Nonlinear Dynamics

, Volume 92, Issue 1, pp 97–106 | Cite as

Analysis of the lateral dynamics of a vehicle and driver model running straight ahead

  • Fabio Della Rossa
  • Gianpiero Mastinu
Original Paper


In this paper, we show that even an extremely simple nonlinear vehicle and driver model can show complex behaviors, like multi-stability and sensible dependence on the initial condition. The mechanical model of the car has two degrees of freedom, and the related equations of motion contain the nonlinear characteristics of the tires. The driver model is described by a single (nonlinear) equation, characterized by three parameters that describe how the driver steers the vehicle. Namely such parameters are the gain (steering angle per lateral deviation from desired path), the preview distance, and the reaction time delay. Bifurcation analysis is adopted to characterize straight ahead motion at different speeds, considering separately the two cases of understeering or oversteering cars. In the first case, we show that at suitable speeds the model can have three different attracting oscillating trajectories on which the system can work and that are reached due to different disturbances. In the second case, we confirm that instability arises if the forward speed is too high. The final results of the paper, bifurcation diagrams, can be used for many considerations critical both from the theoretical and from the practical viewpoints.


Vehicle system dynamics Driver Multi-stability Bifurcation analysis Nonlinear systems 


  1. 1.
    Andrzejewski, R., Awrejcewicz, J.: Nonlinear Dynamics of a Wheeled Vehicle, vol. 10. Springer Science & Business Media, Berlin (2006)zbMATHGoogle Scholar
  2. 2.
    Apel, A., Mitschke, M.: Adjusting vehicle characteristics by means of driver models. Int. J. Veh. Des. 18, 583–596 (1997)Google Scholar
  3. 3.
    Della Rossa, F., Mastinu, G., Piccardi, C.: Bifurcation analysis of an automobile model negotiating a curve. Veh. Syst. Dyn. 50, 1539–1562 (2012)CrossRefGoogle Scholar
  4. 4.
    Plochl, M., Edelmann, J.: Driver models in automobile dynamics application. Veh. Syst. Dyn. 45(7–8), 699–741 (2007)CrossRefGoogle Scholar
  5. 5.
    Della Rossa, F., Gobbi, M., Mastinu, G., Piccardi, C., Previati, G.: Bifurcation analysis of a car and driver model. Veh. Syst. Dyn. 52, 142–156 (2014)CrossRefGoogle Scholar
  6. 6.
  7. 7.
  8. 8.
    Lozia, Z.: Modelling and simulation of a disturbance to the motion of a motor vehicle entering a skid pad as used for tests at driver improvement centres. In: Archiwum Motoryzacji vol. 69 PIMOT, Warsaw University of Technology (2015)Google Scholar
  9. 9.
    Pacejka, H.: Tire and Vehicle Dynamics, 2nd edn. Elsevier, Amsterdam (2006)Google Scholar
  10. 10.
    Mastinu, G., Plöchl, M.: Road and Off-Road Vehicle System Dynamics Handbook. CRC, Boca Raton (2014)Google Scholar
  11. 11.
    Mitschke, M., Wallentowitz, H.: Dynamik der Kraftfahrzeuge, 4th edn. Springer, Berlin (2004)CrossRefGoogle Scholar
  12. 12.
    Gillespie T.D.: Fundamentals of vehicle dynamics. SAE International (1992)Google Scholar
  13. 13.
    Abe, M.: Vehicle Handling Dynamics. Elsevier, Oxford (2015)Google Scholar
  14. 14.
    Jazar, R.N.: Vehicle Dynamics: Theory and Application. Springer, Berlin (2008)CrossRefGoogle Scholar
  15. 15.
    Andrzejewski, R., Awrejcewicz, J.: Nonlinear Dynamics of a Wheeled Vehicle. Springer, Berlin (2005)zbMATHGoogle Scholar
  16. 16.
    Guiggiani, M.: The Science of Vehicle Dynamics. Springer, Berlin (2013)Google Scholar
  17. 17.
    Genta, G.: The Automotive Chassis. Springer, Berlin (2016)Google Scholar
  18. 18.
    Reimpell, J., et al.: The Automotive Chassis. Butterworth-Heineman, Oxford (2001)Google Scholar
  19. 19.
    Crolla, D.: Encyclopedia of Automotive Engineering. Wiley, Chichester (2015)Google Scholar
  20. 20.
    Dixon, J.: Tires. Suspension and Handling. Society of Automotive Engineers, Warrendale (1996)Google Scholar
  21. 21.
    Della Rossa, F., Sukharev, O., Mastinu, G.: Straight ahead running of a nonlinear car and driver model. In: AVEC, Munich (2016)Google Scholar
  22. 22.
    Dhooge, A., Govaerts, W., Kuznetsov, Y.A.: MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw. 29, 141–164 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Doedel, E.J., Champneys, A.R., Fairgrieve, T., et al.: AUTO-07P: continuation and bifurcation software for ordinary differential equations. Concordia University, Montreal (2007)Google Scholar
  24. 24.
    Kuznetsov, YuA: Elements of Applied Bifurcation Theory. Springer, New York (2004)Google Scholar
  25. 25.
    Mastinu, G., Della Rossa, F., Piccardi, C.: Nonlinear dynamics of a road vehicle running into a curve. Appl. Chaos Nonlinear Dyn. Sci. Eng. 2, 125–153 (2012)MathSciNetGoogle Scholar
  26. 26.
    Pacejka, H.: Simplified analysis of steady-state turning. Veh. Syst. Dyn. 21, 269–296 (1973)Google Scholar
  27. 27.
    Gobbi, M., Mastinu, G., Previati, G., De Filippi, R., Lunetta, I., Moscatelli, D.: Optimal design of an electric power steering system for sport cars. In: 23rd International Symposium on Dynamics of Vehicles on Roads and Tracks, 19–23 Aug Qingdao, China (2013)Google Scholar
  28. 28.
    True, H.: Multiple attractors and critical parameters and how to find them numerically: the right, the wrong and the gambling way. Veh. Syst. Dyn. 51(3), 443–459 (2013)CrossRefGoogle Scholar
  29. 29.
    Liu, Z., Payre, G., Bourassa, P.: Stability and oscillations in a time-delayed vehicle system with driver control. Nonlinear Dyn. 35, 159–173 (2004)CrossRefzbMATHGoogle Scholar
  30. 30.
    Liu, Z., Payre, G., Bourassa, P.: Nonlinear oscillations and chaotic motions in a road vehicle system with driver steering control. Nonlinear Dyn. 9, 281–304 (1996)CrossRefGoogle Scholar
  31. 31.
    Tang, T., Li, C., Huang, H., Shang, H.: A new fundamental diagram theory with the individual difference of the driver’s perception ability. Nonlinear Dyn. 67, 2255–2265 (2012)CrossRefGoogle Scholar
  32. 32.
    Yi-Rong, K., Di-Hua, S., Shu-Hong, Y.: A new car-following model considering driver’s individual anticipation behavior. Nonlinear Dyn. 82, 1293–1302 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Wen, H., Rong, Y., Zeng, C., Qi, W.: The effect of driver’s characteristics on the stability of traffic flow under honk environment. Nonlinear Dyn. 84, 1517–1528 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Politecnico di MilanoMilanItaly

Personalised recommendations