Vehicle Dynamics Control and Controller Allocation for Rollover Prevention Brad Schofield, Tore H¨ agglund and Anders Rantzer Department of Automatic Control, Lund University Box 118, SE 221 00 Lund, Sweden {brad, tore, rantzer}@control.lth.se Abstract—Vehicle rollover accidents are a particularly dan- gerous form of road accident. Commercial vehicles are espe- cially prone to rollover accidents due to their high centres of gravity. A nonlinear control strategy is presented which guarantees asymptotic tracking of a yaw rate reference while bounding the roll angle, thus preventing rollover. A new computationally–efficient control allocation strategy is used to map controller commands to braking forces, taking into account actuator constraints. Simulations show that the strategy is capable of preventing rollover of a commercial van during various standard test manoeuvres. I. INTRODUCTION The use of active safety systems in road vehicles is widespread, and most modern passenger vehicles are equipped with Anti–lock Braking Systems (ABS) to improve braking performance. Electronic Stability Programs (ESP), used to prevent skidding are also widely available. These sys- tems help to prevent accidents by reducing braking distances and stabilizing yaw motion. However, so–called ‘untripped’ rollover accidents, caused by extreme manoeuvring at high speeds, require a new form of active safety system. Several algorithms have been suggested for different vehicle classes; for example, the case of Sports Utility Vehicles (SUVs) is studied in [1] and [2]. A common theme with these algorithms is the assumption that wheel lift-off is the critical situation which should be avoided. However, wheel lift-off may occur on at least one axle without a resulting loss of roll stability. In this paper a new approach is presented, based on limiting the roll angle while following a yaw rate reference. The actuators are the individual braking forces, so the strategy may easily be implemented on real vehicles equipped with braking systems allowing individual brake force assignment. II. VEHICLE MODELS Vehicle models typically consist of two components, a chassis model which describes the dynamics of the vehicle, and a tire model which describes the forces generated at the contact point between the tire and the road. A. Tire Models In order for a tire to produce a force, slip must occur. Longitudinal force F x is produced by the longitudinal slip λ, and the lateral force F y is produced by the lateral slip α. For small slip values, the relationships are approximately linear. For larger values, the forces saturate. The maximum F y,max F x,max F y F x Fig. 1. The friction ellipse, showing maximum lateral and longitudinal forces, the resultant force and its components achievable longitudinal force is given by F x,max = µF z , where µ is the coefficient of friction between the tire and the road and F z is the normal force on the tire. The maximum available lateral force for a given lateral slip can be described by the so–called Magic Formula [3], which is a function of α and F z . In the case of combined slip, where lateral and longitudinal slip occur simultaneously, a simple model for the resulting force is the friction ellipse, illustrated in Figure 1. The ellipse is described by the equation  Fy Fy,max  2 +  Fx Fx,max  2 =1. B. Chassis Model In order to adequately describe the dynamics of the vehicle, a nonlinear two–track model with roll dynamics can be used. Figure 2 illustrates the model in the vertical plane. The suspension is modelled by a torsional spring–damper system, with roll stiffness C φ and damping K φ . The model is given by [3]:       m 0 -mhφ 0 0 0 m 0 mh 0 -mhφ 0 I z I z θ r - I xz 0 0 mh I z θ r - I xz I x + mh 2 K φ 0 0 0 0 1             ˙ v x ˙ v y ¨ ψ ¨ φ ˙ φ       (1) =       F xT + m ˙ ψv +2mh ˙ φ ˙ ψ F yT - m ˙ ψu + mh ˙ ψ 2 φ M T - mhv ˙ ψφ -mhu ˙ ψ +(mh 2 + I y - I z ) ˙ ψ 2 φ - (C φ - mgh)φ ˙ φ      