A Study on the Control of Handling and Stability of a Four Wheel Independent Steering Electric Vehicle

Abstract

The path-tracking control of electric vehicles with four wheel independent steering (FWIS) is crucial for enhancing vehicle stability. This paper aims to address the issue of multi-actuator redundancy by coordinating the control allocation of multiple actuators, thereby improving the handling stability and path tracking performance of FWIS vehicles. First, a model of FWIS electric vehicle is developed, taking into account both the nonlinear tyre and motor actuator characteristics. Subsequently, a path tracking model is established and a hierarchical control architecture is designed. The upper-level controller computes the generalized tracking force and the lower-level control force is distributed based on the principle of optimal tyre utilization rate. Finally, the simulation results demonstrate the effectiveness of the proposed control scheme in terms of tracking accuracy and handling and stability.

1 Introduction

Vehicle electrification has not only provided a solution for reducing transportation emissions, but also new possibilities for enhancing the handling and stability performance. With advancements in electric motor technology and x-by-wire, the e-corner module, which consists of in-wheel motors, brake-by-wire, steer-by-wire, and active suspension, has been proposed as a next-generation modular platform for electric vehicles. Vehicles equipped with e-corner modules can realize FWIS and even special functions such as zero turn. Against the backdrop of a boom in skateboard platforms for EV designs, how to ensure stability at high speeds is a great challenge.

In recent decades, there has been a growing focus on the steering stability of FWIS systems. The intricate steering characteristics and over-actuated of these vehicles necessitate an effective control strategy to coordinate the actuators and ensure system stability, particularly at high speeds. In addition to traditional front wheel steering, various control strategies such as direct yaw moment control [2], torque vectoring control [3], active rear wheel steering [4], and four wheel steering [5] have been proposed for FWIS and four wheel independent driving vehicles. However, these control strategy studies often overlook the influence of the actuators on control effectiveness. Consequently, ensuring stability in FWIS electric vehicles with e-corner modules at high speeds remains a challenging problem. Furthermore, most studies utilize linear tyre models and lack further analysis of tyre and motor characteristics [6].

The remainder of this paper is organized as follows. The vehicle path tracking model is established in Sect. 2. The modeling and design of the controllers have been fully studied in Sect. 3. Simulation and discussion are presented in Sect. 4 and the conclusion is drawn in Sect. 5.

2 Vehicle Path Tracking System Dynamics Modeling

The study focuses on a fully electric vehicle equipped with four drive motors and four steering motors. The objective is to effectively manage the torque and steering angle of each motor to coordinate the chassis control system and optimize tyre utilization. Figure 1 presents a top view of the vehicle, illustrating the yaw moment M_z and wheel forces F_x/F_y.

Fig. 1.

Path tracking model and FWIS vehicle

2.1 FWIS Vehicle Dynamics Model

Referring to Fig. 1, the FWIS vehicle model is described as:

$$ m\left( {\dot{v}_{x} - v_{y} \dot{\psi }} \right) = \sum\limits_{i = 1}^{4} {\left( {F_{x,i} \cos \delta_{i} - F_{y,i} \sin \delta_{i} } \right)} - F_{{{\text{res}}}} $$

(1)

$$ m\left( {\dot{v}_{y} + \dot{v}_{x} \dot{\psi }} \right) = \sum\limits_{i = 1}^{4} {\left( {F_{x,i} \sin \delta_{i} + F_{y,i} \cos \delta_{i} } \right)} $$

(2)

$$ \begin{aligned} I_{z} \ddot{\psi } & = \sum\limits_{{i = 1}}^{2} {l_{{\text{f}}} \left( {F_{{x,i}} \sin \delta _{i} + F_{{y,i}} \cos \delta _{i} } \right)} - \sum\limits_{{i = 3}}^{4} {l_{{\text{r}}} \left( {F_{{x,i}} \sin \delta _{i} + F_{{y,i}} \cos \delta _{i} } \right)} \\ & + \sum\limits_{{i = 1}}^{4} {( - 1)^{i} d\left( {F_{{x,i}} \cos \delta _{i} - F_{{y,i}} \sin \delta _{i} } \right)} \\ \end{aligned} $$

