Research ArticleCalculation Algorithm of Tire-Road Friction Coefficient Basedon Limited-Memory Adaptive Extended Kalman Filter

Bin Huang,1 Xiang Fu ,1 SenWu,2 and Song Huang2

1Hubei Key Laboratory of Advanced Technology for Automotive Components, Wuhan University of Technology, Wuhan, China2Hubei Collaborative Innovation Center for Automotive Components Technology, Wuhan, China

Correspondence should be addressed to Xiang Fu; fuxiang@whut.edu.cn

Received 1 February 2019; Revised 11 April 2019; Accepted 2 May 2019; Published 14 May 2019

Academic Editor: Alessandro Palmeri

Copyright © 2019 Bin Huang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, a limited-memory adaptive extended Kalman Filter (LM-AEKF) to estimate tire-road friction coefficient is proposed.By combining extended Kalman filter (EKF) with the limited-memory filter, this algorithm can reduce the effects of oldmeasurement data on filtering and improve the estimation accuracy. Self-adaptive regulatory factors were introduced to weighcovariance matrix of evaluated error. Meanwhile, measured noise covariance matrix was adjusted dynamically by fuzzy inferenceto accurately track the breaking status of system.Therefore, problems, including large filter error and divergence caused by incorrectmodel, can be solved. Joint simulation was conducted for the proposed algorithm with Carsim and Matlab/Simulink. Under thedifferent road conditions, real-vehicle road tests were conducted in various working conditions for contrast with traditional EKFresults. Simulation and real-vehicle road tests show that this algorithm can enhance the filter stability, improve the estimationaccuracy of algorithm, and increase algorithm robustness.

1. Introduction

With continuous development and application of informa-tion technology in automotive field, people have higherrequirement for the active safety of automobile. Four-wheelindependent driven electric vehicle can realize active safetycontrol by independent control over four-wheel driventorque. It is necessary to obtain the current tire-road fric-tion coefficient to make control system rationally distributemotor torque and fully utilize the friction limit of tire.Therefore, wheel slip or lock can be avoided to ensure theactive safety control of the whole vehicle [1–3]. The useof instruments to directly measure pavement informationcan achieve relatively accurate results, but such sensorshave very high requirements on the working environmentand are expensive, which limits their application. Therefore,using low-cost on-board sensors to measure vehicle stateparameters, and then estimating the state of vehicle dynamicsbased on the vehicle dynamics equation, and then real-timeestimation of pavement information has become a research

focus [4, 5]. Domestic and foreign scholars have made a largeamount of studies on tire-road friction coefficient estimationbased on 𝜇-s curve [6–11]. However, a large amount ofdata is required for data fitting because this method hasproblems of slow response time, weak timeliness, and heavydependence of identification correctness on model accuracy.As a strong nonlinear complex system, vehicle may causean inevitable error between the vehicle dynamics model andthe real system. Such error restrains the model reflectingthe real physical process and fails the matching betweenobservation value and the model. Therefore, the problems,such as large filter error and even divergence, may be caused[12, 13].

Based on the vehicle sensor information fusion technol-ogy, this paper designs the road surface adhesion coefficientestimator based on the nonlinear vehicle dynamics modeland uses the simplified Dugoff tire model to extract the wheeladhesion coefficient variables, using the extended Kalmanfilter (EKF) theory. By combining EKF with the limited-memory filter to reduce the effects of old measurement

HindawiMathematical Problems in EngineeringVolume 2019, Article ID 1056269, 14 pageshttps://doi.org/10.1155/2019/1056269

2 Mathematical Problems in Engineering

data on filtering and improve the estimation accuracy, self-adaptive regulatory factors were introduced to weigh covari-ance matrix of evaluated error for solving the problems ofexcessive or even fading due to inaccurate model.

Traditional EKF has high estimation accuracy, but it isunlimitedly growing memory filter as standard Kalman filter(KF). When estimating filter at any time, all the data beforethis moment should be used, causing the insufficient use ofcurrent sensor measurement. When there is model error andunknown time-varying noise in the system, new observationdata only plays a small role in correcting state estimation.Therefore, errormay accumulate to cause large filter error andeven divergence [14–17]. Based on it, this work introducedlimited-memory filter based on traditional EKF to increasethe role of new observation data and reduce the harmfuleffects of old measurement data on filter. The performanceand accuracy of estimation algorithm can be improved, andthe stability of filter can be enhanced. Meanwhile, with fuzzylogic inference, the noise measurement matrix was adjustedonline in real time according to the difference between actualvariance and theoretical variance of real-time information toimprove the self-adaption of algorithm.

