Changsong Zheng， Yichun Chen and Ran Jia

（1. School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China；2. Collaborative Innovation Center of Electric Vehicle in Beijing, Beijing 100081, China）

Abstract: In the steering process of tracked vehicle with hydrostatic drive, the motion and resistance states of the vehicle are always of uncertain and nonlinear characteristics, and these states may undergoe large-scale changes. Therefore, it is significant to enhance the steering stability of tracked vehicle with hydrostatic drive to meet the need of future battlefield. In this paper, a sliding mode control algorithm is proposed and applied to achieve desired yaw rates. The speed controller and the yaw rate controller are designed through the kinematics and dynamics analysis. In addition,the nonlinear derivative and integral sliding mode control algorithm is designed, which is supposed to efficiently reduce the integration saturation and the disturbances from the unsmooth road surfaces through a conditional integrator approach. Moreover, it improves the response speed of the system and reduces the chattering by the derivative controller. The hydrostatic tracked vehicle module is modeled with a multi-body dynamic software RecurDyn and the steering control strategy module is modeled by MATLAB/Simulink. The co-simulation results of the whole model show that the control strategy can improve the vehicle steering response speed and also ensure a smooth control output with small chattering and strong robustness.

Key words: tracked vehicle；hydrostatic drive；steer control；nonlinear derivative and integral sliding mode control

Hydrostatic high-speed tracked vehicles are widely used in agricultural, mining, construction machinery and military vehicles due to their good traverse ability and cross-country maneuverability[1]. Despite these merits, steering control is a difficult issue in driving tracked vehicles. Hydrostatic high-speed tracked vehicles need to control the output torque and speed of the hydraulic pump-motor system on both sides to achieve vehicle driving control[2]. Because the vehicle hydraulic transmission system is a time-varying nonlinear system, there are large parameter changes and time-varying load interferences, so the steering control algorithm has an important impact in traceability, anti-disturbance and antisaturation[3].

In recent years, a lot of research has been conducted on steering and hydraulic control systems of tracked vehicles both at home and abroad. From the concept of vehicle steering dynamics and kinematics, Yang L proposed that the output torque of the motor is controlled by giving the target torque of the inside and outside motors, and the theoretical analysis of this torque control method can be achieved[4-5]. But this only provides a theoretical basis for making a torque control strategy, without any in-depth study on the torque control strategy. Do H T[6]proposed an adaptive fuzzy sliding mode controller to cope with the nonlinearities of a hydrostatic transmission system. Experiments were carried out on the condition of disturbance load to evaluate the effectiveness of the proposed controller and the results demonstrated that the proposed controller was excellent from the standpoint of performance and stability for the velocity control of hydrostatic transmission systems.Nonetheless, the torque control was not taken into consideration in the study. More importantly,Zeng Q H designed a decoupling control structure to resolve the whole control system into two independent subsystems in order to reduce the strong coupling problem and improve the vehicle steering yaw rate control by adopting an equivalent conditional integral sliding mode control algorithm[7-8]. Although this method has been proved to be effective in the dual-motor tracked vehicle, the change of displacement of hydrostatic drive system was not studied. In addition,based on the structural features of hydrostatic driving wheels of both sides, Zhang H L[9-10]introduced a steering control strategy where the outer motor is controlled by pressure and engine speed, and the inside motor is controlled by a neutral PID controller to follow the outside motor. However, the authors did not consider the influence of the changes in the adhesion coefficient. Since the adhesion coefficient is not constant, the method cannot be fully considered as effective.

In order to solve the above problems, we propose a steering controller based on a decoupling control structure for a tracked vehicle with a hydrostatic drive which includes a speed controller and a yaw rate controller. A sliding mode control algorithm is designed to ensure the precise yaw rate control of vehicle steering for the yaw rate controller, which has the advantages of wide tracking range, high precision and strong anti-interference ability. The co-simulation model of RecurDyn and MATLAB/Simulink shows that the control algorithm can reduce torque input and increase the yaw rate to the ideal state rapidly.

