Design of an Interacting Multiple Model-Cubature Kalman

Research ArticleDesign of an Interacting Multiple Model-CubatureKalman Filter Approach for Vehicle Sideslip Angle andTire Forces Estimation

SuoJun Hou12 Wenbo Xu1 and Gang Liu 1

1College of Vehicle and Transportation Engineering Henan Institute of Technology XinXiang China2State Key Laboratory of Automotive Simulation and Control Jinlin University Changchun 453000 China

Correspondence should be addressed to Gang Liu gliu14mailsjlueducn

Received 2 December 2018 Revised 14 April 2019 Accepted 2 May 2019 Published 20 June 2019

Academic Editor Andras Szekrenyes

Copyright copy 2019 SuoJun Hou et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Vehicle states estimation (eg vehicle sideslip angle and tire force) is a key factor for vehicle stability control However the accuratevalues 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 thispaper Firstly the accuracy of the reference model is improved by building two different models one is 7-degree-of-freedom (7DOF) vehicle model with linear tire model and the other is 7 DOF vehicle model with nonlinear Dugoff tire model Secondly thedifferent models are switched by IMM-CKF to match different driving condition Thirdly the lateral acceleration correction forsideslip angle estimation is considered because the sensor of lateral acceleration is easy to be influenced by the gravity on bankedroad 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-CKFmethodology can provide accurate estimationvalues of vehicle states parameters

1 Introduction

With the development of the electronic and automotivetechnology the vehicle stability control is also undergoingcontinuous progress Vehicle stability control plays an impor-tant role in active vehicle safety control The performanceof vehicle stability control is determined by the accuracy ofvehicle state (eg vehicle sideslip angle and tire force) becausethe vehicle state parameters show the potential of vehiclestability However the vehicle state parameters such as thesideslip angle and tire force cannot be measured directly forboth technical and economic reasons Therefore the vehiclestate parameter must be estimated by a variety of algorithmmethods

Recently many estimation approaches based on standardESP sensors (eg steering angle yaw rate longitudinal andlateral acceleration and wheel-speed) have been proposed inthe literature to estimate vehicle state parameters [1] Theseapproaches can be divided into two types according to thevehicle dynamic models Type 1 uses the linear vehicle model

to estimate the sideslip angle and tire-road forces Based onthe linear vehicle model known as the single-track modelwith a linear tire-force model the estimation approach ofvehicle state works well under normal driving condition [2ndash4] because the vehicle state depends on the tire force and theeffect of tire force can change the trajectory of the vehicleFigure 1 shows the relation between the tire force and thewheel slip angle Under the normal driving condition thevalue of wheel slip angle is small Therefore the curve whichreflects the relationship between the tire force and the wheelslip angle is initially linear Many algorithms such as thevariable forgetting factor recursive least squares algorithm[5] Luenberger linear Observer [6] 119867infin filtering [7] andKalman filters [8 9] based on the linear model have beenused to estimate the vehicle state parameters However thevehicle cannot always be under normal driving conditionUnder extreme driving conditions the value of vehicle statetends to nonlinear growth Figure 1 shows that as the wheelslip angle grows the tire force gradually enters into thenonlinear region which is caused by the limited friction on

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 6087450 13 pageshttpsdoiorg10115520196087450

2 Mathematical Problems in Engineering

0 10Wheel slip angle (deg)

0

5La

tera

l for

ce (K

N)

