Yaw stability control for distributed drive electric vehicle based on AFS and DYC integrated control

Yong ZHANG, Feng LI, Ke-lin REN,Shen YANG, Hai-qin ZHU

(1Key Laboratory of Advanced Manufacturing Technology for Automobile Parts, Ministry of Education， Chongqing University of Technology，Chongqing 400054，China)(2College of Vehicle Engineering， Chongqing University of Technology，Chongqing 400054，China)

Abstract: To improve the driving stability of distributed-drive electric vehicles, a model stability control method is proposed based on model predictive control, which uses the saturated output torque of the drive motor and actuators as control input constraints and yaw rate as output constraints. Establish a 2-DOF vehicle state-space model as a prediction model, calculate the additional yaw moment required to track the expected yaw angular velocity, and consider the constraints of the motor and hydraulic brake system to design three control allocation algorithms, The above-mentioned algorithm is applied to a four-wheel-drive 7-DOF vehicle model for simulation. The results show that the optimal control allocation method with the minimum variance of the square of the tire load rate and the weighted mean value as the goal can effectively improve the stability of the distributed drive electric vehicle.

Key words: Electric vehicle, Model predictive control, Yaw stability, Hydraulic braking, Torque distribution

1 Introduction

According to the summary of the impact of each active safety control system proposed by Ford on the vehicle’s directional motion, the direct yaw moment control (DYC) can effectively improve the active safety of the vehicle[1]. Also, the active front steering system (AFS) based on lateral force control and the electronic stability program (ESP) based on direct yaw moment control has strong coupling, which can avoid the dangerous working conditions faced by the vehicle by accurately controlling the yaw moment of the vehicle[2]. The driving torque of each wheel of the traditional automobile can’t be controlled separately, they can only be controlled by yaw moment generated from the differential braking of the vehicle, but the intervention of the braking system will often influence the driving experience of the driver and affect the acceleration performance of the vehicle[3]. However, on a distributed drive electric vehicle, the driving/braking torque of each wheel can be controlled independently, and the in-wheel motor has the advantages of fast response, high control accuracy, and good controllability, etc., and it is more conducive to the realization of the yaw stability control of the vehicle[4].

At present, the commonly used control methods for yaw moment control of distributed-drive electric vehicles include Linear Quadratic Regulator (LQR), Sliding Mode Control (SMC), Generalized Predictive Control, and Adaptive Control, etc. In existing research of yaw stability control methods, the vehicle body and wheel dynamics are considered separately, which is not convenient to analyze the stability of the system[5]. Or the model used is too simple to analyze the influence of non-modelling characteristics on control performance [6]. The predictive control method is most suitable when the system has a high order model as well as state and control constraints at the same time. Besides, predictive control enables the system model to predict the state of the system in a short time in the future, so measures can be taken in advance to correct the vehicle’s yaw status to avoid accidents [7].

This paper designs a hierarchical integrated controller based on the model predictive control theory for the yaw stability control of distributed-driven electric vehicles. The upper controller is used to solve the optimization problem satisfying the objective function and various constraints obtain additional yaw moment and the additional front-wheel steering angle required to satisfy the vehicle stability demand. Under the condition of considering the actuator constraint and tire force coupling, the lower controller distributes the additional yaw moment to the actuator motors and hydraulic brake system of the four wheels, to improve the yaw stability of the vehicle under the extreme operating condition.

2 Vehicle dynamics model

2.1 7-DOFs vehicle dynamics model

The vehicle stability is mainly shown as lateral dynamic control of the vehicle and eventually shown as the control of the yaw movement and the lateral movement. In the development process of the vehicle dynamics control system, due to the limitation of the site and the difficulty in predicting the dangerous conditions, to verify the relevant theories, the model-based research method is often used. In this paper, a 7-DOFs vehicle dynamics model is built[8], which includes the longitudinal, lateral and yaw motions of the vehicle body, as well as the rotation motion of four wheels, as shown in Fig.1.

Fig.1 7-DOFsvehicle dynamics model

The vehicle dynamics equation is as follows:

(1) Differential equation of longitudinal motion:

(Fyfl-Fyfr)sinδ+Fxrl+Fxrr

(1)

(2) Differential equation of lateral longitudinal motion:

(Fyfl+Fyfr)cosδ+Fyrl+Fyrr

(2)

(3) Differential equation of yaw motion:

(3)

2.2 Tire model

In this paper, the “magic formula” tire model proposed by Professor H.B. Pacejka is used. The formula is a semi-empirical tire model based on test data, features fewer input parameters and high fitting accuracy. Its general expression is as follows[9]:

y(x)=Dsin(Carctan{Bx-E(Bx-arctan(Bx))})

(4)

Y(X)=y(x)+Sv

(5)

x=X+Sh

(6)

Where:Bis the stiffness factor;Cis the curve shape factor;Dis the peak factor;Eis the curve curvature factor;Xis the sideslip angle or longitudinal slip rate;Yis the longitudinal force, lateral force or aligning torque of tire;Svis the vertical drift of curve;Shis the horizontal drift of curve.

