Study on Chassis Leveling Control of a Three-Wheeled Agricultural Robot

Abstract

Three-wheeled agricultural robots possess the advantages of high flexibility, strong maneuverability, and low cost. They can adapt to various complex terrains and operational environments, making them highly valuable in the fields of crop planting, harvesting, irrigation, and more. However, the horizontal stability of the three-wheeled agricultural robot chassis is compromised when working in harsh terrain, significantly impacting the overall operational quality and safety. To address this issue, this study designed a leveling system based on active suspension and proposed a stepwise leveling method based on an adaptive dual-loop composite control strategy (ADLCCS-SLM). Firstly, in the overall control of the three-wheeled chassis, a stepwise leveling method (SLM) was introduced. This method allows for rapid leveling by incrementally adjusting one or two suspensions, effectively avoiding the complex interactions between suspension components encountered in traditional methods involving the simultaneous linkage of three suspensions. Next, in terms of suspension actuator control, an adaptive dual-loop composite control strategy (ADLCCS) was proposed. This strategy employs a dual-loop composite control both internally and externally and utilizes an improved adaptive genetic algorithm to adjust critical control parameters. This adaptation optimizes the chassis leveling performance across various road conditions. Finally, the effectiveness of the proposed ADLCCS-SLM was validated through simulation and experimental testing. The test results showed that the control effect of the proposed method was significant. Compared to the traditional multi-suspension linkage leveling method based on PID, the peak values of pitch angle and roll angle were reduced by 31.8% and 33.3%, respectively.

1. Introduction

Three-wheeled agricultural robots, employing a three-wheel drive system, offer advantages such as flexible turning and high maneuverability, enabling them to adapt to irregular field shapes and complex terrains. Additionally, these robots are characterized by low costs and easy operation. They can perform various agricultural tasks, including tilling, sowing, fertilizing, and harvesting, thereby presenting broad application prospects in the agricultural sector.

When operating in complex terrains such as hills, wheeled agricultural robots need to maintain a level chassis in order to enhance their stability and maneuverability in agricultural environments, thereby enabling them to perform agricultural tasks more effectively. However, agricultural robots with three-wheeled chassis face greater challenges in maintaining chassis levelness compared to their counterparts with four or more wheels. Due to having only three wheels for support, a three-wheeled chassis is more susceptible to uneven ground when leveling on rough terrain or under complex working conditions. This can result in inadequate leveling or the inability to achieve complete balance, which affects the efficiency and quality of mechanical operations. Therefore, research on chassis leveling control is of paramount importance for the development of three-wheeled agricultural robots. By implementing adaptive leveling mechanisms in the chassis, the adaptability of these robots to diverse terrains can be enhanced, thereby improving work efficiency and stability and providing better support for agricultural production.

The chassis leveling system has significant application value not only in agricultural machinery but also in construction machinery and specialized vehicles. It helps maintain vehicle stability and improve operational efficiency on varying road conditions. The current research literature mainly focuses on innovative design of system structures and improvements in control methods, covering different types of vehicles and various leveling techniques.

In the design of leveling system structures, researchers have developed various types of systems based on the operating environment and working conditions of vehicle chassis. For instance, Federico et al. [1] designed a two-level cascade regulator to control the roll and pitch angles of a combine harvester chassis. Hu et al. [2] developed a four-point adjustable track chassis for combine harvesters to address body tilt issues. He et al. [3] proposed a fuzzy sliding-mode variable structure control algorithm for leveling systems in tractors operating in hilly areas. Lü et al. [4] designed a controllable adaptive leveling mechanism to enhance the automatic leveling performance of tractors in hilly and mountainous regions.

In terms of leveling control methods, researchers primarily focus on active control for single-wheel suspension and overall vehicle leveling strategies. Applying active control to suspension systems is the most direct and effective method for vehicle leveling. For single-wheel suspension control, the main research areas include nonlinear system modeling and adaptive control strategies for unknown road disturbances, including modeling nonlinear systems with multi-parameter uncertainties [5], designing nonlinear adaptive control strategies [6], optimizing active suspension parameters based on road estimation [7], implementing pressure-independent methods to suppress vibrations [8], and estimating road conditions and optimizing active suspension parameters based on hydraulic system pressure [9]. For overall vehicle active suspension leveling control strategies, due to higher hardware costs and control computation requirements, researchers focus on coordinated control strategies for full-vehicle suspension, efficient dimension-reduced vehicle models, and low-cost vehicle control. Liu et al. [10] proposed a multi-suspension obstacle-crossing strategy based on feedforward- and feedback-coordinated control for four-wheel off-road vehicles tackling step-road obstacles. Zhang et al. [11] introduced a dimension-reduced vehicle model based on network topology structure and coupled constraints to balance active suspension control system performance and computational efficiency. Hamza et al. [12] used artificial neural networks (ANN) to control low-cost active dampers, significantly reducing vibrations for patients and stretchers in ambulances. Designing active suspension controllers generally involves linear and nonlinear control strategies. Linear strategies include optimal control [13,14] and H∞ control [15,16], while nonlinear strategies encompass sliding-mode control [17,18,19] backstepping control [20], and adaptive control [21].

In summary, the research on chassis leveling system design and control is progressing towards intelligent, high-efficiency, and highly adaptable directions. Current research focuses include improving system response speed, leveling accuracy, and self-learning adjustment capabilities. Additionally, there is ongoing exploration of new sensor technologies, control algorithms, and simulation platforms to enhance overall system performance. Future research is expected to further integrate electromechanical and hydraulic control, ensuring safety while improving operational efficiency and driving comfort in complex environments.

However, existing research on vehicle chassis leveling technology primarily focuses on four-wheeled and multi-wheeled vehicles, with limited studies addressing the leveling systems and control methods for three-wheeled chassis. Although four-wheeled chassis leveling systems and control strategies can theoretically be applied to three-wheeled chassis, significant challenges remain in terms of design compatibility, stability, cost, efficiency, and safety. Therefore, it is essential to develop simple and efficient leveling control methods specifically for three-wheeled chassis. To address this need, this paper designed a three-wheeled chassis leveling system based on active suspension and proposed a stepwise method based on an adaptive dual-loop composite control strategy (ADLCCS-SLM). This method only requires adjusting the minimum number of suspensions based on the chassis’ pitch and roll angle data, specifically by step-by-step adjustment of one or two suspensions, to achieve the leveling goal. Additionally, the method employs inner and outer dual-loop composite control, with key control parameters adjusted using an improved adaptive genetic algorithm (IAGA) to adapt to various road conditions and optimize chassis leveling performance.

This study designed a three-wheeled chassis leveling system based on active suspension. A 6-DOF mathematical model for the entire chassis and a 2-DOF mathematical model for a single suspension, which consider the nonlinearities of the hydraulic system, were then established. A comprehensive analysis of the chassis posture of the three-wheeled agricultural robot during operation was conducted, resulting in the identification of 19 different attitudes. Based on these 19 attitudes, five categories of leveling conditions were summarized, thereby simplifying the workload of leveling decisions. On this basis, a stepwise leveling method based on an adaptive dual-loop composite control strategy (ADLCCS-SLM) was proposed. Finally, the effectiveness of this method was verified through simulations and experiments. The definitions of all symbols in the manuscript are shown in

Table 1. Notation.

