Straight running – stability analysis with a driving simulator

  • Danilo Biggio
  • Fabio della Rossa
  • Marco Fainello
  • Giampiero Mastinu
Conference paper
Part of the Proceedings book series (PROCEE)


The straight running of the system composed by a car plus driver is studied. Straight running is an important case study for analysing stability. Despite the lateral slip angles of the tyres are small, the system is highly non linear, due essentially to the driver action. Following the simple model of McRuer, later developed by Mistschke and revised by many other authors, we have developed a mathematical model of a car plus driver. The dynamic behaviour of the mathematical model has shown the presence of limit cycles generated by so called Hopf-bifurcations. The mathematical model predicts that, despite the understeering vehicle is globally stable, the driver can make the whole system (car plus driver) unstable. This occurs in case an external disturbance is sufficiently strong. If the external disturbance is small, the understeering vehicle plus driver remains stable. There is a speed above which the understeering car plus driver is unstable, usually such a speed is much greater than the maximum speed of the car on high grip surface. The statements introduced above have been validated by employing the driving simulator of Danisi Engineering, Nichelino, Italy. We experimentally saw that limit cycles do exist and that the driver can make the understeering vehicle model of the simulator quite unstable. We were able to validate the mathematical model by including two humans in the driving loop. One driver was a professional driver, the other one was a novice. The same non linear behaviours were highlighted for the two drivers, however, the amplitudes of the limit cycles and the ability of controlling the car were higher for the professional driver. A question arises whether an electronic power steering (EPS) may reduce or cancel instability. The answer is that there are a number of possible solutions for ESP to counteract the effect of unstable limit cycle.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1] Danisi engineering, via ippolito nievo 62, nichelino (to) { italy.
  2. [2] Vi-grade: Driving simulation | dynamic driving simulator.
  3. [3] C. Bobier-Tiu, C. Beal, J. Kegelman, R. Hindiyeh, and J. Gerdes. Vehicle control synthesis using phase portraits of planar dynamics. Vehicle System Dynamics, 2018.Google Scholar
  4. [4] B. Catino, S. Santini, and M. Di Bernardo. Mcs adaptive control of vehicle dynamics: an application of bifurcation techniques to control system design. In 42nd IEEE International Conference on Decision and Control, volume 3, pages 2252-2257. IEEE, 2003.Google Scholar
  5. [5] F. Della Rossa, M. Gobbi, G. Mastinu, C. Piccardi, and G. Previati. Bifurcation analysis of a car and driver model. Vehicle System Dynamics, 52(sup1):142- 156, 2014.Google Scholar
  6. [6] F. Della Rossa and G. Mastinu. Analysis of the lateral dynamics of a vehicle and driver model running straight ahead. Nonlinear Dynamics, 92(1):97-106, 2018.Google Scholar
  7. [7] F. Della Rossa and G. Mastinu. Straight ahead running of a nonlinear car and driver model-new nonlinear behaviours highlighted. Vehicle system dynamics, 56(5):753-768, 2018.Google Scholar
  8. [8] F. Della Rossa, G. Mastinu, and C. Piccardi. Bifurcation analysis of an automobile model negotiating a curve. Vehicle System Dynamics, 50(10):1539-1562, 2012.Google Scholar
  9. [9] A. Dhooge, W. Govaerts, and Y. A. Kuznetsov. Matcont: a matlab package for numerical bifurcation analysis of odes. ACM Transactions on Mathematical Software (TOMS), 29(2):141-164, 2003.Google Scholar
  10. [10] T. D. Gillespie. Fundamentals of vehicle dynamics. Technical report, SAE, 1992.Google Scholar
  11. [11] J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, 1983.Google Scholar
  12. [12] K. Guo and H. Guan. Modelling of driver/vehicle directional control system. Vehicle System Dynamics, 22(3-4):141-184, 1993.Google Scholar
  13. [13] Z. Hao, L. Xian-sheng, S. Shu-ming, L. Hong-fei, G. Rachel, and L. Li. Phase plane analysis for vehicle handling and stability. International journal of computational intelligence Systems, 4(6):1179-1186, 2011.Google Scholar
  14. [14] H. Hu, Z. Gao, Z. Yu, and Y. Sun. An experimental driving simulator study of unintentional lane departure. Advances in Mechanical Engineering, 9(10):1687814017726290, 2017.Google Scholar
  15. [15] H. Hu and Z. Wu. Stability and hopf bifurcation of four-wheel-steering vehicles involving driver’s delay. Nonlinear Dynamics, 22(4):361-374, 2000.Google Scholar
  16. [16] Y. A. Kuznetsov. Elements of applied bifurcation theory, volume 112. Springer Science & Business Media, 2013.Google Scholar
  17. [17] D.-C. Liaw, H.-H. Chiang, and T.-T. Lee. Elucidating vehicle lateral dynamics using a bifurcation analysis. IEEE Transactions on Intelligent Transportation Systems, 8(2):195-207, 2007.Google Scholar
  18. [18] Z. Liu and G. Payre. Global bifurcation analysis of a nonlinear road vehicle system. Journal of Computational and Nonlinear Dynamics, 2(4):308{315, 2007.Google Scholar
