Proceedings of the 36th Chinese Control Conference July 26-28, 2017, Dalian, China
Road Friction Coefficient Estimation Based on BP Neural Network Tao Song1, Hongliang Zhou2, Haifeng Liu3 1. Automotive Electronics Engineering Center, Harbin Institute of Technology, Harbin 150001 E-mail:
[email protected] 2. Foundation and Interdisciplinary Science Research Institute, Harbin Institute of Technology, Harbin 150001 E-mail:
[email protected] 3. Automotive Electronics Engineering Center, Harbin Institute of Technology, Harbin 150001 E-mail:
[email protected] Abstract: In the development process of automotive chassis control system, it is difficult to estimate road parameters accurately. Based on vehicle force analysis and Dugoff tire model, the method of road friction coefficient estimation using BP neural network is proposed based on the motion states in the vehicle running process. Finally, the simulation of road friction coefficient estimation method is carried out a high precision vehicle dynamics simulation software veDYNA. The results show that the algorithm can estimate road friction coefficient well. Key Words: Road friction coefficient, neural network, vehicle dynamic simulation
1
method of estimating the slope of the
Introduction
With the development of automotive electronics and control technology, vehicle stability control in these years has been rapid development.
Precise control relies on accurate road parameters such as road friction coefficient [1] . Therefore, it is very important to estimate road friction coefficient to improve the performance of active safety system. In the moving process of vehicle, especially in the process of turning, accessing to road information can help make a better stability control strategy. Factors that affect friction coefficient between road and tire is divided into three categories: vehicle parameters, such as wheel positioning angle; tire parameters, such as tire material and type, tire pressure, tread depth; road parameters, such as road conditions (dry road, wet road, snow and ice road)[2] and road type (asphalt pavement, cement pavement, gravel pavement, etc.)[3] . In recent years, many scholars have done a lot of research on the estimation of road friction coefficient. Foreign researchers research earlier in the field of road friction coefficient estimation. According to the test methods and measurement parameters, the estimating methods can be divided into the following methods: instrument measuring method, tire changing measuring method, P -O curve based method and so on. Instrument measurement method is the analysis of ground reflected light using optical sensors, to determine what material is on the road, and estimates road friction coefficient [4] . This method needs additional sensors, which are high cost, and it just has good estimation accuracy for trained pavement, which has bad effect for untrained road. Germany's joint research team developed a smart tire system by sensing the deformation of the outer contour of the tire to estimate road friction coefficient [5] . This study is still in progress. Hori Y et al. of University of Tokyo put forward a *
This work is supported by National Natural Science Foundation (NNSF) of China under Grant 61304135 and U1564207.
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P -O curve
[6]
. By
estimating the slope of the P -O curve, they could judge whether the tire force is in linear region or non-linear region. However, the exact value of road friction coefficient cannot be obtained. Laura R. Rvay used the statistical theory to recognize road friction coefficient. Based on the 8-DOF vehicle dynamics model, extended Kalman filter [7] was designed to estimate the vehicle states. Based on Dugoff tire model, Bayes law was used to obtain the estimated road friction coefficient. The algorithm model is complex and real-time poor. The domestic research on the estimation method of road surface friction coefficient started late. Estimation methods are mainly the following: analytical method mostly based on the modified Dugoff tire model, multi-sensor data fusion method, and estimation method based extended Kalman filter. Professor Luo Yugong of Tsinghua University proposed an algorithm based on modified Dugoff model [8] . Based on quarter-vehicle model and modified tire model, the friction coefficient was obtained by segmentation method and inverse solution method. The form of the analytic solution is simple, but the estimation error is large. Professor Song Jian of Tsinghua University, who proposed a multi-sensor data fusion method [9] used vehicle sensor data and estimated road friction coefficient in real time. Prof. Zong Changfu of Jilin University put forward an estimation method based on extended Kalman filter. By constructing the vehicle dynamics model and modified HSRI tire model, the extended Kalman filter has been used to estimate the vehicle states and road friction coefficient [10] . However, the method is limited in its working condition and can only be estimated accurately in steady state. At present most of the research is based on analyzing the relationship between road friction coefficient and vehicle states and establishing models to estimate. The modeling method has the following difficulties: (1) The relationship between vehicle states and road friction coefficient is difficult to describe accurately. Only through tire force formula, can road friction coefficient be
associated with vehicle states. However, the form is very complex [11] . (2) It is difficult to describe the dynamic characteristics of road friction coefficient. Usually, the dynamic equation describing road friction coefficient is random walk model. This method cannot guarantee the accuracy of estimation when the road condition is complicated. At present it is hard to analyze the dynamic characteristic from the mechanism, and improve the model. To solve this problem, this paper presents a feasible solution, which is based on BP neural network estimation method. Based on Dugoff tire model and vehicle longitudinal and lateral force analysis, it can be deduced that twelve vehicle states and road friction coefficient exist certain functional relationship. On the basis of the functional relation, the simulation data under different working conditions are extracted from the vehicle dynamics simulation software veDYNA. Using the 12 vehicle states as inputs and road friction coefficient as output, BP neural network can be well-trained. The well-trained BP neural network can be used to estimate road friction coefficient.
