Design and Simulation of Steering Control Strategy for Four-Wheel Steering Hillside Tractor

Abstract

Steering control is vital for hillside tractors, and four-wheel-steering technology significantly enhances stability and flexibility. However, existing methods often overlook the slope–stability relationship. In order to address this issue, considering the impact of tractor wheels on the steering characteristics, a nonlinear three-degree-of-freedom dynamic model based on the basic requirements within a 15° range of slope angles was derived. Subsequently, a four-wheel steering control strategy for hillside tractors was designed based on this model, incorporating a fuzzy PID control algorithm. The dynamics model was validated on the high-fidelity Carsim/Simulink co-simulation platform, and relevant experiments were conducted in Matlab/Simulink. Results showed that fuzzy PID control reduced the yaw rate’s average settling time from 0.36 s to 0.1 s and the response value from 0.28 rad/s to 0.038 rad/s. In addition, the lateral velocity and sideslip angle responses closely matched ideal values. Thus, the tractor with four-wheel steering under fuzzy PID control exhibited improved wheel angle flexibility and higher tracking accuracy. Finally, hardware-in-the-loop-experiments were conducted, confirming the algorithm’s effectiveness in ensuring stability and meeting the requirements of semi-physical simulation scenarios. This research provides a foundation for potential applications in tractor manufacturing to improve control performance on hilly terrains.

1. Introduction

Hilly areas are important production bases for grain and specialty agricultural products in China, and there are about 700 million acres of arable land in China’s hilly areas, with the agricultural population and the sown area of crops accounting for about half of the country’s total [1,2]. Promoting agricultural mechanization in hilly areas is of great significance for building modern agriculture, promoting poverty alleviation, and accelerating rural revitalization in hilly areas [3].

Hillside tractors are agricultural machinery designed to adapt to the complex terrain of hills and mountains. They have high mobility, strong traction, and good stability, making them suitable for steep slopes and rugged working environments, and they are vital for the improvement of agricultural mechanization in hilly areas [4]. But nowadays, most tractor companies only conduct application research on tractors for plain operations, while the research on tractors with four-wheel steering is still in the laboratory stage and there are no mature product applications. When tractors are operated in hilly areas, due to the characteristics of a short path, small working plot, and discrete operation, the tractor needs to steer frequently and has high requirements for the turning radius [5,6]; in addition, because hillside tractors not only operate on horizontal ground, but also need to work on angled slopes, they have higher requirements for the stability of the tractors [7,8]. It has been shown that four-wheel steering technology can reduce the center of gravity of the vehicle to a certain extent, reduce the steering radius of the vehicle, and improve the stability of the steering process [9,10]. At present, the application of four-wheel steering technology in the field of hillside tractors is still in its infancy, and the study of four-wheel steering control strategies for hillside tractors can improve the operational efficiency and performance of agricultural machinery and enrich the theoretical research in the field of tractors with four-wheel steering.

In recent years, four-wheel steering control has been widely used for its high flexibility and stability. Tian et al. [11] proposed a new hierarchical control method based on the design of vehicles with four-wheel steering, which uses the sideslip angle, yaw rate feedback, and a linear three-degree-of-freedom (N3DOF) model to describe uncertainty. The proposed hierarchical controller achieves the same performance between the nonlinear 4WS vehicle and the reference model, showing good robustness. Zhang et al. [12] used linear quadratic control (LQC) theory to track the desired yaw rate to achieve zero sideslip angle response and proposed a numerical method to evaluate steering characteristics and handling stability. The simulation results show that the 4WS system with the LQC algorithm performs better in path tracking accuracy and active safety performance. Wang et al. [13] proposed a tractor rollover control method based on a single universal joint control torque gyroscope, established a nonlinear time-varying dynamics model considering factors such as gradient, road conditions, and centrifugal force, and designed a gyroscope rotor-based feed angular velocity controller. In order to improve the handling stability of cars with four-wheel steering (4WS), Zhang et al. [14] proposed an optimal control method for the 4WS system based on the quadratic optimal control theory. Through yaw rate feedback control, state feedback control, and optimal control, a vehicle with 4WS was controlled based on the yaw angle and center-of-mass turning angle using MATLAB/Simulink simulation. The results showed that the proposed control method can completely eliminate the center-of-mass turning angle, thereby further enhancing the vehicle’s handling performance. Men et al. [15] used fuzzy control technology to study the proportional feedforward and fuzzy feedback control strategy of vehicles with four-wheel steering. The simulation results show that fuzzy control has a good effect on the response of rear-wheel steering. It can improve the vehicle’s low-speed maneuverability, reduce the turning radius, and improve the high-speed handling stability. He et al. [16] used the fuzzy control technique to study the proportional feedforward plus fuzzy feedback control strategy for a two-degree-of-freedom four-wheeled steering vehicle, and the results show that fuzzy control can improve the responsiveness of rear-wheel steering, increase the maneuverability of the vehicle at low speeds, and reduce the turning radius. Yang et al. [17] comprehensively controlled the rear wheels of a car with 4WS by using a zero-center-of-mass sideslip angle feedforward controller and a yaw angular velocity fuzzy PID feedback controller. The vehicle model and control algorithm model were built using CarSim and MATLAB/Simulink. The joint simulation experiment was conducted under step conditions. The experiment shows that the 4WS joint control strategy can significantly improve the vehicle handling stability at different vehicle speeds and has good comprehensive control effects. Partono et al. [18] designed a prototype four-wheel steering control system based on fuzzy-Mamdani to control the yaw rate response and vehicle sideslip response, reducing the effects caused by oversteering and understeering. Concerning the steering control issue of hillside tractors, Yang et al. [19] designed an adaptive hillside tractor chassis; the steering system is designed with a disconnected trapezoidal structure, and two steering modes of front-wheel steering and four-wheel steering are adopted, which can realize full hydraulic four-wheel out-of-phase steering, which solves the problem of difficult steering in the field and is easy to be compacted in the working plot. However, its steering system lacks adaptive steering system control and cannot adapt to the operation of slope plots. The above research proves the superiority of four-wheel steering technology, but the current four-wheel steering control methods are mainly concentrated in the automotive field. The steering control strategy for hilly mountain tractors is less developed and it mostly uses automotive dynamics modeling analysis or multibody dynamics models to conduct related analysis and research.