1 System Overview and Modelling

Fig.1 shows the diagram of the hydrostatic drive tracked vehicle used in this study. The power transmitted from the engine to the hydraulic pump, and then to the hydraulic motors on both sides, from which, it reaches the drive wheel of the vehicle to drive the whole vehicle[11].The hydrostatic drive system generally adopts the form of variable displacement pump and motor, which does not only have a wide range of speed regulations, but also speed regulation characteristics adapted to have load requirements and higher work efficiency.

Fig.1 Structure diagram of hydrostatic drive system

Assuming that the vehicle is running on a flat horizontal ground, a simplified dynamic model of a tracked vehicle is elaborated, as shown in Fig.2, where L and B denote the length and width of the tracked vehicle, respectively. R denotes the steering radius of the vehicle. v1and v2denote the velocities of the track on both sides.Also, v is the speed and ω is the yaw rate of the vehicle. Fr1and Fr2denote the ground rolling resistance of both sides of the track. In addition, F1and F2denote the drive forces of the hydraulic motors on both sides of the track. Mμrepresent the steering resistance moment.

According to the vehicle dynamic balance,the vehicle forces and dynamics equations can be expressed as follows[12]

Fig.2 Dynamic model of the tracked vehicle

where μmaxis the maximum steering resistance coefficient; μ denotes the steering resistance coefficient; f is the ground deformation resistance coefficient on both sides; m is the vehicle mass; icis the transmission ratio of the hydrostatic motor output shaft to the driving wheel; r is the radius of the vehicle sprocket; m is the mass of the vehicle and Izis the yaw inertia of the vehicle; η is the transmission efficiency.

From Eq. (1), the vehicle speed and yaw rate can be expressed in a matrix form as

where the coefficient matrix G is

From Eq. (2), the expected torque can be calculated as follows

It can be seen from Eq. (3) that the load torque is affected by parameters, such as the load condition, vehicle dynamic state and load torque.The road load fluctuation of one side will lead to steering of the vehicle. At this time, the ground steering resistance moment will act on both sides of the track, so the load torque change of one side of the track will affect the torque of the other side. The aim of steering control is to change the output torque on the both sides of the vehicle so that the speed and the yaw rate of the vehicle can reach its desired values.

2 Design of Control Strategy for the Hydrostatic Drive

In order to achieve an effective control of vehicle speed and yaw rate, it is necessary to design a closed-loop control algorithm to accurately calculate the desired torque on both sides of the vehicle. Therefore, the dynamic controller is designed to realize the longitudinal motion and steering control of the vehicle, including a speed controller and a yaw rate controller[13-14]. The diagram of the control structure is shown in Fig.3.

In addition to the dynamic controller, we also need to analyze and define the expected signal of each subsystem according to the combination of driver’s control signals and actual vehicle condition.

The vehicle desired speed v*is determined by the displacement of acceleration pedal

where α is the displacement of acceleration pedal;vmaxis the maximum speed of vehicle; αmaxand α0are the maximum and initial displacements of acceleration pedal.

Fig.3 Vehicle control strategy structure

Also, we considered only one direction of rotation due to the symmetric nature of the steering wheel. The vehicle desired yaw rate ω*is determined by the input angle of the steering wheel displacement:

where θ is the input angle of the steering wheel displacement; ωmaxis the maximum yaw rate of vehicle, the value of which is linear to the realtime speed of vehicle; θmaxand θ0are the maximum and initial angles of the steering wheel displacement.