2
Estimation Method Based on BP Neural Network
When four wheels of the vehicle are not fully operating in the linear region, the twelve vehicle states including wi ,V i , r, ax , ay , G and road friction coefficient
P constitute a certain functional relationship. The symbol description is shown in table 1.
Vi
Slip rate of four wheels(-)
r ax
Yaw rate( rad / s )
ay
G P
2.1
Vehicle Dynamics
Vehicle force analysis[13] is shown in Figure 1.
Gi
Fi , x
v E
Fi , y
lf
J
l
lr b Fig. 1: Analysis of Vehicle Force
1 [( Fx1 cos G Fy1 sin G ) ( Fx 2 cos G m Fy 2 sin G ) Fx 3 Fx 4 Faero ]
(5)
1 [( Fx1 sin G Fy1 cos G ) ( Fx 2 sin G m Fy 2 cos G ) Fy 3 Fy 4 ]
(6)
ax
ay
From (1) - (5), the vehicle longitudinal force equation can be simplified as M =A f (O1 ) B f (O2 ) (7) C f (O3 ) D f (O4 ) (M, A, B, C, D are values associated with vehicle states) Assume that four wheels are all working in the non-linear region, M =A O1 (2 O1 ) B O2 (2 O2 ) (8) C O3 (2 O3 ) D O4 (2 O4 ) Among them, O1 d 1 , O2 d 1 , O3 d 1 , O4 d 1 .
Table 1: Symbol Description Angular velocity of four wheels( rad / s )
2.2
Analyze the longitudinal and lateral force,
Before using BP neural network, the following conclusion should be deduced.
wi
When O ! 1 , the tire force is located in the linear region, and when O d 1 , the tire force is located in the non-linear region.
Longitudinal acceleration( m / s 2 ) Lateral acceleration( m / s 2 ) Steering wheel angle ( deg )
From (4), it can be derived that 2 CV2 V x21 CD2 tan 2 D1 0 P d Fz1 (1 V x1 )
Road friction coefficient (-)
Tire Model
Quantitative analysis is performed based on Dugoff tire model. For a single wheel, in the tire coordinate system, the lateral and longitudinal force formula is expressed as follows [12] . Vx Fx CV f (O ) (1) 1 V x tan(D ) Fy CD f (O ) (2) 1 V x Where, (2 O )O O d 1 f (O ) ® (3) O !1 ¯ 1 P Fz (1 V x ) O (4) 2 CV2 V x2 CD2 tan 2 D
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(9)
0 P d
2 CV2 V x22 CD2 tan 2 D 2 Fz 2 (1 V x 2 )
(10)
0 P d
2 CV2 V x23 CD2 tan 2 D3 Fz 3 (1 V x 3 )
(11)
0 P d
2 CV2 V x24 CD2 tan 2 D 4 Fz 4 (1 V x 4 )
(12)
Define that Pi
2 CV2 V xi2 CD2 tan 2 D i (i 1, 2,3, 4) Fzi (1 V xi )
(13)
Pi is a parameter regarding vehicle states and parameters. Therefore, friction coefficient satisfies such conditions, 0 P d min( P1 , P2 , P3 , P4 ) (14)
In summary, the longitudinal force equation can be simplified as M A ( P P1 ) 2 B ( P P2 ) 2 (15) C ( P P3 ) 2 D ( P P4 ) 2 where 0 P d min( P1 , P2 , P3 , P4 ) .
