Design and Verification of Crab Steering System for High Clearance Self-Propelled Sprayer

Abstract

A crab steering system aiming at improving the steering flexibility and operation efficiency of the high clearance sprayer was designed. First, a steering transmission mechanism of the high clearance sprayer was designed according to the operational and structural characteristics of the sprayer. Meanwhile, a crab steering hydraulic system based on load sensing was designed, and a mathematical model of the steering transmission mechanism and hydraulic system was established to describe the working characteristics of the crab steering system. On the basis of analyzing the operating environment of the sprayer and the operating characteristics of crab steering mode, a control strategy and algorithm of the crab steering system were proposed. The simulation model of the crab steering control system was built according to the established mathematical model, and the simulation analysis of the crab steering system was carried out. The simulation results showed that under the excitation of step signal, the average deviation of the two front wheels is 0.063°, and the absolute value of the maximum deviation is 2.75°, which is within the allowable range of deviation and meets the crab steering demand of the sprayer. Additionally, in order to verify the effectiveness of the designed system, an actual vehicle test platform of the crab steering system was built based on the 3WPG-3000 high clearance self-propelled sprayer independently developed by the research group, and the field test results revealed that the average deviation of the four wheels was 0.285°, the maximum absolute deviation was 2.587°, the rotation deviation was small, within the allowable and reasonable range. Altogether, the results of the simulation and field test verify the accuracy, stability and practicability of the crab steering system, which effectively improved the maneuverability of the sprayer.

1. Introduction

With the development of precision agriculture and modern agricultural production mode, there is a growing demand for high clearance self-propelled sprayers which can be used for middle and late stage pest control of cash crops such as soybeans, corn and sorghum [1,2,3,4]. The high clearance self-propelled sprayer has been widely promoted and applied for the advantages of high efficiency, good uniformity and environmental friendliness [5,6,7].

However, the high clearance self-propelled sprayer is limited by the length and width of the body, which leads to problems such as difficulty in changing lines or crop damage due to the excessive steering radius during the operation. Crab steering is a mode with four wheels having the same rotation direction and angle, which can realize the oblique walking of the sprayer. It can be used in field obstacle avoidance and rapid alignment in the field process, which effectively improves the flexibility and efficiency of the sprayer. At the same time, the characteristics of small steering radius effectively reduce the crop damage loss rate (crushed by the wheel) of the sprayer during the operation. Therefore, it is of great significance to study the crab steering system suitable for high clearance self-propelled sprayers to improve the flexibility and efficiency of the sprayer.

At present, the crab steering system is mainly used in aircraft towing vehicles [8,9,10], construction machinery [11,12,13], motor driven orchard operation machinery [14,15,16] and small mobile platforms [17,18]. Guo [8] designed a hydraulic steering system of an aircraft tractor that can realize front wheel steering, rear wheel steering, all wheel steering and crab steering, which improves the flexibility of the vehicle. Sun [9] designed a steering system of an aircraft tractor that can rotate independently through four proportional valves with high control accuracy and response speed. Zhang et al. [11] developed a hydraulic crab steering system with locking function for a double drum roller, which can realize the crab movement of wheels. Donaldson [14] designed a mobile elevator that can realize front wheel steering, rear wheel steering, all wheel steering and crab steering modes. Ye et al. [15,16] designed a four wheel independent steering system for orchard management machinery, which can realize Ackerman steering, active front wheel steering, crab steering and in situ steering, so as to complete the movement management of orchard fruit boxes. Zhang [17] designed a hydraulic steering system for a high clearance self-propelled platform, which can realize three steering modes: two wheel steering, four wheel steering and crab steering, and verified the superiority of crab steering mode under obstacle avoidance conditions by simulation. Tu [18] designed a four wheel steering vehicle controlled by a pneumatic steering cylinder, which can realize four steering modes: front wheel steering, rear wheel steering, crab steering and cooperative steering. The system is composed of a steering motor and encoder, which can select different steering modes according to the operation environment, effectively improving the mobility of the vehicle. Zong et al. [19] built a dynamic simulation model by MATLAB/Simulink for a four wheel independent drive/steering electric vehicle, and simulated and analyzed the dynamic response characteristics of the crab steering, oblique travel and in situ steering conditions.