2.1 Speed controller

Vehicle speed control focuses on the speed and accuracy of speed tracking, overcoming the pavement structure and effects of parameter changes. In general, the speed controller usually uses the PID method as a control strategy to adjust the longitudinal traction torque until the vehicle speed reaches the expected value.However, it is very difficult to track the speed of the vehicle during the steering process, especially in high speed conditions. In addition, the traditional PID parameters cannot meet the objectives and requirements of all aspects of the control process, so it always affects the performance in practical applications. Therefore, a speed control strategy based on the fuzzy PID is designed for hydraulic tracked vehicle to improve the tracking speed ability and adapts to the nonlinear system. Obviously, the fuzzy PID control design objective is to control the longitudinal traction torque Tvin order to eliminate the error difference between the vehicle actual speed v and the reference speed v0. The fuzzy PID combines the fuzzy concept with the conventional PID control. It uses the error of vehicle speed and its rate of change to calculate the adjustment of the proportion coefficient Kp, integration coefficient Kiand differentiation coefficient Kdin real time,and achieves the adaptive effect that the conventional PID control algorithm does not have through real-time updating of parameters. The fuzzy control algorithm can express human accumulated experiences and common sense. It can also be used to express group language, and the expected regulation rules can be concisely incorporated into the strategy. Through the adjustment of fuzzy controller parameters, the fuzzy rules for the speed tracking controller can be shown in Fig.4.

2.2 Yaw rate controller

Fig.4 Fuzzy rules for the speed tracking controller

The steering process of hydrostatic tracked vehicle is nonlinear, which brings about uncertainties. The torques on both sides of the sprocket may lead to large load torque disturbances,hydraulic pressure, motor displacement and fluctuations in other parameters. For the hydrostatic vehicle steering system, a traditional PID control parameter tuning process is very complex,and the fixed PID parameters are difficult to satisfy the need of steering characteristics under different working conditions. Therefore, the sliding mode control algorithm is widely used to control the yaw rate. In order to reduce the steady-state error, the integral item of tracking error is added to the conventional sliding mode control to form the integral sliding surface. In addition, because the controller is composed of discontinuous functions, the system will have the problem of chattering. Hence, the application of this method will be limited in the field of engineering if this problem cannot be fundamentally solved. Therefore, a nonlinear derivative controller is added into the conventional controller to eliminate the chattering of the system and achieve the accurate tracking control of the yaw rate.

According the dynamic equation of vehicle steering, the equilibrium relationship can be expressed as[15]

The control law is designed to track the reference yaw rate. The tracking error e is defined as

Introducing the integral component, we can get the integral sliding mode surface

where k＞0 is the integral coefficient; g(e) is a smooth non-linear function with the features of“small error amplification, large error saturation”.

Without considering disturbance and uncertainty, the equivalent controller ueqfor the sliding mode control can be obtained by taking s=0.We obtain

Further, in order to ensure that the sliding mode condition is reached, the switching controller is designed as

where β denotes the switch gain, and the saturation function sat(s/ε) can be formulated as

where ε is the width of sliding mode switch boundary layer.

The integral sliding mode control law is designed as follows

We use a Lyapunov function candidate

Differentiating Eq. (10) with respect to time as

On the basis of traditional integral sliding mode control algorithm, the derivative controller is taken into consideration to solve the problem of chattering and improve tracking accuracy.

The derivative controller is designed as follows

where δ ＞0, h ＞ 0.

The overall derivative and integral sliding mode variable structure controller takes the following form

So we have

Now, we can obtain the following equations

3 Modeling and Simulation Analysis

In order to verify the performance of the proposed controller algorithm, a co-simulation model is designed based on the control software MATLAB/Simulink and multi-body software RecurDyn. The driver model, controller model,and hydraulic drive system model are established in MATLAB/Simulink, and the vehicle dynamic model and the ground resistance are established in RecurDyn[16]. The data exchanges are carried out through the software interface to realize the integrated simulation of mechanical, hydraulic and control systems. Vehicle dynamics model includes vehicle body and moving device,in which the vehicle moving device model is established by using TrackHM module of Recur-Dyn, as shown in Fig.5.

Fig.5 Dynamic simulation model of tracked vehicle

In this work, the major parameters of the tracked vehicle are listed in Tab.1. In order to verify the effectiveness of the proposed controller,simulations were carried out while considering various conditions. The proposed controller is compared with a traditional PID controller at three different speeds and yaw rates of the vehicle.

