Page - 51 - in Maximum Tire-Road Friction Coefficient Estimation

3 Vehicle model zg xg yg Og Ofr Ofl Orl y b x b z b Ob Figure 3.5.: Global coordinate system {Og,xg,yg,zg} (inertial), vehicle-fixed coordi- nate system {Ob,xb,yb,zb} and wheel-fixed horizonted coordinate systems {Oi,xi,yi,zi} for wheel index i = {fl,fr,rl,rr} (with wheel i = rr not displayed) based on ISO 8855, [fSI11]; graphic representation modified from Hirschberg, [HW12, p.160]. Figure 3.5 shows all coordinate systems used, which are based on ISO 8855, [fSI11]. The position of the vehicle with respect to the global coordinate systemOg is described with the coordinates xg and yg, and the orientation of the vehicle’s longitudinal axis with respect to the xg axis with the yaw angleψ. The relationship between gy˙ and bz reads gy˙=T g b(gy) · bz, (3.4) with the rotation matrix Tgb=     cosψ −sinψ 0 0 0 0 0 sinψ cosψ 0 0 0 0 0 0 0 1 0 0 0 0     . (3.5) The resulting equation of motion reads M · z˙+k=q (3.6) and includes the mass matrix M, gyroscopic and centrifugal forces k and the vector of applied forces q. The equation of motion applies in the moving coordinate systemOb which is located in the vehicle’s centre of gravity, cf. ISO 8855, [fSI11]. The mass matrix M in Equation 3.6, which includes the vehicle massmb and the moments of inertia of 51