Research on differential performance of four-wheel independent steering of a hydraulic wheel-driving off-road vehicle

: Steering differential performance is an important factor affecting the driving performance of the vehicle. A hydraulic control method is proposed for the differential system of the hydraulic wheel-driving off-road vehicle, and the driving force equalisation technique is adopted to realise the adaptive steering differential under the different speed modes. Based on the analysis of the working principle of static pressure driving steering differential, the mathematical model of the steering differential system is derived by using kinematics and dynamics theory. The hydraulic driving system and multibody dynamics model are established by AMESim and LMS Virtual.Lab Motion, respectively, and joint simulation is carried out. The experimental scheme is designed for the typical working conditions in the simulation, and the rotation speed and the pressure differential of the motor in the steady state are analysed. The experimental results are consistent with the theoretical analysis and simulation results, which indicate that the system can achieve a better steering differential performance. This research provides a new idea for the study of the steering performance of the hydraulic wheel-driving walking system, which is of great practical significance to improve the driving performance of off-road vehicles under the complex terrain.


Introduction
When a vehicle turns, the relative slip of the inner and outer wheels with the ground causes tire wear and power loss [1,2], and even worse, and the steering and braking performance of the vehicle may deteriorate [3]. The steering differential mainly refers to the fact that the linear velocity of the wheel rotation should be coordinated with the wheel centre velocity to avoid slipping or dragging [2]. The conventional drive axle vehicles often use a mechanical differential device for steering, but the entire structure is complicated by the need of the mechanical transmission mechanism and the mechanical differential device. The four-wheel independent electric vehicle adopts electric differential instead of mechanical differential function [4]. The torque and rotation speed of each driving motor are adjusted by the vehicle controller to control the yaw moment [5], which has good steering differential performance and handling stability, but the differential control strategies and algorithms are more complex. At the same time, the increase of the unsprung mass by the hub motor affects the tire grounding and ride comfort of the vehicle [6]. The wheel hydraulic driving power flow is transmitted by the hydraulic pipeline [7], which can, respectively, control the individual wheels in a precise way to achieve the coordination of the control [8], and also solve the problem of the layout of the special vehicle transmission system. With respect to the wheel electric drive, the wheel hydraulic drive can meet the requirements of greater braking strength [9], which is widely used in the construction machinery [10,11]. The turning radius of the wheel hydraulic drive vehicle is small, also the adaptability to the road and driving flexibility of the vehicle is great [12][13][14], but the inner and outer tire differential needs to be set reasonably. At present, there is little research on the steering differential system of this type of vehicle [15].
In this paper, the hydraulic wheel-driving off-road vehicle for the complex unstructured terrain is used as the carrier. Based on the theory of vehicle steering differential and static pressure drive, the four wheels are directly driven by four hydraulic motors. The hydraulic system and liquid resistance can automatically adapt to the change of pressure, so that the flow of the motor can be accurately adjusted, thereby accurately controlling the steering differential of the inner and outer tires, and independently adjusting the torque of each wheel while turning. The system greatly simplifies the mechanical transmission structure of the vehicle with a flexible layout and high transmission efficiency [16].

Principle analysis of the static pressure drive system
The layout of the off-road vehicle running system studied in this paper is arranged as shown in Fig. 1. The engine is the power source, and the hydraulic transmission system is composed of an axial variable displacement piston pump, a diverter, a control valve group, a quantitative cycloid motor, a liquid resistance, and so on.
The hydraulic transmission principle of the system is shown in Fig. 2. Each wheel is driven directly by a quantitative cycloidal motor. In the system diagram, the solenoid valve 15 controls the liquid resistance 21, 22 for on-off, the liquid resistance 21, 22 is cut off at the neutral position, the liquid resistance 21 is connected when the electromagnet 3Y1 is energised, and the liquid resistance 22 is connected when the 3Y 2 is energised. The values of resistance of the liquid resistance 21 and 22 are different, and they are connected in parallel with each other between motors. They are manually selected according to the magnitude of the system pressure, so that the driving force equalisation control can be realised. The diverter 14 uses the C oil inlet, and the A and B oil outlets divide the inlet flow rate equally. By switching the electromagnets of the change valves 13,15,16,17,18,19, 20 in the control valve units, the drive modes of low, medium, and high velocities can be realised. The marched route of each wheel of the off-road vehicle is different when passing the uneven road surface or steering, and the vertical load distribution of each wheel is uneven when loading and transporting, at this time, part of the flow rate can be differentially controlled by flowing through the liquid resistance, thereby avoiding dragging caused by the forced shunting.