Symbol	Description
Zb	Vehicle body centroid vertical displacement
α	Roll angle of the three-wheeled agricultural robot
β	Pitch angle of the three-wheeled agricultural robot
ZA1	Sprung mass displacement at the left front wheel
ZB1	Sprung mass displacement at the right front wheel
ZC1	Sprung mass displacement at the rear wheel
ZA2	Wheel displacement of the left front wheel
ZB2	Wheel displacement of the right front wheel
ZC2	Wheel displacement of the rear wheel
wA	Road excitation of the left front wheel
wB	Road excitation of the right front wheel
wC	Road excitation of the rear wheel
FA1	Force of the left front wheel suspension on the chassis
FB1	Force of the right front wheel suspension on the chassis
FC1	Force of the rear wheel suspension on the chassis
FA2	Controllable forces provided by the left front wheel suspension hydraulic actuators
FB2	Controllable forces provided by the right front wheel suspension hydraulic actuators
FC2	Controllable forces provided by the rear wheel suspension hydraulic actuators
cA	Left front wheel suspension damping coefficients
cB	Right front wheel suspension damping coefficients
cC	Rear wheel suspension damping coefficients
ksA	Left front wheel suspension spring stiffness
ksB	Right front wheel suspension spring stiffness
ksC	Rear wheel suspension spring stiffness
ktA	Left front wheel tire stiffness
ktB	Right front wheel tire stiffness
ktC	Rear wheel tire stiffness
Ms	Sprung mass of the three-wheeled chassis
Mw	Unsprung mass of the three-wheeled chassis
MA1	Sprung mass at the left front wheel
MB1	Sprung mass at the right front wheel
MC1	Sprung mass at the rear wheel
MA2	Unsprung mass at the left front wheel
MB2	Unsprung mass at the right front wheel
MC2	Unsprung mass at the rear wheel
lf	Distance between the two front wheel suspensions
lb	Distance between the center of the two front wheel suspensions and the rear wheel suspension
lbf	Distance between the vehicle body centroid and the center of the two front wheel suspensions
lbr	Distance between the vehicle body centroid and the rear wheel suspension
Ix	Pitch moment of inertia
Iy	Roll moment of inertia
Mx	Pitch moment of the vehicle body
My	Roll moment of the vehicle body
Xv	Valve spool displacement
Ps	Supply pressure
P1	Pressure in the rodless chamber
P2	Pressure in the rod chamber
ρ	Fluid density
ω	Valve port area gradient
Cd	Flow coefficient of the throttling orifice
V1	Volume of the rodless chamber of the hydraulic cylinder
V2	Volume of the rod chamber of the hydraulic cylinder
A1	Effective piston area of the rodless chamber
A2	Effective piston area of the rod chamber
βe	Effective bulk modulus

.

2. The Structure of the Three-Wheeled Chassis Leveling System

The chassis layout of the three-wheeled agricultural robot is shown in Figure 1. Both the front left and front right wheels are tracked, while the rear wheel features a pair of symmetrically arranged dual tires. To ensure effective leveling for the chassis, all three wheels are equipped with independent suspension. The chassis leveling system of the three wheeled agricultural robot utilizes the three-point leveling principle, which controls the horizontal orientation of the chassis by adjusting the height of three support points. This leveling system offers advantages such as a simple structure, simple control strategy, and rapid leveling response. Consequently, it is suitable for agricultural tasks in situations with poor road conditions that require quick leveling.

Firstly, the rear suspension is briefly introduced. Figure 2 illustrates the structure of the rear wheel suspension. This suspension employs a single oblique arm independent suspension. By adjusting the suspension actuator, the relative position between the rear suspension support seat and the single oblique arm can be changed, thus altering the height of the chassis support point from the rear wheel.

The mechanism principle of the rear wheel suspension is shown in Figure 3. The detailed derivation of the mathematical model is provided in Appendix A.

Then, the front suspension structure is briefly introduced. The front suspension of the three-wheeled agricultural robot chassis employs a double-wishbone independent suspension, as shown in Figure 4. The front wheel suspension actuator can adjust the relative position of the front suspension seat and the double wishbone, thereby altering the distance between the chassis support point and the wheel.

Figure 5 shows the mechanism schematic diagram of the front wheel double-wishbone independent suspension. The detailed derivation of the mathematical model is provided in Appendix A.

3. Mathematical Modelling

3.1. Force Model of the Three-Wheeled Chassis

Based on the mechanism principle of the active suspension of the three-wheeled chassis, the force analysis model of the chassis was established, as shown in Figure 6. For the force model, the physical meaning of each variable symbol is described below. Z_b represents the vertical displacement of the vehicle body centroid. α and β indicate the roll angle and pitch angle, respectively. Z_A1, Z_B1, and Z_C1 indicate the sprung mass displacement at the left front wheel, right front wheel, and rear wheel suspension, respectively. Z_A2, Z_B2, and Z_C2 represent the wheel displacement of the left front wheel, right front wheel, and rear wheel. w_A, w_B, and w_C indicate the road excitation of each wheel. F_A1, F_B1, and F_C1 indicate the force exerted by the left front wheel, right front wheel, and rear wheel suspension on the chassis. F_A2, F_B2, and F_C2 represent the controllable forces provided by the hydraulic actuators of the left front wheel, right front wheel, and rear wheel suspension. c_A, c_B, and c_C indicate the damping coefficients of the left front wheel, right front wheel, and rear wheel suspension. k_sA, k_sB, and k_sC indicate the spring stiffness of the left front wheel, right front wheel, and rear wheel suspension. k_tA, k_tB, and k_tC indicate the tire stiffness of the left front wheel, right front wheel, and rear wheel. M_s stands for the mass of the vehicle body of the three-wheeled chassis. M_A1, M_B1, and M_C1 represent the sprung mass at the left front wheel, right front wheel, and rear wheel suspension, while M_A2, M_B2, and M_C2 represent the unsprung mass at the left front wheel, right front wheel, and rear wheel suspension. After the force analysis of the chassis, the following formula can be obtained. The vibration model of the chassis can be given by:

$$\begin{array}{c}{M}_{x2}\stackrel{••}{Z_{x2}}+k_{sx}\left(Z_{x2}-Z_{x1}\right)+k_{tx}\left(Z_{x2}-w_x\right)+c_x\left(\stackrel{•}{Z_{x2}}-\stackrel{•}{Z_{x1}}\right)+F_{x2}=0\\ \begin{array}{l}{F}_{x1}+k_{sx}\left(Z_{x1}-Z_{x2}\right)+c_x\left(\stackrel{•}{Z_{x1}}-\stackrel{•}{Z_{x2}}\right)-F_{x2}=0\\ M_s\stackrel{••}{Z_b}-F_{\mathrm{A}1}-F_{\mathrm{B}1}-F_{\mathrm{C}1}=0\end{array}\end{array}\begin{array}{ccc}& & (x=\mathrm{A},\mathrm{B},\mathrm{C})\end{array}$$

(1)

3.2. Attitude Model of the Three-Wheeled Chassis

The attitude analysis parameters of the chassis mainly include the body vertical displacement, Z_b, pitch angle, β, and roll angle, α. Each wheel of the three-wheeled agricultural robot experiences different road excitations, resulting in different sprung mass displacement of each suspension. The superposition effect of these three different displacement amounts causes changes in the chassis attitude. The relationship between the sprung mass displacement of the three suspensions and the chassis attitude parameters can be expressed as follows:

$$Z_{\mathrm{A}1}={\displaystyle \frac{l_{\mathrm{f}}}{2}}\sin \alpha -l_{\mathrm{bf}}\sin \beta +Z_{\mathrm{b}}$$

(2)

$$Z_{\mathrm{B}1}=-{\displaystyle \frac{l_{\mathrm{f}}}{2}}\sin \alpha -l_{\mathrm{bf}}\sin \beta +Z_{\mathrm{b}}$$

(3)

where l_f is the distance between the two front wheel suspensions, l_bf is the distance between the vehicle body centroid and the center of the two front wheel suspensions, and l_br is the distance between the vehicle body centroid and the rear wheel suspension.

