Page - 74 - in Maximum Tire-Road Friction Coefficient Estimation

4 Sensitivity Analysis are the most sensitive for the driving state, the influences of the different state variables within one investigated region have to be made comparable. Thus, it is necessary to normalise the resulting sensitivities in order to achieve physical equivalence. One possi- bility is to normalise the sensitivities pl(t) to the value of the respective state variable zl(t). As µ max itself is dimensionless, the unit of the sensitivity ∂zl/∂µ corresponds to the unit of the respective state variable zl. The aforementioned sensitivity normalised to zl(t) is then dimensionless, but it does not provide the same orders of magnitude within the sensitivities of the different state variables. Thus, this method of normalisation is suitable when assessing the change in sensitivity of a state variable over different driving states, but it isnotusefulwhencomparingthesensitivityofdifferent statevariables fora specific driving state. Normalising the sensitivities to the maximum state variables (e.g. the maximum speed of the investigated vehicle) or the maximum sensitivities during the simulations would result in sensitivities within the same order of magnitude (especially the latter normalisation). Nevertheless, small sensitivities of small state variables (e.g. the vehicle’s yaw rate) would appear larger than can be physically explained. This also applies for the first normalisation method mentioned above. Thus, the following ap- proach based on kinematic relations is used in this work to ensure physical equivalence of the different state variables and their sensitivities. 4.4.1. Kinematic relation for normalisation The longitudinal velocity vx is used as the reference value for the other state variables in order to achieve physical equivalence within their sensitivities. As the sole variable, it is normalised to its maximum value1 vmaxx . With v max x not depending on µ max, as every velocity can be reached within enough time even on low-friction surfaces, the normalisation of the state variable y1 = vx is directly applicable to p1, as shown in Equation 4.15. v¯x= vx vmaxx → ∂v¯x ∂µ = 1 vmaxx · ∂vx ∂µ (4.15) To make the lateral velocityvy physical equivalent to the reference valuevx, the side slip angle characteristics for a single-track model and for small values of the side slip angle given by β=β0 +∆β= arctan vy vx ≈ vy vx (4.16) 1Reference value vmaxx was taken from the technical data sheet, [AG08]. 74