Burckhardtasphalt wetBurckhardtasphalt dry

Burckhardt cobblestone wetBurckhardtice

LinearLinear

5

Figure 1 Lateral force curve

the road surface especially on a low friction road Conse-quently the parameters estimated based on linear vehiclemodel are error due to the tirersquos inherent nature Comparedto type 1 type 2 uses the estimation approach based onthe nonlinear vehicle model [10 11] which is more close toreflect the vehicle dynamics nature to predict the vehiclestate under extreme driving conditions A lot of literatureshave been proposed to estimate the parameters based on thenonlinear vehicle model Liang Li and Gang Jia proposed avariable structure extended Kalman filter with the sideslipangle rate feedback to estimate the sideslip angle [12] Theirsimulations and vehicle winter test show that the proposedapproach can provide accurate value of sideslip under lowfriction road conditions Yu andKhGuo proposed a reduced-order sliding mode observer for vehicle state estimation [1314] The precise nonlinear tire model ldquoUniTirerdquo is used toimprove the accuracy of vehicle states parameters estima-tion A novel approach is proposed to estimate parametersby using a mathematical tool which includes a nonlinearMagic tire model [15] BLBoada proposed a novel observerbased on adaptive Neuro-Fuzzy Inference System (ANFIS)combined with Unscented Kalman Filter in order to estimatethe sideslip angle and tire force [16] Wei L proposed anestimation approach based on the Extended Kalman Filter(EKF) combing with the Minimum Model Error (MME)criterion for vehicle state estimation [17] However theestimation based on nonlinear vehicle model is not suitablefor parameters estimations under normal driving conditionsbecause of the excessive computational power in embeddedsystems In sum the time to choose an optimal model forvarious driving conditions is also important besides thechoice of vehicle model The multiple model (MM) methodtakes advantage of several models to express possible vehiclemodel under different driving conditions which can achievemore accurate vehicle state parameters by combined filtersbased on corresponding vehicle model of a limited numberof different models compared with single vehicle modelAmong several MM estimation algorithms the interacting

12

21

22

Fx12 Fx12Fy22Fx22

11

Fy21Fy11 Fx11

Fx21

x

y

gBf

Lr Lf

Figure 2 7-DOF vehicle model

MM (IMM) has attractedwide attention because it can obtainaccurate calculation results with low computational load [18]

In this paper we use the interacting multiple model(IMM) to choose an optimal model for various driving con-ditions In the IMM method two filters are used in parallelto estimate sideslip angle and tire force One filter is CKFfilter based on four-wheel nonlinear vehicle dynamics modelwith linear tire model for normal driving conditions andthe other is based on four-wheel nonlinear vehicle dynamicsmodel with nonlinear Dugoff tire model for extreme drivingconditions The sideslip angle and tire force predicted by thevariable structure IMM-CKF filter aremore accurate than thesingle filter for various driving conditions

This paper is organized as follows In Section 2 thedynamics models of the vehicle linear tire and nonlinearDugoff tiremodel are presented respectively In Section 3 thedetailed structure of IMM-CKF is represented In Section 4the key dynamics variable is given in detail In Section 5 thesimulations are presented to validate the performance of theestimation method

2 Design of Vehicle Dynamics Model

21 Vehicle Model In Figure 2 the 7-DOF vehicle dynamicsmodel is proposed to estimate the vehicle state parametersThe 7-DOF vehicle model includes longitudinal motionlateral motion yaw motion and 4-wheel rotation [13]

The equation of longitudinal motion is expressed asfollows

119898(V119909 minus V119910) = 2sum119894119895=1

119865119909119894119895= (11986511990911 + 11986511990912) cos 120575 + (11986511990921 + 11986511990922)minus (11986511990911 + 11986511990912) sin 120575 = 119898119886119909

(1)

The equation of lateral motion is expressed as follows

119898(V119910 + V119909) = 2sum119894119895=1

119865119910119894119895= (11986511991011 + 11986511991012) cos 120575 + (11986511991021 + 11986511991022)minus (11986511991011 + 11986511991012) sin 120575 = 119898119886119910

(2)

Mathematical Problems in Engineering 3

The equation of yaw motion is expressed as follows

119868119911 = (11986511991011 + 11986511991012) 119871119891 cos 120575 + (11986511991011 minus 11986511991012) 119871119903 sin 120575minus (11986511991021 + 11986511991022) 119871119903 minus (11986511990911 minus 11986511990912) 119871119891 sin 120575minus (11986511990911 minus 11986511990912) 1198611198912 cos 120575 minus (11986511990921 minus 11986511990922) 1198611198912

(3)

The equation of wheel motion is expressed as follows119869119908119894119895 = 119879119894119895 minus 119901119894119895119870119875119879119894119895 minus 119865119909119894119895119877 (4)

The equation about sideslip angle can be expressed as follows

119898( 120573 + ) V119892 = (11986511991011 + 11986511991012) cos (120573 minus 120575)+ (11986511991011 + 11986511991012) cos120573minus (11986511990911 + 11986511990912) sin (120573 minus 120575)minus (11986511990911 + 11986511990912) sin120573

(5)

