In recent years, the rapid advancement of intelligent and electric vehicle technologies has positioned trajectory tracking control as a critical research focus. As a leading player in the global electric vehicle market, China EV industry has witnessed significant growth, driving innovations in autonomous driving systems. Distributed electric vehicles, characterized by independent four-wheel drive and flexible control capabilities, serve as an ideal platform for developing advanced control algorithms. However, trajectory tracking in extreme driving conditions faces challenges due to the nonlinear and time-varying nature of tire cornering stiffness, which directly impacts vehicle stability and control accuracy. Traditional control methods often assume constant cornering stiffness, leading to inaccuracies in path tracking when tires operate near adhesion limits. This study addresses this issue by proposing an enhanced trajectory tracking controller that dynamically estimates tire cornering stiffness using an optimized neural network approach.
The core of our work involves developing a high-fidelity vehicle dynamics model and integrating a real-time cornering stiffness estimator into a model predictive control (MPC) framework. We focus on distributed electric vehicles due to their independent torque distribution capabilities, which allow for precise handling and stability control. The electric vehicle paradigm, particularly in China EV developments, emphasizes the importance of adaptive control systems for safe autonomous navigation. Our contributions include a modified radial basis function neural network (RBFNN) optimized with k-means clustering and least mean square (LMS) algorithms for accurate stiffness estimation, and an MPC-based trajectory tracking controller that incorporates these estimates to improve performance under varying road conditions.
To establish a foundation for control design, we first derive a seven-degree-of-freedom (7-DOF) vehicle dynamics model. This model captures essential motions, including longitudinal, lateral, yaw, and four-wheel rotations, while neglecting suspension effects, road slope, and aerodynamic forces for simplicity. The equations of motion in the vehicle coordinate system are as follows:
$$m(\dot{v}_x – \omega v_y) = (F_{x,lf} + F_{x,rf})\cos\delta_f – (F_{y,lf} + F_{y,rf})\sin\delta_f + (F_{x,lr} + F_{x,rr})$$
$$m(\dot{v}_y + \omega v_x) = (F_{y,lf} + F_{y,rf})\cos\delta_f + (F_{x,lf} + F_{x,rf})\sin\delta_f + (F_{y,lr} + F_{y,rr})$$
$$I_z \dot{\omega} = [(F_{x,rf} – F_{x,lf})\cos\delta_f + (F_{y,lf} – F_{y,rf})\sin\delta_f]\frac{w_b}{2} + [(F_{x,lf} + F_{x,rf})\sin\delta_f + (F_{y,lf} + F_{y,rf})\cos\delta_f]l_f + (F_{x,rr} – F_{x,lr})\frac{w_b}{2} – (F_{y,lr} + F_{y,rr})l_r$$
where $m$ is the vehicle mass, $v_x$ and $v_y$ are longitudinal and lateral velocities, $\omega$ is the yaw rate, $\delta_f$ is the front steering angle, $I_z$ is the yaw moment of inertia, $w_b$ is the wheel track, $l_f$ and $l_r$ are front and rear axle distances from the center of gravity, and $F_{i,j}$ represents tire forces with $i$ indicating direction (x or y) and $j$ denoting wheel position (lf: left-front, rf: right-front, lr: left-rear, rr: right-rear). The transformation between vehicle and inertial coordinates is given by:
$$\dot{X} = v_x \cos\phi – v_y \sin\phi$$
$$\dot{Y} = v_x \sin\phi + v_y \cos\phi$$
where $X$ and $Y$ are inertial positions, and $\phi$ is the yaw angle. Wheel dynamics are described by:
$$J_f \dot{\omega}_1 = T_{d1} – T_{b1} – F_{x1} r_{e1}$$
$$J_f \dot{\omega}_2 = T_{d2} – T_{b2} – F_{x2} r_{e2}$$
$$J_r \dot{\omega}_3 = T_{d3} – T_{b3} – F_{x3} r_{e3}$$
$$J_r \dot{\omega}_4 = T_{d4} – T_{b4} – F_{x4} r_{e4}$$
Here, $J_f$ and $J_r$ are front and wheel moments of inertia, $T_{di}$ and $T_{bi}$ are drive and brake torques, $F_{xi}$ is longitudinal force, and $r_{ei}$ is the effective rolling radius for each wheel $i$.
