Page - 77 - in Maximum Tire-Road Friction Coefficient Estimation

4 Sensitivity Analysis µmax. It has to be mentioned that high values of bax and bay cannot be reached at low µmax. For this reason, these conditions cannot be displayed. 10.5 0 1 2 3 4 5 6 x10 -4 µmax 0.5 1 0 0.5 1 1.5 2 x10 -3 µmax b a x =1.5, b a y =0 b a x =2, b a y =0 0.5 1 -4 -3 -2 -1 0 1 2x10 -11 µmax 0.5 1 0 0.5 1 1.5 2 x10 -3 µmax 0.5 1 -1 -0.5 0 0.5 1 1.5 2 x10 -12 µmax 0.5 1 -8 -6 -4 -2 0 µmax x10-2 0.5 1 -8 -6 -4 -2 0 µmax x10-2 Figure 4.7.: Relation between non-normalised sensitivity of state variables and the fric- tion potentialµmax for two selected longitudinal accelerations. The units of the shown sensitivities are the same as for their respective state variable. Asexpected, atall fourwheels theabsolutevalueof thesensitivitiesof thewheel speed is higher for lowµmax and decreases with increasingµmax. This can be explained by the higherwheel’s rotationalaccelerationsat lowerµmax at the same bax. Theabsolutevalue of the sensitivities for the front wheels is higher than for the rear wheels. For a front- wheel drive (FWD), this can be explained by the high driving torque on the front axle and the comparably low rolling resistance torque on the rear axle. The different signs on frontandrearwheels follows fromthe signof the sumof torques. Thesensitivity∂vx/∂µ for the longitudinal speed shows the same tendency as those of the rear wheels, but the absolute value is several orders of magnitude lower. This is due to the fact that the inertia of the chassis is higher than that of the wheels, which results in a slower response to dynamic changes. This already indicates that the longitudinal velocity is not the best indicator for a change of µmax for pure longitudinal accelerations. As expected, during purely longitudinal excitation, the sensitivities of the lateral velocity ∂vy/∂µ and the yaw rate ∂bωz/∂µ can be omitted due to the small contribution. Figure 4.8 shows the non-normalised sensitivities of the state variables with respect to µmax for two combined longitudinal and lateral accelerations. The sensitivities for the state variables vx, ωfl, ωfr, ωrl and ωrr show a similar behaviour as the purely longitudinal acceleration manoeuvre in Figure 4.7, although the orders of magnitude 77