119898V119892 = (11986511990911 + 11986511990912) cos (120573 minus 120575)+ (11986511990921 + 11986511990922) cos120573+ (11986511991011 + 11986511991012) sin (120573 minus 120575)+ (11986511991011 + 11986511991012) sin120573

(6)

where m represents the mass of the vehicle 120573 denotes thesideslip angle 120575denotes the front steering angle V119909 representsthe longitudinal velocity V119910 is the lateral velocity of thevehicle 119865119909119894119895 denotes the longitudinal tire force of 4 wheelsrespectively 119865119910119894119895 denotes the lateral tire force of 4 wheelsrespectively 119871119891 and 119871119903 are the distances from the front andrear axle to the gravity center respectively 119861119891 representstrack width of the vehicle 119869119908 denotes the wheel inertia 119879119894119895denotes driving torque 119901119894119895 denotes the brake pressure ofwheels 119870119875119879119894119895 denotes the coefficient of the brake pressure120596119894119895 denotes the speed of 4 wheels respectively 119868119911 is the yawmoment of inertia R is the wheel radius 120593 is yaw angle

22 TireModel When the vehicle is turning the longitudinalacceleration and the later acceleration will have an effect oneachwheel because of the load transferTherefore the verticalload of 4 wheels can be expressed as follows

11986511991111 = 1198981198921198711199032119871 minus 119898ℎ1198881198921198861199092119871 minus ℎ119888119892119871119903119871119861119891 11989811988611991011986511991112 = 1198981198921198711199032119871 + 119898ℎ1198881198921198861199092119871 + ℎ119888119892119871119891119871119861119891 11989811988611991011986511991121 = 1198981198921198711198912119871 + 119898ℎ1198881198921198861199092119871 + ℎ119888119892119871119891119871119861119891 11989811988611991011986511991122 = 1198981198921198711199032119871 + 119898ℎ1198881198921198861199092119871 minus ℎ119888119892119871119891119871119861119891 119898119886119910

(7)

where 119865119911119894119895 denotes the vertical force of 4 wheels ℎ119888119892 denotesthe distance between the gravity center and the ground 119886119910

denotes the lateral acceleration 119886119909 denotes the longitudeacceleration L denotes the distances between the front axleand rear axles

As discussed in Section 1 in case of vehicle driving undernormal driving conditions the relationship between thelateral force and the tire slip angle is linear On the other handunder extreme driving conditions the lateral acceleration ofvehicle tends to be big and the tire operates in the nonlinearregion Therefore in this paper linear tire model and thenonlinear tire model are established

221 Linear Tire Model The linear tire model proposed bythe authors [19] is adopted in this paper The tire slip anglescan be formulated as follows

[[[[[[

12057211120572121205722112057222

]]]]]]= [[[[[[

12057512057500

]]]]]]minus [[[[[[

12058511120585121205852112058522

]]]]]]

(8)

where 120585119894119895 can be expressed as

[[[[[[

12058511120585121205852112058522

]]]]]]= tanminus1

[[[[[[[[[[[[[[[

V119910 + 119871119891V119909 minus 1198611198912V119910 + 119871119891V119909 + 1198611198912V119910 minus 119871119903V119909 minus 1198611199032V119910 minus 119871119903V119909 + 1198611199032

]]]]]]]]]]]]]]]

asymp

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

(9)

At small slip angle the lateral tire force is proportion to thetire slip angleThe equation of lateral tire force can be writtenas follows

119865119910119894119895 = minus119862119910119894119895120572119894119895 (10)where 119862119910119894119895 represents cornering stiffness of 4 wheels respec-tively

The equation of longitudinal tire force can be expressedas

119865119909119894119895 = 119896120583120582119894119895119865119911119894119895 (11)where 120582119894119895 denotes the slip ratio of 4 wheels respectively 119896120583 isthe proportional coefficient

222 Nonlinear Tire Model Under the extreme drivingconditions such as a low friction road surface the tire is easyto enter into nonlinear state even at low lateral accelerationThus the Dugoff nonlinear tiremodel is used to represent thetire forces in nonlinear region [20]

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

119865119910119894119895 = 119862119910119894119895 tan1205721198941198951 minus 120582119894119895 119891 (119878) (12)

119865119909119894119895 = 1198621199091198941198951205821198941198951 minus 120582119894119895 119891 (119878) (13)