At present, the crab steering system scheme adopted by mobile vehicles mainly includes the following two types: (1) motor driven steering mode. This scheme is mainly based on the feature that the motor can control the forward and reverse rotation, and the controller can control the motor rotation direction and angle of each wheel independently, so as to complete the rotation of four wheels at the same angle. This kind of scheme is mainly applied to the steering system of small and medium-sized vehicles; (2) hydraulic steering mode. In this scheme, multiple groups of hydraulic valves are used to control the hydraulic cylinders to push the wheels to complete the same direction rotation of the front and rear wheels, which can realize crab steering under small angle conditions. According to the Ackerman theorem and the geometric relationship of the steering transmission mechanism, when the wheel angle is within 10°, the steering angle of the two front wheels is basically equal. When it exceeds 10°, with the increase in the wheel angle, the deviation value of the two front wheel angles gradually increases and, taking the steering system designed in this paper as an example, under the non-crab mode, when the outside wheel angle is 25°, the outside wheel steering angle reaches 36° [17,20]. According to references [21,22], the steering angle error of the sprayer should be less than 3° and, if the steering angle error is too large, this may lead to increased wheel sideslip and tire wear of the sprayer in the large angle steering mode.

In the current application of the crab steering system, the motor driven steering cannot meet the steering requirements of heavy vehicles, and the hydraulic driven steering solution can only realize small angle crab steering, but cannot complete the crab steering operation under the large angle mode.

In this study, aiming at solving the problems existing in the crab steering system of the high clearance mobile platform, the steering system of the high clearance self-propelled sprayer was taken as the research object, and a hydraulic crab steering system was designed, which can achieve the crab steering of the sprayer under any working condition. The system can realize fast obstacle avoidance and line changing operations based on the crab steering, effectively improving the maneuverability and operation efficiency of the sprayer. At the same time, according to the structural characteristics of the sprayer with high clearance and the characteristics of crab steering, the control strategy and algorithm of the high clearance self-propelled sprayer were proposed, and the stability and response speed of the designed crab steering control system were verified by experiments.

2. Materials and Methods

2.1. Design of Crab Steering System

2.1.1. Principle of Crab Steering

Crab steering is a variant of four wheel steering, and the traditional four wheel steering means that the rotation direction of the front wheel is opposite to the rotation direction of the rear wheel, which can realize the same imprint steering of the front and rear wheels. In the crab steering mode, the steering angle and rotation direction of the four wheels are the same, which can realize the oblique walking of the vehicle, which is convenient for small field operations. The steering principle diagram is shown in Figure 1.

The steering angles of all wheels meet the following requirements:

$${\alpha }_1={\beta }_1={\alpha }_2={\beta }_2,$$

(1)

where α₁, α₂, β₁ and β₂ are the steering angle of the four wheels; N is the wheelbase; K is the track width.

2.1.2. Structural Design of Crab Steering Transmission System

According to the structural characteristics and operating characteristics of the high clearance self-propelled sprayer, a steering transmission mechanism scheme of the sprayer is designed, with the front wheel steering trapezoid located at the front and the rear wheel steering trapezoid located at the rear. The overall structure layout of the steering system of the sprayer (WUZHENG Inc., Rizhao, Shandong province, China) is shown in Figure 2, and the steering transmission mechanism is shown in Figure 3.

The movement process of the steering transmission mechanism is as follows: the driver turns the steering wheel to control the flow of hydraulic oil into the steering cylinder through the hydraulic redirector, the steering hydraulic cylinder pushes the steering arm to rotate, the steering arm and the air spring bracket are fixed together, which in turn drives the air spring bracket to rotate, the air spring bracket is connected with the vertical shaft and the motor protective case by splines, and the tire and motor protective case are fixed together, so the air spring bracket can drive the wheels to rotate coaxially, thereby completing the steering of the sprayer.

2.1.3. Hydraulic System Design of Crab Steering

The principle diagram of the hydraulic system of crab steering is shown in Figure 4. The hydraulic system of crab steering is mainly composed of a crab steering compensation valve, steering compensation hydraulic cylinder, hydraulic redirector, four wheel steering control valve, steering hydraulic cylinder and other components. The system is a load pressure sensitive hydraulic steering system, which can realize three steering modes: two wheel steering, four wheel steering and crab steering. Four wheel steering function is realized by the four wheel steering control valve controlling the two rear wheels to follow the corresponding front wheels’ rotation [20]. The realization of the crab steering function is to add a crab travel compensation valve on the basis of the four wheel steering system, and the crab steering compensation valve is controlled to compensate the angle difference between the two front wheels during steering.

