Approach to estimation of vehicle-road longitudinal friction coefficient

Song Xiang Li Xu Zhang Weigong Chen Wei Xu Qimin

(School of Instrument Science and Engineering, Southeast University, Nanjing 210096, China)

With the implementation of active safety control systems, vehicles have become safer to drive with less involvement in fatal accidents. These active safety control systems can greatly profit from being made road-adaptive; i.e., the control algorithms can be modified to account for the external road conditions if the actual tire-road friction coefficient information is available in real time. The longitudinal tire-road friction coefficient is an essential parameter for the vehicle longitudinal active safety control systems. For example, in an adaptive cruise control (ACC) system, road condition information from the friction coefficient estimation can be used to adjust the longitudinal spacing headway from the preceding vehicle that the ACC vehicle should maintain.

The tire-road friction coefficient must be estimated in real-time to meet the requirements of the vehicle longitudinal active safety control systems under normal driving conditions. So the method of tire-road friction coefficient estimation based on vehicle longitudinal dynamics is most feasible.

The relationship between the normalized longitudinal tire force and the slip ratio is different under different road conditions, which is the basis of utilizing the vehicle longitudinal dynamics to estimate the tire-road friction coefficient[1]. The most well known research in this area is on the use of slip-slope for friction coefficient identification[2-5]. In this method, the normalized longitudinal force is considered proportional to the slip ratio at low slip ratios. The slope of the relationship between the normalized longitudinal force and the slip ratio at low slip ratios is called slip-slope. The basic idea behind the use of slip-slope for friction coefficient estimation is that at low slip ratios, the tire-road friction coefficient is proportional to slip-slope. Thus, by estimating slip-slope, the tire-road friction coefficient can be estimated. But this method is only suitable for the condition of low slip ratios. The parameter estimation method is another commonly used method[6-7].But only at the large slip ratios, the estimation results will be close to the true value. Domestic researches[8-9]are based on the above two methods,the drawbacks as mentioned above also exist. Shim et al.[10]assumed a tire-road friction coefficient, and then the response of the vehicle is estimated based on the vehicle dynamics model. According to the differences between the estimated response and the actual vehicle response, the tire-road friction coefficient can be calculated. But the method is difficult to apply to complex road conditions since it requires a lot of experience.

As mentioned above, the main problem of the tire-road friction coefficient estimation algorithms is that the algorithms cannot be applied to both high and low slip ratios simultaneously. To solve this problem, the recursive least squares (RLS)method with the forgetting factor and the extended Kalman filter (EKF) algorithm are employed to estimate the longitudinal tire-road friction coefficient in this paper. The method utilizes the relationship between the normalized longitudinal tire force and the slip ratio to identify the longitudinal tire-road friction coefficientμ, which can be applicable to for both the high and the low slip ratios, and the effectiveness and feasibility are verified by simulation.

1 Proposed Method

If only the longitudinal motion is considered and the lateral force is ignored, the normalized longitudinal tire forceφand the slip ratiosat each wheel can be represented as

(1)

(2)

whereωis the angular wheel speed;ris the effective tire radius;vis the vehicle’s absolute velocity;Fxis the longitudinal force from ground to wheel; andFzis the normal force.

Fig.1 shows a typical relationship betweensandφf(shuō)or various values of the tire-road friction coefficient.μis the tire-road friction coefficient.

Fig.1 s-φ curves with different friction coefficients

In this paper, the friction coefficient is assumed to be the same at each wheel of the vehicle. By calculatingsandφ, the longitudinal tire-road friction coefficientμcan be estimated by the RLS method with the forgetting factor, which is based on the simplified magic formula tire model. Then the estimatedμand the tire model parameters are used as extended states. The EKF algorithm is employed to filter out the noise and adaptively adjust the tire model parameters. Then the final road longitudinal friction coefficient is accurately and robustly estimated. The flowchart of the estimation method is shown in Fig.2.

Fig.2 Flowchart of estimation method

2 Vehicle and Tire Models

The longitudinal vehicle dynamics model can be written as

max=Fx-Dav2-Crollmg

(3)

wheremis the mass of the vehicle;axis the vehicle longitudinal acceleration;Dais the air resistance coefficient;Crollis the rolling resistance coefficient; andgis the acceleration of gravity.

A simplified magic formula tire model[11]is adopted in this paper.

φ=μsin[Carctan(Bs)]

(4)

whereBandCare the model parameters.

3 Road Friction Coefficient Preliminarily Estimated based on RLS

3.1 Longitudinal slip ratio calculation

The effective tire radiusris calculated as

(5)

whereruis the undeformed radius of the tire;rsis the static tire radius and it can be described asrs=ru-Fz/kt,ktis the vertical tire stiffness. The longitudinal slip ratio can be calculated by Eq.(1).

3.2 Normalized longitudinal tire force calculation

Eq.(3) can be rewritten as

Fx=Fxf+Fxr=max+Dav2+Crollmg

(6)

whereFxfandFxrare the traction forces of the front and the rear wheels. The total vehicle longitudinal forceFxcan be obtained by Eq.(6).

The normal forces at the front and rear tires can be calculated as follows:

(7)

whereFzfandFzrare the normal forces at the front and the rear tires;aandbare the distances from the center of gravity to the front and the rear axles.

