Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 7, No. 3, 2023 

 

131 

Soft	Measurement	of	Road	Adhesion	Coefficient	
Volumetric	Kalman	Filtering	
Penghao Ji1, *, Jiquan Wu1 

1 School of Automotive and Transportation, Tianjin University of Technology and Education, Tianjin, China 
* Corresponding author: Penghao Ji (Email: jipenghao0831@163.com) 

 

Abstract: In order to solve the problem that the road adhesion coefficient is not extensive enough and difficult to obtain in 
practice, this paper designs a soft measurement method of road adhesion coefficient based on the volumetric Kalman filter theory. 
A three-degree-of-freedom vehicle estimation model using the Dugoff tyre model is established, and the validity and accuracy 
of the algorithm are verified with the help of the joint simulation platform of Carsim and Matlab/Simulink. The results show that 
the estimation error of the proposed estimation algorithm is within 5% under the three road adhesion coefficients, and the initial 
design can be completed well in terms of accuracy and real-time performance. 

Keywords: Road adhesion coefficient, Kalman filter, Volumetric Kalman filter. 
 

1. Introduction 
The purpose of this paper is to develop a soft measurement 

algorithm for road surface adhesion coefficient estimation 
through experimental simulation, through which a higher 
computational accuracy and faster response speed can be 
achieved, which is conducive to further development and 
research in the automotive industry. Meanwhile, the real-time 
estimation of the road surface adhesion coefficient is an 
important part of the safety control of the vehicle, and is also 
a necessary prerequisite for the vehicle to carry out a series of 
stability control strategies. There are many methods and types 
of soft measurement of road surface adhesion coefficient in 
the present research, which can be divided into two categories 
of Cause-based and Effect-based according to different data 
acquisition and measurement methods[1]. The Cause-based 
approach applies past experience to estimate the current state 
of the road attachment coefficient, and generally applies 
conditions that have a large impact on the road attachment 
coefficient to the calculations and predictions[2]. For example: 
Optical sensors work by recognising water and other 
substances that reduce the coefficient of adhesion of the road 
surface, while Mika Morii and Hiroyuki Yasuo use a laser 
beam with a maximum power of 200W to illuminate the road 
surface for estimation[3]. Effect-based methods are carried 
out to analyze the response of the tyre or vehicle to estimate 
the road adhesion coefficient, tyre noise and tread 
deformation are mainly referred to collectively as the tyre 
response, and longitudinal dynamics and lateral dynamics are 
mainly referred to collectively as the vehicle response. Due to 
practical conditions, sensors are generally expensive and less 
robust, and easily affected by environmental factors, while the 
Cause-Based method requires additional sensors and is 
therefore not widely utilized. The Effect-Based method can 
accurately estimate road adhesion coefficients, and has the 
advantage of being more robust compared to the Cause-Based 
method. Cause-Based method has the advantages of being 
more robust and less expensive, but its computation is prone 

to hysteresis, real-time is not high, and the amount of 
computation is generally large[4]. Therefore, Kalman filter 
and its many improved forms are being widely used in the 
study of vehicle driving state and vehicle parameter 
estimation, which is a relatively effective estimation method 
and estimation means. Among them, the theory of Extended 
Kalman Filter (EKF) must be used to expand the nonlinear 
system with Taylor's expansion formula to be linearised, 
otherwise it will lead to dispersion of the filter output data and 
non-convergence. Unscented Kalman Filter (UKF) has the 
advantage of high computational accuracy, but it is based on 
the standard Kalman filter, which is an infinite-growth 
memory filter, and when the filter estimation is performed at 
any moment, all the data prior to the current moment are used, 
leads to underutilized of current sensor measurements. 
Cubature Kalman filter (Cubature Kalman Filter) is derived 
from the Bayesian estimation of Gaussian function 
assumption, using the third-order spherical-radial volume 
criterion, the nonlinear system filtering into a Gaussian 
probability density function product of the integral solution of 
the problem, which has a rigorous mathematical proof of 
numerical integration through the third-order volume law 
approximates the weighted Gaussian integral. It has 2n 
volume points with equal weights (n is the number of system 
state dimensions), and it has been proved that its accuracy of 
approximating the probability distribution after the nonlinear 
transformation is better than that of the UKF, and it is simple 
to implement, and it has a high scalability, so it has a very 
broad application value[5]. 