The changes in the chassis pitch angle and roll angle reflect the variations in the external moments acting on the vehicle body. The change in pitch angle results from the combined pitching moment acting on the vehicle body, while the change in roll angle is due to the combined rolling moment. Therefore, based on the rigid body rotational dynamics equations, the following two equations can be derived:

$$M_x=I_x\stackrel{••}{\beta }=-\left(F_{\mathrm{A}1}+F_{\mathrm{B}1}\right)l_{\mathrm{bf}}+F_{\mathrm{C}1}{l}_{\mathrm{br}}$$

(4)

$$M_y=I_y\stackrel{••}{\alpha }=\left(F_{\mathrm{A}1}-F_{\mathrm{B}1}\right){\displaystyle \frac{l_{\mathrm{f}}}{2}}$$

(5)

where I_x and I_y represent the pitch moment of inertia and the roll moment of inertia, respectively. M_x denotes the pitch moment of the vehicle body and M_y denotes the roll moment of the vehicle body.

3.3. Single-Wheel Suspension System Model

3.3.1. Single-Wheel Suspension Model

The single-wheel suspension model of the chassis of the three-wheeled agricultural robot is shown in Figure 7. In this simplified model, since only one of the A, B, and C suspensions is considered for modeling, the symbols used in the full model in Section 3, such as M_x1 (where x = A, B, C), are simplified to M₁ for clarity. Other similar symbols are simplified in the same way.

In this paper, the distances from the hinge point, S, to the hydraulic actuator and the wheel are defined as S_b and S_a, respectively. Thus, the motion equation of the single-wheel suspension model can be derived, as shown in the following equation.

$$\begin{array}{l}{M}_1{\ddot{Z}}_1-F_2+k_s(Z_1-Z_2)+c\left(\stackrel{•}{Z_1}-\stackrel{•}{Z_2}\right)=0\\ M_2{\ddot{Z}}_2+k_t(Z_2-w)+F_2+k_s(Z_2-Z_1)+c\left(\stackrel{•}{Z_2}-\stackrel{•}{Z_1}\right)=0\end{array}$$

(6)

3.3.2. Hydraulic Servo System Model of the Suspension Actuator

The hydraulic actuator system is the primary energy absorption and output component in independent active suspension systems. Its dynamic response performance is a crucial factor affecting the damping and leveling capabilities of the active suspension system. As a nonlinear system, the hydraulic actuator can be simplified to a valve-controlled cylinder system, as illustrated in Figure 8.

Based on the valve flow equation, hydraulic cylinder flow equation, and hydraulic cylinder force balance equation, the state equations for hydraulic system pressures P₁ and P₂ and suspension displacements Z₁ and Z₂ are derived as follows (The detailed derivation of the hydraulic servo system model of the suspension actuator is provided in Appendix B):

$$\begin{array}{l}{\dot{P}}_1={\displaystyle \frac{{\beta }_e}{V_1}}\left\{C_d\omega X_v\sqrt{{\displaystyle \frac{2}{\rho }}}\left[S\left(X_v\right)\sqrt{P_s-P_1}+S\left(-X_v\right)\sqrt{P_1}\right]-A_1\left(\stackrel{•}{Z_1}-\stackrel{•}{Z_2}\right)\right\}\\ {\dot{P}}_2=-{\displaystyle \frac{{\beta }_e}{V_2}}\left\{C_d\omega X_v\sqrt{{\displaystyle \frac{2}{\rho }}}\left[S\left(X_v\right)\sqrt{P_2}+S\left(-X_v\right)\sqrt{P_s-P_2}\right]-A_2\left(\stackrel{•}{Z_1}-\stackrel{•}{Z_2}\right)\right\}\end{array}$$

(7)

$$S(x)=\{\begin{array}{l}1\begin{array}{cc}& x\ge 0\end{array}\\ 0\begin{array}{cc}& x<0\end{array}\end{array}$$

(8)

$$\begin{array}{l}{\ddot{Z}}_1={\displaystyle \frac{A_1{P}_1-A_2{P}_2+k_s(Z_2-Z_1)+c\left(\stackrel{•}{Z_2}-\stackrel{•}{Z_1}\right)}{M_1}}\\ {\ddot{Z}}_2={\displaystyle \frac{A_2{P}_2-A_1{P}_1+k_s(Z_1-Z_2)+c\left(\stackrel{•}{Z_1}-\stackrel{•}{Z_2}\right)+k_t(w-Z_2)}{M_2}}\end{array}$$

(9)

where X_v represents the valve spool displacement, P_s represents the supply pressure, P₁ represents the pressure in the rodless chamber, P₂ represents the pressure in the rod chamber, ρ represents the fluid density, ω represents the valve port area gradient, C_d represents the flow coefficient of the throttling orifice, V₁ represents the volume of the rodless chamber of the hydraulic cylinder, V₂ represents the volume of the rod chamber of the hydraulic cylinder, A₁ represents the effective piston area of the rodless chamber, A₂ represents the effective piston area of the rod chamber, and β_e represents the effective bulk modulus.

3.3.3. State Equation of the Single-Wheel Suspension

Choose the following state variables: $x_1=Z_1-Z_2$, $x_2={\dot{Z}}_1$, $x_3=Z_2$, $x_4={\dot{Z}}_2$, $x_5=w$. Based on the dynamics equations of the single-wheel suspension system, the state-space equations can be derived as follows:

$$\begin{array}{l}{\dot{x}}_1=x_2-x_4\\ {\dot{x}}_2={\displaystyle \frac{\tilde{u}}{M_1}}\\ {\dot{x}}_3=x_4\\ {\dot{x}}_4=-{\displaystyle \frac{1}{M_2}}\left[K\left(x_3-x_5\right)+\tilde{u}\right]\\ {\dot{x}}_5=-2\mathsf{\pi }{f}_{0}w\left(t\right)+2\mathsf{\pi }\sqrt{G_{0}v}r\left(t\right)\end{array}$$

(10)

Taking the suspension actuator control force as the virtual control input, $\tilde{u}$, and selecting the vertical acceleration, ${\ddot{Z}}_1$, of the vehicle body, the suspension dynamic stroke ($Z_1-Z_2$), and the tire dynamic displacement ($Z_2-w$) as the system outputs, Equation (10) can be rewritten as follows:

$$\begin{array}{l}\dot{\mathit{X}}=\mathit{A}\mathit{X}+\mathit{B}\tilde{\mathit{U}}+\mathit{G}\mathit{r}\\ \mathit{Y}=\mathit{C}\mathit{X}+\mathit{D}\tilde{\mathit{U}}\end{array}$$

(11)

where X represents the state variables, r represents the disturbance input, and A, B, C, D, and G represent the coefficient matrices.