In this paper, we proposed a limited-memory adaptiveextended Kalman Filter (LM-AEKF) to estimate tire-roadfriction coefficient. By combining EKF with the limitedmemory filter, this algorithm can reduce the effects of oldmeasurement data on filtering and improve the estimationaccuracy. Self-adaptive regulatory factors were introducedto weigh covariance matrix of evaluated error. Meanwhile,measured noise covariance matrix was adjusted dynamicallyby fuzzy inference to accurately track the breaking statusof system. Therefore, problems including large filter errorand divergence caused by incorrect model can be solved. Insummary, our contributions are as follows:

(1) An algorithm for enhancing estimation accuracy androbustness is designed.

(2) The paper establishes a 3-DOF nonlinear vehicledynamics model to better reflect the response characteristicsof real vehicles.

(3) By combining EKF with the limited-memory filter,this algorithm can reduce the effects of old measurementdata on filtering and improve the estimation accuracy. Self-adaptive regulatory factors were introduced to weigh covari-ance matrix of evaluated error.

(4) The algorithm has good estimation accuracy andstability, the response speed of the algorithm is improved, andthe adaptability is enhanced.

In recent years, study was widely conducted on real-time estimation of road-vehicle friction coefficient basedon the information fusion technology of vehicle sensor. Asimultaneous online estimation of the vehicle’s state andmax-imum friction coefficient is presented using a joint UnscentedKalman Filter in [18]. Online estimation of the boundedmaximum friction coefficient based on serial sensors usinga sensitivity-based joint unscented Kalman filter is presentedin [19]. The coefficient was initially simulated and estimatedbased traceless Kalman filter algorithm in [20]. Interac-tive multimodel algorithm was used to estimate the speedand tire-road friction coefficient in [21]. Tire-road friction

coefficient was identified with the least square method ofregression and auxiliary variable method in [22]. [23, 24] usethe dual unscented Kalman filter theory to estimate the vehi-cle state and the road surface adhesion coefficient, so that thetwo-estimators' information is exchanged and corrected, andfinally the identification of the current road surface adhesioncoefficient is realized.

In summary, scholars have carried out multimethodresearch on the road surface adhesion coefficient estimationalgorithm and achieved certain results. However, the con-vergence speed and accuracy of these algorithms need to beimproved, and the robustness needs to be enhanced.

2. Nonlinear Vehicle Dynamic Model

This section analyzes the 3-DOF vehicle dynamics model andthe wheel rotation dynamics model, revealing the couplingrelationship between the vehicle operating state parametersand the tire-road adhesion coefficient. The purpose is toprovide theoretical basis and technical support for the designof the parameter estimator.

2.1. Vehicle Model. This work mainly focuses on the tire-road friction coefficient estimation of vehicle in horizontaldirection. The proposed estimation algorithm is based onthree-degree freedom vehicle model in vertical (see Figure 1),lateral, and horizontal directions.

The equations in these three movement directions are asfollows:

𝑎𝑥 = 1𝑚 [(𝐹𝑥1 + 𝐹𝑥2) cos 𝛿 + 𝐹𝑥3 + 𝐹𝑥4− (𝐹𝑦1 + 𝐹𝑦2) sin 𝛿 − 𝐹𝑤]

(1)

𝐹𝑤 = 12𝐶𝐷𝜌𝐴V2𝑥 (2)

𝑎𝑦 = 1𝑚 [(𝐹𝑥1 + 𝐹𝑥2) sin 𝛿 + (𝐹𝑦1 + 𝐹𝑦2) cos 𝛿 + 𝐹𝑦3+ 𝐹𝑦4]

(3)

𝑟 = 1𝐼𝑧 {𝑎 [(𝐹𝑥1 + 𝐹𝑥2) sin 𝛿 + (𝐹𝑦1 + 𝐹𝑦2) cos 𝛿]− 𝑏 (𝐹𝑦3 + 𝐹𝑦4)− 𝑡𝑓2 [(𝐹𝑥1 − 𝐹𝑥2) cos 𝛿 − (𝐹𝑦1 − 𝐹𝑦2) sin 𝛿]− 𝑡𝑟2 (𝐹𝑦3 − 𝐹𝑦4)}

(4)

where ax, ay, and r refer to the vertical accelerated velocity,lateral accelerated velocity, and horizontal angular velocity,respectively; m is the vehicle total mass; Fxi and Fyi are

Mathematical Problems in Engineering 3

u

v

C.m.

y

x

V

1

2

3

4

Fy1

Fy2

Fy3

Fy4

Fx1

Fx2

Fx3

Fx4

Figure 1: Three-degree freedom vehicle model.

longitudinal force and lateral force, respectively (i=1, 2, 3, 4refer to the left front wheel, right front wheel, left rear wheel,and right rear wheel); CD is the coefficient of air resistance;𝜌 is the density of the air; a and b are the distance betweenfront and rear axial and vehicle centroid, respectively; tf andtr are tracks of front wheel and rear wheel, respectively; 𝛿 isthe front wheel steering angle (assuming the steering anglesof both front wheels are the same); Iz is the moment of inertiawith respect to the roll axis;A the longitudinal windward areaof vehicle.

