Research on coupled bifurcation and stability of tractor semi-trailer under combined braking and steering conditions

Existing stability analyzes for the tractor semi-trailer cannot fully explain the instability phenomenon during combined steering and braking conditions, under which the tire lateral forces are affected by the longitudinal tire forces due to adhesive force ellipse. In response to the problem, this article obtains the coupled bifurcation and the stability area of the tractor semi-trailer under different deceleration and steering angles. First, the 8 degrees of freedom (8 DOF) tractor semi-trailer nonlinear dynamical model considering the nonlinear tire characteristics is built. The constrained continuation method is used, and the inertial force caused by braking is equivalent to two external forces and added to the tractor semi-trailer model. Thus, the tractor semi-trailer system can be seen as being in an equivalent equilibrium state. Next, the equilibrium points under varying steering angles and different decelerations are solved by a hybrid method combining the Runge-Kutta method with the gradient descent method. The bifurcation diagrams denote the changes in the equilibrium brought by different conditions and reveal the influence of different decelerations on the stable steering angle range. Further, the stable regions under different conditions are illustrated to give an understanding of the effect of the braking operations on the stability of the tractor semi-trailer. Compared with the existing literature, it can be proved that in addition to the fixed parameters such as road adhesion coefficient, the deceleration under braking conditions will also have an impact on the stability regions of the tractor semi-trailer. The stable boundary of the tractor semi-trailer under combined braking and steering conditions can be obtained by off-line calculation, and the results can serve on the theoretical considerations on the tractor semi-trailer braking stability control.