Page - 99 - in Integration of Advanced Driver Assistance Systems on Full-Vehicle Level - Parametrization of an Adaptive Cruise Control System Based on Test Drives

B.2. Non-linear Tyre Model Thetyrestiffnessequalsthe initial slopeofthetyrecharacteristicscy=dF0, as illustrated in fig. B.2(a). The transformationof the front tyre forces fromtheWf to thevehicle-coordinate system can be done using the formulation[ vfFx vfFy ] = [ cosδ −sinδ sinδ cosδ ] ︸ ︷︷ ︸ Tv,wf=T −1 wf,v [ wfFx wfFy ] . (B.6) Using assumption 4, the transformation described in eq. (B.6) can be simplified using cosδ≈1 and sinδ≈ δ. The constant longitudinal velocity of simplification 5 means vv˙x= 0, and therefore the first row of eq. (B.1) is omitted. The state variable vvy in eq. (B.1) is substituted by the side slip angle in the CG, which reads β= arctan ( vvy vvx ) . (B.7) Simplifications3and4 leadtosmall side slipangles. Thereforeeq. (B.7)canberedefined toβ≈ ( vvy vvx ) forβ 1rad. All these assumptions lead to the linear STM, which reads[ β˙ ω˙z ] ︸︷︷︸ x˙ =   −2(fcy+rcy)m vvx 2(rcy lr−fcy lf)m vv2x −1 2(rcy lr−fcy lf) vIzz −2(fcy l 2 f+rcy l 2 r) vIzz vvx   ︸ ︷︷ ︸ A [ β ωz ] ︸︷︷︸ x + [ 2fcy m vvx 2fcy lf vIzz ] ︸ ︷︷ ︸ b δ︸︷︷︸ u . (B.8) B.2. Non-linear Tyre Model The TMsimple tyre model is used to calculate the forces in the contact patch between tyre and road. The model described in this chapter is based on the publication [Hir09] of Hirschberg. The tyre forces are a function of the slip between tyre and road surface in the contact patch. The longitudinal tyre force is a function of the longitudinal slip, which reads sx= cωy re−wvx max(wvx, cωy re, wvxε) , (B.9) where cωy is the rotational speed of the tyre, re describes the effective tyre radius, and wvx is the speed of theW-point in the x-direction of the tyre coordinate system. The tyre parameter wvxε avoids division by zero when cωy= 0 and wvx= 0, by limiting the denominator to a minimum value ofwvxε. The lateral slip is defined in eq. (B.3). 99