Page - 68 - in Maximum Tire-Road Friction Coefficient Estimation

4 Sensitivity Analysis of the vehicle model, i.e. the four wheel torques and the front wheels’ steering angles, to reach different combinations of longitudinal and lateral accelerations within the pa- rameter space. For the longitudinal and the lateral desired acceleration profile, separate proportional-integral-derivative controllers (PID) was used. This enabled reproducible results of driving states for different friction potentials µmax within the physical lim- its as well. A wide range of possible accelerations are covered in the analyses. Thus, manoeuvres are defined that start with either pure longitudinal or lateral acceleration and then traverse the parameter space of combined accelerations in ellipses. Figure 4.1 shows the manoeuvre to cover positive bax and bay by starting with a pure longitudinal acceleration until a predefined value is reached. Then, the wheels’ torques and the front wheels’ steering angles are controlled such that an elliptic acceleration characteristics is achieved, as shown inFigure4.1 (left). Both thepredefinedvalueafterpure longitudinal or lateral acceleration and the ellipses are varied by a factor fS in order to reach many conditions in the parameter space, see gray lines in Figure 4.1 as two examples. With a scaling factor fS = 1, the maximum accelerations are simulated as defined by the Krempel diagram, as shown in Figure 2.5 with the maximum values presented in Table 4.1. The temporal representation of the desired values of bax and bay for the controller are shown in Figure 4.1 (right). These manoeuvres are simulated for both positive and negative bax. To include manoeuvres with values of bay while bax ≈ 0, a manoeuvre similar to the proposed one was added, whereby bay is first increased while bax remains at zero, and then elliptic behaviour towards positive or negative bax is reached. 4.3.2. Validity and limitations Both the validity and the limitations of the results of the sensitivity analysis depend on the proposed procedure, including the simulated manoeuvres. To find variable sets sensitive toµmax thatonlydependonthecurrentdrivingstates,firstofall the sensitivity of the state variable must not depend on the time when the driving states are reached (no time-dependence). In addition, the influence of the initial longitudinal velocity vx,0 at which the simulation is started has to be known. Finally, it has to be assured that the tendencies of the sensitivities of the manoeuvres that start with longitudinal excitation deliver comparable results to those that start with lateral excitation, in order to be able to compare the results. 68