The relationship betweensandφf(shuō)or the front and rear tires can be written as

(8)

(9)

3.3 Preliminary estimates of μ

Assuming that the front and rear tires are under the same road surface condition, which is true for many driving situations, the total longitudinal force is

Fx=Fxf+Fxr=φf(shuō)Fzf+φrFzr=

μ{Fzfsin[Carctan(Bsf)]+Fzrsin[Carctan(Bsr)]}

(10)

Eq.(10) can be rewritten into a standard parameter identification format as

y(k)=φT(k)θ(k)+e(k)

(11)

wherekdenotes the discrete time;y(k)=Fxis the system output;θ(k)=μis the unknown parameter of interest;φ(k)={Fzfsin[Carctan(Bsf)]+Fzrsin[Carctan(Bsr)]} is the measured regression vector;e(k) is the identification error. Then the only unknown parameterθ(k)=μcan be identified in real-time using the RLS method with the forgetting factor as follows:

1) Measure the system outputy(k) and calculate the regression vectorφ(k).

2) Calculate the identification errore(k),

e(k)=y(k)-φT(k)θ(k-1)

3) Calculate the updated gain vectorK(k) as

And calculate the covariance matrixN(k)by

The parameterλis called the forgetting factor, which is used to effectively reduce the influence of old data which may no longer be relevant to the model, and, therefore, prevents a covariance wind-up problem.

4) Update the parameter estimate vectorθ(k),

θ(k)=θ(k-1)+K(k)e(k)

The road friction coefficientμcan be preliminary estimated in real-time.

4 Longitudinal Tire-Road Friction Coefficient Identification based on EKF

In the tire-road friction coefficient estimation process described above, the model parametersBandCare assumed to be known and constant. However, during vehicle operation,BandCcannot be directly measured and they are time-varying, which may affect the accuracy of the estimation of the tire-road friction coefficient. In order to real-time updateBandC, and filterμ, the EKF model is established based on the longitudinal dynamic model using Eq.(3).

The discretized state equation and measurement equation can be written as

(12)

wherekrefers to the discrete-time step; the state vectorX={v,μ,B,C}T; the measurement vectorZ={ax,v,μ}T;WandVare the system and measurement noise vectors, respectively;f(·) andh(·) are the nonlinear system and measurement functions which can be deduced from Eq.(3).

Assuming that the system and measurement noises to be Gaussian with a zero mean and their covariance matrices areQandR, respectively, the EKF process consists of the following two phases.

1) Time update:

P(k,k-1)=A(k,k-1)P(k-1)A′(k,k-1)+Q(k-1)

2) Measurement update:

K(k)=

P(k,k-1)H′(k)[H(k)P(k,k-1)H′(k)+R(k)]-1

P(k)=[I-K(k)H(k)]P(k,k-1)

whereIis the identity matrix;AandHare the Jacobian matrices of the system functionf(·) and the measurement functionh(·) with respect toX; i.e.,

The model parametersBandC, estimated by the EKF, are feedbacks to the tire model, so the estimated values by the RLS can be updated in real-time. Therefore, the estimation accuracy of the tire-road friction coefficient can be improved, and the estimated values can respond to the road state changes. Theμoutput by the EKF is the final estimation result.

5 Simulation Results and Discussion

To evaluate the performance of the proposed estimation method of the longitudinal friction coefficient, numerical simulations are performed using Carsim in Matlab/Simulink. According to Ref.[12], the initial values of model parametersBandCare 14 and 1.3, respectively. The forgetting factorλis set to be 0.995. The proposed algorithm is validated under the high and the low slip ratio conditions with the tire-road friction coefficient changing, and the estimation results are compared with the conventional slip-slope algorithm. Simulation results show that the proposed algorithm can be applied to both the high and the low slip ratios; the estimation results are accurate and robust, and they can quickly respond to the changes in road conditions.

5.1 Simulation under low slip ratio condition

The main vehicle parameters used in the simulations are:kt=230 N/mm,m=1220 kg,rs=310.8 mm,rw=304 mm,a=1.04 mm,b=1.56 mm. Fig.3 and Fig.4 are the simulation results. The figures show that the values of the slip ratio are small, and the proposed method can quickly identify the road friction coefficient with high accuracy; the error is less than 0.1. From Fig.4, we can see that the proposed method can converge to the true value within 2 s when the tire-road friction coefficient jumps, which meets the real-time requirements.

Fig.3 Simulation results of low slip ratios. (a) Slip ratio; (b) Tire-road friction coefficient

Fig.4 Simulation results of low slip ratios with friction coefficient changing. (a) Slip ratio; (b) Tire-road friction coefficient

5.2 Simulation under high slip ratio condition

The conventional slip-slope algorithm is no longer suitable for the high slip ratio condition because the relationship betweensandφis not linear. Fig.5 and Fig.6 are the simulation results. The figures show that estimation results by the slip-slope algorithm produce a great error. The proposed method can quickly identify the road friction coefficient with high accuracy at high slip ratios and quickly respond to the changes in road conditions.

6 Conclusion

Simulation results show that the proposed algorithm can quickly and accurately estimate the tire-road friction coefficient under both the high and the low slip ratio conditions, which can meet the requirements of the vehicle longitudinal active safety system. And the proposed method only needs the existing sensors in commercial vehicles, so the proposed method is suitable for on-board applications with low computational complexity.

The key of the proposed algorithm is to obtain an accurates-φcurve. Thes-φcurve can be obtained by the bench test, but the friction conditions on an actual road is different from the bench test, and the accuracy of the real-time tire-road friction coefficient is also reduced due to the high dynamic characteristics and noises. So the further work must focus on buildings-φrelationships in different roads by a lot of vehicle tests on the common road, and then the proposed method can be applied to practice and achieves mass-market applications.

Fig.5 Simulation results of high slip ratios. (a) Slip ratio; (b) Friction coefficient estimated by the proposed method; (c) Friction coefficient estimated by the slip-slope method

Fig.6 Simulation results of high slip ratios with friction coefficient changing. (a) Slip ratio; (b) Friction coefficient estimated by the proposed method; (c) Friction coefficient estimated by the slip-slope method

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