Page - 110 - in Maximum Tire-Road Friction Coefficient Estimation

6 Results and conclusion 5 5.5 6 6.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Reference µmax(fixed particles), no resampling Time in s Figure 6.3.: Estimate of µmax for a driving manoeuvre with a longitudinal acceleration of bax≈ 4.5 m/s2 with fixed particles ofµmax for all time steps. The mean absolute error (MAE) of the estimate is 0.1725 and the maximum absolute error of 0.7860 occurs around second 5, see also Table 6.1. 6.1.2. Results with resampling step With the resampling step described in Section 5.2 and the particles moving towards the most probable condition shown in Figure 5.1, convergence towards the most likely state can be achieved by smoothing short-term outliers. Figure 6.4 shows how the particles converge with time (top) for the manoeuver shown in Figure 6.2 and how, in contrast, the particles of the particle filter without the resampling step (fixed particles, bottom) stay constant over time. It has to be mentioned that regardless of whether a particle is fixed or variable, its PDF changes with time. Figure 6.5 shows three exemplary results of the estimation with µmax for the same measurement input. As the resampling strategy is subject to chance, the outcome of every experiment or run on the same measurement input is slightly different. Thus, three different outcomes are shown as examples. It can be seen that the resampling step delivers a smooth and accurate estimate. When includingaresamplingstep, themaximumdeviationsbetweenthereferenceand the estimatedµmax are small. For the examples shown in Figure 6.5, the mean absolute error (MAE) is less than 0.08, and the absolute error is still less than 0.15, see Table 6.1. There is a reason that all three examples converge to different values even though 110