Mathematical model establishment of the turning differential
The wheeled off-road vehicle studied in this paper adopts the fourwheel steering mode. Compared with the front-wheel steering, the front and rear wheels deflect in opposite directions during the lowspeed steering, which can effectively reduce the steering radius and enable the vehicle to achieve steering in a small space. During the high-speed steering, the front and rear wheels are deflected in the same direction, which avoids the influence of the large lateral acceleration on the steering stability [17].
In order to meet the requirements of the smooth steering of a four-wheeled off-road vehicle in a narrow space, two steering trapezoids are used to realise four-wheel steering. As this paper mainly studies the longitudinal driving performance of the vehicle, the following assumptions are made when establishing the mathematical model: the vehicle ignores the influence of the lateral force during the steering process; the walking system is the rigid system; the four-wheeled off-road vehicle acts on the hard road surface, ignoring the ground deformation; the four-wheel orientation angle of the off-road vehicle is zero.

Establishment of the kinematics model
The four-wheel steering kinematics model is shown in Fig. 3.
It can be seen that the front steering centre is at the same point O with the rear steering centre, and the deflection angles of the four wheels are in accordance with the following equation: where c 1 is the deflection angle of the outer front wheel, b 1 is the deflection angle of the inner front wheel, L 1 is the distance from the front wheel axis to the steering centre, c 2 is the deflection angle of the outer rear wheel, b 2 is the deflection angle of the inner rear wheel, and L 2 is the distance from the rear wheel axis to the steering centre. The angle relationship among the deflection angles of the four wheels can be obtained by substituting L 1 + L 2 = L into (1) and (2): When the rear wheel steering structure is consistent with the frontwheel steering structure, there are equations The steering radius of the vehicle is In the same way, the steering radius can be obtained when the front wheel is turned: It can be seen that, in the case of the same steering mechanism, the steering radius of the four-wheel steering using the symmetrical steering trapezoid is smaller than that of the front-wheel steering corresponding to the same deflection angle. The steady-state tangential velocities of the four wheels are expressed as: where Ω is the angular velocity matrix of the vehicle around the point o and E is the distance between the wheel kingpin and the ground centre of the tire. It can be seen from (7) that the longitudinal speed relationship of the four wheels is (v o2 = v o4 ) > (v o1 = v o3 ); that is, the longitudinal speed of the outer wheel is larger than that of the inner wheel.
The angular velocity of the four wheels is thus obtained as shown in (8).
where i is the vehicle slip ratio and r is the tire radius.

Establishment of the kinetics model
Substitute the traction F x = μ p W − ((λ(μ p W − K 1 i) 2 )/2lK 1 i) into the steering torque equation; then, the steering torque of each wheel can be obtained by where K 1 = k t λl, l is the tire-grounded length, μ p is the peak value of the ground-adhesion coefficient, λ is the longitudinal deformation coefficient of the tread grounded front end, and F x is the tangential reaction force acting on the wheel by the ground, and W is the normal load acting on the wheel. The torque balance equation of the vehicle-to-point O: where M f is the driving resistance torque on four wheels and the vehicle around O. The traction power of the vehicle is expressed as If the traction of the four wheels is equal during the steering process, the four wheels are in the same ground state and the slip ratio is equal, that is, Simulation study of the steering differential

Simulation model establishment
The four-wheeled off-road vehicle involves many fields such as the mechanical dynamics, the tire ground mechanics, and the oil-flow characteristics. AMESim simulation software is powerful in the field of hydraulic and control; however, in the mechanical structure and the tire ground mechanics, only the equivalent load can be used to simulate, which cannot reflect the true dynamic interactions; LMS Virtual.Lab Motion is software designed to simulate the real load distribution and motion of the mechanical systems. It can build a vehicle simulation model. By setting the parameters of the tire and ground, it can reflect the interaction between the mechanical structures and between the tire and ground as much as possible. Therefore, this paper uses AMESim to model the engine and the hydraulic transmission system, and its terminal is four motors. According to the ground load characteristics of the tire, the multi-body dynamics model is established by LMS Virtual.Lab Motion software and its terminal is the tire. The torque of the wheel motor is applied to the tire in the dynamics model through the communication interface, and the rotation speed of the tire is fed back into the hydraulic driving system to realise the joint simulation.
The driving system AMESim model is shown in Fig. 4. During the simulation, it is necessary to adjust the throttle signal, the gear signal, and the liquid resistance. The simplified model of the multibody dynamics is shown in Fig. 5.

