Page - 54 - in Maximum Tire-Road Friction Coefficient Estimation

3 Vehicle model δ fr δ fl F x,flF y,fl F x,rl F y,rl F x,rr F y,rr β r F y,fr F x,fr F z,r F z,f G F A Figure 3.7.: Kineticquantitiesofthetwo-trackvehiclemodel includingthehorizontaltire forcesFx,i andFy,i, the aerodynamic forceFA, the weight forceG=mb ·g and the vertical forces Fz,j, based on Eichberger, [Eic11, p.148]; graphic depiction modified from Hirschberg, [Hir13, p.43, 69]. The rotation matrix Tbi reads Tbi=     cosδi −sinδi 0 sinδi cosδi 0 0 0 1     (3.12) and contains the wheel’s steering angles δi, see Section 3.2.4. The linear momentum in the longitudinal direction of the vehicle also involves the aerodynamic resistance force FA, which reads FA= 1 2 ·cD ·Ap ·ρa ·vx · |vx|. (3.13) Theaerodynamic force showsaquadraticdependenceonthevehicle’s longitudinal speed vx and considers the air drag coefficient cD, the projected frontal areaAp of the vehicle andtheairdensityρawhich isa functionofair temperatureandairpressure. Additional wind forcesFW,x andFW,y actingonthevehiclebodycanalsobeconsidered. Theweight forceG, which is given byG=mb ·gwith the vehicle’s massmb and the gravitational acceleration g, contributes to the linear momentum in the longitudinal direction and is directly proportional to the sine of the road slope βr. The third row of q contains the angular momentum of the vehicle body, which reads Oe ∑ Mz= ∑ i     rx,i ry,i 0     ×     b Fx,i bFy,i 0     (3.14) 54