The working method of the crab steering system is as follows: the crab steering compensation valve is opened, the pressure compensation valve is in the right position communication state, a compensation oil circuit is connected between the rod cavity series oil circuit of the front wheel steering hydraulic cylinder, to compensate the oil difference required for the steering angle difference between the left and right front wheels caused by the steering transmission mechanism under crab steering mode, and the steering compensation hydraulic cylinder is used to ensure that the pressure of the access oil circuit is the same as that of the front wheel steering hydraulic cylinder.

When carrying out the crab steering operation, the controller judges whether the vehicle is turning left or right through the displacement of the front wheel steering hydraulic cylinder measured by the displacement sensor installed on the steering hydraulic cylinder. The displacement of the inner wheel hydraulic cylinder is taken as the reference, and the displacement required to move the hydraulic cylinder on the other side when the steering angles of the two front wheels are the same is calculated according to Ackerman steering theory. There are two working conditions: (1) if the required displacement is larger than that of the hydraulic cylinder at the current moment, it needs to be replenished with oil, that is, the differential pressure reducing valve is opened, and the two-position three-way proportional valve is in the right position, so the hydraulic oil flows to the outer hydraulic cylinder; (2) if the required displacement is smaller than the current displacement of the hydraulic cylinder, the oil in the outer hydraulic cylinder needs to be drained, that is, the differential pressure reducing valve is closed, the two-position three-way proportional valve is located at the left position, and the two-position three-way proportional valve is opened, and the oil is returned to the hydraulic oil tank, which ensures that the steering angles of the two front wheels are the same. At the same time, through the rear wheel steering control system [20], the two rear wheels are controlled to follow the rotation of the front wheel, so as to ensure that the steering angles of the four wheels are the same, and then the crab steering operation of the sprayer is completed.

The model and manufacturer of key hydraulic components in the crab steering system are shown in

Table 1. Model list of key hydraulic components.

Hydraulic Components	Model	Manufacturer
Load sensing closed center full hydraulic steering gear	102S-5T-200-10-E	ZHENJIANG HYDRAULICS Co., Ltd. (Zhenjiang, China)
Load sensing pressure compensation flow priority valve	EC10-42-0-N-150	HYDRAFORCE (Changzhou, China)
Differential pressure reducing valve	EC10-32-0-N-80EC1	HYDRAFORCE
Two way hydraulic control check valve	DC08-40-0-N-25	HYDRAFORCE
Two-position three-way proportional valve	SPCL10-32-0-N-12DR	HYDRAFORCE
Two-position two-way proportional valve	SP10-20-0-N-12DR	HYDRAFORCE
Three-position five-way load sensing electro-hydraulic proportional valve	SP10-58D-0-N-12DR	HYDRAFORCE
Relief valve	RV08-20A-0-N-33	HYDRAFORCE
Steering hydraulic cylinder	HSG63/35-300×584-WX	HEFEI CHANGYUAN HYDRAULIC Co., Ltd. (Hefei, China)

.

2.2. Mathematical Model of Crab Steering System

2.2.1. Mathematical Model of Steering Transmission Mechanism

According to the overall arrangement structure of the sprayer steering system and the structure of the steering transmission mechanism, combined with the operation characteristics of the sprayer, ignoring the influence of steering positioning parameters and tire cornering characteristics and establishing the actual rotation angle relationship model of the front wheel according to the geometric structure of the steering transmission mechanism, the motion analysis diagram is shown in Figure 5.

The structure of the steering system of the sprayer is symmetrical, and the rod chambers of the two front wheel steering hydraulic cylinders are connected in series by the hydraulic oil pipe. According to the motion diagram, if the compressibility of the hydraulic oil is ignored, there are:

$$l_{\mathrm{xp}0}^2=l_{\mathrm{sa}1}^2+l_{\mathrm{s}}^2-2l_{\mathrm{sa}1}{l}_{\mathrm{s}}\cos {\gamma }_{\mathrm{s}0},$$

(2)

$$l_{\mathrm{s}}=\sqrt{l_{\mathrm{ls}}^2+l_{\mathrm{cs}}^2},$$

(3)

$$l_{\mathrm{xpFL}}^2={\left(l_{\mathrm{xp}0}+x_{\mathrm{pFL}}\right)}^2=l_{\mathrm{sa}1}^2+l_{\mathrm{s}}^2-2l_{\mathrm{sa}1}{l}_{\mathrm{s}}\cos \left({\gamma }_{\mathrm{s}0}-{\gamma }_{\mathrm{FL}}\right).$$