Steering-wheel-angle sensors

Yaw-rate sensors

Accelerationsensor

Wheel-speed sensors

The CKF filter based on vehicle mode of linear tire

The SRCKF filter based on vehicle mode of non-linear tire

Sideslip angle lateral tire force

IMM algorithm

11 12

21 22

ax ay

ℎ

Figure 3 Scheme of the IMM-CKF method for vehicle state estimation

119878 = 120583119865119911119894119895 (1 minus 120576V119909radic1205821198941198952 + tan2120572119894119895)

2radic11986211989411989521205821198941198952 + 1198621198941198952tan2120572119894119895 (1 minus 1205821198941198952) (14)

119891 (119878) = 1 119878 gt 1119878 (2 minus 119878) 119878 lt 1

120582119894119895 = 119877119908120596119894119895 minus V119909max (119877119908120596119894119895 V119909)

(15)

where 120576 denotes roll steer coefficient

223 Modified Tire-Road Force Considering the time lagof tire force Reference [21] proposed a relaxation length torepresent the transient behavior of tires The modified tire-road force can be expressed as

119910119894119895 = V119892120590119894119895 (minus119865119910119894119895 + 119865119910119894119895) (16)

119909119894119895 = V119892120590119894119895 (minus119865119909119894119895 + 119865119909119894119895) (17)

where 119865119910119894119895 denotes the quasi-static lateral Dugoff tire force120590119894119895 denotes the relaxation coefficient

3 IMM-CKF

The algorithm structure of vehicle state estimation proposedin this paper is shown in Figure 3 The whole algorithmstructure can be regarded as hierarchical strategy and it canbe divided into 2 layers The first layer is signal layer In thefirst layer the signals obtained by the sensors or derived fromother systems are transmitted to next layers for estimationand these signals include the longitudinal acceleration 119886119909

the lateral acceleration 119886119909 yaw rate longitudinal speedV119909 speed of each 120596119894119895 steer angle 120575ℎ and break pressure ofeach wheel cylinder 119901119894119895 The second layer is responsible forcalculation ofweight according to the road friction coefficientand vehicle lateral acceleration

In the second layer the IMM-based estimation layercalculates the model switch probabilities and integrates theCKF estimation of each model by stochastic process to adaptto various driving conditions The further algorithm aboutIMM-CKF can be found in Reference [22]

It is appropriate that the filter of 7-DOF vehicle modelbased on linear tire model is used under the normal workconditions because the relationship of tire slip angle andlateral force is linear and there is a small amount of computa-tions for embedded system In contrast the relationship of thetire slip angle and the lateral force is no longer linear underextreme driving conditionsThefilter of 7-DOF vehiclemodelbased on nonlinear tire model can provide better predictionperformance The process of IMM-CKF can be described inFigure 4 The IMM-CKF can be broken down into 4 steps

Step 1 (interaction reinitialization) Compute the mixedprobabilities and the initial condition in first step Initialmean and covariance for each CKF filter model can beexpressed as follows

119909(119894)119896minus1|119896minus1 = 119864 (119909119896minus1 | 119898(119894)119896 119885119896minus1) =119903sum119894=1

119909(119894)119896minus1|119896minus1120583(119894119895)119896minus1|119896minus1 (18)

119901(119894)119896minus1|119896minus1 = 119903sum119894=1

[119875(119894)119896minus1|119896minus1+ (119909(119895)119896minus1|119896minus1

minus 119909(119895)119896minus1|119896minus1

) (119909(119895)119896minus1|119896minus1


)119879]sdot 120583(119894119895)119896minus1|119896minus1

(19)


Previous Step

Initialconditions

Model 1The CKF filter

based on vehicle mode of linear

tire

Modeprobability

Update


based onvehicle mode ofnon-linear tire

1kminus

11kp minus

2| 1kminus

21kP minus

1|2k

2|1k

( ) ( )1

1i j ik ij k

Cπ minus=

1|1k

2|2k

21kx minus

11kx minus

11kp minus

21kp minus

1| 1k kminus

1| 1k kp minus

2| 1k kminus

2| 1k kp minus

1k1kp2k2kp

Estimation Fusion

kkp

Step 2 Step 3 Step 4 Step 1

x

x

x x

xx

x1k

2k

Figure 4 The process of IMM-CKF

where 120583(119894119895)119896minus1|119896minus1

= 119875(119898(119894)119896minus1

| 119898(119894)119896119885119896minus1) = (1119862)120587119894119895120583(119894)119896minus1 119894 119895 =1 2 119903120583(119894119895)