2.3 Motor model

The torque response of the hub motor is a fast dynamic process[10], so it can be simplified as a first-order response system without delay, and its transfer function is as follows:

(7)

Where,Toutindicates the actual output torque of the motor;Tdindicates the target output torque of the motor;τindicates the time constant of the motor, which can be obtained through experiments.

3 Design of automobile stability controller

Vehicle stability control mainly involves trajectory maintenance and stability control. Generally speaking, the trajectory can be described by the sideslip angle, while the stability is described by the yaw rate. These two control variables are coupled with each other [10]. The research object of this paper is the distributed drive electric vehicle with an active front-wheel steering system. According to the characteristics of the research object and the goal of stability integrated control system, a hierarchical integrated controller is designed, as shown in Fig.2. The upper controller is the motion tracking layer, it calculates and tracks the expected state of the vehicle based on the model predictive control theory, so to protect the additional yaw moment and the additional front-wheel steering angle required by the vehicle stability control. The lower controller is the torque distribution layer, it directly applies the additional front-wheel steering angle on the active front-wheel steering system, and distributes the additional yaw moment to the actuator motor and hydraulic brake system of the four wheels with consideration of actuator constraints and tire force coupling.

Fig.2 The hierarchical control structure for Vehicle stability

3.1 Linear 2-DOF vehicle model

As shown in Fig.3, this paper selects the linear 2-DOF vehicle model as the reference model to obtain the desired yaw rate and the expected sideslip angle.

Fig.3 Linear 2-DOF vehicle model

The linear 2-DOF vehicle model has only two degrees of freedom of lateral and yaw motion, but it contains the most important parameters of vehicle mass and tire cornering lateral stiffness, which reflect the most basic characteristics of vehicle curve motion. Its differential equation is as follows:

(8)

Where:k1andk2are the cornering lateral stiffness of the front and rear axles respectively.

3.2 Model predictive controller

MPC is an optimization control method that has the ability to handle various soft and hard constraints including various system response and actuator actions. Its principle and characteristics can be summarized as a predictive model, rolling optimization, and feedback correction. As the MPC itself can carry out closed-loop feedback correction, it has good robustness, without relying on an accurate prediction model, this paper selects the linear 2-DOF vehicle model with additional front-wheel steering angle and additional yaw moment as the prediction model, its state space expression is as follows:

(9)

Where,

In which,δdis the front wheel angle input by the driver,δais the additional front wheel steering angle and ΔMis the additional yaw moment.

To better limit the side slip of the vehicle and maintain the consistency of the driving direction, the ideal sideslip angle is set to 0[11]. Considering the limitation of road adhesion, a modification was made to obtain the ideal yaw rate[12].

βd=0

(10)

(11)

(12)

Where:βdindicates ideal sideslip angle of centroid;ωrdindicates the ideal yaw rate;gindicates the gravity acceleration;Kindicates the stability factor;μindicates the road adhesion coefficient.

3.2.1 Prediction equation

Model predictive control usually takes the current state of the controlled system as the initial condition and predicts the future state of the controlled system based on the predictive model. If the discrete state-space model is changed to incremental form, we get:

(13)

Predict the future dynamics of the system according to equation (13), the prediction time domain is set top, the control time domain isq, andq≤p, in this paper,p=10,q=2. If the control quantity does not change outside the control time domain, then Δu(k+i)=0,i≥m.x(k) andy(k) at currentktime are known conditions, and the state in the time domain fromk+1 tok+pcan be predicted with equation (13). The predicted output vector sequence of the futurepstep of the system is expressed as follows:

Yp(k+1|k)=SxΔx(k)+Γy(k)+SuΔU(k)

(14)

3.2.2 Constrained optimization problem

The ultimate purpose of this paper is to hope that the controller output is close to the reference output and at the same time do not want the control action to change too much. The objective function is selected as:

(15)

Where:R(k+1) is the expected output state in the prediction time domain, i.e. the expected values of yaw rate and sideslip angle of the centroid.τyis the weighted coefficient matrix of predicted controlled output deviation,τuis the weighted coefficient matrix of controlled output increment.