(4)

It can be deduced from the above three equations that:

$${\gamma }_{\mathrm{FL}}={\gamma }_{\mathrm{s}0}-\arccos \frac{l_{\mathrm{sa}1}^2+l_{\mathrm{s}}^2-{\left(l_{\mathrm{xp}0}+x_{\mathrm{pFL}}\right)}^2}{2l_{\mathrm{sa}1}{l}_{\mathrm{s}}},$$

(5)

$${\gamma }_{\mathrm{s}0}=\arccos \frac{l_{\mathrm{sa}1}^2+l_{\mathrm{s}}^2-l_{\mathrm{xp}0}^2}{2l_{\mathrm{sa}1}{l}_{\mathrm{s}}},$$

(6)

$${\gamma }_{\mathrm{FR}}=\arccos \frac{l_{\mathrm{sa}1}^2+l_{\mathrm{s}}^2-{\left(l_{\mathrm{xp}0}+x_{\mathrm{pFR}}\right)}^2}{2l_{\mathrm{sa}1}{l}_{\mathrm{s}}}-{\gamma }_{\mathrm{s}0},$$

(7)

where l_ls and l_cs are the horizontal longitudinal and lateral distances from the connection point between the steering hydraulic cylinder and the sprayer body to the corresponding steering column axis; l_s is the distance from the connection point of the steering hydraulic cylinder barrel and the sprayer body to the corresponding steering column axis; l_sa1 is the steering arm length; l_xp0 is the distance from the hinge point of the steering hydraulic cylinder to the hinge point of the piston rod when the vehicle is traveling in a straight line; γ_s0 is the angle between O_FL-O_pFL and the steering arm O_FL-A_FL when the vehicle runs straight; x_pFL and x_pFR are the displacement of the left and right front wheel steering hydraulic cylinder pistons relative to the straight position of the vehicle, the outward extension is the positive direction; γ_FL and γ_FR are the deflection angle of the left and right front wheels relative to the position of the vehicle in a straight line, the clockwise rotation is the positive direction.

2.2.2. Mathematical Model of Crab Steering Hydraulic System

The crab steering compensation valve is mainly composed of a differential pressure reducing valve, two-position three-way proportional valve, two-position two-way proportional valve, ball shuttle valve and fixed orifice. The structural diagrams of the two-position two-way proportional valve and two-position three-way proportional valve are shown in Figure 6 and Figure 7.

When the two-position two-way proportional valve is energized, the pilot spool moves to the left and the pilot valve port opens, the hydraulic oil in the pilot valve returns to the hydraulic oil tank through the valve port and the pressure inside the valve cavity of the pilot valve drops. The spool of the main valve moves to the left under the action of the pressure difference at both ends, and the hydraulic oil returns to the hydraulic oil tank through the throttle port of the main valve, the opening of the main valve orifice is controlled by the displacement of the pilot spool.

Part of the hydraulic oil flowing into the main valve cavity flows out through the pilot valve and the main valve, and the other part is used to compensate the volume change of the valve cavity and the compression of the hydraulic oil. The flow continuity equation is as follows:

$$\{\begin{array}{l}{q}_{\mathrm{xpv}0}-q_{\mathrm{xpv}}=\frac{V_{\mathrm{xp}0}-\frac{\pi (d_{\mathrm{xz}1}^2-d_{\mathrm{xpv}}^2)}{4}{x}_{\mathrm{xs}}}{{\beta }_{\mathrm{e}}}{\dot{p}}_{\mathrm{xpv}}-\frac{\pi (d_{\mathrm{xz}1}^2-d_{\mathrm{xpv}}^2)}{4}{\dot{x}}_{\mathrm{xz}}\\ q_{\mathrm{xs}}-q_{\mathrm{xpv}0}-q_{\mathrm{xz}}=\frac{V_{\mathrm{xz}0}+\frac{\pi (d_{\mathrm{xz}1}^2-d_{\mathrm{xz}}^2)}{4}{x}_{\mathrm{xpv}}}{{\beta }_{\mathrm{e}}}{\dot{p}}_{\mathrm{xs}}+\frac{\pi (d_{\mathrm{xz}1}^2-d_{\mathrm{xz}}^2)}{4}{\dot{x}}_{\mathrm{xz}}\end{array},$$

(8)

where q_xs is the inlet flow of the main valve cavity; q_xz is the outlet flow of the main valve cavity; q_xpv is the outlet flow of the pilot valve chamber; q_xpv0 is the inlet flow of the pilot valve; p_xs is the pressure of the main valve inlet; p_xpv is the pressure of the pilot valve chamber; d_xz is the orifice spool diameter of the main valve; d_xz1 is the diameter of the guide part of the main valve spool; d_xpv is the diameter of the guide part of the pilot valve spool; β_e is the bulk modulus of the elasticity of oil; V_xp0 is the initial volume of the pilot valve oil chamber; V_xz0 is the initial volume of the main valve cavity; x_xz is the displacement of the main valve spool.