(3)

where m is the total vehicle mass. v_x and v_y are the longitudinal and lateral speeds of the vehicle. $\psi$ is the yaw angle. F_x and F_y are the longitudinal and lateral forces. i = fl(front left), fr(front right), rl(rear left) and rr(rear right). $\delta$ is the wheel steering angle. F_res is the sum of aerodynamic resistance force and rolling resistance. I_z is the yaw moment of inertia. d is one-half of the tread width. The nonlinear tyre forces are described by means of the Pacejka Magic-Formula [1], where the independent variables are the tyre slip rate and tyre slip angle, as well as the dependent variables are the longitudinal and lateral tyre forces. The longitudinal wheel speed v_w,i and tyre slip rate λ_i are defined as

$$ v_{w,i} = \left( {v_{x} \mp d\dot{\psi }} \right)\cos \delta_{i} + \left( {v_{y} + l_{{\text{f}}} \dot{\psi }} \right)\sin \delta_{i} ,i = 1,2 \, v_{w,i} = \left( {v_{x} \mp d\dot{\psi }} \right)\cos \delta_{i} + \left( {v_{y} - l_{{\text{r}}} \dot{\psi }} \right)\sin \delta_{i} ,i = 3,4 $$

(4)

$$ \lambda_{i} = \frac{{\omega_{w,i} r_{w} - v_{w,i} }}{{\omega_{w,i} r_{w} }}, \, a > 0 \, \lambda_{i} = \frac{{\omega_{w,i} r_{w} - v_{w,i} }}{{v_{w,i} }}, \, a < 0 $$

(5)

where ω_w,i and r_w are the rotational velocity and effective radius of the wheel. α_i is the tyre slip angle defined as:

$$ \alpha_{i} = \arctan \left( {\frac{{v_{y} + l_{{\text{f}}} \dot{\psi }}}{{v_{x} \mp d\dot{\psi }}}} \right) - \delta_{i} ,i = 1,2 \, \alpha_{i} = \arctan \left( {\frac{{v_{y} - l_{{\text{f}}} \dot{\psi }}}{{v_{x} \mp d\dot{\psi }}}} \right) - \delta_{i} ,i = 3,4 $$

(6)

In order to describe the tyre force characteristics under the combined conditions, the weight functions G_xα and G_yλ are introduced. By multiplying the weight functions with Eq. (4), the tyre force output can be obtained. The vertical forces F_z,i are dependent on steady-state component of the load transfer which is a function of longitudinal and lateral accelerations:

$$ F_{zi} = \frac{m}{{2\left( {l_{{\text{f}}} + l_{{\text{r}}} } \right)}}\left( {l_{{\text{r}}} g - h_{{{\text{cg}}}} \dot{v}_{x} \mp \frac{{l_{{\text{r}}} h_{{{\text{cg}}}} \dot{v}_{y} }}{d}} \right),i = 1,2 \, F_{zi} = \frac{m}{{2\left( {l_{{\text{f}}} + l_{{\text{r}}} } \right)}}\left( {l_{{\text{f}}} g + h_{{{\text{cg}}}} \dot{v}_{x} \mp \frac{{l_{{\text{f}}} h_{{{\text{cg}}}} \dot{v}_{y} }}{d}} \right),i = 3,4 $$

(7)

where h_cg is the height of the center of gravity. Wheels’ dynamic equation as follows:

$$ I_{w} \dot{\omega }_{w,i} = T_{w,i} - F_{x,i} r_{w} - F_{z,i} fr_{w} $$

(8)

where T_w is the drive motor output torque. f is the rolling resistance coefficient.