For f (P )
A ( P P1 ) 2 B ( P P2 ) 2
C ( P P3 ) 2 D ( P P4 ) 2
f (P )
(16)
is a quadratic function. The four additions
A (P P1 )2
B (P P2 )2
,
,
C (P P3 )2
,
D (P P4 ) must be monotonic(monotonically increasing or monotonically decreasing) in the interval . 0 P d min( P1 , P2 , P3 , P4 ) Therefore, f ( P ) A ( P P1 ) 2 B ( P P2 ) 2
2
C ( P P3 ) 2 D ( P P4 ) 2
must also be a monotonic function in the interval 0 P d min( P1 , P2 , P3 , P4 ) .
company TESIS, focusing on real-time application and off-line research. It can guarantee the stability and efficiency of the simulation. At the same time, most of the parameters can be modified in real time, such as vehicle quality, tire longitudinal and lateral stiffness, road adhesion coefficient and so on. The use of the software can effectively improve the development efficiency and reduce development costs. Therefore, the simulation in this paper is based on the vehicle dynamics model constructed in veDYNA. Bmw_325i vehicle model is chosen in this paper, which is based on BMW 325i's design parameters and kinetic test parameters. 3.1
In order to basically cover all vehicle driving conditions, simulation conditions are set as shown in Table 2. Table 2: Simulation Conditions of Training Sample Friction coefficient 0.2
Therefore, according to the equation M
f (P )
(17)
There are two possible solutions: z There is no real root. z There is a unique real root. Since the parameters in the equation are taken from veDYNA, the equation for friction coefficient P must have a unique real root. The above proof is based on the hypothesis that four wheels of the vehicle are all working in the nonlinear region. When only one wheel, two wheels or three wheels work in the nonlinear region, it can be proved in the same way. Taking slip rate formula, wheel center velocity formula, and wheel slip angle formula into Dugoff tire force formula, the complete expression of longitudinal and lateral force formula is as follows. ax ay
f1 ( wi , vx , v y , G , P , r )
f 2 ( wi , vx , v y , G , P , r )
(18)
Steering angle(deg) 0
Torque(Nm) 125
Initial velocity(km/h) 25
0.5
±90
300
80
0.7
±210
500
125
0.9
±360
A total of 13 parameters wi , V i , r, ax , ay , G , P are collected from veDYNA. Simulation last for 5 seconds in each group, and sampling frequency is 1000Hz. Thus, each group can get 5000 samples. Each group selects 49 samples of data averagely. A total of 252 groups are obtained. Therefore, 12348 samples of data can be obtained. 3.2
Structure of BP Neural Network
The input values are wi , V i , r, ax , ay , G and the target value is P . A three-layer BP neural network is adopted, which consists of two hidden layers and one output layer. The network has 12 inputs, 1 output, 1 output layer and 2 hidden layers, which contain 10 neurons each. BP neutral Vehicle states
network
Road friction coefficient
Input
Thus, when there is a unique set of vehicle parameters ax , ay , wi , vx , vy , G , r (a reasonable value
Output
Training
extracted from veDYNA), the equation for the friction coefficient must have a unique solution. Friction coefficient P and ax , ay , wi , vx , vy , G , r constitute a certain functional relationship. P f (wi , vx , vy , ax , ay , G , r )
Acquisition of Training Samples
Vehicle data Fig. 2: BP neural network structure diagram
wi ,V i , r, ax , ay , G also holds.
In the choice of transfer function, the hidden layer 1 and hidden layer 2 select “tansig” function which is a sigmoid function. The output layer chooses linear function “purelin” as transfer function. In addition, the iterative termination condition parameters and weight learning algorithm parameters also need to be set.
3
3.3
(19) Slip rates of four wheels V i contain the information of
wi , vx , vy , r , so the functional relationship for P and
Realization of Estimation Method Based on BP Neural Network
Analysis of Training Effectiveness
After inputting 12,348 samples, the BP neural network iterates for 421 times in total. When iterative termination condition is met, the training is terminated, which costs 12:08, as shown in Fig3.