Analysis of the simulation results of the four-wheel steering condition
The Simulation conditions: the off-road vehicle with no-load, forward with low-speed gear, the full throttle opening, the left steering and the steering radius of 10 m. The simulation results of the four-wheel steering and the two front-wheel steering are shown in Figs. 6 and 7, respectively. In Fig. 6a, the curves 1, 2, 3, and 4 represent the pressure differential of the left front, the right front, the left rear, and the right rear motors, respectively; in Fig. 6b, 1  and 3 are the flow of the left and right front motors, respectively, and 2 and 4 are the flow of the front and rear motors, respectively. The meaning of each curve in Fig. 7 is the same as Fig. 6a.
The simulation data on the stabilisation phase is summarised in Table 1. Due to the two steering trapezoidal symmetrical arrangements, the two motors on the right side have a consistent flow rate of ∼20.8 l/min, which is slightly larger than the left motor. The four motor pressure differential is 1.7 MPa, which is smaller than the motor pressure differential (∼2.5 Mpa) of the twowheel steering under the same working condition. This is because during the four-wheel steering, the inner and outer two motors are in the same force state and the inner and outer two wheels are in the same rotating path, which determines that the four motors have the constant pressure differentials and good differential performance. This result is consistent with the analysis of the establishment of the four-wheel steering mathematical model, and the rationality of the power matching of the walking driving system and the correctness of the walking system are preliminarily verified.

Experimental study on the differential of the four-wheel steering condition
In order to further determine the rationality of the power matching, the theoretical analysis, the simulation research, and the experimental exploration form the relationship of verification and guide, according to the theoretical analysis and the simulation conditions, the experiment is designed to detect the actual rotation speed of the pump and the actual pressure differentials of the four motors. The experimental conditions are as follows: the selfdeveloped static pressure-driven four-wheeled off-road vehicle is used as the experimental prototype, as shown in Fig. 8. It is adjusted to the forward low-speed gear and the four-wheel steering mode, started on the straight cement road and accelerated quickly to the pump speed around 2300 r/min. Turn to right with the uniform speed along the mark with a radius of 10 m and decelerate to stop after a certain period of time.
The experimental results: The results of the four-wheel steering and two front-wheel steering experiments are shown in Figs. 9 and 10, respectively.
The experimental data of the pump rotation speed and the motor pressure differential on the steady state are summarised in Table 2, and the front-wheel steering experimental data under the same working condition is summarised in the table for comparison. The pump rotation speed is stable at around 2350 r/min, and the pressure differential between the left motor and the right motor is similar. The four-wheel steering system pressure is lower than that of the front-wheel steering, which is consistent with the simulation results.   For the independent static pressure driving system designed for the non-actuated articulated wheeled off-road vehicles, the motors on both sides are connected in parallel to realise the function of the differential, and the flow of the left and right wheels is automatically distributed to prevent steering or the dragging slide of one or more wheels passing through the bumpy road then to realise the automatic steering differential. It is found that the experimental results of the four-wheel steering system under the same turning radius are highly consistent with the theoretical analysis and the simulation studies. Comparing the four-wheel steering with the front-wheel steering, it is found that the system pressure is less than that of the two front-wheel steering when using the four-wheel steering under the same steering radius. In this paper, the research on the walking system is based on the hard road. In order to further explore and optimise the performance of the off-road vehicle, the multi-body dynamics model and the road surface need to be further refined, and the walking system of the off-road vehicle driven on the swamp road and the unstructured is developed to study.

Acknowledgments
The research in this paper was funded by the National Key Research and Development Program, Project Number 2016YFC08029.