119896minus1|119896minus1is mixing probabilities of model i the model

switch probabilities of the IMMfilter between the twomodelsare due to the Markov Process which is determined bythe transition probability matrix 120587119894119895 The model transitionprobability matrix 120587119894119895 denotes the transition probability ofvehicle model from model 119895 to model 119894 The parameter 120587119894119895can be expressed as

120587119894119895 = [099 001002 098] (20)

where 119895119894 denote the 7-DOF vehicle model with linear tiremodel and 7-DOF vehicle model with nonlinear Dugoff tiremodel respectively These values of model switch probabilitymatrix were obtained by a statistical method which related tothe sampling time interval under the real driving condition[23ndash26]

Step 2 (mode-filtering using CKF and then computationof the model probability update) In this step the CKF isused to obtain each model state 119909119896|119896 and covariance 119875119896|119896which is based on the previous mixing state 119909(119894)

119896minus1|119896minus1and

the covariance 119901(119894)119896minus1|119896minus1 The overall flowchart of the CKF forvehicle model 119894 is presented in Figure 5The state equation ofthe vehicle model with the linear tire model is similar to thatof the vehicle model with nonlinear Dugoff tire model Thestate equation of the vehicle model with the linear tire modelincludes equations (1) (2) (3) (4) (5) (6) (9) (10) and (11)The state equation of the vehiclemodel with nonlinearDugofftire model includes equations (1) (2) (3) (4) (5) (6) (9)(12) and (13) Both state equations can be expressed as

(119905) = 119891 (119909 (119905) 119906 (119905)) + 119882 (119905)119910 (119905) = ℎ (119909 (119905) 119906 (119905)) + 119881 (119905) (21)

Figure 5 HIL simulation platform

The input vector of both models can be written as

119906 = [120575ℎ 12059611 12059612 12059621 12059622] (22)

The output of both models can be described as

119910 = [119886119909 119886119910 12059611 12059612 12059621 12059622] (23)

The state of both models can be expressed as

119909 = [ V119892 120573 11986511991011 11986511991012 11986511991021 11986511991022 11986511990911 11986511990912 11986511990921 11986511991022] (24)

Hence the function f() of equation (17) can be written as

1198911 = 1119868119911 [(1199094 + 1199095) 119871119891 cos 1199061 + (1199094 minus 1199095) 119871119903 sin 1199061minus (1199096 + 1199097) 119871119903 minus (1199098 minus 1199099) 119871119891 sin 1199061minus (1199098 minus 1199099) 1198611198912 cos 1199061 minus (11990910 minus 11990911) 1198611198912 ]


1198912 = 1119898 [((1199098 + 1199099) cos (1199093 minus 1199061) + (1199096 + 1199097) cos1199093+ (1199094 + 1199095) sin (1199093 minus 1199061) + (1199094 + 1199095) sin1199093]

1198913 = 11198981199092 [(1199094 + 1199095) cos (1199093 minus 1199061) + (1199094 + 1199095) cos1199093minus (1199098 + 1199099) sin (1199093 minus 1199061) minus (1199098 + 1199099) sin1199093] minus 1199091

1198914 = 119909212059011 (minus1199094 + 11986511991011) 1198915 = 119909212059012 (minus1199095 + 11986511991012) 1198916 = 119909212059021 (minus1199096 + 11986511991021)1198917 = 119909212059022 (minus1199097 + 11986511991022) 1198918 = 119909212059011 (minus1199098 + 11986511990911) 1198919 = 119909212059012 (minus1199099 + 11986511990912)11989110 = 119909212059021 (minus11990910 + 11986511990921) 11989111 = 119909212059022 (minus11990911 + 11986511990922)

(25)

The measurement function h() of equation (21) can bewritten as

ℎ1 = 1119898 [(1199098 + 1199099) cos 1199061 + 1199096 + 1199097minus (1199098 + 1199099) sin 1199061]

ℎ2 = 1119898 [(1199094 + 1199095) cos 1199061 + 11990910 + 11990911minus (1199094 + 1199095) sin 1199061]

ℎ3 = 1199091ℎ4 = 1199062ℎ5 = 1199063ℎ6 = 1199064ℎ7 = 1199065

(26)

