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Four-wheel independent drive electric vehicle was used as the research object to discuss the lateral stability control algorithm, thus improving vehicle stability under limit conditions. After establishing hierarchical integrated control structure, we designed the yaw moment decision controller based on model predictive control (MPC) theory. Meanwhile, the wheel torque was assigned by minimizing the sum of consumption rates of adhesion coefficients of four tires according to the tire friction ellipse theory. The integrated simulation platform of Carsim and Simulink was established for simulation verification of yaw/rollover stability control algorithm. Then, we finished road experiment verification of real vehicle by integrated control algorithm. The result showed that this control method can achieve the expectation of effective vehicle tracking, significantly improving the lateral stability of vehicle.

With independent controllable drive/brake moment, rapid moment response, and measurable torque and speed, four-wheel independent drive electric vehicle has advantages in improving vehicle stability. When the vehicle has the risk of instability or has lost stability, it is difficult for the ground to maintain vehicle tracking and attitude adjustment with sufficient lateral force. Meanwhile, the vehicle based on active steering has little effect on driving direction control [

Domestic and foreign scholars have made lots of researches on yaw stability control based on distributed structure electric vehicle. The developed control strategy can be divided into yaw moment decision and wheel torque distribution control layers. References [

For the vehicles with high centroid, when they run on the high adhesion road with high speed, the emergency avoidance operation can easily cause the risk of rollover because of large lateral acceleration. The rollover stability control (RSC) is as important as the yaw stability control (YSC). Fujimto and Hori’s team of University of Tokyo have done a lot of related research on their developed small test vehicle. In [

In the work, MPC-based four-wheel braking/driving torque coordinated control strategy was proposed to improve the driving stability of four-wheel independent electric drive off-road vehicle. According to MPC system structure, we established predictive control-oriented vehicle dynamics model. After that, the work improved vehicle yaw and rollover stability under limit conditions by taking driver’s input as the reference and road adhesion as constraint. Vehicle braking/driving force was allocated by MPC to minimize rollover index LTR, the deviations of yaw rate and side slip angle. Then, we used Carsim and Simulink joint simulation platform including professional vehicle and road models for yaw stability control under low adhesion coefficient and double lane-change conditions as well as rollover stability control algorithm validation under high adhesion coefficient and hook conditions. At last, real vehicle test verification was performed under high speed double lane-change condition.

The control goal of lateral stability control strategy enables the vehicle to run in the control of the driver without the risk of rollover, achieving convenient operation and stable driving. The lateral stability of the vehicle includes both the yaw and the roll stabilities. Scholars take linear yaw characteristic deduced from two-degree-of-freedom vehicle model as the ideal characteristic of the vehicle. In the work, we also select it as the control target of ideal yaw stability of the vehicle. For rollover stability, it is impossible to obtain the ideal rollover characteristic by establishing a simplified vehicle model. The current rollover condition of the vehicle is characterized only by selecting the appropriate rollover index. Besides, a suitable predicted value is selected to measure the possibility of rollover. In the rollover stability control, the expected rollover possibility should be as small as possible.

The yaw stability is characterized by state variables including side slip angle and yaw rate. Therefore, we determine the yaw stability of vehicle, and whether to carry out yaw stability control according to the two state variables.

When the side slip angle of vehicle satisfies the following equation, the vehicle is in a stable zone; otherwise, it is in an unstable zone, and the stability control must be added.

According to (

Vehicle stability judgment phase diagram based on the side slip angle.

If the side slip angle and yaw rate of the vehicle satisfy the following equation, then the vehicle is in a stable state; otherwise, the vehicle is in an unstable state.

In order to prevent the control system from frequent operation, the yaw stability control module is activated only when the deviation reaches a certain value. The control threshold is subtracted to determine effective deviation of additional yaw moment.

