The integrated longitudinal-lateraland yaw rate dynamics of the vehicle are simultaneously considered to improve thetracking accuracy and system stability when navigating under critical
Steering control formulation
The AFS control system is engineered to accurately track the reference path by adjusting the steering angle, thereby reducing lateral deviation Lateral tracking error is defined as ey, representing the difference between the actual and desired paths over time.
Let(x 1y ,x 2y ) T = (e y ,e˙y) T , the following system can be obtained: ˙ x 1y =x 2y ˙ x 2y =F ky +G ky δ+D y (3.2) where
Fˆx f sinδ−C α non f v y +l f ψ˙ v x cosδ−C α non r vy−lrψ˙ v x cosψ
+vxψ˙ cosψ+v˙xsinψ−vyψ˙ sinψ−y¨ d
Dy =∆Fx f sinδcosψ+∆Fy fcosδcosψ+∆Fyrcosψ
Assuming that the two desired tracking references are given as x 1yd and x˙ 1yd , re- spectively The tracking error is defined as: z 1y =x 1y −x 1yd z 2y =x 2y −x˙ 1yd (3.3)
To improve lateral control performance, the steering controller is designed to keep the tracking error within specific bounds, ensuring that the lateral position \(x_{iy}(t)\) remains within the defined limits \(k_{nyi}(t) < x_{iy}(t) < k_{pyi}(t)\) for \(i = 1, 2\) This research focuses on a steering controller that directly manages the angles of the front steering wheels Additionally, to facilitate a smooth transition, the steering angle \(\delta(t)\) and its rate of change \(\dot{\delta}(t)\) are constrained to remain within established limits, ensuring optimal performance and safety during operation.
The original lateral dynamic model is enhanced by incorporating the second derivative of the steering angle, enabling the conversion of the input constraint into a state constraint problem This transformation is represented by the equations: \( \dot{x}_{1y} = x_{2y} \), \( \dot{x}_{2y} = F_k y + G_k y \delta + D_y \dot{\delta} = \omega_y \), and \( \dot{\omega}_y = U_y \).
Then, the control signalU y will be designed to stabilize this extended lateral sys- tem and guarantee the satisfaction of the constraints one y ,δ,δ˙.
In normal driving conditions, the forces Fx and Fy are constrained by the limitations of road surface adhesion and tire friction, leading to the assumption that the lateral force Dy is also bounded, specifically |Dy| ≤ λy < ∞, given that the steering angle δ remains relatively small.
Longitudinal and direct yaw control formulation
The longitudinal controller is responsible for generating the driving and braking force \( F_l \) to achieve the desired speed, while the DYC controller creates a yaw moment \( M_z \) for vehicle stabilization By defining the state vector \( x_s = [v_x, \dot{\psi}]^T \) and the tracking reference \( x_{sd} = [v_{xd}, \dot{\psi}_d]^T \), we can derive the equation \( \dot{x}_s = F_k s + G_k s T_s + D_s \) as shown in (3.5).
Fks= [F kx Fk γ] T ,Gks= [G kx Gk γ] T ,Ds= [D x D γ ] T ,
∆Fy f l +∆Fy f r cosδ−lr ∆Fy rl +∆Fy rr
In addition, for safety reasons, it is important to ensure that x s is constrained byx s ∈Ω s1 x s (t)∈R 2 |k ns1 (t)