Vehicle Roll Dynamics

  • Reza N. Jazar


The roll vehicle dynamic model is well expressed by four kinematic variables: the forward motion x, the lateral motion y, the roll angle φ, and the yaw angle ψ, plus four equations for the dynamics of each wheel. In the roll model of vehicle dynamics, we do not consider vertical movement z and pitch motion θ. The model of a rollable rigid vehicle is more exact and more realistic compared to the vehicle planar model. Using roll dynamic model, we are able to analyze the roll behavior of a vehicle in maneuvers. Angular orientation of the vehicle is expressed by three angles: roll φ, pitch θ , and yaw ψ, and the vehicle angular velocities are expressed by their rates: roll rate p, pitch rate q, and yaw rate r.

A rolled vehicle introduces new reactions in the tires of the vehicle that must be considered in the development of the dynamic equations of motion. The most important reactions are:
  1. 1–

    Roll-thrust F. An extra lateral force appears because of the vehicle roll. Tire roll-thrust is assumed to be proportional to the vehicle roll angle φ.

  2. 2–

    Roll-steer angle δφ.


An extra steer angle appears because of the wheel roll. The roll steer is a result of suspension mechanisms that provide some steer angle when the vehicle rolls and the mechanism deflects. The wheel roll steering is assumed to be proportional to the vehicle roll angle φ. Therefore, the actual steer angle δa of such a tire will be δa = δ + δφ.

In this chapter we introduce bicycle as well as four-wheel roll models with independent in-wheel motors. The four-wheel roll vehicle model is the best practical vehicle mathematical model. This model provides us with in-wheel torques Ti, tire slips αi, si, βi, tire and vehicle forces Fx, Fy, \(F_{x_{i}}\), \(F_{y_{i}}\), \(F_{z_{i}}\), velocity components of the vehicle vx, vy, ωi, as well as yaw and roll angular variables φ, ψ, p, r. This model is an extension to the two-wheel roll vehicle model to include the lateral weight transfer as well as roll effects on vehicle dynamics. The four-wheel roll vehicle model is an excellent model to simulate drifting of vehicles.


Tire force system Vehicle roll dynamics Bicycle roll model Four-wheel roll vehicle model Drift model In-wheel motors vehicle dynamics 


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Reza N. Jazar
    • 1
  1. 1.Aerospace, Mechanical and Manufacturing EngineeringRMIT UniversityMelbourneAustralia

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