The pressure–flow equation of the oil outlet orifice of the main valve is:

$$q_{\mathrm{xz}}=C_{\mathrm{d}}{A}_{\mathrm{xz}}\sqrt{\frac{2}{\rho }\left(p_{\mathrm{xs}}-p_{\mathrm{xz}}\right)},$$

(9)

where C_d is the flow coefficient; ρ is the hydraulic oil density; p_xz is the main valve outlet pressure.

The cross-sectional area A_xz of throttle port of the main valve of the two-position two-way proportional valve can be expressed as:

$$A_{\mathrm{xz}}=\{\begin{array}{l}0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{c}\begin{array}{c}0\le x_{\mathrm{xz}}\le x_{\mathrm{xzs}0}\end{array}\end{array}\\ n{B}_{\mathrm{xz}}(x_{\mathrm{xz}}-x_{\mathrm{xzs}0})\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \begin{array}{c}{x}_{\mathrm{xzs}0}\le x_{\mathrm{xz}}\le x_{\mathrm{xzs}0}+L_{\mathrm{xz}}\end{array}\\ n{B}_{{}_{\mathrm{xz}}}{L}_{\mathrm{xz}}+\pi d_{\mathrm{xz}}(x_{\mathrm{xz}}-x_{\mathrm{xzs}0}-L_{\mathrm{xz}})\hspace{1em}\hspace{1em}\begin{array}{c}{x}_{\mathrm{xz}}\ge x_{\mathrm{xzs}0}+L_{\mathrm{xz}}\end{array}\end{array},$$

(10)

where x_xzs0 is the overlap of the main valve; B_xz is the opening width of the main valve; L_xz is the opening length of the main valve.

The pressure–flow equation for the inlet orifice of the pilot valve is:

$$q_{\mathrm{xpv}0}=\frac{C_{\mathrm{d}}\pi d_{\mathrm{xpv}0}^2}{4}\sqrt{\frac{2}{\rho }\left(p_{\mathrm{xs}}-p_{\mathrm{xpv}}\right)}.$$

(11)

The pressure–flow equation at the outlet orifice of the pilot valve is:

$$q_{\mathrm{xpv}}=C_{\mathrm{d}}\pi d_{\mathrm{xpv}}\left(x_{\mathrm{xps}}-x_{\mathrm{xz}}\right)\sin {\theta }_{\mathrm{xz}}\sqrt{\frac{2}{\rho }\left(p_{\mathrm{xpv}}-p_{\mathrm{xz}}\right)},\begin{array}{c}\end{array}{x}_{\mathrm{xps}}-x_{\mathrm{xz}}\ge 0,$$

(12)

where x_xps is the displacement of the pilot valve spool; θ_xz is the half cone angle of the pilot valve spool.

According to the structure diagram of the two-position three-way proportional valve, it is mainly composed of a two-position two-way proportional valve, an oil outlet check valve and a load feedback oil port.

The flow continuity equation of the main valve cavity is:

$$q_{\mathrm{xz}}+q_{\mathrm{xpv}}-q_{\mathrm{x}1}-q_{\mathrm{Ls}}=\frac{V_{\mathrm{T}0}+\frac{\pi d_{x1}^2}{4}{x}_{\mathrm{x}}}{{\beta }_{\mathrm{e}}}{\dot{p}}_{\mathrm{xz}}+\frac{\pi d_{\mathrm{x}1}^2}{4}{\dot{x}}_x,$$

(13)

where q_x1 is the flow rate of the outlet check valve orifice; q_Ls is the flow rate of the load feedback port; V_T0 is the initial volume of the oil return chamber; x_x is the spool displacement of the oil outlet check valve; d_x1 is the outlet check valve seat hole diameter.

