Page - 96 - in Maximum Tire-Road Friction Coefficient Estimation

5 Tire/road friction estimator re-sampling p(z(k) x(k))— x h (k) x h (k) p(x(k-1)) + —b) Particles: a) PDF: moving samples x Figure 5.1.: a)Probabilitydensity functions(PDF)ofthe initial statep(x(k−1)) inblack and the measurement likelihood p(z(k)|x(k)) in gray, b) Particles x−h(k) distributed based onp(x(k−1)) before re-sampling and new particlesx+h(k) randonmly generated during re-sampling based on q¯h, which depends on p(z(k)|x(k)). Graphic representation based on Watzenig, [Wat06, p.68, 71]. 5.3. Choice of observer model The following sections describe the application of the theoretical considerations on state estimation from Sections 5.1 and 5.2 on the estimation of the friction potential. As a first step, a comparison is given of the state of the art on estimators for the friction potential that are based on observers within the Bayesian framework. In a second step, bothasuitable state estimationmethodandasuitablemodel tobeobservedare selected based on the results of the sensitivity analysis (cf. Section 4). In existing works, observers within the Bayesian framework have already been suc- cessfully implemented for non-series application and have shown promising results for certain driving states (e.g. braking, cornering) or certain road conditions (e.g. only road surfaceswith low frictionpotential). Anoverview isgiven inSection2.2.2. Inparticular, the approach used by Ray, [Ray97], which can be described as a particle filter without the re-sampling step, has attracted attention due to its ability to deal with non-linear systems while still having the advantages of treating state models under uncertain mea- surements (e.g. as with a Kalman filter). Within Ray’s proposed approach, see also Section 2.2.2, the tire forces are estimated using an extended Kalman filter. These es- timated tire forces are then treated as the measurement input for the particle filter, in whichµmax is then estimated, [Ray97]. This means that the measurement input for the particle filter is also an uncertain estimate. The results from the sensitivity analysis 96