, 25-369 Kielce, Poland

Funding

Slovak Research and Development Agency · National Science Center (NCN), Poland

References (42)

Show source-provided references
1. Jankowski, K.P., Gidlewski, M., and Jemioł, L. (2007, January 8–10). Comparative study of vehicle absorbed energy determination for road accident reconstruction. Proceedings of the XVI EVU Annual Meeting, Kraków, Poland.
2. Wąsik, M., and Skarka, W. (2016, January 3–7). Simulation of crash tests for electrically propelled flying exploratory autonomous robot. Proceedings of the Transdisciplinary Engineering: Crossing Boundaries—Proceedings of the 23rd ISPE Inc. International Conference on Transdisciplinary Engineering, Curitiba, Brazil.
3. Sharma, S., Brophy, J., and Choi, E. (2007, January 18–21). An overview of NHTSA’s crash reconstruction software WinSMASH. Proceedings of the 20th International Conference on the Enhanced Safety of Vehicles, Lyon, France.
4. Siddall, D., and Day, T. (1996). Updating the Vehicle Class Categories, Society of Automotive Engineers. SAE Technical Paper 960897.10.4271/960897 ↗
5. Foret-Bruno, J.Y., Trosseille, X., Le Coz, J.Y., Bendjellal, F., Steyer, C., Phalempin, T., Villeforceix, D., Dandres, P., and Got, C. (1998). Thoracic Injury Risk in Frontal Car Crashes with Occupant Restrained with Belt Load Limiter, Society of Automotive Engineers. SAE Technical Paper 983166.10.4271/983166 ↗
6. Campbell, B.J. (1969, January 24–28). The traffic accident data project scale. Proceedings of the Collision Investigation Methodology Symposium, Warrenton, VA, USA.
7. Cromack, J.R., and Lee, S.N. (1974). Consistency Study for Vehicle Deformation Index, Society of Automotive Engineers. SAE Technical Paper 740299.10.4271/740299 ↗
8. Hight, P., Fugger, T., and Marcosky, J. (1992). Automobile Damage Scales and the Effect on Injury Analysis, Society of Automotive Engineers. SAE Technical Paper 920602.10.4271/920602 ↗
9. Pulse shape analysis and data reduction of real-life frontal crashes with modern passenger cars10.1080/13588265.2015.1057005 ↗
10. Car size and injury risk: A model for injury risk in frontal collisions10.1016/S0001-4575(01)00003-3 ↗
11. Reconstruction of occupant kinematics and kinetics for real-world accidents10.1533/ijcr.2003.0217 ↗
12. Simplified method for evaluating energy loss in vehicle collisions10.1016/j.aap.2009.02.012 ↗
13. Evaluation of energy loss in motorcycle-to-car collisions10.1080/13588265.2014.899072 ↗
14. Adaptive inertial shock-absorber10.1088/0964-1726/25/3/035031 ↗
15. Modelling the effect of forming history in impact simulations10.1080/13588260801976120 ↗
16. Integration of forming effects into vehicle front rail crash simulation10.1080/13588265.2015.1091170 ↗
17. Wach, W., Gidlewski, M., and Prochowski, L. (2016, January 5–7). Modelling reliability of vehicle collision reconstruction based on the law of conservation of momentum and Burg equations. Proceedings of the 20th International Scientific Conference TRANSPORT MEANS, Juodkrante, Lithuania.
18. The use of energy methods for calculation of vehicle impact velocity
19. Real world car crash investigations–A new approach10.1533/ijcr.2003.0245 ↗
20. McHenry, R. (1973). Computer Program for Reconstruction of Highway Accidents, Society of Automotive Engineers. SAE Technical Paper 730980.10.4271/730980 ↗
21. Wach, W., and Unarski, J. (2006). Determination of Vehicle Velocities and Collision Location by Means of Monte Carlo Simulation Method, Society of Automotive Engineers. SAE Technical Paper 2006-01-0907.10.4271/2006-01-0907 ↗
22. The analysis of a car motion path after collision with a concrete barrier
23. Analysis of the influence of passenger position in a car on the risk of injuries during an accident
24. Analysis of motion of the body of a motor car hit on its side by another passenger car10.1088/1757-899X/148/1/012039 ↗
25. Work of non-elastic deformation against the deformation ratio of the subcompact car class using the variable correlation method10.1016/j.forsciint.2018.03.033 ↗
26. Kubiak, P., Wozniak, M., Karpushkin, V., Jozwiak, P., Ozuna, G., Madziara, S., Najbert, M., and Szosland, A. (2015, January 1–3). The method of determining velocity by measuring the vehicle-body deformation plane approximation method. Proceedings of the International Conference on Engineering, London, UK.10.1007/978-981-10-1088-0_4 ↗
27. Nonlinear Optimization in Vehicular Crash Reconstruction10.4271/2015-01-1433 ↗
28. An inverse reconstruction approach considering uncertainty and correlation for vehicle-vehicle collision accidents10.1007/s00158-019-02231-9 ↗
29. Uncertain inverse traffic accident reconstruction by combining the modified arbitrary orthogonal polynomial expansion and novel optimization technique10.1016/j.forsciint.2022.111213 ↗
30. A Bayesian Neural Network approach to estimating the Energy Equivalent Speed10.1016/j.aap.2005.08.008 ↗
31. Moravcová, P., Bucsuházy, K., Zůvala, R., Semela, M., and Bradáč, A. (2024). What should I use to calculate vehicle EES?. PLoS ONE, 19.10.1371/journal.pone.0297940 ↗
32. Restrained occupants on the nonstruck side in lateral collisions10.1016/0001-4575(93)90054-Z ↗
33. McHenry, B. (2025, December 24). The algorithms of CRASH. Available online: https://www.researchgate.net/publication/237300782_THE_ALGORITHMS_OF_CRASH.
34. Nelson, W.D. (1981). The History and Evolution of the Collision Deformation Classification, Society of Automotive Engineers. SAE Technical Paper 810213.10.4271/810213 ↗
35. Neptune, J.E. (1998). Crush Stiffness Coefficients, Restitution Constants, and a Revision of CRASH3 & SMAC, Society of Automotive Engineers. SAE Technical Paper 980029.10.4271/980029 ↗
36. Analysis of seasonal and regional variations in accident hazard in Poland10.2495/UT170381 ↗
37. Predicting Vehicle Crashworthiness: Validation of Computer Models for Functional and Hierarchical Data10.1198/jasa.2009.ap06623 ↗
38. Turoboś, F., Koneczny, P., Stajuda, Ł., Levchenko, D., Siczek, K., Kubiak, P., Mrowicki, A., and Madziara, S. (2024, January 24–26). Innovative precrash velocity calculation for forward-facing collisions using Gaussian process regression and modern vehicle data. Proceedings of Automotive Safety 2024: Proceedings of the 14th International Science and Technical Conference Automotive Safety, Sandomierz, Poland.10.1201/9781003465959-13 ↗
39. Determining vehicle pre-crash speed in frontal barrier crashes using genetic algorithm model adjustment techniques for intermediate car class10.1080/13588265.2021.1889237 ↗
40. Determining vehicle pre-crash speed in frontal barrier crashes using artificial neural network for intermediate car class10.1016/j.forsciint.2020.110179 ↗
41. De Boor, C. (2001). Practical Guide to Splines: With 32 Figures, Springer.
42. A comparison between filtering approach and spline approximation method in smoothing flight data10.1007/s42401-023-00201-0 ↗