Page - 33 - in Maximum Tire-Road Friction Coefficient Estimation

2 Estimation of the friction potential Table 2.2.: Mathematical and physical classification of vehicle dynamics based methods to estimate the friction potential with selected examples based on Lex et al., [LEH11] Slip based (long., lat.) Other physical quantities Algebraic Holzinger, [Hol92, p.18-34] Holzinger, [Hol92, p.35-46] Rajamani et al., [RPLG06] Villagra et al., [VdFM11] Statistical Ray, [Ray97] (Bayesfilter) Gustafsson, [Gus97] Boßdorf-Zimmer et al., [BZFHK07] Observer Ahn et al., [APT09] Hsu et al., [HLGG06] Optimization Uchanski, [Uch01] Ivanov et al., [ISAA10] Lee et al., [LHY04] Ding et al., [DT10] Svendenius, [Sve07] Lex et al., [LKE13a] Dieckmann showedthat for small valuesofsx, the slipneeded inorder to transmit the same longitudinal tire force Fx is higher for lower µ max, [Die92, p.32-45]. This means that he empirically proved a correlation between the initial slip slope k|sx=0 and the friction potentialµmax, which can be exploited to estimateµmax for small values of the longitudinal slip sx. Dieckmann also proposed an approach for calculating so-called micro-slip values of sx< 0.1 % using only wheel speed sensors by summing-up the cal- culated slip over several wheel revolutions, [Die92, p.19-22, 110]. For a reference value of vx, the wheel speed sensors of non-driven wheels are used. This can only be done when there is no wheel torque (e.g. no wheel slip), on one axle, which is only true for vehicles withonedrivenaxle andwhennobraking torque is applied. Inaddition, the stateof the velocity should not change quickly to make it possible to observe the wheel speed differ- ences on the front and rear axles for several revolutions, [Uch01, p.132]. Additionally, the implementation in practice is difficult due to measurement noise and uncertainty, since this effect is very small. Gustafsson addressed this problem by implementing a Kalman filter, [Gus97]. He used a linear relation between the initial slip slope k|sx=0 and the demanded coefficient of frictionµD in the form ofµD=k|sx=0 ·(sx+δ). This relation includes the longitudinal slip sx and a measurement offset δ that is estimated at the same time as k|sx=0. Applying Dieckmann’s findings that the friction potential is a function of the estimated k|sx=0, the relevantµmax can be assigned when additional a priori knowledge (e.g. a look-up table) is available for µmax (k|sx=0). Unfortunately, 33