Page - 76 - in Maximum Tire-Road Friction Coefficient Estimation

4 Sensitivity Analysis ities with the minimum turning circleRmin such that ω¯z = Rmin vmaxx · bωz and (4.21) ∂ω¯z ∂µ = Rmin vmaxx · ∂bωz ∂µ . (4.22) Accordingly, the values of the four wheel angular speeds ωi are normalised by the ref- erence effective tire radius re,i. For zero longitudinal slip at the wheel, the longitudinal velocity of the wheel hub is proportional to the effective tire radius and the wheel’s angular speed. Thus, the normalised ω¯i is physically equivalent to vx, and its sensitivity reads ω¯i = ωi ·rr,i vmaxx and (4.23) ∂ω¯i ∂µ = rr,i vmaxx · ∂ωi ∂µ . (4.24) Thereferenceeffective tire radiusrr,i is calculated for static tire loadsbasedonEquation 3.23. Finally, the normalised state variables and their normalised sensitivities read z¯ = N ·z and (4.25) ∂z¯ ∂µ = N · ∂z ∂µ , (4.26) where thenormalisationmatrixN is adiagonalmatrixwhoseprincipaldiagonal consists of the elements 1vmaxx · [ 1 1( lr Rmin +∆βmax ) Rmin rr,fl rr,fr rr,rl rr,rr]. 4.5. Results and discussion of sensitivity analysis As discussed in Section 4.1.1, the sensitivity p shows the relation between the change of the state variable z with respect to the change of the investigated parameter in the investigated case µmax. This means that the sign of the resulting sensitivity takes into account the relation of the changes in both z andµmax for the investigated data point. Figure 4.7 shows how the non-normalised sensitivities of all state variables change with µmax for two different longitudinal accelerations during a manoeuvre with pure longitudinal acceleration. With increasing longitudinal acceleration, the absolute value ofthesensitivity increases. This is true forallof thestatevariableszandforall simulated 76