$$\begin{gathered} \mathit{A}=\left[\begin{array}{ccccc}0& 1& 0& -1& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& -{\displaystyle \frac{K}{M_a}}& 0& {\displaystyle \frac{K}{M_a}}\\ 0& 0& 0& 0& -2\mathsf{\pi }{f}_0\end{array}\right] \\ \mathit{B}={\left[\begin{array}{ccccc}0& {\displaystyle \frac{1}{M_b}}& 0& -{\displaystyle \frac{1}{M_a}}& 0\end{array}\right]}^{\mathrm{T}} \\ \mathit{C}=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 1& 0& -1\end{array}\right] \\ \mathit{D}={\left[\begin{array}{ccc}0& {\displaystyle \frac{1}{M_b}}& 0\end{array}\right]}^{\mathrm{T}} \\ \mathit{G}={\left[\begin{array}{ccccc}0& 0& 0& 0& 2\mathsf{\pi }\sqrt{G_{0}v}\end{array}\right]}^{\mathrm{T}} \end{gathered}$$

(12)

4. Chassis Leveling Control

The horizontal stability performance of the chassis in a three-wheeled agricultural robot significantly impacts the overall work efficiency, operational safety, and precision of the machine during agricultural tasks. In terms of ride smoothness, the agricultural robot doesn’t need to consider ride comfort; the main goal is to reduce mechanical wear. Therefore, for the chassis attitude control of the three-wheeled agricultural robot, priority should be given to the leveling performance, while also considering the control performance smoothness. Considering the need for chassis attitude control, this paper introduces a new leveling method: a stepwise leveling method based on an adaptive dual-loop composite control strategy (ADLCCS-SLM).

Firstly, in terms of overall machine attitude control, this paper proposes a rapid leveling method suitable for three-wheeled chassis, referred to in this paper as the stepwise leveling method (SLM). The SLM achieves the vehicle’s leveling goal by adjusting the vehicle suspension in stages, rather than adjusting all the suspensions simultaneously. Specifically, the SLM requires only the pitch and roll angle data of the chassis. By adjusting the minimum number of suspensions, gradually adjusting one or two suspensions, the leveling objective can be achieved. The simplicity and efficiency of this method lie in its incremental approach to adjustments, rather than making simultaneous adjustments to all suspensions. Compared to traditional suspension control methods, the SLM avoids the complex interactions between suspension components that occur when various suspension mechanisms work together. By gradually optimizing adjustments, it is possible to more effectively achieve the vehicle’s balanced state, while also simplifying the design and implementation process of the control system.

In terms of suspension system control, this paper proposes a new method called the adaptive dual-loop composite control strategy (ADLCCS). This strategy employs an inner and outer dual-loop composite control approach, with key control parameters adjusted using an improved adaptive genetic algorithm (IAGA) to adapt to different road conditions and optimize chassis leveling performance. The dual-loop control combines the advantages of both inner and outer loop controls. The outer loop utilizes the linear quadratic Gaussian model (LQG) to derive the optimal virtual control law, effectively enhancing the system’s robustness and response speed. The inner loop employs the backstepping algorithm to handle nonlinear terms, providing the capability to manage complex nonlinear systems. IAGA is used to select the optimal variable parameter combinations for the inner and outer loops under different working conditions, allowing for adaptive adjustments based on actual road conditions, thereby improving the system’s adaptability and leveling performance. The leveling control logic diagram of the three wheeled chassis is shown in Figure 9.

4.1. Chassis Stepwise Leveling Method

Figure 10 illustrates the coordinate system diagram of the three-wheeled chassis. Points A, B, and C represent the support points of the left front suspension, right front suspension, and rear suspension, respectively. OXYZ is the horizontal reference coordinate system, which only translates relative to the absolute coordinate system. OX₁Y₁Z₁ is the rotational coordinate system, which rotates with the vehicle body relative to the horizontal reference coordinate system, OXYZ. The angle of rotation around the X-axis is the roll angle, α, and the angle of rotation around the Y-axis is the pitch angle, β. Both coordinate systems share the same origin, O, located at the midpoint of the AB line segment. The proposed stepwise leveling method sets the chassis reference point at the origin, O, instead of the centroid, simplifying the control process and increasing the speed of leveling.

The chassis attitudes of the three-wheeled agricultural robot during operation were analyzed and classified based on the surface excitations experienced by each wheel. In the end, 19 different chassis attitudes were identified, as shown in

Table 2. The 19 attitudes of the three-wheeled chassis.

Item	∆A	∆B	∆C	α	β	Chassis Attitude Classification Based on Angles α and β
1	0	0	0	0	0	α = 0, β = 0
2	0	0	−1	0	+	α = 0, β > 0
3	0	0	1	0	−	α = 0, β < 0
4	−1	0	0	−	−	α < 0, β < 0
5	1	0	0	+	+	α > 0, β > 0
6	0	−1	0	+	−	α > 0, β < 0
7	0	1	0	−	+	α < 0, β > 0
8	−1	−1	0	0	−	α = 0, β < 0
9	1	1	0	0	+	α = 0, β > 0
10	1	−1	0	+	0	α > 0, β = 0
11	−1	1	0	−	0	α < 0, β = 0
12	−1	0	−1	−	+	α < 0, β > 0
13	−1	0	1	−	−	α < 0, β < 0
14	1	0	−1	+	+	α > 0, β > 0
15	1	0	1	+	−	α > 0, β < 0
16	0	−1	−1	+	+	α > 0, β > 0
17	0	−1	1	+	−	α > 0, β < 0
18	0	1	−1	−	+	α < 0, β > 0
19	0	1	1	−	−	α < 0, β < 0

.

(1)

When the chassis maintains a level attitude while moving, α = 0 and β = 0, as shown in item 1 of the table.

(2)

When only one wheel is subjected to road excitation, the chassis has six attitudes, which are considered the basic attitudes of the chassis, as shown in items 2 to 7.

(3)

By combining any two of the aforementioned six basic chassis attitudes and then excluding the combinations that result in the horizontal attitude, an additional 12 chassis attitudes can be derived, as shown in items 8 to 19.

It should be noted that in the table, ΔA, ΔB, and ΔC represent the road excitations experienced by the wheels corresponding to the chassis support points A, B, and C, respectively. The values in the table are not the actual road excitation values but indicate three states of road excitation: 0 indicates no excitation, 1 indicates upward road excitation, and −1 indicates downward road excitation.

To simplify the process of recognizing chassis attitudes, this paper classifies 19 types of chassis attitude into five categories based on the angles α and β: ① α = 0 or β = 0; ② α > 0, β > 0; ③ α > 0, β < 0; ④ α < 0, β > 0; ⑤ α < 0, β < 0. In Table 2, these five categories are marked with gold, yellow, brown, green, and blue backgrounds, respectively. Based on the characteristics of these five categories, a rapid leveling method applicable to all five types of attitudes, called the stepwise leveling method, is proposed. This method involves first adjusting the relatively flexible rear wheel suspension, followed by the front wheel suspension. By adjusting one or two suspensions in a stepwise manner, the leveling goal can be achieved. This method features a small number of suspension adjustments, high precision in phased leveling, and a simple leveling strategy, making it suitable for three-wheeled chassis. The basic principles of the stepwise leveling method will be introduced next.

The first category of chassis attitudes (α = 0 or β = 0) includes seven forms, as indicated by items 1, 2, 3, 8, 9, 10, and 11 in Table 2. Excluding item 1, the remaining six chassis attitudes require adjustment of only one angle, making the leveling process relatively simple. The control logic flowchart for their leveling process is shown in Figure 11.