2.2. Wheel Model. This work mainly studied parameter esti-mation. Taking algorithm complexity, timelines, and modelaccuracy into consideration, Dugoff tire model was used.Longitudinal force and lateral force of tire are expressed withthe following formula:

𝐹𝑥 = 𝜇𝐹𝑧𝐶𝑥 𝑠1 − 𝑠𝑓 (𝐿)𝐹𝑦 = 𝜇𝐹𝑧𝐶𝑦 tan 𝛼1 − 𝑠 𝑓 (𝐿)

(5)

𝑓 (𝐿) = {{{(2 − 𝐿) 𝐿, 𝐿 < 11, 𝐿 ≥ 1 (6)

𝐿 =(1 − 𝑠) (1 − 𝜀V𝑥√(𝐶𝑥𝑠)2 + (𝐶𝑦 tan 𝛼)2)

2√(𝐶𝑥𝑠)2 + (𝐶𝑦 tan 𝛼)2(7)

whereCx is the longitudinal rigidity of tire;Cy is the corneringstiffness of tire; 𝛼 is the slip angle of tire; s is the longitudinalslip (shift) rate of tire; when boundary value L > 1, the tire isin the linear interval; when boundary value 𝐿 ≤ 1, the tire isnot in the linear interval; 𝜀 is the influential factor of speedwhich corrects the effect of sliding velocity on tire force; 𝜇is the tire-road friction coefficient; Fz is the vertical load oftire.

Here, Cx and Cy can be acquired by searching the table;𝛼 and s can be expressed as functions of parameters 𝛿, vx,

vy, 𝜔i, ax, and ay (see the specific equations in [25]). For theconvenience of realizing estimation algorithm, Formula (5)can be simplified as the following normalized form:

𝐹𝑥 = 𝜇𝐹0𝑥 = 𝜇𝐹𝑧𝐶𝑥 𝑠1 − 𝑠𝑓 (𝐿)𝐹𝑦 = 𝜇𝐹0𝑦 = 𝜇𝐹𝑧𝐶𝑦 tan 𝛼1 − 𝑠 𝑓 (𝐿)

(8)

where 𝐹𝑥0 and 𝐹𝑦0 are normalized force of longitudinal andlateral tires, respectively, which are irrelevant with tire-roadfriction coefficient.

3. Limited-Memory Adaptive ExtendedKalman Filter

3.1. Design of Estimator. For normal nonlinear system, theestimation model of status can be expressed with the follow-ing state-space equation:

𝑥 (t) = 𝑓 (𝑥 (𝑡) , 𝑢 (𝑡)) + 𝑤 (𝑡)𝑧 (𝑡) = ℎ (𝑥 (𝑡) , 𝑢 (𝑡)) + V (𝑡) (9)

where the random variables wk ∈ Rn and vk ∈ Rm are processnoise and measurement noise that are independent of eachother with Gaussian distribution p(w)∝N (0,Q) and p(v)∝N (0,R), with Q ∈ Rn×n and R ∈ Rm×m being the process andmeasurement covariance matrices, respectively.

This workmainly studied the real-time estimation of tire-road friction coefficient. With the friction coefficients of fourwheels as the state variables of system, x=[𝜇1, 𝜇2, 𝜇3, 𝜇4]T wasdefined. With axial acceleration, lateral acceleration, and yawrate of car measured by sensor (gyroscope) as observationalvariables of system, z=[ax, ay, r]T was defined. With frontwheel steering angle and longitudinal and lateral normalizedforce of each wheel as input variables for control, u=(𝛿,𝐹𝑥𝑖0, 𝐹𝑦𝑖0)T was defined. By simultaneous Formulas (1)-(8),the following state formula and observation formula can beobtained:

4 Mathematical Problems in Engineering

𝑥 (𝑡) = 𝜙𝑥 + 𝑤 (𝑡) = [[[[[

1 0 0 000

10

01

000 0 0 1]]]]]

[[[[[[

𝜇1𝜇2𝜇3𝜇4

]]]]]]+ 𝑤 (𝑡) (10)

𝑧 (𝑡) = 𝐻𝑥 + 𝜓𝐹𝑤 + V (𝑡) =[[[[[[[

𝐹0𝑥1 cos 𝛿 − 𝐹0𝑦1 sin 𝛿𝑚𝐹0𝑥2cos𝛿 − 𝐹0𝑦2 sin 𝛿𝑚 𝐹0𝑥3𝑚 𝐹0𝑥4𝑚𝐹0𝑥1 sin 𝛿 + 𝐹0𝑦1 cos 𝛿𝑚𝐹0𝑥2 sin 𝛿 + 𝐹0𝑦2 cos 𝛿𝑚