  19. [19] Z. Liu, G. Payre, and P. Bourassa. Nonlinear oscillations and chaotic motions in a road vehicle system with driver steering control. Nonlinear Dynamics, 9(3):281-304, 1996.Google Scholar
  20. [20] Z. Liu, G. Payre, and P. Bourassa. Stability and oscillations in a time-delayed vehicle system with driver control. Nonlinear Dynamics, 35(2):159-173, 2004.Google Scholar
  21. [21] C. C. Macadam. Understanding and modeling the human driver. Vehicle System Dynamics, 40(1-3):101-134, 2003.Google Scholar
  22. [22] G. Mastinu, F. Della Rossa, and C. Piccardi. Nonlinear dynamics of a road vehicle running into a curve. In Applications of Chaos and Nonlinear Dynamics in Science and Engineering-Vol. 2, pages 125-153. Springer, 2012.Google Scholar
  23. [23] G. Mastinu, A. Lattuada, and G. Matrascia. Straight-ahead running of road vehicles analytical formulae including full tyre characteristics. Vehicle System Dynamics, in press, 2018.Google Scholar
  24. [24] G. Mastinu and M. Ploechl. Road and off-road vehicle system dynamics handbook. CRC press, 2014.Google Scholar
  25. [25] M. Mitschke and H. Wallentowitz. Dynamik der kraftfahrzeuge, volume 4. Springer, 1972.Google Scholar
  26. [26] E. Ono, S. Hosoe, H. D. Tuan, and S. Doi. Bifurcation in vehicle dynamics and robust front wheel steering control. IEEE transactions on control systems technology, 6(3):412-420, 1998.Google Scholar
  27. [27] H. Pacejka. Tire and vehicle dynamics. Elsevier, 2005.Google Scholar
  28. [28] M. Ploechl and J. Edelmann. Driver models in automobile dynamics application. Vehicle System Dynamics, 45(7-8):699-741, 2007.Google Scholar
  29. [29] G. Reymond, A. Kemeny, J. Droulez, and A. Berthoz. Role of lateral acceleration in curve driving: Driver model and experiments on a real vehicle and a driving simulator. Human factors, 43(3):483-495, 2001.Google Scholar
  30. [30] S. Shen, J. Wang, P. Shi, and G. Premier. Nonlinear dynamics and stability analysis of vehicle plane motions. Vehicle System Dynamics, 45(1):15-35, 2007.Google Scholar
  31. [31] S. Shi, L. Li, X. Wang, H. Liu, and Y. Wang. Analysis of the vehicle driving stability region based on the bifurcation of the driving torque and the steering angle. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 231(7):984-998, 2017.Google Scholar
  32. [32] S. H. Strogatz. Nonlinear Dynamics and Chaos with Student Solutions Manual: With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press, 2018.Google Scholar
  33. [33] S. Tousi, A. Bajaj, and W. Soedel. Finite disturbance directional stability of vehicles with human pilot considering nonlinear cornering behavior. Vehicle System Dynamics, 20(1):2155, 1991.Google Scholar
  34. [34] H. True. On the theory of nonlinear dynamics and its applications in vehicle systems dynamics. Vehicle System Dynamics, 31(5-6):393-421, 1999.Google Scholar
  35. [35] X. Wang and S. Shi. Analysis of vehicle steering and driving bifurcation characteristics. Mathematical Problems in Engineering, 2015, 2015.Google Scholar
  36. [36] X. Wang, S. Shi, L. Liu, and L. Jin. Analysis of driving mode e ect on vehicle stability. International journal of automotive technology, 14(3):363-373, 2013.Google Scholar
  37. [37] D. H. Weir. Models for steering control of motor vehicles. In Proceedings, Fourth Annual NASA-University Conference on Manual Control, 1968, pages 135-169. US Government Printing Office 1968Google Scholar
  38. [38] A Lattuada, G Mastinu, G Matrascia, Tire Ply-Steer, Conicity and Rolling Resistance – Analytical Formulae for Accurate Assessment of Vehicle Performance during Straight Running, Technical Paper 2019-01-1237 ISSN 0148-7191, DOI:, Published April 2, 2019 by SAE International in United States
  39. [39] Lindenmuth BE. Tire conicity and ply steer effects on vehicle performance. SAE Technical Paper 740074. 1974Google Scholar
  40. [40] Topping RW. Tire induced steering pull. SAE Technical Paper 750406. 1975Google Scholar
  41. [41] Matyja FE. Steering pull and residual aligning torque. Tire Science and Technology. 1987 Jul-Sep;15(3):207–240Google Scholar
  42. [42] Y. A. Kuznetsov.Elements of applied bifurcation theory, volume 112. Springer Science &Business Media, 2013Google Scholar
  43. [43] A. Dhooge, W. Govaerts, and Y. A. Kuznetsov. Matcont: a matlab package for numerical bifurcation analysis of odes. ACM Transactions on Mathematical Software (TOMS),29(2):141–164, 2003.Google Scholar
  44. [44] K Yoshimoto, X Tang, Research of Driver Assistance System for Recovering Vehicle Stability from Unstable States, SAE Technical Paper2001-01-1276, DOI:

Copyright information

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020

Authors and Affiliations

  • Danilo Biggio
    • 1
    • 2
  • Fabio della Rossa
    • 1
  • Marco Fainello
    • 3
  • Giampiero Mastinu
    • 4
  1. 1.Politecnico di MilanoMilanoItaly
  2. 2.Ferrari GESMaranelloItaly
  3. 3.Danisi EngineeringModenaItaly
  4. 4.Department of Mechanical EngineeringPolitecnico di MilanoMilanoItaly

Personalised recommendations