When α ≠ 0 and β ≠ 0, the chassis attitudes fall into categories 2 to 5, which are more complex and comprise 12 forms (as indicated by items 4, 5, 6, 7, 12, 13, 14, 15, 16, 17, 18, and 19 in Table 2). The following text will detail their leveling principles. The control logic flowchart for their leveling process is shown in Figure 12. First, determine whether the rear wheel suspension needs adjustment based on the condition specified in Equation (13).

$${\displaystyle \frac{l_{\mathrm{AB}}\sin \left|\alpha \right|}{2}}=l_{oc}\sin \left|\beta \right|$$

(13)

If α and β satisfy this angular relationship, the rear wheel suspension does not need to be adjusted. Instead, adjust one of the front wheel suspensions to raise or lower the corresponding support point. The adjustment height, $\Delta h^{\prime }$, for the support point is calculated using Equation (14).

$$\Delta h^{\prime }=\mathrm{sgn}\alpha ·\mathrm{sgn}\beta ·l_{\mathrm{AB}}\sin \left|\alpha \right|$$

(14)

If $\Delta h^{\prime }>0$, adjust the left front wheel suspension. The direction of the suspension adjustment is determined by the sign of α (sgn(α)): if sgn(α) is positive, the suspension adjusts downward; if sgn(α) is negative, the suspension adjusts upward. If $\Delta h^{\prime }<0$, adjust the right front wheel suspension, with the direction of adjustment similarly determined by sgn(α): if sgn(α) is positive, the suspension adjusts upward; if sgn(α) is negative, the suspension adjusts downward.

If α and β do not satisfy this angular relationship, the rear wheel suspension must be adjusted first to raise or lower the support point C by Δh. The calculation for Δh is given by Equation (15).

$$\Delta h=\mathrm{sgn}\beta ·l_{oc}\sin \left|\beta \right|-{\displaystyle \frac{\mathrm{sgn}\alpha ·l_{\mathrm{AB}}\sin \left|\alpha \right|}{2}}$$

(15)

Δh > 0 indicates that the support point C should be raised; Δh < 0 indicates it should be lowered. After this, adjust one of the front wheel suspensions according to Equation (14).

Taking the second category of chassis attitudes (α > 0, β > 0) as an example, the paper will introduce the principles of the stepwise leveling method. According to Table 2, the second category of chassis attitudes includes the three forms indicated by items 5, 14, and 16, as shown in Figure 13.

First, determine whether the rear wheel suspension needs adjustment. If α and β satisfy Equation (13), no adjustment to the rear wheel suspension is necessary; otherwise, adjustment is required.

When α and β satisfy Equation (13), specifically $\beta ={\sin }^{-1}({\displaystyle \raisebox{1ex}{$l_{OA}$}\!\left/ \!\raisebox{-1ex}{$l_{OC}$}\right.}·\sin \alpha )$, no adjustment to the rear wheel suspension is necessary. Instead, adjust the left front wheel suspension to lower support point A by $\Delta {h^{\prime }}_2$. $\Delta {h^{\prime }}_2$ is calculated as:

$$\Delta {h^{\prime }}_2=l_{\mathrm{AB}}·\sin \alpha$$

(16)

When α and β do not satisfy Equation (13), specifically $\beta \ne {\sin }^{-1}({\displaystyle \raisebox{1ex}{$l_{OA}$}\!\left/ \!\raisebox{-1ex}{$l_{OC}$}\right.}·\sin \alpha )$, first adjust the rear wheel suspension to raise support point C by $\Delta h_2$. Then, adjust the left front wheel suspension to lower support point A by $\Delta {h^{\prime }}_2$. $\Delta h_2$ is calculated as:

$$\Delta h_2=l_{\mathrm{OC}}·\sin \beta -{\displaystyle \frac{l_{\mathrm{AB}}·\sin \alpha }{2}}$$

(17)

The proposed stepwise leveling method is specifically designed for three-wheeled chassis, integrating the unique structure and power distribution characteristics of three-wheeled vehicles. By individually adjusting the front two wheels and the rear wheel, it better meets the practical requirements of three-wheeled vehicles. Traditional leveling methods adjust all suspensions simultaneously based on the chassis’ pitch and roll angles, which leads to dynamic coupling issues among the suspensions. In contrast, the stepwise leveling approach not only resolves the complex control challenges arising from complex interactions between suspension components but also streamlines the implementation process of leveling control.

4.2. Adaptive Dual-Loop Composite Control Strategy (ADLCCS)

To enhance the control performance of suspension actuators, this paper proposes a new control strategy, namely, the adaptive double-loop composite control strategy (ADLCCS). First, a composite controller combining inner-loop and outer-loop control was designed. The outer loop uses the linear quadratic Gaussian (LQG) model to obtain the optimal virtual control law, while the inner loop employs the backstepping algorithm, which excels in handling nonlinear systems, to address the nonlinear components of the system. The structural principle of the composite controller is shown in Figure 14. Then, an improved IAGA was introduced to optimize the key control variables of the inner and outer loop controllers, in order to enhance the adaptability of the composite controller in actual road conditions.

4.2.1. Design of the Outer-Loop Controller

The design of the LQG optimal controller adheres to the separation principle and can be divided into two main parts: firstly, optimal state estimation is conducted using a Kalman filter; secondly, the optimal control law is derived based on optimal control theory.

During vehicle operation, certain state variables such as vertical ground displacement are difficult to measure directly. Although previous optimal controllers typically assume a full-state feedback system, acquiring these variables directly often poses a challenge in practice. Therefore, employing a Kalman filter for state estimation becomes essential, using optimal estimation methods to predict key state variables like vertical displacement. Let the measurable state output equation be:

$${\mathit{Y}}_{\mathit{c}}\left(\mathit{t}\right)={\mathit{C}}_{\mathit{m}}\mathit{X}\left(\mathit{t}\right)+\mathit{v}\left(\mathit{t}\right)$$

(18)

where C_m represents the measurement matrix and v(t) represents the measurement noise.

$${\mathit{C}}_m=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0\end{array}\right]$$

(19)

Assume that the model noise and measurement noise are uncorrelated and satisfy the following relationship:

$$\begin{array}{l}E\left[\mathit{w}\left(t\right)\mathit{w}\left(t+\tau \right)\right]={\mathit{R}}_1\mathbf{\delta }\left(t\right)\\ E\left[\mathit{v}\left(t\right)\mathit{v}\left(t+\tau \right)\right]={\mathit{R}}_2\mathbf{\delta }\left(t\right)\end{array}$$

(20)

According to the Kalman filter equation, the following equation can be obtained:

$$\begin{array}{cc}\hfill \dot{\widehat{\mathit{X}}}\left(t\right)& =\mathit{A}\widehat{\mathit{X}}\left(t\right)+\mathit{B}\tilde{\mathit{U}}\left(t\right)+\mathit{L}\left[{\mathit{Y}}_c\left(t\right)-{\mathit{C}}_m\widehat{\mathit{X}}\left(t\right)\right]\hfill \\ & =\left(\mathit{A}-\mathit{L}{\mathit{C}}_m\right)\widehat{\mathit{X}}\left(t\right)+\mathit{B}\tilde{\mathit{U}}\left(t\right)+\mathit{L}{\mathit{Y}}_c\left(t\right)\hfill \end{array}$$

(21)

The gain matrix of the optimal state observer is:

$$\mathit{L}=\widehat{\mathit{P}}{{\mathit{C}}_m}^{\mathrm{T}}{{\mathit{R}}_2}^{-1}$$