3.1 Small radius steering

Initially, the vehicle accelerated to 20 km/h for 2 s and then remains constant in a straight direction. After that, the vehicle undergoes steering for 5 s while the yaw rate of the vehicle is 0.6 rad/s. Figs. 6a-6c show the trajectory,sprocket speed and sprocket torque of the vehicle. It can be seen that the vehicle steering response was much faster under the proposed controller taking into consideration the trajectory and sprocket speed is 3 s faster to reach a steady state. Furthermore, the sprocket torque was lower than the traditional PID control torque. This signifies an improvement in the stability of the vehicle. Fig.6d illustrates the displacement of hydrostatic pump and motor. It is observed that the displacement of the pump remained at its maximum and the displacement of the motor changes with the load at high speed.The speed and yaw rate of the vehicle is shown in Figs. 6e-6f. It was observed that, the difference in the vehicle speed was insignificant, because the same control strategy was used for the speed control. Although both control strategies can reach the desired yaw rate, the proposed controller achieved stability more rapidly without fluctuation.

Tab. 1 Major parameters of the tracked vehicle

3.2 Medium radius steering

The vehicle accelerated to 40 km/h for 2 s and then remains constant in a straight horizontal path. Then it steered for 5 s until the yaw rate reaches 0.4 rad/s. Figs. 7a-7c show the trajectory, sprocket speed and sprocket torque of the vehicle. It is obvious to note that the vehicle steering response speed increase is 1 s faster to reach a steady state with the reduction in sprocket torque and showed better manoeuvrability compared to the PID control. Fig.7d shows the displacement of the hydrostatic pump and motor.The displacement of the motor exhibited a continuous and smooth change. Figs. 7e-7f show the speed and yaw rate of the vehicle. Increasing the vehicle speed causes the stability of the vehicle steering process to become unstable. In contrast,the proposed controller showed a stable yaw rate control while the PID control fluctuated at the beginning of steering.

3.3 Large radius steering

For large radius steering, the vehicle accelerated to 55 km/h for 2 s and keeps a straight horizontal direction. Then, the vehicle started to steer for 5 s and until the yaw rate of the vehicle reached 0.24 rad/s. Figs. 8a-8c show the trajectory, sprocket speed and sprocket torque of the vehicle. The simulation results of vehicle trajectory show that the proposed controller is similar to the PID controller but have a smaller steering radius and faster reaction speed. More importantly, it can be seen that the proposed controller showed a significant difference in the sprocket speed than the PID controller at the beginning of steering. This signifies an improvement in the steering speed. Nonetheless, the sprocket torque was not as smooth as the PID controller, but it can be considered as efficient because the aim of a torque fluctuation is to balance the load during the vehicle steering. Fig.8d represents the displacement of the hydrostatic pump and motor.The displacement of the inner hydrostatic motor appears to fluctuate. However, this phenomenon can be solved by adding a hydraulic accumulator.Figs. 8e-8f show the speed and yaw rates of the vehicle. The change in the vehicle speed was insignificant during the steering process due to the same control strategy used for the vehicle speed.From Fig.8f, it can be seen that the proposed controller has more stability than the PID controller, and the vehicle has great ability to overcome nonlinear disturbances.

Fig.6 Simulation results of center steering

Fig.7 Simulation result of small radius steering

Fig.8 Simulation results of large radius steering

4 Conclusion

In this paper, a steering stability control strategy based on the nonlinear derivative and integral sliding mode control for the tracked vehicle with hydrostatic drive was proposed to improve the steering stability and steering response rate. The system stability was verified based on Lyapunov stability theory. A co-simulation based on MATLAB/Simulink and Recur-Dyn was carried out to test the effectiveness of the proposed controller under different steering conditions. Results show that the proposed controller can overcome the random disturbances of road steering resistance, nonlinear change and the influence of motor saturation and realize the accurate tracking of yaw rate under the condition of multiple steering. Compared with the PID controller, the steering tracking fluctuation is obviously reduced and the higher steering and return response speed are achieved, respectively,improving the stability control performance of vehicle steering in the full speed range.