2.2 Motor Model

The drive system employs direct drive with electric motors. The primary function of the drive motor system is to deliver the specified torque to each wheel as dictated by the controller. The simplified ideal closed-loop dynamics of the motor can be described as follows:

$$ T_{w} = \frac{1}{{\tau_{1} s^{2} + \tau_{2} s + 1}}T_{c} $$

(9)

where τ₁ and τ₂ are close-loop response times and represent the control characteristic of the motor’s field-oriented controller. T_c is motor torque command.

To simplify the FWIS actuator, the dynamic equation of the steer-by-wire (SBW) system can be expressed as follows:

$$ I_{eq} \ddot{\delta }_{i} + B_{eq} \dot{\delta }_{i} + T_{e} + T_{f} = i_{g} T_{m,i} $$

(10)

where equivalent moment of inertia of the SBW system I_eq = I_w + i_g²I_m. i_g is the steering gear ratio. I_w and I_m are the moments of inertia of the wheel and steering motor. Equivalent damping coefficient of SBW system B_eq = B_w + i_g²B_m. B_w and B_m are damping factors of wheel steering and steering motor. T_m,i is the steering motor output torque. T_e and T_f are the internal friction torque of the SBW system and aligning torque.

2.3 Path Tracking Model

The path-following model is depicted in Fig. 1. A linear two-degree-of-freedom (2-DOF) vehicle dynamic model is employed to generate the desired yaw rate for maintaining vehicle stability. The state space equation can be given as follows:

$$ {\varvec{\dot{X}}} = {\varvec{AX}} + {\varvec{BU}} $$

(11)

where $\dot{\user2{X}} = \left[ {\begin{array}{*{20}c} \beta & {\dot{\psi }} & \theta & Y \\ \end{array} } \right]^{{\text{T}}}$ is the state variable set. β is the vehicle body side-slip angle. θ is the heading angle. U = δ_f is the input variable.

$$ A = \left[ {\begin{array}{*{20}c} { - \frac{{2C_{\text{f}} + 2C_{\text{r}} }}{mv_x }} & { - \frac{{2C_{\text{f}} l_{\text{f}} - 2C_{\text{r}} l_{\text{r}} }}{mv_x^2 } - 1} & 0 & 0 \\ { - \frac{{2C_{\text{f}} l_{\text{f}} - 2C_{\text{r}} l_{\text{r}} }}{I_z }} & { - \frac{{2C_{\text{f}} l_{\text{f}}^2 + 2C_{\text{r}} l_{\text{r}}^2 }}{v_x I_z }} & 0 & 0 \\ { - \frac{{2C_{\text{f}} + 2C_{\text{r}} }}{mv_x }} & { - \frac{{2C_{\text{f}} l_{\text{f}} - 2C_{\text{r}} l_{\text{r}} }}{mv_x^2 }} & 0 & 0 \\ {v_x \cos \theta } & 0 & {v_x \cos \theta - v_x \beta \sin \theta } & 0 \\ \end{array} } \right],B = \left[ {\begin{array}{*{20}c} {\frac{{2C_{\text{f}} }}{mv_x }} \\ {\frac{{2C_{\text{f}} l_{\text{f}} }}{I_z }} \\ {\frac{{2C_{\text{f}} }}{mv_x }} \\ 0 \\ \end{array} } \right] $$

(12)

3 Multi-Controller Integrated Control Strategy

Figure 2 depicts the overall control scheme, including a path tracking controller and a hierarchical controller. It provides the ability to accurately assess the impact of coordinated control.

Fig. 2.

Control system architecture.

