Page - 60 - in Maximum Tire-Road Friction Coefficient Estimation

3 Vehicle model tion. Accordingly, the basis valueFby is calculated using the transformed slip quantity |s| ·Gs. Fbx Fby 0 pi/2 F β s s y FF b x FbyF x F y β s F F s s x Figure 3.12.: Combined tire force interpolation according to Hirschberg, with character- istics Fx and Fy for either pure longitudinal or lateral tire forces. The resulting combined tire contact force F is assumed to act in the opposite direction as the slip vector s and is thus defined by βs. The magnitude of the combined force F = |F| depends on the basis values Fbx and Fby, [HRW07]. The angle βs describes the ratio between the longitudinal and the lateral combined slip and is given by βs = arctan ( sy sx ) . With the basis values of the tire forces, the combined value of the tire force is interpolated using the cosine function F= |F|= 1 2 [ Fbx+F b y+(F b x−Fby) ·cos(2 ·βs) ] , (3.27) which is dependent on βs. It has to be noted that the resulting combined tire contact force F is assumed to act in the opposite direction as the slip vector s (see Figure 3.12, left). Agraphical interpretationof the interpolationbasedonβs is shown inFigure3.12, right. Finally, the resulting tire force vector is given by F= [ Fx Fy ] =F · [ cosβs sinβs ] . (3.28) 3.3.2. Lateral tire dynamics The tire model presented to this point is valid for steady-state conditions. However, the horizontal tire forces react to a change in the slip quantities with a time delay. As the longitudinalstiffnesscT,xofthetire is typicallyabouttwiceas largeasthe lateral stiffness 60