Before using the CKF to estimate the state vector 119909(119905)equation (21) should be discretized as follows

119909119896 = 119891 (119909119896minus1 119906119896minus1) + 119908119896minus1119911119896 = ℎ (119909119896 119906119896) + V119896

(27)

where 119909119896 isin R119899119909 denotes the state of the vehicle system atdiscrete time k 119911119896 isin R119899119909 denotes measurement 119906119896 isin R119899119909

denotes the input 119908119896minus1 isin R119899119909 and V119896 isin R119899119909 denote inde-pendent process andmeasurement Gaussian noise sequencesassumed to be in white independent and with covariance 119876119896and 119877119896 respectively The CKF can be expressed as follows

(i) Time Update

119878119896minus1|119896minus1 = 119878119881119863 (119875119896minus1|119896minus1)120594119896minus1|119896minus1 = 119878119896minus1|119896minus1120585 + 119909119896minus1|119896minus1120594lowast119896|119896minus1 = 119891 (120594119896minus1|119896minus1)119909119896|119896minus1 = 1119898

119898sum119894=1

120594lowast119894119896|119896minus1119875119896|119896minus1 = 1119898

119898sum119894=1

120594lowast119894119896|119896minus1120594lowast119879119894119896|119896minus1 minus 119909119896|119896minus1119909119879119896|119896minus1 + 119876119896

(28)

(ii) Measurement Update

119878119896|119896minus1 = 119878119881119863 (119875119896|119896minus1)120594119896|119896minus1 = 119878119896|119896minus1120585 + 119909119896|119896minus1119885119896|119896minus1 = ℎ (120594119896|119896minus1)119911119896|119896minus1 = 1119898

119898sum119894=1

119885119894119896|119896minus1119875119911119911119896|119896minus1 = 1119898

119898sum119894=1

119885119894119896|119896minus1119885119879119894119896|119896minus1 minus 119911119896|119896minus1119911119879119896|119896minus1 + 119877119896119875119909119911119896|119896minus1 = 1119898

119898sum119894=1

120594119894119896|119896minus1119885119879119894119896|119896minus1 minus 119909119896|119896minus1119911119879119896|119896minus1 + 119877119896119870119896 = 119875119909119911119896|119896minus1119875minus1119911119911119896|119896minus1119909119896|119896 = 119909119896|119896minus1 + 119870119896 (119911119896 minus 119911119896|119896minus1)119875119896|119896 = 119875119896|119896minus1 minus 119870119896119875119911119911119896|119896minus1119870119879119896

(29)

where SVD denotes the matrix singular value decompositionmethod S represents the square-root of the covariancematrixP m=2nm denotes the total number of Cubature points 120585 =radic1198982[1]119894 120594119894 is the Cubature point which is generated by thestates equations 119885119894 is the Cubature point which is generatedby measurements

Step 3 (model probability update) In this step the likelihoodfunction of each mode can be expressed as

Λ(119894)119896 = 119875 ((119894)119896 | 119898(119894)119896 119911119896minus1)= 100381610038161003816100381610038162120587119878(119894)119896 10038161003816100381610038161003816minus12 exp minus12 ((119894)119896 )119879 (119878(119894)119896 )minus1 (119894)119896

(30)

120583(119894)119896 = 119875 (119898(119894)119896 | 119911119896) = 1119888Λ(119894)119896 119888119895 (31)

119888 = 119903sum119894=1

Λ(119894)119896 119888119895 (32)


where 119878(119894)119896

= 119867(119894)119896119875(119894)119896|119896minus1

(119867(119894)119896)119879 + 119877(119894)

119896 119878(119894)119896

denotes theinnovation covariance c denotes the normalization factor

Step 4 (estimation fusion) After calculation of each modersquosprobabilities the vehicle state parameters prediction and thecovariance can be calculated according to Gaussian mixtureequationThe equation of vehicle state parameters predictionand its covariance can be respectively expressed as

119909119896|119896 = 119903sum119894=1

119909(119894)119896 120583(119894)119896 (33)

119875119896|119896 = 119903sum119894=1

[119875(119894)119896|119896 + (119909119896|119896 minus 119909(119894)119896|119896) (119909119896|119896 minus 119909(119894)119896|119896)119879] 120583(119894)119896 (34)