To ensure the optimization of control performance, according to the actual situation, the constraint conditions of the actuator must be taken into consideration, and constrain the control amount and control increment of the additional front-wheel steering angle and additional yaw moment:

umin(k+i)≤u(k+i)≤umax(k+i)i=0,1,…,q-1

(16)

Δumin(k+i)≤Δu(k+i)≤Δumax(k+i)

i=0,1,…,q-1

(17)

3.2.3 Constrained optimization problem solution

The model predictive control optimization problem with constraints is also a quadratic programming problem. According to equation 15～17, it is transformed into a standard quadratic problem for solution, we get:

(18)

Solve the optimization problem in each sampling cycle, finally, the optimal control input increment sequence in the control time domain will be obtained:

(19)

Then, the first item of its optimal control input increment sequence is applied to the controlled system, obtain the additional front-wheel angle and additional yaw moment required for the current time control.

(20)

Repeat the above optimization process at the next sampling time.

3.3 Torque optimal distribution algorithm

Torque distribution belongs to the bottom layer of vehicle stability control. Its function is to coordinate control multiple actuators and convert the generalized force into the output torque of each actuator. The advantage of in-wheel motor drive is that each wheel is independently controllable and its response is fast. But the disadvantage is that the peak power of the existing in-wheel motor at high speed is limited, and it offend cannot meet the torque requirements of stability control under extreme conditions [3]. In combined with a hydraulic braking system, it can provide a larger longitudinal force, and design the torque distribution controller. As shown in Fig.4, it is a vehicle stress analysis diagram only considering the longitudinal force of tires.

Fig.4 Vehicle force analysis diagram

The total longitudinal force and yaw moment of the vehicle are:

(21)

Where:Fxdis the required longitudinal resultant force;ΔMis the required additional yaw moment.

When the front wheel angle is small, cosδ≈1, i.e. the influence of the front wheel angle on the longitudinal force and the yaw moment is ignored, the above equation is written in the form of a matrix:

v=Bu

(22)

In which:

If the minimum square sum of tire utilization ratio of each wheel is taken as the optimal control objective, it is actually equivalent to the distribution of tire force according to the principle of “the greater the tire capacity, the greater the role”, so as to realize the greater stability margin of vehicles under different working condition[13]. The optimization objective function is as follows:

(23)

Where:μiis the road adhesion coefficient, which can be obtained through identification;Fziis the vertical load.

Because the lateral force of the wheel can’t be directly controlled in engineering, the longitudinal force of the wheel is mainly controlled here, so the optimization objective function is simplified as:

(24)

If the motor is used as the actuator only, the torque provided by the motor is restricted by the external characteristics of the motor:

-Tmax/r≤Fxi≤Tmax/r

(25)

If the motor and the hydraulic braking system are used as actuators at the same time, the insufficient executive ability of the motor can be compensated by the hydraulic braking system, and the optimization objective function can be expressed as follows:

(26)

Taking the minimum square sum of the tire utilization rate of each wheel as the optimal control objective is equivalent to the optimization of the mean square value of each tire load rate, which can improve the stability margin of the whole vehicle by reducing the load rate of the whole vehicle, but the load of each wheel cannot be guaranteed, which may lead to significant differences in the wear degree of each tire, thus affecting the stability of the vehicle.

If the weighted minimization of the variance and mean value of the square of the load rate of each wheel is taken as the optimal control objective, it can ensure that the adhesion capacity of each wheel can be used uniformly, at the same time, it can reduce the load rate of the whole vehicle, improve the stability margin of the whole vehicle, and thus improve the operating stability of the vehicle. The optimal objective function is as follows:

(27)

Where:εvis the weighted coefficient of variance and mean value.

Ground adhesion and actuator constraints are the same as above. This is a typical nonlinear quadratic programming problem with constraints, which can be solved by using the fmincon function in the optimization toolbox.

4 Simulation analysis

4.1 Test condition

In order to verify the effectiveness of the control system, the steering wheel single cycle sine input test and the steering wheel Sine-with dwell input tests are carried out. The test conditions are shown in Table 1.

Table 1 Simulation test conditions

4.2 Test schemes

Case 1: no yaw stability control.

Case 2: the control method of moment average distribution.

Case 3: the torque optimal distribution method which takes the minimum square sum of tire load rate as the control objective.

Case 4: the torque optimal distribution method which takes the least weighted variance and mean value of the square of tire load rate as the control objective.

4.3 Single cycle sine input experiment of the steering

wheel

From Fig.5 (a) -(d) we can see, when the control is not applied, the responses of the vehicle’s yaw rate, sideslip angle, lateral acceleration, and driving track have greatly deviated from the desired state, and the vehicle appears instability. There are a few oscillations in the three different controlled methods when they follow the desired states of yaw rate, sideslip angle of, lateral acceleration and the driving track, but they can well follow the desired state eventually, which shows that the control can improve the driving stability of the vehicle.