2. Dynamical Model  

2.1. Vehicle Model 
Vehicle dynamic models can be used not only for vehicle 

motion and control, but also for tyre force calculation and 
road adhesion coefficient estimation. As shown in Fig. 1, this 
paper simplifies the vehicle into a 3-degree-of-freedom model 
with only three directions: longitudinal, lateral and 
transverse[6]. 

 



 

132 

 
Figure 1. Vehicle dynamics modelling 

 
The vehicle dynamics equations are: 
 

11 12 21 22 11 12

11 12 21 22 11 12

.

11 12 11 12 21 22

11 12 11 12 21 22

( )cos ( )sin

( )sin ( )cos

( )sin ( )cos ( )

( )sin ( )cos ( )
2 2 2

     


     

      

     


x x x x x y y

y x x y y y y

z x x y y y y

f f r
y y x x y y

ma F F F F F F

ma F F F F F F

I a F F a F F b F F

T T T
F F F F F F

 

 

  

 





(1) 

 
In the above equation, 𝑎 、𝑎  for the longitudinal 

acceleration and lateral acceleration of the vehicle; 𝑣 、
𝑣  for the longitudinal speed and lateral speed of the vehicle; 

𝑎、𝑏 for the distance from the center of mass of the vehicle 
to the front and rear axes; m for the total mass of the vehicle; 
𝑇 、𝑇   for the distance between front and rear wheels; 

𝐹 、𝐹 for the longitudinal and lateral forces on the wheels

（𝑖=1,2 for the front axle and the rear axle, respectively, and 
𝑗=1,2,3,4 for the left-front, right-front, left-rear, and right-rear 
wheels, respectively); 𝛼 is the angle of gyration of the front 

wheels; 𝐼 for the rotating moment of inertia; and .

  for the 

angular acceleration of the transverse swing. 

2.2. Tyre Model 
For the research content of this paper, considering the effect 

of model accuracy, Dugoff tyre model is selected, for each 
wheel, the longitudinal force and lateral force acting on the 
tyre can be expressed by equation (2). 

 

 

 
1
tan

1

xij zij xij

yij zij yij

s
F F C f k

s

F F C f k
s





  

 
 

                   (2) 

 
included among these, 
 

 

2 2 2 2

(2 ), 1

1, 1

1
(1 )

2 tanx y

k k k
f k

k

k s
C s C 








 
  

 
 

               (3) 

 
Do tyre normalized processing of equation (2): 

 

 

 

*

*

1
tan

1

xij xij zij xij

yij yij zij xij

s
F F F C f k

s

F F F C f k
s

 

 







 








              (4) 

 
Where 𝐹  is the vertical load of four wheels; 𝐶 、𝐶  

are the longitudinal stiffness and lateral stiffness of each tyre, 
respectively; s is the longitudinal slip rate; 𝜇  is the road 
adhesion coefficient; k  is the characteristic parameter due to 

the nonlinear change of tyre force caused by tyre slip, 1k 
denotes the linear interval of the tyre, 1k   denotes the tyre 

non-linear interval. 𝐹∗ 、𝐹∗   denote the longitudinal and 
lateral normalized tyre forces of the 4 wheels. From equation 
(4), it can be seen that 𝐹∗ 、𝐹∗ are not correlated with the 
changes in the road adhesion coefficient. 

The calculation of the normalized force becomes the key to 
estimating the road adhesion coefficient. In this paper, the 
vehicle state measured or estimated by known sensors is used 
as input to calculate the normalized force, thus laying the 
foundation for the design of the parameter estimator based on 
Equations (3) and (4). 