4 Key Dynamics Variable of VSIMM-CKF

The sensor of lateral acceleration is a sort of inertial sensorswhich is easily affected by the gravity when the axis ofchassis is not horizontal Therefore the outputs of lateralacceleration sensor need to be corrected Similarly roadadhesion coefficient and cornering stiffness correction arevery important so they are introduced separately in thissection

41 Correction of Lateral Acceleration In case of vehicledriving on the flat and level road the measured results oflateral acceleration are precise But in case of driving onthe slope road the measured results tend to become higherthan those on the flat and level road Because the lateralacceleration sensor is easy to be influenced by the roll angle ofthe vehicle therefore it is necessary to correct the measuredresults before being used in the vehicle state estimation Theequation of correct lateral acceleration can be written as

119886119910 = 119886119910119898 minus 119892120572119903119900119897119897 (35)

where 119886119910 denotes the corrected lateral acceleration 120572119903119900119897119897denotes the roll angle 119886119910119898 denotes the measured results oflateral acceleration

The equation of roll angle can be expressed as

120572119903119900119897119897 = 119898119886119910ℎ119903119900119897119897119870119865 + 119870119877 minus 119898119892ℎ119903119900119897119897 (36)

where ℎ119903119900119897119897 denotes the distance between the gravity and theaxis of rolling 119870119865 denotes the rolling stiffness of front sus-pension 119870119877 denotes the rolling stiffness of rear suspension42 Road Friction Estimation Road friction plays an impor-tant role in vehicle state estimation However it is difficult toacquire its value directly According to the estimationmethodproposed in Reference [27] the value of road friction can becalculated as follows

120583 = 1003816100381610038161003816100381610038161003816100381611988611991011989210038161003816100381610038161003816100381610038161003816 + 120583119890 (37)

where g denotes the gravitational acceleration 120583119890 denotes theevaluated error The value of 120583119890 can be referred in Reference[27]

43 Correction of Cornering Stiffness According to Reference[27] the cornering stiffness can be expressed as a secondorder polynomial of tire force But in the real vehicle dynam-ics system the lateral tire force is always coupled with thelongitudinal tire force In order to minimize the discrepancycaused by simplification the following correction is appliedto cornering stiffness According to Reference [28] Pacejkaand HB proposed the limiting factor with the longitudinaltire force as input variable for the condition where nolongitudinal dynamics is considered

120593119897119891 = radic12058321198652119911119894119895 minus 1198652119909119894119895120583119865119911119894119895 (38)

The limiting factor limits the max lateral tire force to itscertain fraction which conversely redefines the corneringstiffness as follows

119862120572 (120583 119865119911119894119895 119865119909119894119895)= 120593119897119891 119862120572 (120583 119865119911119894119895 119865119909119894119895 = 0) minus 12120583119865119911119894119895+ 12 (120583119865119911119894119895 minus 119865119909119894119895)

(39)

5 Simulation Verification

In order to validate the proposed estimation method thesimulations were carried out The simulations are based onthe Hardware-In-Loop (HIL) and the simulation platformis shown in Figure 5 Vedyna [29] is used to describe thevehicle model and the vehicle model is downloaded into theSimulator of DSPACEThe vehicle state estimation algorithmis realized by MatlabSimulink and the method is alsodownloaded into the MicroAutobox of DSPACE The signalof steering angle which is generated by driver is transmittedto the Simulator And then the vehicle state is changed bythe operation of the driver The MicroAutobox receives thesignal of vehicle state such as the longitudinal acceleration119886119909 the lateral acceleration 119886119910 the yaw rate the steeringangle 120575 and the wheels speed 120596119894119895 by CAN BUS the SimulatorIn order to improve CANBUS delay problem a node isselected as the synchronization node The synchronizationnode sends a synchronization message with ID0 once every300ms interval The transmission intervals of other nodesare set to 5ms 10ms 20ms 30ms 40ms etc When othernodes receive synchronization messages from synchroniza-tion nodes they empty their own counters so that all nodessynchronize once every 300 milliseconds Other nodes usemessages sent by synchronous nodes as the benchmark tosend messages sequentially which can avoid the problem ofcollision between multiple nodes at sending time avoid buscompetition and reduce the possibility of message delay Atlast the vehicle state is calculated by the proposed algorithmin this paper The whole simulation scheme based on HILcan be depicted as Figure 6 The values of the parameters forsimulation are expressed in Table 1 [19 30]