(22)

$\widehat{\mathit{P}}$ is the solution to the following state Riccati equation.

$$\mathit{A}\widehat{\mathit{P}}+\widehat{\mathit{P}}{\mathit{A}}^{\mathrm{T}}-\widehat{\mathit{P}}{{\mathit{C}}_m}^{\mathrm{T}}{{\mathit{R}}_2}^{-1}{\mathit{C}}_m\widehat{\mathit{P}}+\mathit{G}{\mathit{R}}_1{\mathit{G}}^{\mathrm{T}}=0$$

(23)

After estimating the state variables, the next step is to design the optimal controller. Analyzing Equations (11), (12), and (20), it becomes apparent that the system is a typical linear time-invariant system, meeting the basic criteria for optimal controller design. Tire displacement, suspension travel, and the vehicle body’s vertical acceleration are selected as evaluation metrics. Based on these, the system’s objective function is constructed, as shown in the following equation.

$$J=\underset{T\to \infty }{\lim }{\displaystyle \frac{1}{T}}{\displaystyle {\int }_0^T\left[q_1{(Z_1-Z_2)}^2+q_2{{\ddot{Z}}_1}^2+q_3{(Z_2-w)}^2\right]\mathrm{d}t}$$

(24)

Let ${\mathit{Q}}_0=\left[\begin{array}{l}\begin{array}{ccc}{q}_1& & \end{array}\\ \begin{array}{ccc}& q_2& \end{array}\\ \begin{array}{ccc}& & q_3\end{array}\end{array}\right]$ be the weighting function matrix; thus, Equation (24) can be rewritten as follows:

$$J=\underset{T\to \infty }{\lim }{\displaystyle \frac{1}{T}}{\displaystyle {\int }_0^T\left({\mathit{X}}^{\mathrm{T}}\mathit{Q}\mathit{X}+{\tilde{\mathit{U}}}^{\mathrm{T}}\mathit{R}\tilde{\mathit{U}}+2{\mathit{X}}^{\mathrm{T}}\mathit{N}\mathit{U}\right)\mathrm{d}t}$$

(25)

where Q represents the state weighting matrix, and its expression is $\mathit{Q}={\mathit{C}}^{\mathrm{T}}{\mathit{Q}}_0\mathit{C}$; R represents the control weighting matrix, and its expression is $\mathit{R}=r={\displaystyle \frac{1}{{M_1}^2}}$; N represents the correlation weighting matrix, and its expression is $\mathit{N}={\mathit{C}}^{\mathrm{T}}{\mathit{Q}}_0\mathit{D}$.

$$\mathit{Q}=\left[\begin{array}{ccccc}{q}_1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& q_3& 0& -q_3\\ 0& 0& 0& 0& 0\\ 0& 0& -q_3& 0& q_3\end{array}\right]$$

(26)

According to optimal control theory, the optimal control law for state feedback that minimizes the performance index is:

$$\tilde{\mathit{U}}=-{\mathit{K}}_s\widehat{\mathit{X}}$$

(27)

The feedback gain matrix is:

$${\mathit{K}}_s={\mathit{R}}^{-1}{\mathit{B}}^{\mathrm{T}}\mathit{P}$$

(28)

P is the solution to the following state Riccati equation.

$$\mathit{P}\mathit{A}+{\mathit{A}}^{\mathrm{T}}\mathit{P}-\mathit{P}\mathit{B}{\mathit{R}}^{-1}{\mathit{B}}^{\mathrm{T}}\mathit{P}+{\mathit{Q}}_{\mathrm{n}}=0$$

(29)

4.2.2. Design of the Inner Loop Controller

In the inner loop control, the backstepping algorithm is employed to compensate for the nonlinearities in the hydraulic actuator system. To meet performance requirements, the control input, u, must approximate the virtual control, $\tilde{u}$, thereby achieving optimal control output. Additionally, by introducing two extra state variables, x₆ = P₁ and x₇ = P₂, the following equations can be obtained:

$$\begin{array}{l}{\dot{x}}_6={\displaystyle \frac{{\beta }_e}{V_1}}\left\{C_d\omega X_v\sqrt{{\displaystyle \frac{2}{\rho }}}\left[S\left(X_v\right)\sqrt{P_s-x_6}+S\left(-X_v\right)\sqrt{x_6}\right]-A_1\left(x_2-x_4\right)\right\}\\ {\dot{x}}_7=-{\displaystyle \frac{{\beta }_e}{V_2}}\left\{C_d\omega X_v\sqrt{{\displaystyle \frac{2}{\rho }}}\left[S\left(X_v\right)\sqrt{x_7}+S\left(-X_v\right)\sqrt{P_s-x_7}\right]-A_2\left(x_2-x_4\right)\right\}\end{array}$$

(30)

Define the error variable z as shown in the following equation:

$$z=u-\tilde{u}$$

(31)

Construct the Lyapunov function and its derivative, with the specific expressions being $V={\displaystyle \frac{1}{2}}{z}^2$ and $\dot{V}=z\dot{z}$, respectively. Assuming the control input is X_V, to ensure stability of the control system, the actual control law selected is:

$$X_v={\displaystyle \frac{A_1{D}_1(x_2-x_4)+D_2{A}_2(x_2-x_4)+\dot{\tilde{u}}-kz}{D_1{H}_1+D_2{H}_2}}$$

(32)

where

$$\begin{gathered} D_1={\displaystyle \frac{A_1{\beta }_e}{V_1}} \\ {D}_2={\displaystyle \frac{A_2{\beta }_e}{V_2}} \\ {H}_1=C_d\omega \sqrt{{\displaystyle \frac{2}{\rho }}}\left[S\left(X_v\right)\sqrt{P_s-x_6}+S\left(-X_v\right)\sqrt{x_6}\right] \\ {H}_2=C_d\omega \sqrt{{\displaystyle \frac{2}{\rho }}}\left[S\left(X_v\right)\sqrt{x_7}+S\left(-X_v\right)\sqrt{P_s-x_7}\right] \end{gathered}$$

(33)

By choosing k > 0, ensure that $\dot{V}=z\dot{z}=-k{z}^2\le 0$, thereby gradually stabilizing the error, which means the inner loop control strategy achieves asymptotic stability.

4.2.3. Adaptive Optimization of Control Parameters

According to Equation (32), k, as an adjustable gain in the control rate, significantly affects the rate at which the control input, u, approximates the virtual control, $\tilde{\mathit{u}}$, and the ultimate stabilization time in the inner loop control. The actual control rate, X_v, which corresponds to the valve opening, can lead to severe hydraulic cylinder pressure fluctuations and resultant hydraulic system shocks and pipeline vibrations if k is too high. Conversely, if k is too low, it can prolong the stabilization time of the inner loop control, adversely affecting the performance of active suspension control. Therefore, by using an optimization algorithm that considers the rate of change in the virtual control, $\tilde{\mathit{u}}$, selecting the optimal k value for the current conditions can achieve the best control performance.

First, establish the objective function for the inner loop control. Considering the rate of change in the virtual control variable, $\tilde{\mathit{u}}$, along with the adjustment rate of the error variable, $\dot{z}$, and the current error amount, z, the system’s objective function, J_k, is constructed as:

$$J_k=\min f={\displaystyle {\int }_0^T\left[K_1·z^2+K_2·{\left(\dot{z}\right)}^2\right]\mathrm{d}t}$$

(34)

where K₁ is the error weight and K₂ is the weight for the error adjustment rate. Different combinations of K₁ and K₂ reflect the system’s focus on stability and speed in inner loop control under various control modes.