The following equations describing the dynamics of the 
four-degree-of-freedom vehicle model can be obtained 

 

r

1 1

2 2

1 1

2 2

  
       


 

   
 

y
z11,z12 x

f

y
z21,z22 x

a h b h
F mg m ma

T l l

a h a h
F mg m ma

T l l

           (5) 

 

11,12

21,22

arctan
/ 2

b
arctan

/ 2

  
     


 

      





y

x f

y

x f

v ar
α α

v T r

-v r
α

v T r

             (6) 

 


 ij ij

ij
ij

ω r v
λ

v
                   (7) 

 



 

133 

,

,2
r

2

2

  
     

  


        

f
11 12 cog

21 2 cog

T
v v r aβ

T
v v r bβ

              (8) 

 

arctan
 

  
 

y

x

v
β

v
                   (9) 

 

2 2 cog x yv v v                   (10) 

 
where h is the height of the vehicle body, 𝛼  is the tyre 

side deflection angle, ij  is the wheel angular velocity, r is 

the wheel radius, cogv is the vehicle center velocity, and β is 
the center of mass side deflection angle.  

For the calculation of the state variables in the longitudinal 
and lateral normalized forces 𝐹∗ 、𝐹∗   the reference 
equation. At this point, the three-degree-of-freedom vehicle 
model and the tyre model are established. 

3. Design of Volumetric Kalman Filter 
Estimation Algorithm 

3.1. Estimator Design 
According to equations (1) and (4), a mathematical 

relationship between the road adhesion coefficient and the 
vehicle dynamics equations can be seen. In this paper, this 
relationship is be used to estimate the road adhesion 
coefficient using UKF. 

Since most of the studies set the road attachment coefficient 
as a constant, it is usually assumed that 𝜇  takes a value 
between 0.1 and 1. Definition: system state variable 
𝑥 𝜇 , 𝜇 , 𝜇 , 𝜇  , systematic observation variable 𝑧

𝑎 , 𝑎 ,
.

 , input variable 𝜇 𝛼, 𝐹∗ , 𝐹∗ . 𝑤 𝑡  ，𝑣 𝑡  
is the process white noise and measurement white noise. 
Where 𝜇 、𝜇 、𝜇 、𝜇   are the adhesion coefficients 
between the four tyres and the ground, and α is the front wheel 
turning angle. 

Set up the equation of state as: 
 

𝑥 𝑡 1 𝑓 𝑥 𝑡 , 𝜇 𝑡 𝑤 𝑡

1
0
0
0

    

0 0 0
1 0 0
0
0

1
0

0
1

𝑢
𝑢
𝑢
𝑢

𝑤 𝑡   (11) 

 
The observation equation is: 
 

 𝑧 𝑡 ℎ 𝑥 𝑡 , 𝜇 𝑡 𝑣 𝑡                                                                                                                                       

⎣
⎢
⎢
⎢
⎡
𝐹∗ cos 𝛼 𝐹∗ sin 𝛼

𝑚
𝐹∗ cos𝛼 𝐹∗ sin𝛼

𝑚
𝐹∗

𝑚
𝐹∗

𝑚
𝐹∗ sin 𝛼 𝐹∗ cos𝛼

𝑚
𝐹∗ sin𝛼 𝐹∗ cos𝛼

𝑚
𝐹∗

𝑚
𝐹∗

𝑚
𝐽 3,1 𝐽 3,2 𝐽 3,3 𝐽 3,4 ⎦

⎥
⎥
⎥
⎤ 𝜇

𝜇
𝜇
𝜇

 𝑣 𝑡

 12  

Among them: 

𝐽 3,1
∗ ∗ ∗ ∗ /

； 

𝐽 3,2
∗ ∗ ∗ ∗ /

; 

𝐽 3,3
∗ ∗

;  𝐽 3,4
∗ ∗

 

The measurement noise covariance matrix
1 0 0 0

0 1 0 0
R

0 0 1 0

0 0 0 1

 
 
 
 
 
 

 ,the system noise covariance matrix 

4x 4Q I * 0.1  , the system state variable 

 Tx(0) 1 1 1 1 ,and the initial value of the error covariance 

matrix is 4x 4P(0) I . 
The normalized forces are calculated using the Dugoff  

tyre model. The inputs to the model are: front wheel angle of 
rotation, wheel speeds of all four wheels; longitudinal and 
lateral acceleration; lateral deflection angle of the center of 
mass; transverse angular velocity and longitudinal vehicle 
speed. The final normalized tyre force is obtained for all four 
wheels (Including vertical and lateral total of 8), as input to 
the CKF parameter estimator. The following figure shows the 
schematic diagram of the CKF based pavement adhesion 
coefficient estimation algorithm as shown in Fig2. 

