Page - 69 - in Maximum Tire-Road Friction Coefficient Estimation

4 Sensitivity Analysis 0 1 1.5 0 1 2 3 b a x in m2/s f s =0.2 f s =0.1 f s =0.3 -0.5 0 1 1.5 0 1 2 3 4 Time 0.5 Time f s =0.2 f s =0.1 f s =0.3 f s =0.2 f s =0.1 f s =0.3 0.5 Figure 4.1.: Left: Representation of the longitudinal versus the lateral acceleration in the vehicle’s COG with elliptic characteristic. The simulated manoeuvre starts with pure positive longitudinal acceleration, followed by a combined longitudinal and lateral acceleration state. The scaling factor fs is used to scale thedesiredaccelerationprofiles. Topright: Longitudinalacceleration version timetoachieve thedriving states shownonthe left. Bottomright: Lateral acceleration versus time. Time dependence of sensitivity Simulations with different acceleration profiles versus time have to be compared in order to be able to evaluate many states within the parameter space. The higher the desired acceleration is set, the faster a certain state is reached, as shown in Figure 4.2. As mentioned before, the time at which a certain state (in this case a certain longitudinal and lateral acceleration) is reached must not influence the calculated sensitivities p. Thus, the sensitivity p = ∂z/∂µ calculated for a certain ba scaled with fs = 1 at t1 has to be the same as the one for fs = 0.5 at t2. For example, Figure 4.3 shows the driving state with pure bax of 0.45 m/s 2 and bay = 0. It is evident that the selected sensitivities∂vx/∂µand∂ωfl/∂µhavecongruentcharacteristics independenceofµ max at the investigated longitudinal acceleration of 0.45 m/s2, no matter which scaling factor fs is used for the desired acceleration profile during the simulation manoeuvre. Both the tendency and the absolute values at a certain µmax are similar for all simulations. This behaviour is the same for the all sensitivities pl and for all acceleration points not displayed. Thus, time-dependence can be excluded. 69