VeDYNA (Vehicle Dynamics Analysis) is a fast vehicle dynamics simulation software developed by the German
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1.2
Data Fit Y=T
1 0.8 0.6 0.4 0.2 0 -0.2 0
Validation: R=0.99032 Output ~= 0.98*Target + 0.011
Output ~= 0.99*Target + 0.0076
Training: R=0.99353
0.5
1.2
Data Fit Y=T
1 0.8 0.6 0.4 0.2 0 -0.2
1
0
Target
Fig. 3: Training termination interface
During training process, 12348 samples are divided into three categories: z Train samples (70%). z Validation samples (15%). z Test samples (15%). The mean squared errors of the three samples in the iterative process are shown in Fig. 4, where the mean squared error of the sample at step 415 is minimized to 0.001279.
0.8 0.6 0.4 0.2 0 -0.2 0
0.5
1.2
Data Fit Y=T
1 0.8 0.6 0.4 0.2 0 -0.2
1
0
Target
0.5
1
Target
Fig. 5: Correlation coefficient between output value and target value
Best Validation Performance is 0.001279 at epoch 415
1
10
4
Train Validation Test Best
0
Mean Squared Error (mse)
Data Fit Y=T
1
1
All: R=0.99274 Output ~= 0.99*Target + 0.0079
Output ~= 0.99*Target + 0.0066
Test: R=0.99145 1.2
0.5
Target
10
Verification of the estimation method based on BP neural network
In order to verify the accuracy of BP neural network, the following seven groups of simulation conditions are as shown in Table 3. According to simulation conditions, initial velocity, steering wheel angle, torque and other factors should be taken into account, and it is necessary to ensure that the verification space does not coincide with the sample space.
-1
10
-2
10
-3
Table 3: Simulation Conditions of Verification Test
10
State
-4
10
0
50
100
150
200
250
300
350
421 Epochs
Fig. 4: The mean square error in sample iteration
400
Constant velocity Constant velocity Constant velocity Constant velocity Acceleration
The closer to 1 the correlation coefficient R is, the better the correlation between output value and target value of the network will be and the better the effect of network training will be. The correlation coefficient R between output value and target value of the three kinds of samples after the training termination is greater than 0.99, as shown in Fig5. Acceleration Acceleration
Initial velocity(km/h)
Steering angle(deg)
Torque(Nm)
Friction coefficient
77.1
90
150
0.8
56.5
90
120
0.6
87.1
90
120
0.4
75.8
90
100
0.2
60
180
200
0.8
60
180
200
0.2
77.1
90
150
0.8 to 0.2
(1) Constant velocity (77.1km / h) and high friction coefficient (0.8). Road friction coefficient is estimated as shown in Fig6.
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Torque per wheel:100Nm;Steering wheel angle:90°;Initial speed:75.8km/h;Road friction:0.2 0.6
0.9 0.85 0.8 0.75 0.7
0.4 0.3 0.2 0.1
0.65 0.6
Predictive value of road friction coefficient Expected value of road friction coefficient
0.5 Road friction coefficient[-]
Road friction coefficient[-]
Torque per wheel:150Nm;Steering wheel angle:90°;Initial speed:77.1km/h;Road friction:0.8 1 Predictive value of road friction coefficient 0.95 Expected value of road friction coefficient
0
1
2
3
4
5 Time[s]
6
7
8
9
0
10
0
1
2
3
4
5 Time[s]
6
7
8
9
10
Fig. 6: Friction coefficient estimation-Constant velocity (77.1km / h) and high friction coefficient (0.8)
Fig. 9: Friction coefficient estimation-Constant velocity (75.8km / h) and high friction coefficient (0.2)
(2) Constant velocity (56.5km / h) and medium friction coefficient (0.6). Road friction coefficient is estimated as shown in Fig7.