 

 

Figure 2. Schematic diagram of CKF based pavement adhesion coefficient estimation algorithm 
 

3.2. Classical Kalman Filter 
Classical Kalman filter theory is a filter theory applied to 

linear systems. Many physical processes, such as changes in 
room temperature, free-fall, the movement of satellites in 
orbit around the earth, and the GPS navigation of ships, can 
be approximated as a linear system. Consider the following 
state space expression to describe a linear system. 

Equation of state: ( ) ( -1) ( ) ( -1)x k Ax k Bu k w k    

Observation equation: ( ) ( ) ( )y k Hx k v k   
Where, k discrete time, the state of the system at moment 

k; ( )u k  is the control input to the system, ( )y k   is the 

observed signal to the state. ( )w k  is the input white noise, 

also known as process noise, and ( )v k   is the observation 
noise. A is the state transfer matrix, B is the noise distribution 
matrix, and H is the observation matrix. 

( )w k  and ( )v k are are white noises with zero mean. And, 



 

134 

they are independent of each other. Thus the following 
equation can be obtained. 

 

( ) 0

( ) 0

( ) ( ) 0, ,





 


 T

Ew k

Ev k

E w k v j k j

             (13) 

 
The matrices Q and R are the variance arrays 

corresponding to two noises which satisfy: 
 

 
 

( ) ( )

( ) ( )









T
kj

T
kj

E w k w j Q

E v k v j R




               (14) 

 
Where, kjQ =0，k≠j， k jR  takes any positive real number, 

depending on the actual equation. 
The system gives the exact procedure of Kalman filtering: 
State prediction equations: 
 

ˆ( ) ( 1) ( ) ( 1)x k Ax k Bu k w k            (15) 
 

Error covariance prediction equation: 
 

ˆ ( ) ( 1) TP k AP k A Q                  (16) 
 

Gain equations: 
 

   1ˆ ˆ( ) ( ) ( )T TK k P k H HP k H R


          (17) 

 
Filtering equations 
 

 ˆ ˆ( ) ( ) ( )( ( ) ( ))x k x k K k y k Hx k            (18) 
 
Error covariance update equation:  
 

ˆ( ) (1 ( ) ) ( )P k K k H P k                (19) 
 

A total of five formulas that the five core formulas of the 
classical Kalman filter, the above classical Kalman filter is 
introduced. 

3.3. Volumetric Kalman Filter 
CKF transforms the nonlinear filtering problem into a 

problem of solving the product of a nonlinear function and a 
Gaussian probability density function, using the principle of 
volumetric numerical integration to approximate the state 
posterior distribution. The algorithm is highly accurate and 
stable[7]. And with the increase of the dimension of the system, the 
stability and accuracy of CKF will be more than UKF[8], at the same 
time, the computational amount of CKF is smaller than UKF, in view 
of the advantages of CKF, this paper selects CKF for the soft 
measurement of road adhesion coefficient. 

The design of CKF is based on Bayesian filtering principle. 
According to the third-order spherical-Radial law, when the variable 
x obeys the normal distribution, it can be obtained as follows. 

 

 
2

1

( ) ( ) ( ; , )
n

n

N i iR
i

I f f x N x P dx f P   


         (20) 

 

The covariance matrix P satisfies =
T

P P P；

1 / 2i n 
； [ ]i in  ；

1 0 1 0

0 0 0 0
[ ]

0 1 0 1

i

       
       
                
       

        

� � � �

 
The core idea of CKF's algorithm is to implement the 

nonlinear estimation of 2n equally weighted volume points 
calculated by the Spherical-Radial volume rule. The 
algorithmic processing flow of CKF filtering is as follows: 

Based on the estimate-and discrepancy moment k-1, the 
volume point initialisation is calculated using equation (20) 
to generate the volume point: 

 
( 1) chol{ ( 1)}

ˆ ( 1) ( 1) ( 1)

  


     i i

S k P k

X k S k X k
           (21) 

 
Nonlinear Propagation of Volume Points:  
 

 * ˆ( 1) ( 1), ( )i iX k f X k U k                (22) 

 