3.1 Model Predictive Control

The path tracking can be formulated for MPC by the following optimization problem. The cost function consists of two objectives penalizing the deviation from predefined idealized trajectory X_ref,i for the system states X_i and control inputs U_i:

$$ J = \min \left[ {\sum_{i = 1}^{N_p } {\left( {{\varvec{X}}_{\varvec{i}} - {\varvec{X}}_{{\varvec{ref,i}}} } \right)^{\text{T}} } {\varvec{Q}}\left( {{\varvec{X}}_{\varvec{i}} - {\varvec{X}}_{{\varvec{ref,i}}} } \right) + \sum_{j = 0}^{N_c - 1} {{\varvec{U}}_{\varvec{j}}^{\text{T}} {\varvec{RU}}_{\varvec{j}} } } \right] $$

(13)

where Q and R are the weight matrices of the states and control inputs. Subject to

$$ {\varvec{X}}_{\varvec{i}} = f\left( {{\varvec{X}}_{\varvec{i}} ,{\varvec{U}}_{\varvec{i}} } \right) $$

(14)

$$ \delta_{\min } \le \delta_{i} \left( {k + j\left| k \right.} \right) \le \delta_{\max } ,j = 1,2, \cdots ,N_{c} - 1 $$

(15)

3.2 Hierarchical Controller

The upper-level controller determines the desired forces and moment for tracking the desired path. The sliding mode control method is adopted to ensure the vehicle follows the reference states. The sliding surfaces are defined as follows.

$$ S_{1} = v_{x} - v_{xd} ,S_{2} = v_{y} - v_{yd} ,S_{3} = \psi - \psi_{d} , $$

(16)

To ensure that the errors between actual and desired values reach the sliding mode surface within a limited time and attenuate chattering, a combinatorial reaching law is applied in the controller:

$$ \dot{S}_{i} = - \varepsilon_{i} {\text{sat}}\left( {S_{i} } \right) - k_{i} S_{i} $$

(17)

where ε_i and k_i are parameters of the controller. The required forces and moment are determined as follows:

$$ F_{xd} = m\left[ { - \varepsilon_{1} {\text{sat}}\left( {S_{1} } \right) - k_{1} S_{1} + a_{x} } \right]\quad F_{yd} = m\left[ { - \varepsilon_{2} {\text{sat}}\left( {S_{2} } \right) - k_{2} S_{2} + a_{y} } \right] $$

(18)

$$ M_{zd} = I_{z} \left[ { - \varepsilon_{3} {\text{sat}}\left( {S_{3} } \right) - k_{3} S_{3} + \dot{\psi }} \right] $$

(19)

The lower-level controller is used to realize better control performance through optimal tyre force allocation:

$$ J = \min \sum\limits_{i = 1}^{4} {\frac{{F_{x,i}^{2} + F_{y,i}^{2} }}{{\mu^{2} F_{z,i}^{2} }}} $$

(20)

where μ is the coefficient of road adhesion. The total F_x, F_y and M_z satisfy the equation and the linear constraints:

$$ - 0.9\mu F_{z,i} < F_{x,i} ,F_{y,i} < 0.9\mu F_{z,i} - 0.9 \times \sqrt 2 \mu F_{z,i} < F_{x,i} \pm F_{y,i} < 0.9 \times \sqrt 2 \mu F_{z,i} $$

(21)

After the forces of each tyre are allocated, the motor model is used to transform the longitudinal force into drive or brake torque, and the lateral force into steering angle.

4 Simulation

In order to verify the effectiveness of the proposed integrated control strategy, the double line change (DLC) scenery is utilized to test the performance of the FWIS vehicle at different speeds.

Fig. 3.

Simulation result.

The simulation results are shown in Fig. 3a. Further analysis of the simulation results at 20 m/s speed is shown in Figs. 3b–d. The results show that the FWIS coordinated control maintains good speed and path tracking control accuracy form low to high speeds. Although there are some constraints and influences on the actuators, the independent wheels can still maintain coordinated torque and cornering control.

5 Conclusion

A FWIS electric vehicle path tracking system dynamics model considering nonlinear tyre and motor characteristics is proposed. In order to achieve coordinated stabilization of multiple actuators and the vehicle, a multi-controller integrated control strategy is proposed. The longitudinal and lateral forces are optimally distributed to the actuator layer of each wheel. Simulation results show that the method can achieve good path tracking performance at different driving speeds.