Nonlinear Dynamics

, Volume 40, Issue 4, pp 339–365 | Cite as

Stability and Bifurcation of Longitudinal Vehicle Braking

  • B. J. Olson
  • S. W. Shaw
  • G. Stépán


The longitudinal braking dynamics of a two-wheel vehicle model on an incline are considered using techniques from nonlinear dynamics. The model is planar and incorporates the coupled dynamics of two independently braked wheels and the vehicle body, and takes into account the slip dynamics of each wheel. By using the wheel slip values and the vehicle speed as dynamic states, it is shown that the qualitative behavior of the system can be completely captured by studying a relatively simple phase plane problem described in terms of the slip values. A systematic bifurcation analysis is carried out in which the brake torques of the two wheels are varied, and it is shown how the system transitions from stable braking to the possibility of lockup in one or both wheels, to guaranteed lockup in both wheels. In this manner a quite complete picture of the dynamic behavior is obtained as a function of the two brake torques, including regions with multiple possible steady braking outcomes, depending on the initial conditions. This analysis provides new insights into the dynamics of vehicle braking, and it provides a correction to the standard result for the critical values of the brake torques at which the wheels undergo lockup. This approach may also prove useful for evaluating brake proportioning schedules, or for investigating anti-lock braking systems and other methods of traction control.

Key words

bifurcation lockup instability traction vehicle braking wheel slip 



antilock brake system


traction control system


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Olson, B., Shaw, S., and Stépán, G., ‘Nonlinear dynamics of vehicle traction’, Vehicle System Dynamics 40(6), 2003, 377–399.CrossRefGoogle Scholar
  2. 2.
    Gillespie, T., Fundamentals of Vehicle Dynamics, Warrendale, Society of Automotive Engineers (SAE), 1992.Google Scholar
  3. 3.
    Wong, J., Theory of Ground Vehicles, Wiley, New York, 1978.Google Scholar
  4. 4.
    Olson, B., ‘Nonlinear dynamics of longitudinal ground vehicle traction’, MS Thesis, Michigan State University, East Lansing, MI, 2001.Google Scholar
  5. 5.
    SAE Standard, ‘Antilock brake system review’, SAE Paper No. Iaa46, pp. 90–102, 1992.Google Scholar
  6. 6.
    Harned, J., Johnston, L., and Scharpf, G., ‘Measurement of tire brake force Characteristics as Related to Wheel Slip (Antilock) Control System Design’, SAE Paper No. 690214, pp. 909–925, 1969.Google Scholar
  7. 7.
    Goodenow, G., Kolhoff, T., and Smithson, F., ‘Tire-road friction measuring system-a second generation’, SAE Paper No. 680137, pp. 571–579, 1968.Google Scholar
  8. 8.
    Bakker, E., Nyborg, L., and Pacejka, H., ‘Tyre modelling for use in vehicle dynamics studies’, SAE Paper No. 870421, pp. 190–204, 1987.Google Scholar
  9. 9.
    Bakker, E., Pacejka, H., and Lidner, L., ‘A new tire model with and application in vehicle dynamics studies’, SAE Paper No. 890087, pp. 101–113, 1989.Google Scholar
  10. 10.
    Liu, Y. and Sun, J., ‘Target slip tracking using gain-scheduleing for antilock braking systems’, in Proceedings of the American Control Conference, Vol. 2, pp. 1178–1182, Seattle, WA, 1995.Google Scholar
  11. 11.
    Strogatz, S., Nonlinear Dynamics and Chaos, Addison-Wesley, Reading, MA, 1994.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringMichigan State UniversityEast LansingUSA
  2. 2.Department of Mechanical EngineeringUniversity of CaliforniaSanta BarbaraUSA
  3. 3.Department of Applied MechanicsBudapest University of Technology and EconomicsBudapestHungary

Personalised recommendations