The pressure–flow characteristic equation of the oil outlet check valve is:

$$q_{\mathrm{x}1}=\{\begin{array}{l}{C}_{\mathrm{d}}\pi d_{x1}{x}_{\mathrm{x}}\sin {\theta }_{\mathrm{xpv}}\sqrt{\frac{2}{\rho }\left(p_{\mathrm{xz}}-p_0\right)},x_{\mathrm{x}}\ge 0\\ 0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x_{\mathrm{x}}<0\end{array},$$

(14)

where θ_xpv is the half cone angle of the pilot valve spool of the two-position three-way proportional valve; p₀ is the pressure of the return port.

The load feedback port flow can be expressed as:

$$q_{\mathrm{Ls}}=\frac{C_{\mathrm{d}}\pi d_{\mathrm{Ls}}}{4}\sqrt{\frac{2}{\rho }(P_{\mathrm{xz}}-P_{\mathrm{Ls}})},$$

(15)

where d_Ls is the diameter of the load feedback port; P_Ls is the load feedback pressure.

The flow continuity equation of the front wheel steering hydraulic cylinder can be expressed as:

$$q_{\mathrm{FL}}=A_{\mathrm{p}1}{\dot{x}}_{\mathrm{pFL}}+\frac{V_{\mathrm{FL}1}}{{\beta }_e}\frac{\mathrm{d}{p}_{\mathrm{FL}}}{\mathrm{d}t}+C_{\mathrm{it}}\left(p_{\mathrm{FL}}-p_{\mathrm{FR}}\right)+C_{\mathrm{et}}\left(p_{\mathrm{FL}}-p_0\right),$$

(16)

$$q_{\mathrm{FR}}=A_{\mathrm{p}1}{\dot{x}}_{\mathrm{pFR}}+\frac{V_{\mathrm{FR}1}}{{\beta }_e}\frac{\mathrm{d}{p}_{\mathrm{FR}}}{\mathrm{d}t}-C_{\mathrm{it}}\left(p_{\mathrm{FL}}-p_{\mathrm{FR}}\right)+C_{\mathrm{et}}\left(p_{\mathrm{FR}}-p_0\right).$$

(17)

The flow continuity equation of the rod cavity of the two front wheel steering hydraulic cylinders can be expressed as:

$$A_{\mathrm{p}1}\left({\dot{x}}_{\mathrm{pFL}}+{\dot{x}}_{\mathrm{pFR}}\right)-q_{\mathrm{x}}=\frac{\left(V_{\mathrm{FL}2}+V_{\mathrm{FR}2}\right)}{{\beta }_e}\frac{\mathrm{d}{p}_{\mathrm{x}}}{\mathrm{d}t},$$

(18)

where q_x is the oil flow from the crab steering compensation valve into the front wheel steering hydraulic cylinder; p_x is the pressure at the connection between the crab steering compensation valve and the steering hydraulic cylinder (the pressure in the pipeline connected to the rod cavity of the front wheel steering hydraulic cylinder); C_it and C_et are the oil leakage coefficient between the control ports of the two steering hydraulic cylinders and from the control port to the oil return port in the hydraulic redirector. A_p1 and A_p2 are the effective action area of the steering hydraulic cylinder without a rod cavity and with rod chambers; V_FL1, V_FL2, V_FR1 and V_FR2 are the volume of hydraulic oil in the rodless and rod chambers of the left and right front wheel steering hydraulic cylinders.

2.3. Control Strategy and Method

2.3.1. Control Strategy of Crab Steering

The crab steering system is mainly used for obstacle avoidance between rows and fast line changing operations on the ground. According to the Ackerman steering theory, the steering angle of the inner wheel is greater than the steering angle of the outer wheel during the steering process, but the steering angles of the four wheels are required to be the same during crab steering, so the difference of the steering angles generated during steering is required to be compensated. The wheel rotation status is estimated according to the sensor signal, so as to determine the inner wheel and the angle, and then the fuel volume for oil filling or unloading is calculated according to the Ackerman theorem and the geometric relationship of the wheel steering system. In addition, combined with the design and operational requirements of the sprayer, the maximum limit angle of the crab steering mode is set to 25°.

During the crab steering operation, the control strategy is formulated as follows: the controller judges the rotation direction of the vehicle according to the signal of the displacement sensor installed on the steering hydraulic cylinder, takes the steering angle of the inner wheel in the current rotation direction as the target angle, takes the steering angle of the outer wheel as the detection object, takes the difference between the two steering angles as the control signal and takes the crab compensation valve as control object, then the crab steering compensation valve controls the oil flow into or out of the interface between the rod chambers of the two front wheel steering hydraulic cylinders to ensure the same steering angle of the two front wheels, so as to realize the crab steering operation of the sprayer. The control principle of the crab steering system is shown in Figure 8.

