Page - 97 - in Maximum Tire-Road Friction Coefficient Estimation

5 Tire/road friction estimator in Section 4 show that the wheel rotational speeds show the highest sensitivity to a change inµmax. In addition, since the wheel rotational speeds are directly measured in every vehicle with ABS, they do not require an uncertain estimate. Thus, one possible approach could be to use the wheel’s angular momentum as given in Equation 3.15 to estimate µmax. In discrete time notation, the i-th’s wheel’s rotational equilibrium for the time interval ∆t= t(k+1)− t(k) reads ωi(k+1) =ωi(k)+ 1 Ii ·(MD,i(k)−MR,i−rS,i ·Fx,i(k)) ·∆t. (5.10) Equation 5.10 is primarily determined by the difference between the momentum implied by the longitudinal tire force Fx,i which is a function of µ max and the wheel’s torque MD,i. Nevertheless, changes of ωi for typical driving states are smaller or within the same range as the accuracy with which the termMD,i(k)−MR,i−rS,i ·Fx,i(k) can be calculated, and this relation cannot be used directly. Thus, this work uses an approach related to the one proposed by Ray, [Ray97], in which the observed variables comprise the horizontal tire forces. A priori calculated values of the longitudinal tire forceFx,i are used as the measurement input for the particle filter. Within the particle filter al- gorithm, these real values are then compared toN hypothesis of the longitudinal tire forces calculated using a tire model andN different particles of µmax. The real value of the longitudinal tire force Fx,i is calculated using the discrete time notation of the wheel’s angular momentum shown in Equation 5.10 and is given by Fx,i(k) = 1 rS,i · ( MD,i(k)−MR,i−Ii · ( ωi(k+1)−ωi(k) ∆t )) . (5.11) 5.3.1. Overview on observer model With the relation in Equation 5.11, the state model for the observer can now be de- fined using the notation for the state estimation problem given in Equations 5.2 and 5.2. The state vector x is given by [ µmaxfl µ max fr µ max rl µ max rr ]T . The measurement equation z(k) is given by [ Fx,fl Fx,fr Fx,rl Fx,rr ]T and is being calculated for each wheel using Equation 5.11. Within the particle filter, the measurements z(k) are being compared to h(x(k)), assuming a multivariate Gaussian distribution (cf. Equation 5.6). The vector h(x(k)) includes the longitudinal tire forces that are calculated based on the model for combined horizontal tire forces presented in Section 3.3. It depends on the longitudinal slip sx,i, the slip angleαi, the vertical tire loadFz,i and the state vector x, i.e. µmaxi . UsingN particles for µ max i ,N hypothesis for the longitudinal tire force are 97