Therefore, this paper takes the hillside tractor as the research object and builds a nonlinear three-degree-of-freedom model of tractors with four-wheel steering by considering the influences of the operating speed and the slope angle on the tractor’s dynamic characteristics. Based on this, to optimize the steering radius while meeting the steering stability in the range of 15° slope, a feedforward and pendulum angular velocity feedback following control strategy based on a fuzzy PID algorithm is designed. This control strategy combines the adaptability of fuzzy logic controllers and the accuracy of PID controllers, and can adaptively adjust control parameters according to the real-time working environment, thereby achieving stable steering control under complex terrain. Finally, in order to further verify the practicality and accuracy of the control method, a hardware-in-the-loop test was carried out. By combining the control algorithm with the actual hardware, the working state of the hilly mountain tractor under actual working conditions was simulated.

2. Dynamics Modeling of Tractors with Four-Wheel Steering

2.1. Three-Degree-of-Freedom Model Considering the Effect of Slopes

An accurate tractor dynamics model is the basis for carrying out the analysis. For hillside tractors during farmland and road driving, facing slippery roads, variable slope steering, and many other special scenes and conditions, a simple model cannot describe the body movement well; thus, accurate modeling is needed. This paper takes the ET-1004W tractor (Figure 1) as an example to carry out the modeling research on steering control strategy, and the specific parameters of modeling are shown in

Table 1. Tractor with four-wheel steering’s model parameters.

Parameter Name	Value	Unit
Vehicle mass $m$	6000	kg
Mass of tires $m_s$	5500	kg
Distance from the front axle to the center of mass $a$	1.16	m
Distance from the rear axle to the center of mass $b$	1.24	m
Lateral deflection stiffness of front wheel $k_f$	−105,000	N/rad
Rear-wheel lateral deflection stiffness $k_r$	−195,000	N/rad
Wheelbase $B$	1.37	m
Moment of inertia around the axle $I_x$	536.6	kg∙m2

.

In the process of tractor steering, if the steering angles of the front wheels are the same, the instantaneous centers of rotation of the front and rear wheels of the tractor will not be the same, and the wheels will appear to slip sideways, aggravate the tire wear, and increase the steering resistance in the driving process. Therefore, in general, in turning conditions, there is a certain proportional relationship between the wheels of the vehicle, which can make the steering wheel’s inside and outside turning angle conform to the ideal relationship, which is Ackermann’s steering principle [20]. The reasonable setting of the steering angle of the wheels can make the inner wheel rotate more than the outer wheel, thus reducing the difference in steering radius brought by the steering of different wheels and improving the stability and maneuverability of the vehicle.

Under normal operating conditions, the simplified linear two-degree-of-freedom model of the tractor with four-wheel steering established by the Newtonian mechanical system [21] can meet the requirements of the steering dynamics analysis, as shown in Figure 2.

However, the two-degree-of-freedom model only considers the motion of the vehicle body in the yaw and lateral directions during steering. When the tractor operates in hilly terrain, the body may tilt to form a lateral tilt due to the road conditions, as shown in Figure 3. Therefore, a four-wheel steering three-degree-of-freedom model is established based on the traditional two-degree-of-freedom model by considering the body side tilt.

In Figure 3, ${\varphi }_1$ is the body roll angle of the tractor, ${\dot{\varphi }}_1$ is the body roll angle speed, and $m_s$ is the mass above the suspension. Through dynamics analysis, we can obtain

$$\left\{\begin{array}{c}mu\left(\dot{\beta }+{\omega }_r\right)-m_{s}h\ddot{\varphi }=F_f+F_r\\ I_z{\dot{\omega }}_r-I_{xz}\ddot{\varphi }=a{F}_f-b{F}_r\\ I_x\ddot{\varphi }-I_{xz}{\dot{\omega }}_r-m_{s}hu\left(\dot{\beta }+{\omega }_r\right)\cos \varphi =-\left(c_f+c_r\right)\dot{\varphi }-\left(k_f+k_r\right)\varphi +m_{s}gh\sin \varphi \end{array}\right.$$

(1)

where $I_x$ is the body roll moment of inertia; $I_z$ is the body yaw moment of inertia; $I_{xz}$ is the mass inertia product of the body except the wheels; $R_f$ and $R_r$ are the steering roll coefficients of the front and rear axles; $c_f$ and $c_r$ are the roll damping coefficients of the front and rear axles; and $k_f$ and $k_r$ are the front and rear tires’ turning stiffness.

