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Vehicle states estimation (e.g., vehicle sideslip angle and tire force) is a key factor for vehicle stability control. However, the accurate values of these parameters could not be obtained directly. In this paper, an interacting multiple model-cubature Kalman filter (IMM-CKF) is used to estimate the vehicle state parameters. And improvements about estimation method are achieved in this paper. Firstly, the accuracy of the reference model is improved by building two different models: one is 7-degree-of-freedom (7 DOF) vehicle model with linear tire model, and the other is 7 DOF vehicle model with nonlinear Dugoff tire model. Secondly, the different models are switched by IMM-CKF to match different driving condition. Thirdly, the lateral acceleration correction for sideslip angle estimation is considered, because the sensor of lateral acceleration is easy to be influenced by the gravity on banked road. Then, to compare cubature Kalman filter (CKF) estimation method and IMM-CKF estimation method Hardware-In-Loop (HIL) tests are carried out in the paper. And simulation results show that IMM-CKF methodology can provide accurate estimation values of vehicle states parameters.

With the development of the electronic and automotive technology, the vehicle stability control is also undergoing continuous progress. Vehicle stability control plays an important role in active vehicle safety control. The performance of vehicle stability control is determined by the accuracy of vehicle state (e.g., vehicle sideslip angle and tire force) because the vehicle state parameters show the potential of vehicle stability. However, the vehicle state parameters, such as the sideslip angle and tire force, cannot be measured directly for both technical and economic reasons. Therefore the vehicle state parameter must be estimated by a variety of algorithm methods.

Recently, many estimation approaches based on standard ESP sensors (e.g., steering angle, yaw rate, longitudinal and lateral acceleration, and wheel-speed) have been proposed in the literature to estimate vehicle state parameters [

Lateral force curve.

In this paper, we use the interacting multiple model (IMM) to choose an optimal model for various driving conditions. In the IMM method, two filters are used in parallel to estimate sideslip angle and tire force. One filter is CKF filter based on four-wheel nonlinear vehicle dynamics model with linear tire model for normal driving conditions and the other is based on four-wheel nonlinear vehicle dynamics model with nonlinear Dugoff tire model for extreme driving conditions. The sideslip angle and tire force predicted by the variable structure IMM-CKF filter are more accurate than the single filter for various driving conditions.

This paper is organized as follows. In Section

In Figure

7-DOF vehicle model.

The equation of longitudinal motion is expressed as follows:

When the vehicle is turning, the longitudinal acceleration and the later acceleration will have an effect on each wheel because of the load transfer. Therefore, the vertical load of 4 wheels can be expressed as follows:

As discussed in Section

The linear tire model proposed by the authors [

The equation of longitudinal tire force can be expressed as

Under the extreme driving conditions, such as a low friction road surface, the tire is easy to enter into nonlinear state even at low lateral acceleration. Thus, the Dugoff nonlinear tire model is used to represent the tire forces in nonlinear region [

The equations of lateral tire force and the longitudinal tire forces can be expressed as

Considering the time lag of tire force, Reference [

The algorithm structure of vehicle state estimation proposed in this paper is shown in Figure

Scheme of the IMM-CKF method for vehicle state estimation.

In the second layer, the IMM-based estimation layer calculates the model switch probabilities and integrates the CKF estimation of each model by stochastic process to adapt to various driving conditions. The further algorithm about IMM-CKF can be found in Reference [

It is appropriate that the filter of 7-DOF vehicle model based on linear tire model is used under the normal work conditions because the relationship of tire slip angle and lateral force is linear and there is a small amount of computations for embedded system. In contrast, the relationship of the tire slip angle and the lateral force is no longer linear under extreme driving conditions. The filter of 7-DOF vehicle model based on nonlinear tire model can provide better prediction performance. The process of IMM-CKF can be described in Figure

The process of IMM-CKF.

Compute the mixed probabilities and the initial condition in first step. Initial mean and covariance for each CKF filter model can be expressed as follows:

In this step, the CKF is used to obtain each model state

(i) Time Update

HIL simulation platform.

In this step, the likelihood function of each mode can be expressed as

After calculation of each mode's probabilities, the vehicle state parameters prediction and the covariance can be calculated according to Gaussian mixture equation. The equation of vehicle state parameters prediction and its covariance can be, respectively, expressed as

The sensor of lateral acceleration is a sort of inertial sensors which is easily affected by the gravity when the axis of chassis is not horizontal. Therefore, the outputs of lateral acceleration sensor need to be corrected. Similarly, road adhesion coefficient and cornering stiffness correction are very important, so they are introduced separately in this section.

In case of vehicle driving on the flat and level road, the measured results of lateral acceleration are precise. But in case of driving on the slope road, the measured results tend to become higher than those on the flat and level road. Because the lateral acceleration sensor is easy to be influenced by the roll angle of the vehicle, therefore, it is necessary to correct the measured results before being used in the vehicle state estimation. The equation of correct lateral acceleration can be written as

The equation of roll angle can be expressed as

Road friction plays an important role in vehicle state estimation. However, it is difficult to acquire its value directly. According to the estimation method proposed in Reference [

According to Reference [

In order to validate the proposed estimation method, the simulations were carried out. The simulations are based on the Hardware-In-Loop (HIL) and the simulation platform is shown in Figure

Parameters for simulation.

| | | |
---|---|---|---|

| 1.4 | m | Distance c.g. to front axle |

| 1.4 | m | Distance c.g. to rear axle |

| 110000 | N/rad | Front cornering stiffness |

| 95000 | N/rad | Rear cornering stiffness |

| 3500 | Kg/m^{2} | Yaw moment of inertia |

| 1.5 | M | Track width |

| 2000 | Kg | Vehicle mass |

| 1.3 | Kg/m^{2} | Wheel inertia |

R | 0.28 | M | Wheel radius |

The simulation scheme based on HIL.

Two kinds of simulation tests are adopted to validate the estimation method. The first one is double-lane change (DLC) maneuvers on banked 5% road to validate the method of lateral acceleration correction proposed in Section

Figures

Steer angle of steering wheel

The lateral acceleration of vehicle

The estimated sideslip angle on banked 5% road.

Figure

The estimated sideslip angle on banked 10% road.

In this section, as shown in Figure

The simulation model for the sideslip angle estimation.

The yaw rate of vehicle

The lateral acceleration of vehicle

The estimated sideslip angle on flat level road.

Front right lateral tire force

Front left lateral tire force

Front right longitudinal tire force

Front left longitudinal tire force

As shown in Figure

In order to evaluate the correction of cornering stiffness, the steering input of DLC manoeuver on banked 10% road is generated. The simulation is done on ice road (road friction

Estimated sideslip angle with and without cornering stiffness correction

Front cornering stiffness

Rear cornering stiffness

For the sake of the various driving conditions, two vehicle models were built for estimation. The 7-degree-of-freedom (7 DOF) vehicle model with linear tire model was built for normal driving condition, and the other is 7DOF vehicle model with nonlinear Dugoff tire model for the extreme driving condition. Two models were switched by IMM-CKF to match the different driving condition.

To eliminate the estimation error caused by lateral acceleration sensor in case of vehicle driving on the slope road, the lateral acceleration correction for sideslip angle estimation is considered. The overall results from HIL verify that the proposed method can realize accurate estimation about vehicle state parameters in a wide range of road conditions. Meanwhile, the method is only applied on the mild banked road, and the application of the proposed approach under higher banking angle road would be concerned in the future research.

The data used to support the findings of this study are available from the corresponding author upon request.

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

This work is supported by Henan Science and Technology Project (192102210063; 182102210034).