(5) Acceleration and high friction coefficient (0.8). Velocity change is shown in Figure 10. Road friction coefficient is estimated as shown in Fig11. Torque per wheel:200Nm;Steering wheel angle:180°;Initial speed:60km/h;Road friction:0.8 75
Longitudinal velocity[km/h]
Road friction coefficient[-]
Torque per wheel:120Nm;Steering wheel angle:90°;Initial speed:56.5km/h;Road friction:0.6 0.9 Predictive value of road friction coefficient Expected value of road friction coefficient 0.8 0.7 0.6 0.5
70
65
60
0.4 0.3
0
1
2
3
4
5 Time[s]
6
7
8
9
55
10
Fig. 7: Friction coefficient estimation-Constant velocity (56.5km / h) and high friction coefficient (0.6)
2
3
4
5 Time[s]
6
7
8
9
10
Torque per wheel:200Nm;Steering wheel angle:180°;Initial speed:60km/h;Road friction:0.8 1 Predictive value of road friction coefficient Expected value of road friction coefficient 0.9 Road friction coefficient[-]
Torque per wheel:120Nm;Steering wheel angle:90°;Initial speed:87.1km/h;Road friction:0.4 0.7
Road friction coefficient[-]
1
Fig. 10: Velocity change - Acceleration and high friction coefficient (0.8)
(3) Constant velocity (87.1km / h) and medium friction coefficient (0.4). Road friction coefficient is estimated as shown in Fig8. Predictive value of road friction coefficient Expected value of road friction coefficient
0.6
0
0.5
0.8
0.7
0.6
0.4 0.5 0.3
1
2
3
4
5 Time[s]
6
7
8
9
10
Fig. 11: Friction coefficient estimation - Acceleration and high friction coefficient (0.8)
0.2 0.1
0
0
1
2
3
4
5 Time[s]
6
7
8
9
10
(6) Acceleration and low friction coefficient (0.2). Velocity change is shown in Figure 12. Road friction coefficient is estimated as shown in Fig13.
Fig. 8: Friction coefficient estimation-Constant velocity (87.1km / h) and high friction coefficient (0.4)
Torque per wheel:200Nm;Steering wheel angle:180°;Initial speed:60km/h;Road friction:0.2 95
(4) Constant velocity (75.8km / h) and low friction coefficient (0.2). Road friction coefficient is estimated as shown in Fig9.
Longitudinal velocity[km/h]
90 85 80 75 70 65 60 55
0
1
2
3
4
5 Time[s]
6
7
8
9
10
Fig. 12: Velocity change - Acceleration and low friction coefficient (0.2)
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Road friction coefficient[-]
Torque per wheel:200Nm;Steering wheel angle:180°;Initial speed:60km/h;Road friction:0.2 0.5 Predictive value of road friction coefficient Expected value of road friction coefficient 0.4
coefficient and velocity change, the absolute error between estimated value and expected value is within ± 0.1. However, the current research work is still inadequate. Due to the limited number of samples, the network may not be able to perfectly estimate road friction coefficient under any conditions, and there may be steady-state errors. Since this method has 12 inputs, if vehicle sensor fails, the well-trained BP neural network may not be able to accurately estimate road friction coefficient. In addition, for the reason that the sample space of neural network is taken from bmw_325i vehicle model, the neural network is generally valid for certain vehicle model. When the vehicle parameters change, the estimated effect may become bad, and the sample space needs to be retrieved. Therefore, the robustness and generalization ability of the neural network need to be improved in the future. This is the focus of next step.
0.3
0.2
0.1
0
0
1
2
3
4
5 Time[s]
6
7
8
9
10
Fig. 13: Friction coefficient estimation- Acceleration and low friction coefficient (0.2)
(7) Docking road (0.8 to 0.2). Velocity change is shown in Figure 14. Road friction coefficient is estimated as shown in Fig15. Torque per wheel:150Nm;Steering wheel angle:90°;Initial speed:77.1km/h;Road friction:0.8-0.2 85
References
Longitudinal velocity[km/h]
[1]
80
75
0
1
2
3
4
5 Time[s]
6
7
8
9
10
Fig. 14: Velocity change - Docking road (0.8 to 0.2)
Road friction coefficient[-]
Torque per wheel:150Nm;Steering wheel angle:90°;Initial speed:77.1km/h;Road friction:0.8-0.2 1 Predictive value of road friction coefficient Expected value of road friction coefficient 0.8
0.6
0.4
0.2
0
0
1
2
3
4
5 Time[s]
6
7
8
9
10
Fig. 15: Friction coefficient estimation - Docking road (0.8 to 0.2)
From the simulation results, it can be seen that the BP neural network can estimate road friction coefficient when the vehicle is running under steady condition. The estimating error is less than 0.05. Even if it is docking road, road friction coefficient can be quickly adapted and estimated.
5
Conclusion
In this paper, road friction coefficient estimation method based on BP neural network is proposed, which can estimate road friction coefficient well when road friction coefficient and velocity is constant, and it has some estimation error when steering angle is changing or vehicle is accelerating. From simulation results, when road friction coefficient is constant, the absolute error between estimated value and expected value is within ± 0.05. When road friction
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