The wheel exhibits linear characteristics when its sideslip angle is small; therefore, $F_f$ and $F_r$ in the above formula can be expressed as follows:

$$\begin{array}{l}{F}_f=k_f\left(\beta +{\displaystyle \frac{a{\omega }_r}{u}}-{\delta }_f-R_f\varphi \right)\hfill \\ F_r=k_r\left(\beta -{\displaystyle \frac{b{\omega }_r}{u}}-{\delta }_r-R_r\varphi \right)\hfill \end{array}$$

(2)

Then, the dynamics equation after substituting $F_f$ and $F_r$ is presented as follows:

$$\left\{\begin{array}{c}mu\left(\dot{\beta }+{\omega }_r\right)-m_{s}h\ddot{\varphi }=k_f\left(\beta +{\displaystyle \frac{a{\omega }_r}{u}}-{\delta }_f-R_f\varphi \right)+k_r\left(\beta -{\displaystyle \frac{b{\omega }_r}{u}}-{\delta }_r-R_r\varphi \right)\\ I_z{\dot{\omega }}_r-I_{xz}\ddot{\varphi }=a{k}_f\left(\beta +{\displaystyle \frac{a{\omega }_r}{u}}-{\delta }_f-R_f\varphi \right)-b{k}_r\left(\beta -{\displaystyle \frac{b{\omega }_r}{u}}-{\delta }_r-R_r\varphi \right)\\ I_x\ddot{\varphi }-I_{xz}{\omega }_r-m_{s}hu\left(\dot{\beta }+{\omega }_r\right)\cos \varphi =-\left(c_f+c_r\right)\dot{\varphi }-\left(k_f+k_r\right)\varphi +m_s \mathrm{ghsin} \varphi \end{array}\right.$$

(3)

Considering that differential equations are not convenient for Matlab/Simulink modeling, the dynamics equation is converted into the state-space equation, and we can obtain the following:

$$\left\{\begin{array}{c}\dot{X}=\left(A^{-1}B\right)X+\left(A^{-1}C\right)U\\ Y=DX+EU\end{array}\right.$$

(4)

where $A=\left[\begin{array}{cccc}mu& 0& 0& -m_{s}h\\ 0& I_z& 0& -I_{xz}\\ -m_{s}hu& -I_{xz}& 0& I_x\\ 0& 0& 1& 0\end{array}\right]$, $B=\left[\begin{array}{cccc}{k}_f+k_r& {\displaystyle \frac{a{k}_f-b{k}_r}{u}}-mu& -k_f{R}_f-k_r{R}_r& 0\\ a{k}_f-b{k}_r& {\displaystyle \frac{a^2{k}_f+b^2{k}_r}{u}}& -a{k}_f{R}_f+b{k}_r{R}_r& 0\\ 0& m_{s}hu& m_{s}gh-k_f-k_r& -c_f-c_r\\ 0& 0& 0& 1\end{array}\right]$, $C=\left[\begin{array}{cc}-k_f& -k_r\\ -a{k}_f& b{k}_r\\ 0& 0\\ 0& 0\end{array}\right]$, $X=\left[\begin{array}{c}\beta \\ {\omega }_r\\ \varphi \\ \dot{\varphi }\end{array}\right]$, $U=\left[\begin{array}{c}{\delta }_f\\ {\delta }_r\end{array}\right]$, $Y=\left[\begin{array}{c}\varphi \\ \beta \\ {\omega }_r\end{array}\right]$, $D=\left[\begin{array}{llll}1\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill & 0\hfill \end{array}\right]$, $E=\left[\begin{array}{ll}0\hfill & 0\hfill \\ 0\hfill & 0\hfill \\ 0\hfill & 0\hfill \end{array}\right]$.

2.2. Modeling of Tractor Wheel Dynamics

Tires are one of the components of the body in contact with the ground during tractor operation and are responsible for carrying the forces and moments required for driving, braking, and steering. However, as tires are usually made of rubber, they have strong nonlinear characteristics, resulting in complex forces between the tires and the ground, which brings challenges to the study of tire dynamics. Tires largely determine the dynamics of the whole vehicle, which has a very important impact on handling stability, safety, and so on. The Magic Formula model [22] has good applicability by establishing an empirical–semi-empirical tire model through a simplified structure, force variation analysis, and data derivation. Therefore, the Magic Formula model is chosen to analyze the dynamic characteristics of tires in this paper. The input and output of the wheel model are shown in Figure 4.

The general form of the Magic Formula wheel model is as follows:

$$\left\{\begin{array}{c}Y(x)=y(x)+S_v\\ y(x)=D\sin [C\arctan \{Bx-E(Bx-\arctan (Bx))\}]\\ x=X+S_h\end{array}\right.$$

(5)

where $Y\left(x\right)$ is the lateral force, longitudinal force, or backing moment; $x$ is the corresponding wheel sideslip angle or longitudinal slip rate; $B$ is the stiffness factor; $B, C, \mathrm{and} D$ are determined by the vertical load and camber angle of the wheel; $E$ is the pressure inside the wheel; and $S_h$ and $S_v$ represent the horizontal and vertical drifts of the wheel curve.

Among them, parameter B is the stiffness factor, which can be regarded as a slope parameter and which controls the rate of change in the lateral deflection angle in the Magic tire equation; parameter C is the curve shape factor, which can be understood as a shape parameter and which determines the shape of the curve of the lateral deflection angle on the lateral force; parameter D is the peak factor, which can be regarded as a scale parameter and which controls the overall magnitude of the lateral force; and parameter E stands for the air pressure of the tire, which affects the tire’s grounding area and shape of the tire and which affects the lateral friction coefficient and lateral stiffness coefficient. Since the four parameters, B, C, D, and E, of the Magic Formula wheel model directly affect the final data of the tire, to more closely match the actual working conditions, the test data were imported from CarSim and fitted to the relevant parameters in MATLAB, and the parameter fitting results are shown in

Table 2. Wheel parameter fitting results.