When the off-road vehicle with high centroid runs in the high adhesion road with high speed, the emergency avoidance operation can easily cause the risk of rollover because of large lateral acceleration. The tire load transfer rate is the most direct indicator to describe vehicle rollover condition, which can be expressed by

The vehicle rollover state identified by the LTR value reflects the current vehicle state. For the driver, the vehicle rollover is possible to happen before the controller identifies the wheel off the ground by calculating LTR = 1. In the vehicle rollover stability analysis, it is important to predict the rollover state in the coming time.

The system warns the driver in advance to take an efficient operation within sufficient time or enables the rollover control system to take appropriate action by a command, effectively improving the rollover stability of vehicle [

In 2001, Bo-Chiuan Chen from University of Michigan firstly proposed the concept of Time-To-Rollover (TTR), which characterizes the time interval from the current moment to the rollover time [

TTR rollover warning algorithm.

Wherein, the rollover prediction model takes the current vehicle speed, the steering wheel angle, the additional yaw moment calculated at the previous time, and the vehicle sensor measurement value or the vehicle state estimation value as input. During the prediction process, it is assumed that the vehicle speed, steering wheel angle, and additional yaw moment remain unchanged.

The problem of vehicle yaw stability can occur in the road surface with any adhesion condition, and the rollover stability problem mainly occurs in the high adhesion road. Therefore, the yaw stability control takes effect in low adhesion road. In the high adhesion road, both yaw and rollover stability controls are possibly triggered synchronously or asynchronously.

The integrated controller determines the total driving/braking torque demand based on the throttle/brake pedal opening or the difference between the target and actual vehicle speed. YSC yaw moment decision module calculates the required yaw moment ensuring vehicle yaw stability by taking the difference between the actual and ideal values of centroid slip/yaw angle velocity as the input. RSC yaw moment decision module calculates the required yaw moment for rollover stabilization control by taking the LTR value output by the rollover prediction system as the input. The integrated control module of yaw/rollover stability finally determines the total yaw moment based on yaw and rollover state. When the sum of four-wheel target driving/braking torque is equal to the total demand torque, then the total demand torque is assigned to each wheel according to a certain distribution rule. Thus, the longitudinal forces of wheels produce the desired yaw moments. Wherein, the model prediction theory is used for YSC and RSC yaw moment decision modules.

With little computation and good real-time control, the linear two-DOF vehicle model can better describe the driver’s driving intention. The two degrees of freedom contain lateral and yaw directions. Figure

Linear two-DOF vehicle model.

In Figure

The differential equations of motion in lateral and yaw directions are expressed as follows:

It is assumed that the tire of linear two-DOF vehicle model is linear. When the vehicle is in the limit state, the steady-state response of reference model is not suitable as a reference value. Therefore, the reference value is replaced by boundary value [

The boundary values of the side slip angle are as follows:

The boundary values of the yaw rate are as follows:

In Figure

Three-DOF vehicle model.

In general, the more accurate prediction model leads to better control effect of the controller. However, the precise model weakens the real-time performance of controller. At present, most controllers in real vehicle use two or three-DOF model [

According to

The two-stage fold line tire model is used for vehicle prediction (see Figure

Two-stage fold line tire model.

MPC belongs to a discrete control. The state equation of (

First, (

Let

According to the MPC theory, the most recent measured value is taken as the initial condition to predict the future dynamic based on the predictive model. We make the following assumptions in the work:

Within predicted time domain

Similarly, we can deduce the

MPC is an algorithm of determining control strategy by optimization. Firstly, it is necessary to determine

In the control process, the control increment should not have sharp change. Based on this factor, soft constraints are added to optimization performance indexes. Thus, the optimization performance index of moment

Taking

At the moment

The Lyapunov function

For the tire, lower pavement adhesion consumption rate leads to greater adhesion margin and larger distance between the tire and nonlinear saturation zone, indirectly improving the stability of the vehicle. According to tire friction ellipse theory, the work distributes the wheel torque for the purpose of minimum sum of pavement adhesion consumption rates of four tires. Equation (

The vertical load

The objective function in (

When the yaw moment control is not required, the sum of longitudinal forces of four wheels satisfies the total moment demand, which is obtained by analyzing the pedal input. Therefore, constraint equation is expressed as follows:

The nonlinear constrained optimization problem is solved by numerical optimization method to obtain the longitudinal forces for four tires. Then, we derive the drive torques which should be allocated to four wheels by calculation.