Status prediction:
2

*ˆ ( ) ( 1)
n

i iX k X k                  (23) 

 
forecast variance: 
 

2
* *

1

ˆ ˆ ˆ( ) ( 1) ( 1) ( ) ( )
n

T T
i i i

i

P k X k X k X k X k Q


          (24) 

 
Calculate the volume point based on the estimate and 

variance at moment k: 
 

ˆ ˆ( ) chol{ ( )}S k P k                (25) 
 

ˆˆ ˆ( ) ( ) ( )i iX k S k X k               (26) 
 

Calculation of volumetric point values after propagation 
of the observation equation: 

 

 * ˆ( ) ( ), ( )i iY k h X k U k              (27) 

 

Observations forecast: *

1

ˆ( ) ( )


 
m

i i
i

Y k Y k                

(28) 
 
Estimating the new observation covariance: 
 

* *

1

( ) ( ) ( ) ( ) ( )
m

T T
yy i i i

i

P k Y k Y k y k y k R


            (29) 

 
Estimating the covariance: 
 

*

1

ˆ ˆ ˆ( ) ( ) ( ) ( ) ( )
m

T T
xy i i i

i

P k X k Y k X k Y k


          (30) 

 
Calculating Kalman Gain: 
 

1( ) ( ) ( )xy yyK k P k P k              (31) 



 

135 

 
Status Update: 
 

ˆ ˆ( ) ( ) ( )( ( ) ( ))X k X k K k Y k Y k             (32) 
 
Estimated covariance values: 
 

ˆ( ) ( ) ( ) ( ) ( )TyyP k P k K k P k K k               (33) 
 

At this point, the CKF algorithm process ends, and the CKF 
will recursively push in real time to the next point of the 
calculation of the iterative loop has reached the final result. 

4. Simulation Analysis of Road 
Attachment Coefficient Estimation 

This paper addresses the design of soft measurement of 
road adhesion coefficient, while Carsim simulation software 
has a wealth of vehicle dynamics models, including the body, 
tyres, braking system and transmission system. Vehicle 
dynamics model, including the body, tyres, braking system, 
driveline system and other subsystems, the model precision is 
high, and its simulation test application has a strong practical 
application. Its simulation test application has strong practical 
application value. Therefore, this paper adopts the joint 
simulation test of Carsim and simulink, and selects the sdan 
E-class car as the simulation vehicle in this paper, and some 
of its parameters are shown in the table: 

 
Table 1. The part parameters of the vehicle table 

Vehicle parameter Symbol Numerical value Unit 
Overall vehicle mass M 1650 kg 

Wheel 
base 

D 3050 mm 

Center of gravity height H 530 mm 
Effective radius of tires S 353 mm 

Vehicle length L 4868 mm 
 

After setting three different road conditions with different 
road adhesion coefficients in Carsim, the filtered estimates of 
the road adhesion coefficients were compared with the actual 
values as a way to verify that the algorithm proposed in this 
paper is effective. The model is simulated, and the 
relationship between the estimated and given values of the 
road simulation coefficients is obtained as shown in Fig. 3. 
From the simulation results, it can be obtained that the road 
adhesion coefficients of all four wheels converge to the real 

values in a short time. The details are as follows: 
Setting the road adhesion coefficient at 0.9 corresponds to 

a dry concrete road surface, and the vehicle simulation 
condition is: driving at an initial speed of 60 km/h, the 
simulation results using the UKF algorithm are shown in Fig. 
3. Under the high road adhesion coefficient of 0.9, the 
estimated value stabilizes to the set value of 0.9 in about 15s, 
and converges to 0.88-0.92 after 15s. 

 

 
Figure 3. Simulation of high adhesion coefficient pavement 

 
Setting the road adhesion coefficient to 0.6 corresponds to 

the wet concrete road surface, the vehicle simulation 
condition is: driving at an initial speed of 60 km/h, the 
simulation results using the UKF algorithm are shown in Fig. 

4. In the case of the set medium road adhesion coefficient of 
0.6, the estimated value stabilizes to the set value of 0.6 in 
about 10s, and converges to 0.57-0.62 after 10s. 