Position	Parameters
	B	C	D	E
Front axle	0.1706	1.623	6078	−0.6919
Rear axle	0.1687	1.640	4233	−1.2650

.

2.3. Establishment of Nonlinear Three-Degree-of-Freedom Four-Wheel Steering Model

The transverse force on the body in the linear three-degree-of-freedom vehicle model of the tractor is linear, while the nonlinearity of the tractor in the sloping operation state is mainly reflected in the tires. The steering of the tractor in the uphill and downhill process can be approximated as level ground steering, and the lateral body inclination angle in the transverse operation process is much larger than that in the longitudinal operation, so this paper focuses on the force state of the tires of the tractor in transverse driving on a slope. Considering that hillside tractors usually have a body leveling mechanism, the tractor’s lateral force on the slope is shown in Figure 5.

The resultant force of the tractor body along the y direction can be expressed as

$${\displaystyle \sum F_y}=F_f+F_r+mg\sin {\varphi }_t=F_f+F_r-mg{\varphi }_t$$

(6)

Since the wheels are elastic, the resulting roll angle should be ${\varphi }_r=\varphi -{\varphi }_t$. Therefore, its changing rate is $\dot{{\varphi }_r}=\dot{\varphi }-\dot{{\varphi }_t}$. It is assumed that the slope angle remains unchanged, and ${\varphi }_r=0$; $K_{\varphi }$ and $C_{\varphi }$ represent the roll stiffness coefficient and roll damping coefficient of the vehicle, and $h_C$ is the horizontal height of the center of mass. Then, the roll moment generated by the overall wheel can be expressed as

$$M_x=K_{\varphi }\left(\varphi -{\varphi }_t\right)+C_{\varphi }\dot{\varphi }$$

(7)

Considering the role of the leveling mechanism, the formula for the tractor’s rolling motion is

$$\begin{gathered} m\dot{v}=-m\dot{\phi }u+m{h}_{\mathrm{C}}\ddot{\varphi }+\left(F_f+F_r\right)-mg\varphi \\ {I}_x\ddot{\varphi }=m{h}_{\mathrm{C}}\left(\dot{v}+{\omega }_{r}u-h_{\mathrm{C}}\ddot{\varphi }\right)+mg{h}_{\mathrm{C}}\varphi -K_{\varphi }\left(\varphi -{\varphi }_{\mathrm{t}}\right)+C_{\varphi }\dot{\varphi } \end{gathered}$$

(8)

The vehicle dynamics equation considering the slope angle can be obtained as

$$\begin{gathered} I_z\ddot{\phi }-\left({\displaystyle \frac{a{k}_f-b{k}_r}{u}}\right)v-\left({\displaystyle \frac{a^2{k}_f+b^2{k}_r}{u}}\right)\dot{\phi }=-a{k}_f{\delta }_f+b{k}_r{\delta }_r \\ m\dot{v}-m{h}_{\mathrm{C}}\ddot{\varphi }+\left(mu+{\displaystyle \frac{a{k}_f-b{k}_r}{u}}\right)\dot{\phi }-\left({\displaystyle \frac{a+b}{u}}\right)v-mg\varphi =-k_f{\delta }_f-k_r{\delta }_r \\ m{h}_{\mathrm{C}}\dot{v}-\left(I_x+m{h}_{\mathrm{C}}^2\right)\ddot{\varphi }-mu{h}_{\mathrm{C}}\dot{\phi }+C_{\varphi }\dot{\varphi }+\left(K_{\varphi }-mg{h}_{\mathrm{C}}\right)\varphi =K_{\varphi }{\varphi }_{\mathrm{t}} \end{gathered}$$

(9)

We select the status quantity as $\mathrm{\xi }={\left[v,\dot{\phi },\dot{\varphi },\varphi \right]}^T$, the control as $u_1={\left[{\delta }_f,{\delta }_r\right]}^T$, and the additional control as $u_2=\left[{\varphi }_t\right]$. Therefore, Formula (9) is rewritten in the following form:

$$M\stackrel{.}{\mathrm{\xi }}+N\mathrm{\xi }=F_1{u}_1+F_2{u}_2$$

(10)

where $M=\left[\begin{array}{cccc}m& 0& -m{h}_{\mathrm{C}}& 0\\ 0& I_z& 0& 0\\ -m{h}_{\mathrm{C}}& 0& I_x+m{h}_{\mathrm{C}}^2& 0\\ 0& 0& 0& 1\end{array}\right]$, $F_1=\left[\begin{array}{cc}-k_f& -k_r\\ -a{k}_f& b{k}_r\\ 0& 0\\ 0& 0\end{array}\right]$, $F_2=\left[\begin{array}{c}0\\ 0\\ K_{\varphi }\\ 0\end{array}\right]$, $N=\left[\begin{array}{cccc}-\left({\displaystyle \frac{k_f+k_r}{u}}\right)& \left(m{v}_x-{\displaystyle \frac{a{k}_f-b{k}_r}{u}}\right)& 0& mg\\ -\left({\displaystyle \frac{a{k}_f-b{k}_r}{u}}\right)& -\left({\displaystyle \frac{a^2{k}_f+b^2{k}_r}{u}}\right)& 0& 0\\ 0& -m{h}_{\mathrm{C}}u& C_{\varphi }& K_{\varphi }-mg{h}_{\mathrm{C}}\\ 0& 0& -1& 0\end{array}\right]$.

