The sideslip angle plays an extremely important role in vehicle stability control, but the sideslip angle in production car cannot be obtained from sensor directly in consideration of the cost of the sensor; it is essential to estimate the sideslip angle indirectly by means of other vehicle motion parameters; therefore, an estimation algorithm with realtime performance and accuracy is critical. Traditional estimation method based on Kalman filter algorithm is correct in vehicle linear control area; however, on low adhesion road, vehicles have obvious nonlinear characteristics. In this paper, extended Kalman filtering algorithm had been put forward in consideration of the nonlinear characteristic of the tire and was verified by the Carsim and Simulink joint simulation, such as the simulation on the wet cement road and the ice and snow road with double lane change. To test and verify the effect of extended Kalman filtering estimation algorithm, the real vehicle test was carried out on the limit test field. The experimental results show that the accuracy of vehicle sideslip angle acquired by extended Kalman filtering algorithm is obviously higher than that acquired by Kalman filtering in the area of the nonlinearity.
In the recent years, with the improvement of the road conditions and the progress of automobile technology, the speed of the car is also accelerating; however, the proportion of traffic accidents caused by the instability of vehicle is also increasing [
In the low adhesive road, the vehicle is easy to sideslip because of the tire lateral force’s saturation; although the yaw rate remains in a small range, the vehicle has lost the balance [
Kalman filtering method is a linear system estimation method. The process can be divided into two parts [
(i) Time updating:
(ii) Measurement updating:
In this formula,
Kalman filtering algorithm has low precision for some systems with strong nonlinearity. It generally uses extended Kalman filter (EKF) for the state estimation of nonlinear systems such as automobiles. EKF transfers nonlinear model into linear model, so that Kalman filter can be used for nonlinear systems.
The processes of EKF are as shown in Figure
EKF algorithm flowchart.
Usually, the state space equation and the observation equation can be expressed by the following formula:
In the nonlinear state equation and observation equation ((
The state update process of EKF is as follows:
The measurement update process of EKF is as follows:
In the formula,
In the formula,
The extended Kalman filtering estimation algorithm is based on the 2degreeoffreedom vehicle dynamics model (shown in Figure
2dof vehicle dynamic model.
According to the 2degreeoffreedom vehicle model
establish the following formula:
(1) Establish the nonlinear state equation as
Measurement equation:
Put
(2) The linear model (take Jacobian matrix for state equation and measurement equation) is as follows:
Put the discretization of the equation of state “
Parameters and meaning.
Parameter  Meaning 


The vehicle centroid to the front axle 

Vehicle quality 

The longitudinal velocity 

Sideslip angle 

Yaw rate 

Tire longitudinal force 

Vertical force of tire 

The vehicle centroid to the rear axle 

Moment of inertia of vehicle around the lead vertical shaft 

The front wheel sideslip angle 

The rear wheel sideslip angle 

Tire rotation 

Tire lateral force 

Wheelbase 
In order to verify the effect of extended Kalman filtering algorithm sideslip angle estimation, we carried out simulation verification to a car. The simulation adopts a typical double lane condition, in order to verify the accuracy of the extended Kalman filtering sideslip angle estimation and make the vehicle lateral acceleration as fast as possible, so as to make the vehicle into the nonlinear control area.
The main interface of Carsim includes four parts: vehicle characteristics interface, test environment interface, simulation solver interface, and 3D animation and experimental curve drawing interface. The relationship between the four parts and the main interface of Carsim is shown in Figure
Parameters of vehicle settings in Carsim.
Variable symbol  Variable name (unit)  Variable value 


Vehicle quality (kg)  1247 

The vehicle centroid to the front axle (m)  1.016 

The vehicle centroid to the rear axle (m)  1.562 

Moment of inertia about the 
1523 
The main interface and constitution of Carsim.
Take the double lane change simulation of 95 km/h on the simulation of ground with adhesion coefficient of 0.45 and use the extended Kalman filtering technology to recognize the sideslip angle. The result is shown in Figures
Vehicle status (
Lateral acceleration diagram
Yaw rate
Steering wheel angle
The contrast of sideslip angle simulation value and estimated value (
Note from Figures
Note from Figure
When the vehicle enters strongly nonlinear region, in order to verify the effect of the extended Kalman filtering algorithm to estimate the sideslip angle, then take the sideslip angle estimation simulation on low attachment snow ground (the ground adhesion coefficient is 0.2).
Take the double lane change simulation of 55 km/h on the snow ground with round adhesion coefficient of 0.2 and then use the extended Kalman filtering algorithm to identify the sideslip angle. The result is shown in Figures
Vehicle state (
Lateral acceleration diagram
Yaw rate
Steering wheel angle
The contrast of sideslip angle simulation value and estimated value (
In the snow and other low adhesion roads, the vehicle easily enters the nonlinear region. From Figure
In the simulation test, the extended Kalman filtering algorithm has been verified. This section uses the real vehicle test data to estimate the sideslip angle and real vehicle test data to estimate the sideslip angle. Usually, when the vehicle goes into the nonlinear region, the sideslip angle estimation will have a large deviation from the actual value. In order to enter the nonlinear state, we take the double lane change test in vehicle limit test site with low adhesion road in Heihe City. It is easy to get the vehicle state data of nonlinear region.
Figure
Arrangement of sideslip angle measurement system based on GPS.
Double lane change test site.
According to ISO 38881: 1999 to implement the double lane change test, pass the double lane change path with the speed of 67 km/h (±5 km/h). At the same time, record the data in computer through Kvaser and write the time history curve of each measurement variable such as the steering wheel angle, yaw velocity, lateral acceleration, longitudinal velocity, the sideslip angle, and the steering rod force information.
Figure
Vehicle state parameters under line condition.
Steering wheel angle
Lateral acceleration
Yaw rate
Longitudinal speed
Sideslip angle measurement and filtering value
Figure
Sideslip angle identification values compared with the experimental value under line condition.
This paper discusses Kalman filtering and extended Kalman filtering, the two kinds of algorithms of the sideslip angle estimation, and analyzes their characteristics. This paper presents the simulation of the two algorithms on the low adhesion road and carries out the low adhesion road double lane change test. The experimental results show that when the vehicle dynamics enter the nonlinear region, the accuracy of the sideslip angle estimation is higher than the accuracy of Kalman filtering algorithm. We will improve the algorithm to restrain the filtering divergence in the following study.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors acknowledge the support from Natural Science Foundation of Shandong Province (ZR2016EEQ06).