𝐹0𝑦3𝑚𝐹0𝑦3𝑚𝐻(3, 1) 𝐻 (3, 2) 𝐻 (3, 3) 𝐻 (3, 4)

]]]]]]]∙ [[[[[[

𝜇1𝜇2𝜇3𝜇4

]]]]]]+ [[[100]]]𝐹𝑤 + V (𝑡) (11)

where

𝐻(3, 1) = 𝑎 (𝐹0𝑥1 sin 𝛿 + 𝐹0𝑦1 cos 𝛿)𝐼𝑧− 𝑡𝑓/2 (𝐹0𝑥1 cos 𝛿 − 𝐹0𝑦1 sin 𝛿)𝐼𝑧

(12)

𝐻(3, 2) = 𝑎 (𝐹0𝑥2 sin 𝛿 + 𝐹0𝑦2 cos 𝛿)𝐼𝑧+ 𝑡𝑓/2 (𝐹0𝑥2 cos 𝛿 − 𝐹0𝑦2 sin 𝛿)𝐼𝑧

(13)

𝐻(3, 3) = − (𝑏𝐹0𝑦3 + 𝑡𝑟/2𝐹0𝑥3)𝐼𝑧 (14)

𝐻(3, 4) = − (𝑏𝐹0𝑦4 − 𝑡𝑟/2𝐹0𝑥4)𝐼𝑧 (15)

where 𝜙 is state-transition matrix; 𝐻 is the linear Jacobianmatrix; 𝜓 is the input matrix for control.

3.2. EKF Algorithm of Limited Memory. As standard KF,traditional EKF is also an infinitely increasing memory filter.When making the optimal estimation at the moment k, allthe data before the moment k should be used [26]. It meansthat, with the increase of k, the proportion of old data infilter will become larger, while that of new data is smaller.When there is model error and unknown time varying noise,the effect of new observation data is too small in correctingstate estimation. Therefore, the effects on state estimationcannot be inhibited effectively, causing error accumulated toexcessive large and even divergence [27]. For filter divergencecaused by model error, the effect of new observation datashould be enlarged to reduce the effect of old data on filtervalue.

In driving process, it is crucial to accurately estimatethe current tire-road friction coefficient in real time forcalculating tire force and developing the strategy for con-trolling vehicle stability. Particularly, old estimation may benot removed effectively in time, and the weight of newmeasurement may enlarge when estimating coefficient insudden changes. At this time, the state estimation may have a

large error. Such inherent error of model will certainly affectthe estimation of tire-road friction coefficient in nonlinearcondition. Therefore, the effects of current observation datashould be further enlarged.

With limited-memory filter, only the first N-1 observationvalues before the current moment will be used to calculatethe optimal estimation. In other words, the effects of olddata before the N-1th observation value can be completelyavoided. The timeliness of measurement data can be ensuredby confirming the memory length N according to previousknowledge. By combining limited-memory filter with EFKalgorithm, the accumulated error of old observation data canbe eliminated to further deduce the extended Kalman filterformula with limited memory:

(1) Further prediction of system state:

𝑋𝑁(𝑘,𝑘−1) = 𝜙(𝑘,𝑘−1)𝑋𝑁(𝑘−1,𝑘−1) + 𝜓𝑈(𝑘) (16)

(2) Further prediction of error variance matrix:

𝑃𝑁(𝑘,𝑘−1) = 𝜙(𝑘,𝑘−1)𝑃𝑁(𝑘−1,𝑘−1)𝜙𝑇(𝑘,𝑘−1) + 𝑄𝑘−1 (17)

(3) Gain matrix:

𝐾(𝑘) = 𝑃𝑁(𝑘,𝑘)𝐻𝑇(𝑘)𝑅−1𝑘 ,𝐾(𝑘) = 𝐾(𝑘−𝑁) = 𝑃𝑁(𝑘,𝑘)𝜙𝑇(𝑘−𝑁,𝑘)𝐻𝑇(𝑘−𝑁)𝑅−1𝑘−𝑁 (18)

(4) Composition error at moment k-N:

𝑅𝜏−1𝑘−𝑁= 𝑅−1𝑘−𝑁+ 𝐻(𝑘−𝑁){ 𝑘∑

𝑙=𝑘−𝑁+1

(𝜙𝑙−1𝑄𝑙−1𝜙𝑇𝑙−1)}𝐻𝑇(𝑘−𝑁)(19)

(5) Residual sequence:

𝑍(𝑘) = 𝑍(𝑘) − 𝐻(𝑘)𝜙(𝑘,𝑘−1)𝑋𝑁(𝑘,𝑘−1),𝑍(𝑘−𝑁) = 𝑍(𝑘−𝑁) − 𝐻(𝑘−𝑁)𝜙(𝑘−𝑁,𝑘)𝜙(𝑘,𝑘−1)𝑋𝑁(𝑘,𝑘−1)

(20)

Mathematical Problems in Engineering 5

(6) Optimal estimation of system:

𝑋𝑁(𝑘,𝑘) = 𝑋𝑁(𝑘,𝑘−1) + 𝐾(𝑘)𝑍(𝑘) − 𝐾(𝑘−𝑁)𝑍(𝑘−𝑁) (21)

(7) Error variance matrix of system:

𝑃𝑁(𝑘,𝑘) = [𝑃𝑁−1(𝑘,𝑘−1) + 𝐻𝑇(𝑘)𝑅−1𝑘 𝐻(𝑘)− 𝜙𝑇(𝑘−𝑁,𝑘)𝐻𝑇(𝑘−𝑁)𝑅𝜏−1𝑘−𝑁𝐻(𝑘−𝑁)𝜙(𝑘−𝑁,𝑘)]−1

(22)

where Q and R are system noise and observation noise,respectively; N is the length of limited memory.

When k ≤ N, the observation time k is less than memorylength N, and only the extended Kalman formula can be usedwithout limited-memory filter. The calculation starts fromthe initial values 𝑋(0, 0) = 𝐸{𝑋(0)} and 𝑃(0, 0) = var{𝑋(0)}to values 𝑋(𝑁,𝑁) and 𝑃(𝑁,𝑁). However, starting from k= N+1, the limited-memory filter 𝑋(𝑁)(𝑘, 𝑘) will be affectedby 𝑋(0, 0) and 𝑃(0, 0) when taking 𝑋(𝑁)(𝑁,𝑁) = 𝑋(𝑁,𝑁)and 𝑃(𝑁)(𝑁,𝑁) = 𝑃(𝑁,𝑁) as initial values for limited-memory filtering.This does not satisfy limited-memory filter.Therefore, effects of initial values 𝑋(0, 0) and 𝑃(0, 0) on𝑋(𝑁)(𝑘, 𝑘)(k > N) should be avoided. When setting k ≤ N, let𝐻(𝑘−𝑁) = 0. At this time,𝑋(𝑁)(𝑘, 𝑘) is the filter of EKF. Whenk > N, let 𝑃−1(0,0) = 0, 𝑋(0, 0) = 0.3.3. Fuzzy Kalman Filter Algorithm. In actual vehicle sys-tems, process noise and measurement noise are constantlychanging. The process noise is affected by factors such as theoperating conditions of the vehicle itself and environmentalconditions. The measurement noise is affected by the accu-racy of the sensor and the environmental conditions of themeasurement. Vehicles are highly nonlinear systems, and theactual operating conditions of the vehicle are complicated,which makes it difficult to predict and accurately measurevehicle noise. Since the vehicles studied mainly operate inconventional urban working conditions, process noise can bedetermined by the statistical characteristics of noise obtainedthrough experiments. In this case, we assume that processnoise can be considered to be a constant value. On theother hand, considering the accuracy of the accelerationsensor owned by the actual vehicle, it is considered that themeasurement noise has a greater impact on the vehicle systemwe are currently testing. Therefore, it is necessary to focus oneliminating the adverse effect of measurement noise.

In actual driving, the statistical characteristics of obser-vation noise are changing. Covariance matrix of measurednoise Rk is dynamically adjusted by fuzzy interference systemto realize the self-adaptive regulation of filter [28]. The workdefines the observation noise variance Rk at moment k,making it satisfy 𝑅k = 𝑅k−1 + Δ𝑅.

The specific steps of self-adaptive adjustment of fuzzyKalman are as follows.

(1) Calculate the theoretical value of residual covariancein extended Kalman filter algorithm:

𝐾(𝑘) = 𝑃𝑁(𝑘,𝑘)𝐻𝑇(𝑘)𝑅−1𝑘𝑃(𝑘) = [𝐼(n) − 𝐾(k)𝐻(𝑘)] 𝑃(𝑘−1) (23)

That is,

𝑃(𝑘) = [𝐼(n) − 𝑃𝑁(𝑘,𝑘)𝐻𝑇(𝑘)𝑅−1𝑘 𝐻(𝑘)] 𝑃(𝑘−1) (24)

(2) Appropriately calculate the actual residual covariance.Residual error is 𝑟𝑘 = 𝑧𝑘 − 𝐻𝑇𝑘 𝑥𝑘/𝑘−1 and the mean value is𝑟 = (1/𝑛)∑𝑘𝑗=𝑘−𝑛+1 𝑟𝑗, so the actual residual variance is 𝐶𝑘 =(1/𝑛)∑𝑘𝑗=𝑘−𝑛+1 𝑟𝑗𝑟𝑇𝑗 ∘.