Considering that Formula (10) is not convenient for modeling in MATLAB/Simulink, it is transformed into Formula (11). The tractor four-wheel steering model considering the road inclination and leveling mechanism is as follows:

$$\stackrel{.}{\mathrm{\xi }}=A\mathrm{\xi }+B_1{u}_1+B_2{u}_2$$

(11)

where $A=M^{-1}N$, $B_1=M^{-1}{F}_1$, $B_2=M^{-1}{F}_2$. So, $A=\left[\begin{array}{cccc}-\frac{\left(k_f+k_r\right)h_{\mathrm{C}}^2}{I_{x}u}& \frac{h_{\mathrm{C}}^{2}u\left(mu-a{k}_f+b{k}_r\right)-m^2{h}_{\mathrm{C}}^3{u}^3}{I_{X}u}& \frac{h{D}_{\varphi }}{I_x}& \frac{mg{h}_{\mathrm{C}}^2+h_{\mathrm{C}}\left(K_{\varphi }-mg{h}_{\mathrm{C}}\right)}{I_x}\\ \frac{b{k}_r-a{k}_f}{I_{z}u}& -\left(\frac{a^2{k}_f+b^2{k}_r}{I_{z}u}\right)& 0& 0\\ -\frac{h_{\mathrm{C}}\left(k_f+k_r\right)}{I_{x}u}& -\frac{h_{\mathrm{C}}\left(a{k}_f-b{k}_r\right)}{I_{x}u}& \frac{C_{\varphi }}{I_x}& \frac{K_{\varphi }}{I_x}\\ 0& 0& -1& 0\end{array}\right]$, $B_1=\left[\begin{array}{cc}-{\displaystyle \frac{k_f{h}_{\mathrm{C}}^2}{I_x}}& -{\displaystyle \frac{k_r{h}_{\mathrm{C}}^2}{I_x}}\\ -{\displaystyle \frac{a{k}_f}{I_{\mathrm{z}}}}& {\displaystyle \frac{b{k}_r}{I_{\mathrm{z}}}}\\ -{\displaystyle \frac{h_{\mathrm{C}}{k}_f}{I_x}}& -{\displaystyle \frac{h_{\mathrm{C}}{k}_f}{I_x}}\\ 0& 0\end{array}\right]$, $B_2=\left[\begin{array}{c}{\displaystyle \frac{h_{\mathrm{C}}{K}_{\varphi }}{I_x}}\\ 0\\ {\displaystyle \frac{K_{\varphi }}{I_x}}\\ 0\end{array}\right]$.

The three-degree-of-freedom dynamics model of the tractor in a sloping environment is finally derived, but only a preliminary discussion on the traverse angular velocity of the tractor during steering is made. The control of the tractor’s traverse angular velocity is crucial because too fast of a traverse angular velocity is prone to rollover in the sloping ground environment. The influence of the lateral inclination angle on the stability under sloping conditions is large, and the stability of the tractor turning on sloping conditions can be equated with the stability of lateral inclination. Thus, the lateral inclination angle of the tractor is related to the traverse swing angle speed to obtain the maximum traverse swing angle speed of the tractor under the action of the lateral inclination angle, to prevent the tractor from tipping over under sloping conditions.

In summary, vehicles with four-wheel steering are relatively complex nonlinear systems, and their nonlinearity is mainly reflected in the tires; therefore, the model established is divided into the tire module and vehicle module. Combining the nonlinear characteristics of the previously established three-degree-of-freedom model and the tire model, a nonlinear three-degree-of-freedom model of the tractor with four-wheel steering is established as shown in Figure 6.

2.4. Verification of Four-Wheel Steering Three-Degree-of-Freedom Model Based on CarSim

The tires generate forces to steer the vehicle, and the vehicle itself has a center-of-mass lateral deflection angle related to the tire lateral deflection angle, so the tire parameters of this model always change. To verify the accuracy and validity of the model, the tractor vehicle parameters were set up and simulated in CarSim and compared with the nonlinear three-degree-of-freedom tractor model. The joint simulation results of the CarSim model and Matlab/Simulink R2019b are shown in Figure 7.

In Figure 7a, the nonlinear three-degree-of-freedom model simulation obtains the stable value of the yaw rate as 0.101 rad/s, the CarSim model simulation obtains the stable value of the yaw rate as 0.098 rad/s, and the deviation in the yaw rate is 3.06%; in Figure 7b, the stable value of the sideslip angle of the nonlinear three-degree-of-freedom model is 0.047 rad, the stable value of the sideslip angle of the CarSim model is 0.048 rad, and the sideslip angle deviation is 2.08%. In Figure 7c, the stable value of the roll angle obtained by the nonlinear three-degree-of-freedom model simulation is 0.00150 rad, the stable value of the roll angle obtained by the CarSim model simulation is 0.00155 rad, and the roll angle deviation is 3.22%. The theoretical model can be considered effective when the deviation between the theoretical model and the empirical model is less than 10% [6]. Therefore, the simulation results show that the yaw rate, center-of-mass sideslip angle, and roll angle of the nonlinear three-degree-of-freedom model are nonlinearly close to those of the real vehicle.