2.3.2. Design of Crab Steering Controller

On the basis of analyzing the operating environment of the sprayer and the application scenario of the crab steering, the PID algorithm with simple structure and good control effect is selected to control the crab steering system. The control law can be expressed as:

$$u=k_p{x}_t+k_i{\displaystyle \int x_{t}dt+k_d\stackrel{.}{x_t}}.$$

(19)

The PID control principle diagram is shown in Figure 9.

The control process of the crab steering system is: the controller judges the direction of wheel rotation, takes the steering angle of the outer wheel as the target angle and takes the angle difference between the inner wheel and the outer wheel as the controller input signal. The input signal is transformed through the PID control algorithm to obtain the crab valve control input signal, to control the input or output flow of the inner wheel steering hydraulic cylinder by controlling the flow of the compensation hydraulic cylinder, to ensure the same steering angle between the two front wheels.

2.4. Construction of Test Platform

To verify the performance of the crab steering system, a simulation model of the system based on the established mathematical model was developed in MATLAB/Simulink. Further, an experimental platform was built for the crab steering system based on the self-developed 3WPG-3000 high clearance self-propelled sprayer, as presented in Figure 10.

During the construction of the test verification platform, the controller adopts HYDAC HY-TTC60, which has 16 analog inputs and 8 PWM outputs. The displacement sensor adopts an MPS-XS type pull wire displacement sensor, the measurement range is 0~400 mm, the output signal is 0~5 V voltage signal and the measurement accuracy is ±1.2 mm. The displacement signal is collected by the NI/USB-6341 data acquisition card, and transmitted to the LabVIEW acquisition system to display and store the data information.

The parameters of the developed sprayer are listed in

Table 2. Main parameters of high clearance self-propelled sprayer.

Parameter	Unit	Index	Parameter	Unit	Index
Drive mode	/	Four wheel hydraulic drive	Steering mode	/	2WS/4WS/Crab steering
Engine horsepower	hp	205	Wheel track	mm	3200–4000
Vehicle quality	kg	15,000	Wheelbase	mm	4000
Boundary dimension	mm	8200 × 4200 × 4100	Ground clearance	mm	≥2000
Liquid chemical tank volume	L	3000	Service brake	/	Clamp plate hydraulic brake
Spray amplitude	mm	28,000	Parking brake	/	Multi-disc brake in the reducer
Speed	km·h−1	Three speed stepless speed change, the highest: 17; 26; 40	Boom height	mm	800–2800

, and the parameters of the crab steering system are shown in

Table 3. Simulation parameters of crab steering system.

Parameter	Unit	Index	Parameter	Unit	Index
Ap1	m3	3.316 × 10−3	dxpv	m	5.5 × 10−4
Ap2	m3	2.154 × 10−3	Lx	m	1.75 × 10−3
Bxz	/	1.28 × 10−3	Lxz	m	1.24 × 10−3
Cet	m3/(Pa·s)	1 × 10−11	θxpv	rad	1.69
Cit	m3/(Pa·s)	2 × 10−11	θxz	rad	0.67
dxz	m	7.36 × 10−3	xxzs0	m	2.0 × 10−4
dxz1	m	1.066 × 10−2	Vxp0	L	3.03 × 10−7
dx1	m	1.45 × 10−2	Vxz0	L	2.154 × 10−3
dxpv0	m	3.2 × 10−4

[23].

3. Test Verification and Results

Two test methods were designed to verify the effectiveness of the designed crab steering system: the simulation test and the field test. The testing schemes and results are described below.

3.1. Simulation Test

With the objective of verifying the crab steering control strategy and method indoors, the PID control method was applied to test the crab steering performance of the designed system in MATLAB/Simulink under typical conditions according to the established mathematical model. The simulation test scheme and process are as follows: set the steering hydraulic cylinder load to 0, and the initial steering angle to 0, input a step excitation signal of 230° to the steering wheel at 5 s and a step excitation signal of -9 rad to the steering wheel at 10 s, and the crab steering system response characteristics in the steering process are obtained through the Simulink simulation model. The change of the two front wheels under the crab steering mode is shown in Figure 11, and the changes in the angle difference of the two front wheels during the steering process of the two front wheels under the two wheel steering mode and crab steering mode are shown in Figure 12.