Considering the response characteristics of hydraulic systems, using only the control force output by the hydraulic actuator as the input for optimal control could lead to excessive fluctuations in the solved virtual control variable due to pressure variations in the hydraulic system. To minimize the impact of these pressure fluctuations, this paper incorporates both the current output control force of the hydraulic actuator and the solved virtual control variable, $\tilde{u}$, as inputs for optimal control. Additionally, it utilizes variable parameter weights, K_cylinder and K_uhat, as input weights for the current output control force and the virtual control variable, respectively.

Based on the analysis of key parameters for the inner and outer loop controllers discussed previously, it is clear that the controller contains five critical optimizable variables: (1) the adjustable gain, k, for the valve opening, X_v; (2) the error weight, k₁, and the error adjustment rate weight, k₂, for the virtual control quantity, $\tilde{u}$, in the inner loop’s control; (3) the control force input weight, K_cylinder, and virtual control variable input weight, K_uhat, for the hydraulic actuator in the outer loop’s control. The different combinations of these five control variables will affect the chassis leveling performance under various operating conditions, necessitating the use of optimization algorithms to determine the optimal variable combinations for different scenarios.

This paper employs an improved adaptive genetic algorithm to overcome the shortcomings of traditional genetic algorithms. These improvements include assessing the degree of population aggregation and adjusting adaptive crossover mutation probabilities.

Figure 15 shows the flowchart of the adaptive genetic algorithm designed in the paper. The specific steps of the optimization process are as follows:

(1)

Take the change matrix of the virtual control variable, $\tilde{u}$, as the input, generate the initial optimization population, and perform encoding.

(2)

Use the objective function, J_k, to evaluate the fitness of each individual in the current population and calculate the average fitness (f_ave) and the maximum fitness (f_max) of the population.

(3)

Analyze the dispersion of the population to determine the evolutionary direction of the population: if the distribution is dispersed, proceed to step (4); if the distribution is concentrated, proceed to step (5).

(4)

In the case of a dispersed population distribution, generate a new generation of individuals: first perform crossover operations, then perform mutation operations, and finally complete the selection process.

(5)

In the case of a concentrated population distribution, generate a new generation of individuals: first perform mutation operations, then perform crossover operations, and finally complete the selection process.

(6)

Evaluate whether the current population meets the aggregation requirements: if it does, proceed to step (7); if not, return to step (2) for re-evaluation.

(7)

After optimization, output the optimal values of k, K₁, K₂, K_cylinder, and K_uhat corresponding to the change matrix of the current virtual control variable, $\tilde{u}$.

To better utilize the roles of crossover and mutation processes across various population dispersion levels, this paper adopts the nonlinear adaptive calculation formulas for crossover probability and mutation probability in IAGA, as given in Equations (35) and (36).

$$P_c=\{\begin{array}{c}{k}_1{\displaystyle \frac{\arcsin \left(\frac{f_{\mathrm{ave}}}{f_{\max }}\right)}{\frac{\mathsf{\pi }}{2}}}\hspace{1em}\hspace{1em} ,\arcsin \left({\displaystyle \frac{f_{\mathrm{ave}}}{f_{\max }}}\right)<{\displaystyle \frac{\mathsf{\pi }}{6}}\\ k_1\left(1-{\displaystyle \frac{\arcsin \left(\frac{f_{\mathrm{ave}}}{f_{\max }}\right)}{\frac{\mathsf{\pi }}{2}}}\right),\arcsin \left({\displaystyle \frac{f_{\mathrm{ave}}}{f_{\max }}}\right)\ge {\displaystyle \frac{\mathsf{\pi }}{6}}\end{array}$$

(35)

$$P_m=\{\begin{array}{c}{k}_2\left(1-{\displaystyle \frac{\arcsin \left(\frac{f_{\mathrm{ave}}}{f_{\max }}\right)}{\frac{\mathsf{\pi }}{2}}}\right),\arcsin \left({\displaystyle \frac{f_{\mathrm{ave}}}{f_{\max }}}\right)<{\displaystyle \frac{\mathsf{\pi }}{6}}\\ k_2{\displaystyle \frac{\arcsin \left(\frac{f_{\mathrm{ave}}}{f_{\max }}\right)}{\frac{\mathsf{\pi }}{2}}}\hspace{1em}\hspace{1em} ,\arcsin \left({\displaystyle \frac{f_{\mathrm{ave}}}{f_{\max }}}\right)\ge {\displaystyle \frac{\mathsf{\pi }}{6}}\end{array}$$

(36)

The arcsin(f_ave/f_max) ratio serves as a criterion for assessing the dispersion level of a population, determined by the population’s average fitness, f_ave, and the maximum fitness, f_max.

When arcsin(f_ave/f_max) < π/6, it indicates that the population is relatively dispersed. The smaller the value compared to π/6, the easier it is to conclude that the population is dispersed, exhibiting high diversity and richness. Under these conditions, adaptively increase the crossover probability to ensure thorough gene exchange and the evolution of superior new individuals; concurrently, adaptively decrease the mutation probability to minimize the risk of deteriorating high-quality individuals and accelerate convergence.

When arcsin(f_ave/f_max) ≥ π/6, it suggests that the population is relatively concentrated. The larger the value compared to π/6, the more it indicates a concentrated population distribution with less diversity and a more homogeneous population. In this scenario, adaptively decrease the crossover probability to prevent disruption of existing high-quality genes; simultaneously, increase the mutation probability to enhance the population’s global search capabilities and avoid being trapped in local optima.

5. Simulation Study

5.1. Simulation Setup

The simulation model was built based on MATLAB/Simulink. Simulations for chassis leveling were conducted using a convex road surface for road excitations. The effectiveness of the ADLCCS-SLM proposed in this paper is validated by comparison with passive suspension and PID-controlled active suspension. The PID-controlled active suspension in the simulation adopts the traditional multi-suspension linkage leveling method, which differs from the stepwise leveling method proposed in this paper. In the following text, this traditional leveling method based on PID control will be referred to as the PID control method. The main simulation parameters are shown in

Table 3. The main parameters of the simulation model.

Variables	Values	Units	Variables	Values	Units	Variables	Values	Units
Ms	1250	kg	Mw	400	kg	$V_1$	0.00098	m3
ks	20,000	N/m	c	1000	N·s·m⁻¹	$V_2$	0.00035	m3
kt	2 × 105	N/m	h	0.1	m	${\beta }_e$	7 × 108	Pa
$P_s$	1.6 × 107	Pa	$C_d$	0.6	—	L	2	m
Ix	525.5	kg·m2	Iy	625.5	kg·m2	$\rho$	900	kg/m3
$A_1$	0.00196	m2	Lb	1.84	m	lbf	0.55	m
$A_2$	0.0007	m2	Lf	1.62	m	lbr	1.29	m
$v$	1	m/s	$\omega$	0.0015	m

. The simulation structure diagram is shown in Figure 16.

The simulation adopts a convex pavement for the road disturbance, and the road model formula is as follows:

$$z_r=\{\begin{array}{c}{\displaystyle \frac{h}{2}}\left[1-\cos \left({\displaystyle \frac{2\mathsf{\pi }v}{L}}t\right)\right],\hspace{1em}0\le t<L/v\\ 0,\hspace{1em}t\ge L/v\end{array}$$

(37)

where h is the height of the convex pavement, h = 0.2 m; L is the length of the convex pavement, L = 2.5 m; and v is the vehicle speed, v = 10 km·h⁻¹.

