Page - 59 - in Maximum Tire-Road Friction Coefficient Estimation

3 Vehicle model 3.2.4. Steering model The relation between the steering wheel angle δS and the wheel steering angles δi on the four wheels is implemented using measured characteristics. As only the front wheels are steered, the steering angle on the rear wheels remains at zero. 3.3. Tire model The semi-physical tire model TMsimple, developed by Hirschberg, is used to calculate the longitudinal, lateral and combined tire forces under steady state conditions on even road and omitting wheel camber, [Hir09a]. The either pure longitudinal or pure lateral forceY acting on the tire at the wheel bottom pointW (see also Figure 2.2) is described by Y =K sin ( B · ( 1−exp−|X|A ) ·sgn(X) ) . (3.24) The slip quantityX is either the longitudinal slip sx calculated with Equation 2.3 or the lateral slip angleα calculated with Equation 2.4. The coefficientsK,B andA are not just mathematical parameters, but are related to physical quantities. They depend on the peak valueYmax, the saturation valueY∞ and the initial stiffness dY0, as shown in Appendix B.1. 3.3.1. Combined tire forces With longitudinaland lateral slipoccurringsimultaneouslyonthetire, thetransmittable forcescannotexceedthose forpure longitudinalor lateral slip. Amodel forcombinedtire forces based on physical similar slip quantities proposed by Hirschberg is used, [Hir09a]. In a first step, the tire slip angleα is transformed into an equivalent lateral tire slip sy, which is given by sy= α Gs(Fz) . (3.25) In order to achieve the same initial stiffness for longitudinal and lateral tire characteris- tics, the weighting factor is defined as Gs(Fz) = dFx0(Fz) dFy0(Fz) = Ay ·Kx ·Bx Ax ·Ky ·By . (3.26) The combined slip vector reads s = [ sx sy ]T with its absolute value |s|= √ s2x+s 2 y. The basis valueFbx for the longitudinal tire force is calculated using Equation 3.24 with |s| being the respective slip quantity, see Figure 3.12, right, for a graphical interpreta- 59