Stability and Hopf bifurcation of four-wheel-steering vehicles involving driver's delay

A mathematical model is presented for four-wheel-steering vehicles, with the time delay in driver's response and the nonlinearity in lateral tyre forces taken into account. It is proved that the vehicle-driver system has a trivial steady state motion, as well as eight non-trivial steady state motions due to the nonlinearity of tyre forces. The asymptotic stability and Hopf bifurcation of the trivial steady state are analyzed for two control strategies of rear-wheel-steering. It is shown through the numerical simulations that the four-wheel-steering technique based on the bilinear control strategy works better when the driver's response involves time delay.