3. Design of Four-Wheel Steering Control System

3.1. Control Strategy Design

Compared with the steering control strategy design of vehicles with traditional front-wheel steering, the structure of vehicles with four-wheel steering is relatively complex. The control of vehicles with four-wheel steering needs to control the front and rear wheel angles for the operating condition of the vehicle so that the operating condition of the vehicle meets the control requirements. Compared with front-wheel steering, four-wheel steering has the advantages of low-speed steering flexibility and high-speed steering stability. In the theory of vehicle dynamics, the center-of-mass side deviation angle and pendulum angular velocity represent the vehicle trajectory maintenance and stability control, respectively. When the center-of-mass side deviation angle is large, vehicle maneuvering becomes very difficult and unreliable for the driver, and maintaining the trajectory becomes difficult; and when the pendulum angular velocity is large, the stability of the vehicle is difficult to ensure. Therefore, the control objective of the control system of a tractor with four-wheel steering in hilly areas is to control the overall steering radius and steering characteristics based on ensuring stability.

To realize stability under the premise of reducing the turning radius as much as possible, it is necessary to carry out joint control of the traverse angular velocity and the center-of-mass side deflection angle. At present, there are two basic control strategies: feedforward control and feedback control. When the front wheel angle is proportional to the rear wheel angle and the rear wheel angle is not directly related to the vehicle parameters, it is called feedforward control; when the rear wheel angle is proportional to the front wheel angle and the rear wheel angle is related to the vehicle parameters, it is called feedback control. These two control methods are for a single variable control, and cannot realize the tractor’s traverse angular velocity and the center-of-mass side deflection angle of the joint control. Therefore, based on feedforward control and feedback control for a single variable, the feedforward and traverse swing angle speed feedback following the control strategy are used to realize the joint control of the traverse swing angle speed and center-of-mass lateral deflection angle.

To verify the feasibility of the feedforward and pendulum angular velocity feedback control strategy on a tractor with four-wheel steering, modeling and simulation were carried out in MATLAB/Simulink, and the control block diagram is shown in Figure 8. The simulation parameters were set as follows: the tractor is traveling at a uniform speed u = 3 m/s, and the total simulation time is 10 s. A 5° step signal of the front wheel angle is provided at the moment of 1 s, and the responses of the yaw rate, the center-of-mass sideslip angle, and the roll angle are as follows.

From Figure 9a, it can be seen that the traverse swing angular velocity is greatly improved after applying the control strategy, and the steering time is shortened by five times compared with that without applying the control strategy; from Figure 9b, it can be seen that the steering of the tractor after applying the control strategy is similar to the expected driving direction and the actual driving direction, which makes the steering more desirable; the curve in Figure 9c shows that the lateral inclination angle increases after applying the control strategy, but considering that its value is still close to 0, the control strategy is more desirable. In summary, the feedforward and traverse angular velocity feedback following control has a good effect on the control of traverse angular velocity and center-of-mass side deflection angle, which is suitable for the control of stability and steering characteristics of tractors with four-wheel steering in hilly areas.

3.2. Fuzzy Adaptive PID Controller Design

Based on the control effect of fuzzy control on nonlinear systems, adaptive fuzzy PID has the advantages of high control precision and strong adaptability of PID, and shows excellent performance in dynamic system control. The fuzzy PID controller can dynamically adjust the three parameters in the PID controller, thereby improving the control accuracy of the system and optimizing the control effect. The structure of the fuzzy PID controller designed in this paper is shown in Figure 9.

In Figure 10, $c\left(t\right)$ represents the actual output of the control object; $r\left(t\right)$ represents the expected input value of the controlled variable; $e$ represents the deviation value of $c\left(t\right)$ and $r\left(t\right)$, where $e=r\left(t\right)-c\left(t\right)$; $ec$ represents the change rate of system deviation, which is the derivative value of $e$; $\Delta k_P$, $\Delta k_I$, and $\Delta k_D$ represent the PID adjustment parameter; $u\left(t\right)$ represents the actual output value of the fuzzy PID controller; and the dotted line represents the fuzzy controller.

The specific principles and processes of fuzzy PID are as follows:

• The input of the fuzzy controller is the deviation $e$ and deviation change rate $ec$ of the controlled variable;
• After the input is fuzzified by fuzzy rules, the membership degrees of $e$ and $ec$ belonging to different fuzzy sets are obtained. Therefore, using the established fuzzy rules and carrying out fuzzy reasoning, the fuzzy result of the output variables $\Delta k_P$, $\Delta k_I$, and $\Delta k_D$ can be obtained;
• The fuzzy result is defuzzified to obtain the precise value of the output variable;
• Finally, the parameters $P$, $I$, and $D$ of the PID controller are adaptively adjusted, respectively, through $\Delta k_P$, $\Delta k_I$, and $\Delta k_D$, and, finally, the output of the fuzzy PID is obtained as shown in the following formula:

  $$\left\{\begin{array}{l}{K}_P=K_{P0}+\Delta K_P\hfill \\ K_I=K_{I0}+\Delta K_I\hfill \\ K_D=K_{D0}+\Delta K_D\hfill \end{array}\right.$$

  (12)

  $$\begin{array}{l}u(t)=K_{P}e+K_I{\displaystyle {\int }_0^{t}e}dt+K_D{\displaystyle \frac{de}{dt}}\\ =\left(K_{P0}+\Delta K_P\right)e+\left(K_{I0}+\Delta K_I\right){\displaystyle {\int }_0^{t}e}dt+\left(K_{D0}+\Delta K_D\right){\displaystyle \frac{de}{dt}}\\ =\left(K_{P0}e+K_{I0}{\displaystyle {\int }_0^{t}e}dt+K_{D0}{\displaystyle \frac{de}{dt}}\right)+\left(\Delta K_{P}e+\Delta K_I{\displaystyle {\int }_0^{t}e}dt+\Delta K_D{\displaystyle \frac{de}{dt}}\right)\end{array}$$