 



 

136 

 
Figure 4. Simulation of middle adhesion coefficient pavement 

 
Setting the road adhesion coefficient to 0.3 corresponds to 

the rainy road surface, and the simulation condition of the 
vehicle is: driving at an initial speed of 60 km/h, the 
simulation results using the UKF algorithm are shown in Fig. 

5. With the low road adhesion coefficient of 0.3, the estimated 
value stabilise to the set value of 0.3 in about 7s and converges 
to 0.29-0.31 after 7s. 

 

 
Figure 5. Simulation of low adhesion coefficient pavement 

 
The overall convergence of per-wheel recognition is 

slightly slower for medium and high road adhesion 
coefficients compared to low road adhesion coefficients, 
which is in line with the actual driving conditions[9]. This 
shows the effectiveness of applying the UKF filter estimator 
in this paper for the estimation of road adhesion coefficients 
for different road surfaces. 

5. Conclusion  
The paper establishes a three-degree-of-freedom vehicle 

estimation model using Dugoff tyre model, and designs a road 
adhesion estimation algorithm based on the theory of 
volumetric Kalman filtering, the estimation value stabilizes to 
the set value of 0.9 around 15s, and then converges to 0.88-
0.92 after 15s, and stabilizes to the set value of 0.6 around 10s, 
and converges to 0.57-0.62 after 10s. 0.57-0.62, stabilized to 
the set value of 0.3 in about 7s, and converged to 0.29-0.31 
after 7s. Calculations show that the estimation error of the 
estimation algorithm based on the Volumetric Kalman Filter 
is less than 5%, which proves that the algorithm is accurate 
and consistent with the real-time estimation. It is proved that 
the algorithm can complete the initial design well in terms of 
both accuracy and real-time performance to achieve the 
purpose of soft measurement of road adhesion coefficient. 

Acknowledgment 
This paper was supported by the funds. The funds are as 

follows: Tianjin graduate student Scientific research and 
innovation projects (2022SKYZ380); Tianjin graduate 
student Scientific research and innovation projects 
(2022SKYZ388). 

References 
[1] Yu Zhuo-Ping, Zuo Jian-Ling and Zhang Li-Jun, "A Review of 

the Development Status of Pavement Adhesion Coefficient 
Estimation Technology", Automotive Engineering, Vol. 6, pp. 

[2] Yanliang Pei, "Estimation of road surface adhesion coefficient 
for active safety system of electric vehicle", Master, Shenyang 
University of Technology, 2013. Available on: 16 October 
2023. 

[3] M. Morii, H. Yasuo, T. Kashihara, T. Iwamoto, and M. 
Sugimoto, "Pavement Condition Detector Using a High Peak 
Power Fibre Laser," IEEE Transactions on Industry 
Applications, vol. 120, pp. 

[4] Wang Hesen, "Research on Longitudinal Safety Control 
Strategy for Cargo Vehicles Based on the Estimation of Road 



 

137 

Surface Adhesion Coefficient", M.S., Shijiazhuang Railway 
University, 2020.000191. 

[5] Jinxi Chen, "Research on Algorithm of Pavement Adhesion 
Coefficient Estimation Based on Volumetric Kalman Filtering", 
Master's Degree, University of Electronic Science and 
Technology, 17 April 2023. 

[6] G. Li, R.C. Xie, S.Y. Wei and C.F. Zong, "Estimation of 
vehicle state and road surface adhesion coefficient based on 
two-volume Kalman filtering", Science in China: 
Technological Sciences, vol. 45, no. 4, pp. 403-414, 2015. 

[7] C. Fernandez-Prades and J. Vila-Valls, Bayesian Nonlinear 
Filtering Using Orthogonal and Cubic Rules Applied to 
Positioning Sensor Data Fusion, 2010 IEEE International 
Conference on Communications, Cape Town, South Africa: 
May 2010, pp. 1-5. 

[8] Li Jun Tang, "Research on Cubature Kalman Filter and its 
Application in Navigation", PhD, Harbin Engineering 
University, 2012. Available on: 16 October 2023. 

[9] Hou-Long Li, "Research and Simulation on Control Strategy of 
Automatic Emergency Braking System Based on Driving 
Characteristics and Road Surface Adhesion Coefficient", M.S., 
Chang'an University, China, 2021.