The tire model employs the Magic Formula to characterize nonlinear force generation:
$$y = D \sin[C \tan^{-1}(Bx – E(Bx – \tan^{-1}Bx))]$$
where $y$ represents output variables (e.g., lateral force), $x$ is input (e.g., slip angle), and $B$, $C$, $D$, $E$ are stiffness, shape, peak, and curvature factors, respectively. For the electric vehicle powertrain, we model permanent magnet synchronous motors with a second-order transfer function:
$$G(s) = \frac{T_r}{T_i} = \frac{1}{2\xi^2 s^2 + 2\xi s + 1}$$
where $T_i$ is input torque, $T_r$ is output torque, and $\xi$ is a physical constant. The motor torque-speed characteristics are encapsulated in a lookup table for real-time simulation.
Critical to trajectory tracking is the tire cornering stiffness, which varies with vertical load, slip angle, and road conditions. Analysis shows that lateral force increases with vertical load up to a point before declining, and maintains linearity with slip angle only within small ranges. Thus, we design an RBFNN-based estimator with vertical load and slip angle as inputs, and cornering stiffness as output. The standard RBFNN uses Gaussian activation functions:
$$u_i = \exp\left(-\frac{(x – c_i)^T (x – c_i)}{2\sigma_i^2}\right), \quad i=1,2,\ldots,M$$
where $u_i$ is the hidden layer output, $x$ is input vector, $c_i$ is center vector, $\sigma_i$ is width constant, and $M$ is the number of hidden nodes. To overcome initialization uncertainties and slow convergence, we integrate k-means clustering for center selection and LMS for weight adjustment. The k-means algorithm minimizes within-cluster variance:
$$E = \sum_{i=1}^k \sum_{x \in C_i} \|x – \mu_i\|_2^2, \quad \mu_i = \frac{1}{|C_i|} \sum_{x \in C_i} x$$
where $E$ is the sum of squared errors, $C_i$ are clusters, and $\mu_i$ are cluster means. The steps include initializing centers, computing Euclidean distances, finding the closest cluster, and updating centers iteratively. Weight adaptation via LMS follows:
$$\omega = \frac{e}{M c_{\text{max}}^2} (x_p – C_i)$$
where $e$ is error, $c_{\text{max}}$ is maximum center distance, and $x_p$ is input pattern. This optimized RBFNN, termed KRBF, demonstrates improved accuracy and faster convergence in estimating cornering stiffness for electric vehicle applications.