3.2. Field Test

The test site was located in Rizhao, China, which is located at 118°25′~119°39′ E, 35°04′~36°04′ N, and the interior ground of the test site was uneven and had obvious pits and bulges, which is consistent with field road conditions, as shown in Figure 13. After the installation of the test equipment, the driver drives the sprayer into the field test site. The sprayer drives at a constant speed of 5 km/h, the switch of the crab steering compensation valve is turned on and the sprayer starts to enter the crab steering mode. The driver turns the steering wheel to the right until the wheel angle is approximately 5°, the wheel steering angle is kept fixed and then the steering wheel is turned to the left at 18 s, until the wheel angle is approximately 14°, and the wheel steering angle is kept fixed. The displacement of the piston rod of each hydraulic cylinder is collected by the displacement sensor and transmitted to the controller and, according to the geometric relationship of the steering transmission mechanism, the displacement of the piston rod of each hydraulic cylinder is converted into the steering angle of each wheel. Taking the collection time as the abscissa and the steering angle of each wheel as the ordinate, the curve of the wheel steering angle with time in the crab steering mode of the sprayer is obtained, as shown in Figure 14.

4. Discussion

4.1. Simulation Test

The average error and maximum error of the two front wheels in the crab steering mode are shown in

Table 4. Analysis of the angle difference between front wheels under crab steering mode.

Parameter	Average Error (°)	Maximum Error (°)
Value	0.139	2.75

.

When the sprayer is under crab steering mode, when a step signal is input to the steering wheel, the steering angle of the two front wheels is approximately the same during the steering, and the maximum steering angle deviation is 2.75°, which can be seen from Figure 11 and Figure 12. The position where the error is larger is the time when the step signal is input. This is because the crab compensation valve in the steering system has a dead zone when starting, and the steering system can be charged and drained only after the dead zone is overcome during operation, therefore, there is a delay in the follow up of the hydraulic cylinder after the reversing, which leads to a large error in the rotation angle of the two front wheels when the step signal is input. As we can see in Table 3, in the crab steering mode, the average deviation of the two front wheel rotation angles is 0.139° under the excitation of the step signal and the maximum deviation value is 2.75°, which meets the requirements of the sprayer for crab steering.

4.2. Field Test

Taking the change of the left front wheel angle as the benchmark, the real time error of the steering angle of each wheel is obtained as shown in

Table 5. Real vehicle test data analysis of crab steering system.

Parameter	Average Error (°)	Maximum Error (°)
Right front wheel	0.265	2.380
Left rear wheel	0.285	2.587
Right rear wheel	0.260	2.493

.

It can be seen from Figure 13 and Table 4, during the field test, when the sprayer is under crab steering mode, while the vehicle is making a step turn, the steering angle of each wheel rotates in the same phase, and the steering angles of the four wheels are basically the same. The relative error of the four wheel steering angles appears at point C at 33 s in Figure 13, the maximum error is 2.587° and the average value of the steering angle error during the rotation of the four wheels is 0.285°. The relative error of the wheel steering angle is small in the crab steering mode of the sprayer, which meets the needs of the sprayer in steering and line changing operations in the field.

5. Conclusions

In this paper, a crab steering system to improve the working efficiency, steering flexibility and maneuverability of a high clearance self-propelled sprayer is proposed. The system consists of a steering transmission mechanism, a hydraulic crab steering system, a control strategy and a PID control method. A simulation test based on a mathematical model of the crab steering system was conducted to verify the developed system. Next, a field test using the 3WPG-3000 high clearance self-propelled sprayer was carried out to verify the crab steering system.

The result revealed that the developed crab steering system was able to meet the flexible steering requirements of the sprayer during the simulation test and field test. The steering error is small in both test modes. In the simulation test, when the vehicle performs a step steering, the two front wheels have approximately the same rotation angle in the crab mode, the maximum steering angle deviation is 2.75° and the average deviation is 0.139°. In the field test, the maximum error of the four wheels is 2.587°, and the maximum average error is 0.285° during the crab steering process, which is obviously smaller than the steering angle error of the traditional crab steering system under the condition of large angle steering [17,20], and the performance of the designed system satisfied the control requirements during the spraying work [21,22].

The research presented in this paper provides a crab steering system, which can effectively ensure the accuracy of the wheel steering angle during the crab steering process of the sprayer, and can effectively improve the steering efficiency and the working efficiency of the sprayer. Meanwhile, it also provides a foundation for the development of steering systems for high clearance self-propelled sprayers.