Page - 139 - in Maximum Tire-Road Friction Coefficient Estimation

C Vehicle model structure for sensitivity analysis C.1.1. Find maximum out of two values Within the calculation of the longitudinal slip, see Equation 2.3, it is necessary to select the bigger variable out of two variables u andw. This function is discontinuous at the transition betweenu andw. The derivable numerical approximation given by max(u,w) = w 2 · ( 1+ 2 pi ·arctan((w−u) · ) ) + u 2 · ( 1+ 2 pi ·arctan((u−w) · ) ) (C.2) is used instead. The accuracy can be increased by the factor which was set at 105 within the simulation. C.1.2. Absolute value Calculatinganabsolutevalueofavariablew isnecessary forseveralcalculations(e.g. the longitudinal slip with Equation 2.3). As conditional programming (e.g. if statements) are also not supported within Adimat, the formulation |w|= √ w2 (C.3) is used. Although this formulation is still discontinuous at zero, it can be used within Adimat, as it is a weak derivative. 139