For trajectory tracking, we design an MPC controller that incorporates the estimated cornering stiffness. The nonlinear vehicle model is linearized around reference states $( \xi_r, u_r )$ using Taylor expansion:
$$\dot{\xi} = f(\xi_r, u_r) + \frac{\partial f}{\partial \xi} \bigg|_{\xi_r, u_r} (\xi – \xi_r) + \frac{\partial f}{\partial u} \bigg|_{\xi_r, u_r} (u – u_r)$$
where $\xi_{\text{dyn}} = [\dot{v}_y, \dot{v}_x, \phi, \dot{\phi}, Y, X]^T$ and $u_{\text{dyn}} = [\delta_f]$. Defining $\Delta \xi = \xi – \xi_r$ and $\Delta u = u – u_r$, the discrete-time state-space representation is:
$$\xi(k+1) = A_{k,t} \xi(k) + B_{k,t} u(k), \quad \Delta u(k) = u(k) – u(k-1)$$
with $A_{k,t} = I_n + T A(t)$, $B_{k,t} = T B(t)$, and $T$ as sampling period. The augmented state $\hat{\xi}(k) = [\xi(k), u(k-1)]^T$ leads to:
$$\hat{\xi}(k+1|k) = \hat{A}_{k,t} \hat{\xi}(k) + \hat{B}_{k,t} \Delta U(k), \quad \eta(k) = \hat{C}_{k,t} \xi(k)$$
where $\hat{A}_{k,t} = \begin{bmatrix} A_{k,t} & B_{k,t} \\ 0_{m \times n} & I_m \end{bmatrix}$, $\hat{B}_{k,t} = \begin{bmatrix} B_{k,t} \\ I_m \end{bmatrix}$, and $\hat{C}_{k,t} = [C_{k,t} 0]$. Predictions over horizon $N_p$ with control horizon $N_c$ are computed as:
$$Y(k) = \Psi_k \xi(k|k) + \Theta_k \Delta U(k)$$
with $Y(k)$ containing outputs from $k+1$ to $k+N_p$, and $\Psi_k$, $\Theta_k$ constructed from system matrices. The cost function minimizes tracking errors and control efforts:
$$J = \sum_{i=1}^{N_p} \| \eta(t+i|t) – \eta_{\text{ref}}(t+i|t) \|_Q^2 + \sum_{i=1}^{N_c} \| \Delta u(t+i|t) \|_R^2 + \rho \epsilon^2$$
where $Q$ and $R$ are weight matrices, $\rho$ is a slack weight, and $\epsilon$ is a relaxation factor. Constraints include control input bounds, increment limits, output constraints, and stability criteria based on yaw rate and sideslip angle:
$$| \omega | \leq \frac{\mu g}{v_x}, \quad \beta_{\text{min}} \leq \beta = \frac{v_y – l_r \omega}{v_x} \leq \beta_{\text{max}}$$
where $\mu$ is road friction coefficient, $g$ is gravity, and $\beta$ is sideslip angle.
Simulation studies validate the proposed approach under high- and low-adhesion conditions. The KRBF estimator achieves lower mean squared errors compared to standard RBF, as summarized below for an 80 km/h test on a $\mu=0.4$ surface:
| Method | $C_f$ MSE | $C_r$ MSE | $F_{yf}$ MSE | $F_{yr}$ MSE |
|---|---|---|---|---|
| KRBF | $2.81 \times 10^6$ | $6.68 \times 10^5$ | $1.10 \times 10^3$ | $7.69 \times 10^2$ |
| RBF | $5.95 \times 10^7$ | $1.66 \times 10^7$ | $1.20 \times 10^4$ | $2.36 \times 10^3$ |
At 110 km/h, similar improvements are observed, confirming the estimator’s robustness. For trajectory tracking, three controllers are compared: MPC without cornering stiffness adaptation, MPC with RBF estimates (RBF_MPC), and MPC with KRBF estimates (KRBF_MPC). On a high-adhesion road ($\mu=0.85$) at 110 km/h, KRBF_MPC reduces maximum lateral error to 0.115 m, versus 0.152 m for RBF_MPC and 0.22 m for baseline MPC. Steering angle and lateral acceleration oscillations are also minimized, enhancing ride comfort. In low-adhesion scenarios ($\mu=0.3$), KRBF_MPC maintains stability with a maximum error of 0.125 m, outperforming other methods. The sideslip angle and yaw rate remain within safe limits, demonstrating the controller’s efficacy for electric vehicle systems in challenging environments.
The integration of adaptive cornering stiffness estimation into MPC framework significantly improves trajectory tracking accuracy and stability for distributed electric vehicles. This approach addresses the limitations of fixed-parameter models and showcases the potential for real-time implementation in autonomous driving systems. Future work will explore combined longitudinal and lateral control under diverse operating conditions, further advancing the capabilities of China EV technologies in the global market.