The proposed controller was simulated on a 4WID vehicle in the Carsim and MATLAB environment. The vehicle under investigation is the light-duty off-road vehicle with the specifications which are listed in Table

Specifications of the experimental electric vehicle.

Parameter | Value |
---|---|

Mass | 3565 kg |

Yaw moment of inertia | 7016 kg⋅m^{2} |

Distance from CG to front axle | 1.593 m |

Distance from CG to rear axle | 1.707 m |

Tire radius | 0.435 m |

Maximum motor torque | 250 Nm |

Maximum motor power | 55.5 kW |

Maximum motor speed | 6000 rpm |

Planetary reducer ratio | 9.15 |

The proposed controller (MPC) in this paper and its results are compared with the proposed controller (DOB) by [

When the steering wheel reaches certain rotor angle, the vehicle will be in danger of yaw instability on low adhesion road. The effect of control algorithm on vehicle stability is verified by selecting the double lane condition on low adhesion road. The specific simulation conditions are as follows: the road adhesion coefficient is 0.3; the driving speed is 60 km/h. Figure

Double lane-change simulation test on low adhesion road.

Vehicle travel path

Steering wheel angle input

Yaw rate

Side slip angle

Lateral acceleration

Wheel torque

From Figure

The hook condition is conducted on the road with an adhesion coefficient of 0.85. The initial speed is 80 km/h, and the maximum steering angle is 180°. Figure

Hook condition simulation test on high adhesion road.

Steering wheel angle input

Roll

Roll rate

Yaw rate

Side slip angle

Longitudinal velocity

Vehicle travel path

Wheel torque

In hook condition, the vehicle will lose stability and roll over if there is no stability control. After MPC-based or DOB-based stability control, the vehicle does not roll over as the roll is in a reasonable range. It can be seen that the response curves of MPC-based controller are as smooth as DOB-based controller. In addition, it can also be concluded that the stability will achieve a litter better accuracy under the proposed MPC-based stability controller and keep a higher longitudinal velocity. Figure

According to ISO/FDIS 3888-1 standard, we performed double lane-change field test of light-duty off-road vehicle. The piles were arranged on the test field (see Figure

Double lane-change test.

The gyroscope was installed in the vehicle for real-time acquisition of the yaw rate and lateral acceleration of the vehicle. The GPS speed sensor was used to measure the longitudinal and lateral velocities. The angular velocities of four wheels were collected by in-wheel motor speed sensor. We measured the steering wheel angle by steering wheel angle sensor. The test speed of vehicle was 80 km/h (±2 km/h).

Figures

Double lane-change test results with MPC-based stability control.

Steering wheel angle

Yaw rate

Double lane-change test results with DOB-based stability control.

Steering wheel angle

Yaw rate

Double lane-change test results without stability control.

Steering wheel angle

Yaw rate

Double lane-change driving route.

In the work, we proposed the lateral stability control method of in-wheel motor drive off-road vehicle based on MPC. The multiinput and multioutput system predictive control law was designed by establishing vehicle dynamics model for predictive control. The motor braking/driving torque was directly generated to conduct yaw stability control under low adhesion coefficient and double lane-change conditions as well as rollover stability control algorithm validation under high adhesion coefficient and hook conditions. At last, real vehicle test verification was performed under high speed double lane-change condition. This paper only carries out the test under the condition of invariable road surface and constant velocity but does not verify the proposed MPC-based stability control under the condition of changing road surface and time-varying velocity. Subsequent further research, we will verify it.

Consequently, the following conclusions were obtained.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors acknowledge that this paper has been supported by the National High Technology Research and Development Program of China (Grant no. 2011AA11A260).