(3) Use fuzzy interference engine for self-adaptive adjust-ment of 𝑟𝑘. Define the matching rate e (k) between variants𝑃𝑧 and 𝐶𝑘 as 𝑒(k) = 𝑃𝑧 − 𝐶𝑘.

In fuzzy control machine, the difference e(k) betweenactual variance and theoretical variance is taken as input andthe regulating variable of measurement Δ𝑅 as output. Theadjustment amount of Δ𝑅 is obtained by repeating a largenumber of experiments. Figure 2 shows the subordinatingdegree functions of input and output.

The fuzzy subsets of input and output are defined asfollows [29]:

(1) If 𝑒(k) ≈ 0 (which means the actual variance is closeto theoretical variance), 𝑟𝑘 keeps constant; that is, if (k) ∈ ZE,then Δ𝑅 ∈ 𝑀.

(2) If 𝑒(k) < 0 (which means the actual variance is largerthan theoretical variance), reduce 𝑟𝑘; that is, if 𝑒(k) ∈ 𝑁, thenΔ𝑅 ∈ 𝐷.

(3) If 𝑒(k) > 0 (which means the actual variance is lessthan theoretical variance), increase 𝑟𝑘 ; that is, if 𝑒(k) ∈ 𝑃, thenΔ𝑅 ∈ 𝐼.

The defuzzification method used in this method is thecentroid method.

4. Conclusions

All experimental maneuvers are driven with a type of all-wheel drive vehicle. Using an inertial measurement unit,precise measurements of yaw-rate, sideslip angle, longitudi-nal and lateral accelerations, and velocity of the c.o.g. aretaken. The measurements of the wheel’s rotational velocityare obtained reading out the CAN bus. To test the validity ofdesigned estimation algorithm, Matlab/Simulink and Carsimsoftware were used for joint simulation experience.

The system status variable was initially set as 𝑥 =[0.001 0.001 0.001 0.001]𝑇 in the beginning of simulation,while the covariance matrices are initialized as

Q0=diag[10−9;10−9;10−9;10−9],

P0 = diag[10−6;10−6;10−6;10−6],

R0=diag[10−4;10−4;10−4].

The longitudinal and lateral accelerated speed and yawrate output by Carsim were taken as measurement valuesof system with Gaussian white noise; corner of steeringwheel and normalized force of each wheel were taken as thecontrol input to system. Then, the current tire-road frictioncoefficient was estimated in real time; LM-AEKF estimationand traditional EKF estimation were compared with theoutput of Carsim.

6 Mathematical Problems in Engineering

−5 −4 −3 −2 −1 0 1 2 3 4 5

0.5

1.0

0.5

1.0N ZE P

(a) Subordinating degree functions of input

0−0.2 −0.1 0 0.1 0.2

0.5

1.0

0.5

1.0D M I

(b) Subordinating degree functions of output

Figure 2:The subordinating degree functions of input and output.

−2

0

2

4

6

8

10

Ax(Ｇ

/Ｍ2)

1 2 3 4 50Time (s)

Figure 3:The longitudinal accelerated measurement with Gaussian white noise.

4.1. Docking Road Test. The acceleration test with startingspeed of 30 km/h was set for a vehicle on the road of suddenchange with tire-road coefficient between 0.2 and 0.8 inCarsim. The road length with tire-road friction coefficient of0.2 is 25 meters; the openness of accelerator pedal is 50%; thesteering wheel angle in test keeps 0.

Figure 3 shows the longitudinal accelerated measurementwith Gaussian white noise. Figure 4 shows the estimation oftire-road friction coefficient by traditional EKF algorithm,UKF algorithm, and LM-AEKF algorithm.

According to simulation results, both algorithms havegood estimation effects before the sudden change of tire-road friction coefficient. However, when the car enters newroad, severe buffeting occurs to estimation by traditionalEKF algorithm. Moreover, the estimation of tire-road frictioncoefficient tends to have larger error and even be convergentwith time. Compared with EKF algorithm, UKF algorithmestimation results have less fluctuation, and the results tendto converge, which can reach the steady-state value morequickly. On the contrary, LM-AEKF algorithm not only elim-inates the buffeting in estimation, but also rapidly convergesthe real value. Therefore, LM-AEKF has better accuracy andstability.