  (13)

To realize the dynamic adjustment of the optimal parameters, the deviation $e$ and deviation change rate $ec$ are selected as the input parameters of the fuzzy controller in the design of the four-wheel-steering control system. By identifying the fuzzy relationships between $\Delta k_P$, $\Delta k_I$, $\Delta k_D$, $e$, and $ec$ and modifying the adjustment, the optimal $k_P$, $k_I$, and $k_D$ are output to meet the needs of control. The fuzzy subsets of deviation $e$ and deviation change rate $ec$ are $\left\{NB,NM,NS,ZO,PS,PM,PB\right\}$, where $PB$ represents positive great, $NB$ represents negative great, and $ZO$ represents zero. The basic domains for the output variables $\Delta k_P$, $\Delta k_I$, and $\Delta k_D$ are set to [0.25, 0.75], [50, 100], and [−0.1, 0.1], respectively. The fuzzy rules of $\Delta k_P$, $\Delta k_I$, and $\Delta k_D$ are established as shown in

Table 3. Δk_P fuzzy rule.

	e	NB	NM	NS	ZO	PS	PM	PB
ec
NB		PB	PB	PM	PM	PS	PS	ZO
NM		PB	PM	PM	PS	PS	ZO	NS
NS		PM	PM	PS	PS	ZO	NS	NS
ZO		PM	PS	PS	ZO	NS	NS	NM
PS		PS	PS	ZO	NS	NS	NM	NM
PM		PS	ZO	NS	NS	NM	NM	NB
PB		ZO	NS	NS	NM	NM	NB	NB

,

Table 4. Δk_I fuzzy rule.

	e	NB	NM	NS	ZO	PS	PM	PB
ec
NB		NB	NB	NM	NM	NS	NS	ZO
NM		NB	NB	NM	NS	NS	ZO	PS
NS		NB	NM	NS	NS	ZO	PS	PM
ZO		NM	NM	NS	ZO	PS	PS	PM
PS		NM	NS	ZO	PS	PS	PM	PM
PM		NS	NS	ZO	PS	PS	PM	PB
PB		NS	ZO	PS	PS	PM	PB	PB

and

Table 5. Δk_D fuzzy rule.

	e	NB	NM	NS	ZO	PS	PM	PB
ec
NB		PS	NS	NB	NM	NB	NM	NS
NM		PS	NS	NB	NM	NM	NS	ZO
NS		ZO	NS	NM	NM	NS	NS	ZO
ZO		ZO	NS	NS	NS	NS	NS	ZO
PS		ZO	ZO	ZO	ZO	ZO	ZO	ZO
PM		PB	PS	PS	PS	PS	PS	PS
PB		PB	PM	PM	PM	PS	PS	PS

. The basic domains for $e$ and $ec$ are [−6, 8] and [−2, 2], with proportional coefficients of 10, 100, and 10. The Mamdani method is used for fuzzy reasoning, and the gravity method is employed for defuzzification [23]. After defuzzification, the surface plots for $k_P$, $k_I$, and $k_D$ are obtained as shown in Figure 11.

4. Simulation Analysis and HIL Test Verification

4.1. Simulation Analysis Based on Fuzzy PID Control Effect

To analyze the control effectiveness of the four-wheel steering fuzzy PID controller designed in this paper on the tractor steering process (as shown in Figure 12), it was simulated based on Matlab/Simulink and compared with the ideal four-wheel steering controller (4WS), the front-wheel steering controller (FWS), and the feedforward four-wheel steering controller (QK). In the simulation test, the front wheel steering angle input was set to 5°, the starting time was 1 s, and the vehicle traveling speed was 3 m/s. The control effects of the tractor yaw rate, center-of-mass sideslip angle, and roll angle are shown in Figure 13, Figure 14 and Figure 15.

As shown in Figure 13, the front-wheel steering vehicle has a relatively small yaw rate, while the ideal four-wheel steering exhibits greater flexibility at low speeds. In contrast, the feedforward-controlled four-wheel steering system produces a large yaw rate, which leads to the unstable performance of the tractor in hilly and mountainous environments and increases the risk of rollover. Under the fuzzy PID control, the yaw rate of the tractor with four-wheel steering increases compared to that with front-wheel steering, but the increase is small, and the following effect on the ideal value is good; although there is a slight deviation, it is still in the safe range, at which time the yaw rate is in a stable state.

As shown in Figure 14, the center-of-mass sideslip angle is relatively large under four-wheel steering control, and increases nearly five times compared to feedforward control and fuzzy PID control. However, the center-of-mass sideslip angle under fuzzy PID control is larger than that under feedforward control, but the difference is not significant, and the values of both are close to 0. Therefore, the fuzzy PID controller is more effective at controlling the center-of-mass sideslip angle.

As can be seen in Figure 15, the roll angle is relatively large under the four-wheel steering control and front-wheel steering control. As the stabilized value of the yaw rate on the inclined ground decreases, the flexibility of the vehicle is also affected. The four-wheel steering system based on fuzzy PID control can better balance the control objectives of optimizing the steering radius and ensuring the stability of the vehicle and is suitable for the steering control of tractors in hilly mountainous hillside environments.

