Page - 100 - in Maximum Tire-Road Friction Coefficient Estimation

5 Tire/road friction estimator Observed system (vehicle) Observer d(k) z(k) z(k) . x(k)ˆ x(k)=f(x(k-1),u(k),d(k)) z(k)=h(x(k),u(k),v(k)) x(k)=g(x(k-1),w(k), z(k),z(k),u(k)) ˆ ˆ u(k) .δfr δ fl F x,flF y,fl F x,rl F y,rl F x,rr F y,rr F y,fr F x,fr Figure 5.3.: Observed system (vehicle) and non-linear observer with the vector u(k) of the driver’s input (steering wheel angle δS and wheel torqueMD,i) and the disturbance vector d(k) that are input to the real vehicle, and its outputs z(k) and z˙(k), which are measured and fed into the observer model; based on Simon, [Sim06]. Within this model, the internal state xˆ(k) is estimated. estimated based on the longitudinal tire forces, as shown in Section 5.3. ForN particles ofµmax, the hypotheses of the longitudinal tire forces are calculated at each wheel iwith therespective tire loadFz,i, the longitudinal slipsx,i andtheside slipangleαidepending onthecurrent inputsat timestepk. Comparing this toa longitudinal forcederived from the wheel’s angular momentum with the inputs at time step k, see Equation 5.11, the most likelyestimateofµmaxi foreverywheel is calculatedwiththeparticlefilterdescribed inSection5.2. Inaddition, aglobalµmax canbecalculatedusingEquation2.8. This can bedonebydirectlyapplyingEquation2.8 to theprior estimatedµmaxi orbyobservingall four wheel states at the same time, i.e. setting h(x(k)) = [ Fx,fl Fx,fr Fx,rl Fx,rr ]T as a function of oneµmaxg rather than four wheel-individualµ max i . The results from the two approaches do not vary significantly. 5.4.1. Measurement inputs For an estimation method based on the vehicle dynamics, the theoretical achievable accuracy of an estimate of µmax depends on the dynamic excitation during a manoeu- vre. The closer to the physical limits of road adhesion, the more accurate an estimate of µmax can be, see Section 6.1. Apart from this theoretical accuracy, the effectively 100