4.2. Double Lane Change Test. The double lane change testwith constant speed of 85 km/h was set for a vehicle on the

road with tire-road coefficient of 0.8 in Carsim. Figures 5 and6 show the measurement of lateral accelerated speed and yawrate with Gaussian white noise, respectively. Figure 7 showsthe estimation of tire-road friction coefficient by traditionalEKF algorithm, UKF algorithm, and LM-AEKF algorithm.According to simulation results, the vehicle is in the nonlineararea at about 4 s. By traditional EKF algorithm, the estimationgreatly changes with the most severe buffeting near nonlineararea. Converging slowly, there is a large difference betweenstable value and real value. UKF estimation results alsoshow obvious fluctuations. However, compared with EKFalgorithm, UKF estimation results can rapidly converge andreach steady-state values more quickly. On the contrary,by LM-AEKF algorithm, the estimation converges rapidly,greatly improving the overall accuracy and stability.

4.3. Bisectional Road Test. In Carsim, the vehicle was drivenstraightly on a flat bisectional road at a uniform speed of 40km/h, wherein the roadside adhesion coefficients of the leftand right sides were 0.2 and 0.8, respectively.

Figure 8 shows the measurement of longitudinal accel-erated speed with Gaussian white noise. Figures 9 and 10respectively show the estimation of tire-road friction coeffi-cient by traditional EKF algorithm, UKF algorithm and LM-AEKF algorithm. Among them, the left wheels are driven ona road surface with a tire-road friction coefficient of 0.2, and

Mathematical Problems in Engineering 7

CarsimLM-AEKF

EKFUKF

1 2 3 4 50Time (s)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Road

fric

tion

coeffi

cien

t

Figure 4: The estimation of tire-road friction coefficient.

−5

−4

−3

−2

−1

0

1

2

3

4

Ay(Ｇ

/Ｍ2)

2 4 6 8 100Time (s)

Figure 5: Lateral accelerated speed with Gaussian white noise.

2 4 6 8 100time (s)

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Yaw

rate

(rad

/s)

Figure 6: Yaw rate with Gaussian white noise.

8 Mathematical Problems in Engineering

CarsimLM-AEKF

EKFUKF

2 4 6 8 100Time (s)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Road

fric

tion

coeffi

cien

t

Figure 7: The estimation of tire-road friction coefficient.

−0.3

−0.2

−0.1

0

0.1

0.2

Ax(Ｇ

/Ｍ2)

3 60 41 2 5Time (s)

Figure 8: The longitudinal acceleratedmeasurement with Gaussian white noise.

the right wheels are driven on a road surface with a tire-roadfriction coefficient of 0.8 According to simulation results,by traditional EKF algorithm, the estimation results greatlychange with the most severe buffeting and converge slowly.Obviously, there is a large difference between stable valueand real value. By UKF algorithm, the estimation value canapproach the steady-state value more quickly, the vibrationamplitude is relatively gentle, and the effect is improved.On the contrary, by LM-AEKF algorithm, the estimationconverges rapidly, greatly improving the overall accuracy andstability.

5. Vehicle Test

To further verify the effectiveness of LM-AEKF algorithm totire-road friction coefficient estimation, this work conducted

vehicle test based on a type of four-wheel independent drivelight-duty off-road vehicle. Steering wheel angle, rotationalspeed of each tire, and longitudinal velocity required in thistest were collected from the sensor of steering wheel angle,sensor of wheel speed, and GPS. In addition, the longitudinaland lateral accelerated speeds and yaw rate were collectedby gyroscope in sampling time of 10 ms. Lateral speedcannot be directly measured, while error accumulation maybe caused in direct measurement of lateral accelerated speed.Therefore, standard Kalman filter processing was used forlateral accelerated speed in this work. Above values measuredwith instrument can be loaded to CAN bus and transferred toupper computer software, controldesk, through D/A modelof dSPACE by I/O interface. Therefore, the data monitoredin real time can be collected and saved. Figures 11 and 12show the testing vehicle, equipment, and data processing flow.

Mathematical Problems in Engineering 9

CarsimLM-AEKF

EKFUKF

1 2 3 4 5 60Time (s)

0.00

0.05

0.10

0.15

0.20

0.25

Left

Fron

t-Whe

el:R

oad

fric

tion

coeffi

cien

t

Figure 9: The estimation of tire-road friction coefficient of left front-wheel.

CarsimLM-AEKF

EKFUKF

1 2 3 4 5 60Time (s)

0.0

0.2

0.4

0.6

0.8

1.0

Righ

t Fro

nt-W

heel

:Roa

d fr

ictio

n co

effici

ent

Figure 10: The estimation of tire-road friction coefficient of right front-wheel.

Figure 11: The testing vehicle.

10 Mathematical Problems in Engineering

GPS Sensor gyroscope

dSPACE and PC

steering wheelangle sensor

vehicledynamic model

LM-AEKF/EKF

comparativeanalysis

actual estimated

t i h l

x,y

3,4

1,2

, rax,ay

Figure 12: The testing equipment and data processing flow.

2 4 6 8 10 12 140Time (s)

3

5

7

9

11

13

Vx

(m/s

)

Figure 13: Longitudinal velocity.