4.2. Experimental Validation Based on the HIL Platform

4.2.1. HIL Platform

To verify the control performance of the four-wheel steering fuzzy PID controller in actual steering scenarios, we built a real-time hardware-in-the-loop (HIL) test bench, which mainly includes NI real-time timing, a rapid prototyping controller, CAN analysis tool, myDAQ collector, steering wheel, pedal, etc. Among them, the NI real-time timing is the PXIe-1071 real-time platform, the controller selects the STM32F103ZET6 of the point atom, the CAN device is the USBCAN-2E-U of the far-reaching idea, and the computer is the Windows system. The test signal is collected in real time through the myDAQ collector, as shown in Figure 16. In the MATLAB/Simulink environment, a hill tractor model based on a fuzzy PID control algorithm is constructed. The control code is compiled and burned into the controller through the rapid prototyping development tool. After the power supply is completed, the control algorithm transmits the tractor’s state information to the STM32F103 controller through the CAN card communication module during the entire HIL test process. The controller processes this information and outputs the target control steering to achieve closed-loop control.

4.2.2. Results and Discussion

After the display speed of the host PC interface tends to be stable, one should turn the steering wheel to the right about 10° and keep one’s foot on the accelerator pedal as stably as possible. As shown in Figure 17, the actual output signal of the pedal is a voltage signal, that is, the distance of the pedal is linked to the resistance of the potentiometer, and the voltage of the pedal signal does not change abruptly when the pedal signal is output, which is reflected in the acceleration stage of the pedal during analog acquisition (Figure 17a). However, in actual production, the general operating state is a steady-state speed, and there is no acceleration phase. Therefore, the corresponding value when the speed is stable is selected as the excitation in the HIL test, so the speed input curve in the actual HIL test is as shown in Figure 17b.

Figure 18a shows that the yaw rate partially overshoots at the beginning. The reason may be that the controller has just performed fuzzy reasoning at this time, and its reasoning speed is slow, that is, the controller has a partial delay at the beginning of the calculation, resulting in a partial overshoot of the response. There is a slight fluctuation at 1.7–2.1 s, and the fluctuation range is relatively small. The fluctuation at 3.5–4.3 s is more severe. It is inferred that this may be due to the failure of the force control when the pedal is stepped on, resulting in a large change in speed. In addition, the later data are stable as a whole, there is no large-scale fluctuation, and the basic floating change is above and below the ideal yaw rate. In general, the control of the yaw rate is effective.

From Figure 18b, it can be seen that the center-of-mass sideslip angle changes from 0 to negative and rises rapidly to positive in the first 0.5 s. The reason may be that the center of mass fails to follow the change in steering in time at the beginning of steering. In the next 3 s, the sideslip angle remains relatively stable, with little change. However, between 3.8 s and 4.2 s, the center-of-mass sideslip angle increases significantly, which may be consistent with the reason for yaw rate fluctuation, that is, it fluctuates due to the change in tractor speed. Overall, the designed control strategy can effectively control the sideslip angle of the center of mass, making it close to 0, achieving a good target trajectory tracking effect, and providing a relatively good driving experience. It can be seen from Figure 18c that after the introduction of a fixed 0.052 rad roll angle, the roll angle of the whole machine still shows a strong correlation with the yaw rate, the roll angle generated by the vehicle body itself is small, and the safety is high.

In summary, the four-wheel steering control strategy designed in this paper is feasible. The control of the vehicle’s sideslip angle reaches the expected control target, and the control of the yaw rate is in line with expectations to a certain extent. On the whole, the control algorithm can run in the simulation of the actual situation and achieve the expected control goal, that is, to optimize the steering radius under the premise of ensuring stability. Fuzzy PID control can show a better control effect in the case of a semi-physical test. The control of steering behavior at a certain speed is in line with the expected control target and has high practicability.

5. Conclusions

A nonlinear three-degree-of-freedom dynamics model of a tractor is established by considering the yaw motion, lateral motion, roll motion, and tire lateral characteristics for the characteristics of hilly terrain, and a joint simulation is carried out between the designed model and the CarSim dynamics model. The deviations in the yaw rate, the center-of-mass sideslip angle, and the roll angle are 3.06%, 2.08%, and 3.22%, respectively, and the deviation is controlled within the standard of 10%, which verifies the accuracy of the model.

Based on the control objective of optimizing the steering radius under the premise of stability, a control strategy based on the fuzzy PID algorithm of feedforward and yaw rate feedback following control is designed, and the three control methods of feedforward control, feedback control, and following control are simulated, compared, and analyzed in MATLAB/Simulink. The results show that compared with feedforward control and feedback control, the maximum amplitude of the yaw rate reaches 0.28 rad/s and 0.31 rad/s under step steering conditions and double lane-change steering conditions, and the steering time is reduced by 10%. The deviation between the response of the sideslip angle and the ideal value is small, and the control effect is ideal. The value of the roll angle is close to 0, which ensures that the driver can better adapt to the hilly and mountainous terrain, and reduces the instability of the vehicle due to terrain changes. The following control effect is better than that of the previous wheel feedforward control and feedback control, which greatly improves the handling stability of the mountain tractor.

Aiming at the inadequacy of model simulation, such as insufficient safety inspections and unrealistic working condition simulation, a hardware-in-the-loop real-time test platform is built based on NI real-time simulator and rapid prototyping controller. The semi-physical simulation test is carried out by inputting the speed and steering wheel angle of the tractor in the actual steering state. The experimental results show that the designed control strategy meets the stability and steering flexibility requirements of the hillside tractor, providing theoretical guidance for the research on four-wheel steering control and control strategies for tractors in hilly terrain.