Page - 98 - in Maximum Tire-Road Friction Coefficient Estimation

5 Tire/road friction estimator calculated for each wheel. Section 5.4 provides a more detailed description of the observer and the necessary measurements as well as vehicle and tire parameters. To calculate the tire forces for theN different hypotheses or particles ofµmax, the horizontal tire model for combined forces presented in Section 3.3 is used. 5.3.2. Friction potential and longitudinal tire force Before presenting the observer structure in Section 5.4, it is important to mention some reflectionsonthe influenceofµmax onthe longitudinal tire forceFx,i. Asdiscussed inSec- tion 2.2.2, Boßdorf-Zimmer estimated both the slip angleα andµmax at the same time, [BZ07, p.84-86]. To implement the non-linear tire behaviour in an extended Kalman filter, he needed its derivatives to both α and µmax. He showed the lateral tire forces and their derivatives versus bothα andµmax, [BZ07, p.84-86]. In the present work, the longitudinal tire forces are needed rather than the lateral tire forces within the observer model, see Section 5.4. Thus, Figure 5.2 shows the non-linear behaviour ofFx,i versus both the longitudinal slip sx and the friction potential µ max. Longitudinal tire force Fx,iwas calculated using the tire model shown in see Section 3.3 and parametrised with longitudinal tire data, see Table D.3. Figure 5.2 (above) shows Fx,i and its change ∆Fx,i/∆sx with respect to sx for two µmax. It can be seen that the initial slope for both Fx,i and for ∆Fx,i/∆sx does not depend on µmax. This is due to the tire model used, which, like most existing models, does not include any dependence of the initial slip slope onµmax, although a small effect has been empirically proven by Dieckmann, [Die92, p.32-45], see also Section 2.2.2. Only for higher values of sx, does µ max show a distinguishable characteristic up until about 8 %. Figure 5.2 (bottom) shows Fx,i and its change ∆Fx,i/∆µ max with respect to µmax for two sx. In these cases, it can be seen that the behaviour with respect to sx is distinguishable. For small values of sx (e.g. 0.5 % as shown in Figure 5.2), ∆Fx,i/∆µ max is small. In contrast, the change inFx,i is higher for higher sx, as shown for 6 %. According to Boßdorf-Zimmer, this shows that it is possible to simultaneously estimateµmax and sx in an observer because the range of influence onFx,i is a different one. Although very promising, simultaneous estimation of sx was not investigated in this thesis. 98