With information collected in the test, the designed LM-AEKF algorithm can be realized in Matlab/Simulink. Then,the estimation by LM-AEKF algorithm can be compared withthat by UKF algorithm and EKF algorithm.

5.1. Docking Road Test. For vehicle test, watering ABS roadand dry, flat, and clean road of cement concrete were selectedas low-friction road (the friction coefficient is between 0.22and 0.25) and high-friction road (the friction coefficientis between 0.78 and 0.8), respectively, in standard test site(see Figure 11 for road test). To take advantage of roadconditions and highlight the conditional adaptation of algo-rithms, low-friction acceleration test and low-friction andhigh-friction sliding test were continuously conducted fortesting vehicle. Figures 13–15 show the longitudinal velocity,longitudinal acceleration, wheel speed, and other informa-tion.

Figure 16 shows the estimation of algorithms. It can beseen that the estimation results of UKF algorithm and LM-AEKF algorithm are generally stable and accurate. After thevehicle entered high-friction road at about 10 s, UKF algo-rithm and LM-AEKF algorithm responded rapidly, showingthat both two algorithms have good adaptation to sud-den changes. On the contrary, tire-road friction coefficientestimated by traditional EKF algorithm is less stable forthe slower response to sudden changes. After meeting new

200 400 600 800 1000 1200 14000Time (s)

−2

−1.5

−1

−0.5

0

0.5

1

1.5

Ax

Figure 14: Longitudinal acceleration.

conditions, the difference between convergence and actualvalue tends to increase because of the accumulated error.

5.2. Double Lane Change Test. Double lane change testwas conducted for vehicle at constant speed according toISO 3888-1-1999 on the horizontal dry asphalt road in test

Mathematical Problems in Engineering 11

LFRF

LRRR

2 4 6 8 10 12 140Time (s)

50

100

150

200

250

300

350

400

450

Whe

el sp

eed

(r/m

in)

Figure 15: Wheel speed.

2 4 6 8 10 12 140Time (s)

0.0

0.2

0.4

0.6

0.8

1.0

Road

fric

tion

coeffi

cien

t

LM-AEKFEKFUKF

Figure 16: The estimation of tire-road friction coefficient.

field (with friction coefficient of about 0.8) [30]. Consider-ing the security, the testing speed was 45km/h (±1km/h).Figures 17–19 show some measurements recorded by GPSand gyroscope. Figure 20 shows the estimation of tire-roadfriction coefficient. According to the test results, tire-roadfriction coefficients estimated by UKF algorithm and LM-AEKF algorithm are stable with fast convergence and highaccuracy. However, in the initial stage of estimation, thefluctuation of the UKF algorithm is more obvious than theLM-AEKF algorithm, and the convergence speed of the UKFalgorithm is also slightly slower, but the steady-state valueof the UKF estimation result approaches the actual value.On the contrary, the coefficient estimated by traditional EKF

algorithm changes greatly with slow convergence and somedifference from actual value.

6. Conclusions

Vehicle model has errors and cannot obtain statistical char-acteristics of time-varying noise. Therefore, this work pro-posed EKF algorithm that can fade memory by exponentialweighting based on traditional EKF algorithm. Then, tire-road friction coefficient was estimated in real time with thisalgorithm. By comparing the results of simulation tests by twoalgorithms in different conditions, conclusions are drawn asfollows:

12 Mathematical Problems in Engineering

0 2 4Time (s)

6 8 10 12

Vx

(m/s

)

12.6

12.5

12.4

12.3

12.2

12.7

Figure 17: Longitudinal velocity.

0 2 4 6 8 10 12−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (s)

Ay(Ｇ

/Ｍ2)

Figure 18: Lateral acceleration.

0 2 4 6 8 10 12−0.2

−0.1

0

0.1

0.2

0.3

Yaw

rate

(rad

/s)

Time (s)

Figure 19: Yaw rate.

Mathematical Problems in Engineering 13

LM-AEKFEKFUKF

2 4 6 8 10 120Time (s)

0.0

0.2

0.4

0.6

0.8

1.0

Road

fric

tion

coeffi

cien

t

Figure 20:The estimation of tire-road friction coefficient.

(a) The algorithm proposed in this work can betterestimate current tire-road friction coefficient in differentconditions. In the same condition, the proposed algorithmhas better estimation accuracy and stability with fasterresponse speed than traditional EKF algorithm.

(b) LM-AEKF algorithm is easy to be operated withoutaffecting timeliness. This algorithm has high estimationaccuracy and filter stability even if the model has errors andnoise statistics is unknown. It means that this algorithm isself-adaptive.

Data Availability

The data that support the findings of this study are availableon request from the corresponding author. The data are notpublicly available due to their containing information thatcould compromise the privacy of research participants.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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