5.2. Comparison with Baseline Methods

The simulation results are shown in Figure 17. During the suspension leveling process, two important performance indicators are the peak values of vertical displacement, the pitch angle, and the roll angle and the leveling time. The peak values refer to the maximum deviation of the chassis’ center of mass, pitch angle, and roll angle from their initial positions during leveling. This indicator reflects the stability and vibration amplitude of the chassis during the leveling process; the smaller the peak values, the better the stability of the chassis. Leveling time refers to the time required from the start of the suspension leveling to the point where the chassis reaches a stable, level state. The shorter the leveling time, the faster the system’s response speed, indicating higher efficiency of the suspension leveling system.

Under the excitation of a convex road, the chassis based on passive suspension experiences large fluctuations in vertical displacement, the pitch angle, and the roll angle. In contrast, both the traditional PID-controlled active suspension and the ADLCCS-SLM-controlled active suspension exhibit significant leveling effects, with the ADLCCS-SLM-controlled chassis showing smaller fluctuations in vertical displacement, the pitch angle, and the roll angle. The leveling time for the passive suspension is 8 s; for the PID-controlled active suspension it is 6 s; and for the ADLCCS-SLM-controlled active suspension it is only 4 s. Clearly, the ADLCCS-SLM achieves a faster leveling speed.

Table 4. The peak values of chassis attitude parameters for the three leveling methods.

Leveling Method	Vertical Displacement (m)	Roll Angle (rad)	Pitch Angle (rad)
Passive suspension	0.052	0.032	0.028
PID	0.041	0.025	0.022
ADLCCS-SLM	0.029	0.018	0.016

shows the peak values of chassis attitude parameters for the passive suspension, PID-controlled active suspension, and ADLCCS-SLM-controlled active suspension. Compared to the passive suspension, the peak values of vertical displacement, the pitch angle, and the roll angle for the chassis with traditional PID control are reduced by 22.46%, 22.45%, and 22.41%, respectively, while those for the chassis with ADLCCS-SLM control are reduced by 44.84%, 42.79%, and 44.1%, respectively. Overall, compared to the passive suspension and the PID-controlled active suspension, the ADLCCS-SLM-controlled active suspension performs better in leveling the chassis attitude.

5.3. Discussion

In terms of acceleration suppression, both the PID-controlled active suspension and the ADLCCS-SLM-controlled active suspension perform significantly better than the passive suspension. Comparing the two active suspensions, the PID-controlled suspension excels in vertical acceleration suppression, while the ADLCCS-SLM-controlled suspension slightly outperforms in roll angle acceleration and pitch angle angular velocity suppression. Overall, ADLCCS-SLM shows a significant improvement in chassis smoothness control compared to the passive suspension but no notable improvement over traditional PID control. As mentioned earlier, the chassis control of the three-wheeled agricultural robot prioritizes leveling performance, with smoothness control being a secondary consideration. Therefore, the performance of ADLCCS-SLM in smooth control also meets application requirements.

6. Test Verification

6.1. Test Setup

The experimental prototype of the three-wheeled agricultural robot is shown in Figure 18. During testing, its speed was set to 1 km/h, which is the typical operating speed of the prototype. Figure 19 illustrates the convex test road, which is an arched one-sided bridge. Since the test prototype employs an active suspension system, only the PID-based multi-suspension linkage leveling method and ADLCCS-SLM were tested in the leveling method comparison.

6.2. Comparison with Baseline Methods

The response curves for vertical displacement, the pitch angle, the roll angle, and their accelerations of the test prototype under convex road excitation are shown in Figure 20. From Figure 20a,c,e, it can be seen that the fluctuations in vertical displacement, the roll angle, and the pitch angle of the vehicle body are smaller under ADLCCS-SLM control, indicating that the leveling effect of the active suspension controlled by ADLCCS-SLM is significantly better than that of the traditional PID-controlled active suspension.

Table 5. The peak values of chassis attitude parameters in the test.

Leveling Method	Vertical Displacement (m)	Roll Angle (rad)	Pitch Angle (rad)
PID	0.035	0.022	0.018
ADLCCS-SLM	0.020	0.015	0.012

shows the peak values of vertical displacement, the roll angle, and the pitch angle for both leveling control methods. Compared to the traditional PID-based multi-suspension-linked leveling control method, the peak values of vertical displacement, roll angle, and pitch angle under ADLCCS-SLM control are reduced by 42.9%, 31.8%, and 33.3%, respectively. These results indicate that the leveling effect of ADLCCS-SLM is effective.

From Figure 20b,d,f, it can be seen that the peak values of vertical acceleration of the chassis under ADLCCS-SLM control are similar to those under PID control. However, for roll and pitch acceleration suppression, the peak acceleration values under ADLCCS-SLM control are slightly lower than those under PID control. This indicates that ADLCCS-SLM performs slightly better than PID in ride comfort control, though the improvement is limited. This result is consistent with the simulation results. The main reason is that the control objective of ADLCCS-SLM is to maintain a stable horizontal attitude of the chassis, and the ride comfort indices were not used in the controller design, only indirectly affecting ride comfort during the stabilization control process.

6.3. Discussion

In summary, in the active suspension control of three-wheeled agricultural machinery, compared to traditional PID control, ADLCCS-SLM demonstrates good leveling performance during obstacle crossing due to its flexible stepwise control of the front and rear suspensions. It significantly suppresses the peak values of vertical displacement, the roll angle, and the pitch angle. With further optimization of active suspension parameters under various road conditions and speeds, it will adapt to more complex operating conditions, further improving its operational efficiency and quality.

7. Conclusions

This paper designs a three-wheeled chassis leveling system based on active suspension and proposes a stepwise leveling method based on an adaptive dual-loop compound control strategy (ADLCCS-SLM) to improve the horizontal stability of a three-wheeled agricultural robot on rough terrain.

A comprehensive analysis of the chassis attitude of the three-wheeled agricultural robot during movement identified 19 different postures. Based on these, five main leveling conditions were summarized, simplifying the complexity and workload of the leveling decision-making process.

The proposed stepwise leveling method (SLM) relies solely on the vehicle’s roll and pitch angle data. By gradually adjusting one or two suspensions, it achieves rapid leveling and avoids complex interactions between suspension components encountered in traditional methods with three suspension linkages. This simplification enhances the design and implementation of the control system.

The adaptive dual-loop compound control strategy (ADLCCS) combines the advantages of inner and outer loop controls and adjusts key parameters through an improved adaptive genetic algorithm (IAGA) to adapt to different road conditions and optimize chassis leveling performance. The outer loop uses the LQG model to generate an optimal virtual control law, enhancing system robustness and response speed; the inner loop employs the backstepping algorithm to handle nonlinear terms, accommodating complex nonlinear systems. The IAGA selects the optimal parameter combination, enabling adaptive adjustments and improving system adaptability and leveling performance.

Experimental results show that compared to the traditional PID-based multi-suspension-linked leveling method, ADLCCS-SLM achieves significantly better leveling effects, reducing the peak values of pitch and roll angles by 31.8% and 33.3%, respectively. The leveling method proposed in this paper effectively enhances the horizontal stability of the three-wheeled agricultural robot, which is of great significance for the overall operational quality and safety of the machine.