From Fig.5 (e) we can see that when no control is applied, the phase plane diagram finally diverges and the vehicle will be in an unstable state. When the control is applied, the phase plane diagram is convergent, it is a stable process, but the convergence region of the test scheme 4 is the smallest, which can better maintain the stable state of the vehicle.

From Fig.5 (f) we can see that in the integrated control of active front wheel angle and direct yaw moment, there is no limit value of the wheel, and the final effective front-wheel angles under different control methods are different, and the additional yaw moment required to control vehicle stability is reduced.

Fig.5 The steering wheel single cycle sine input experiment

From Table 2 we can see that the mean value of tire load rate of the experimental scheme 3 is the smallest, which is 0.058 4, indicating that the highest average adhesion margin of the whole vehicle, and the vehicle’s anti-stability performance is higher. The tire maximum load rate and load rate variance in test scheme 4 is the smallest, which are 0.391 2 and 9.14×10-7, respectively, which can ensure that the uniform usage of adhesion capacity of each wheel and reduce the load rate of the whole vehicle, and best expand the limit of vehicle stability.

Table 2 Statistical analysis of load rate

4.4 Sine-with dwell input test of the steering wheel

From Fig.6 (a) and Fig.6 (b) we can see that when the control is not applied, there is a large deviation between the response of the vehicle’s yaw rate and the sideslip angle and the expected status. Although it took a long time for both of them to reach a stable state, the vehicle does not lose stability at last. Compared with no control, three different control methods can track the expected state of the yaw rate and the sideslip angle, which shows that the vehicle can follow the driver’s driving intention well and improve the driving stability of the vehicle.

From Fig.6 (c) and Fig.6 (d) we can see that the maximum lateral acceleration of the vehicle without control is 0.8g. At this time, the tire is in a strong non-linear state, even slight disturbance will make the vehicle extremely prone to instability. However, the maximum lateral acceleration of the vehicle under different control modes is only about 0.6g, and the lateral acceleration is effectively suppressed. After steering the steering wheel 1.07 s, the maximum lateral displacement of the vehicle without control is 8.473 m, and the maximum lateral displacement of the vehicle with different control modes is about 2.62 m, which shows that the controlled vehicle has a better lateral response.

From Fig.6 (e) we can see that the phase diagrams of all four test schemes converge and are all a stable process. However, the convergence region of the vehicle without control is the largest, and the possibility of losing stability is also the highest, while the convergence region of the test scheme 4 is the smallest, and it has a higher ability to maintain the stability of the vehicle.

From Fig.6 (f), (g) and (h) we can see that due to large lateral acceleration of the vehicle without control, in order to avoid vehicle instability, the additional front-wheel angle is the largest. At the same time, the limit value of the hub motor appears in the vehicles with three different control methods. In order to ensure that the vehicle can generate the additional yaw moment demand for stability control under the limit condition, the hydraulic braking system will participate in the work for braking force compensation.

Fig.6 The steering wheel Sine-with dwell input experiment

From Table 3 we can see that the mean value of tire load rate in test scheme 3 is the smallest, which is 0.047 6, indicating the highest average adhesion margin of the whole vehicle, and higher anti instability performance of the vehicle. The tire maximum load rate and load rate variance in test scheme 4 is the smallest, which are 0.3 and 2.12×10-6, respectively, which can ensure that the uniform usage of adhesion capacity of each wheel and reduce the load rate of the whole vehicle, and best expand the limit of vehicle stability.

Table 3 Statistical analysis of load rate

5 Conclusion

The vehicle is a highly nonlinear system. If no control is applied under extreme conditions, the responses of the yaw rate and the sideslip angle deviate greatly from the expected state, which is prone to instability.

Taking the distributed drive electric vehicle with an active front-wheel steering system is taken as the research object, the paper designs the vehicle yaw stability control system based on active front-wheel steering and direct yaw moment control. The control system adopts a layered control structure. The upper controller uses the model predictive control theory to get the additional yaw moment and additional front-wheel angle. The lower controller directly acts the additional front-wheel angle on the active front-wheel steering system. At the same time, with consideration to the condition of the coupling of actuator restraint and tire force, the optimal distribution algorithm of tire load rate index is adopted, to distribute the yaw moment to the actuator motor and hydraulic brake system of the four wheels.

We used Matlab/Simulink to set up the coupling simulation platform of distributed drive electric vehicle model and yaw stability integrated controller. Through the simulation experiments under different driving conditions, the yaw stability integrated control system was analysed and evaluated. The simulation test results show that the optimal controlled method aiming at the weighted minimization of the variance and mean value of the square of the load rate can effectively improve the yaw stability of the distributed drive electric vehicle.