Two kinds of simulation tests are adopted to validatethe estimation method The first one is double-lane change


Steering anglesensor

dSpace system(vehicle model simulator)

Yawrate

Vehicle stateestimationalgorithm

11 12

21 22

ax

ay Fy11 Fy12 Fy21

Fy22 Fx12 Fx21 Fy22

Figure 6 The simulation scheme based on HIL

2 4 6 8 10Time (s)

minus60

minus40

minus20

0

20

40

60

Stee

ring

angl

e (de

g)

Steering angle

(a) Steer angle of steering wheel

minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)

Lateral acceleration

2 4 6 8 10Time (s)

(b) The lateral acceleration of vehicle

Figure 7

Table 1 Parameters for simulation

Symbol Value Unit Description119871119891 14 m Distance cg to front axle119871 119903 14 m Distance cg to rear axle119862119910119891 110000 Nrad Front cornering stiffness119862119910119903 95000 Nrad Rear cornering stiffness119868119911 3500 Kgm2 Yaw moment of inertia119861119891 15 M Track widthm 2000 Kg Vehicle mass119869119908 13 Kgm2 Wheel inertiaR 028 M Wheel radius

(DLC) maneuvers on banked 5 road to validate the methodof lateral acceleration correction proposed in Section 4 Thesecond one is DLC maneuvers on flat level road And thesimulation results of estimation method proposed in thispaper are compared with the estimation based on CKF todiscuss the performance of both methods

51 Simulation Results of the DLC Manoeuver on Banked 5Road (Steering Angle Is 55 Degrees at 71kmh) Figures 7(a)and 7(b) show the steering angle lateral acceleration duringDLC test on banked 5 road It is because of the existenceof bank angle that the lateral acceleration is influencedFigure 8 shows the simulation result of the estimated sideslip


0 2 4 6 8 10 12Time (s)

minus3

minus2

minus1

0

1

2

3

Side

slip

angl

e (de

gs)

ActualIMM-CKF without correctionIMM-CKF with correction

Figure 8 The estimated sideslip angle on banked 5 road

10Time (s)

minus3

minus2

minus1

0

1

2

3

4

Side

slip

angl

e (de

gs)

IMM-CKF with correctionIMM-CKF without correctionActual

2 4 6 8


angle on banked 5 road Seen from Figure 8 the sideslipangle estimated by IMM-CKFwithout the lateral accelerationcorrection deviates from the real one largely In contrast thevalue of sideslip angle estimated by IMM-CKF with lateralacceleration correction can predict more accurate value ofsideslip angle as described in Figure 8 Thus the proposedIMM-CKFwith lateral acceleration correction has robustnesson the banked road

Figure 9 shows the simulation results of the estimationmethod of the sideslip angle on banked 10 It can be seenfrom the curve that the error of the estimated sideslip anglebecomes large for the reason of the increase in error of lateralacceleration

52 Simulation Results of the DLC Manoeuver on Flat LevelRoad (Steering Angle Is 55 Degrees at 71kmh) In this sectionas shown in Figure 10 two kinds of simulation tests under

high adhesion coefficient (120583 = 08) are adopted to comparethe performance among the proposed estimator (IMM-CKF) cubature Kalman filter (CKF) unscented Kalman filter(UKF) and extended Kalman filter (EKF) The simulationresults which include the estimated sideslip angle and theestimated tire-road force are described in Figures 11ndash13Figure 12 shows the simulation results of sideslip angle Thevalues estimated by other filtermethods havemore error thanthe value estimated by IMM-CKF The same phenomenoncan be observed in Figure 13The IMM-CKF algorithm showsmore accuracy than othermethodsThere is a small differencebetween the actual value and the value estimated by IMM-CKF which maybe neglects the effect of the suspensionmodel

As shown in Figure 12 during the time intervals (5-6s and7-8s) the estimated sideslip and tire force are not consistentwith the true value The error is due to longitudinal load


CKFEKFUKF of7DOF

vehicle model with non-linear tire model

IMM-CKFscope

Figure 10 The simulation model for the sideslip angle estimation

minus15

minus10

minus5

0

5

10

15

Yaw

rate

(deg

s)

Yaw rate

2 4 6 8 10Time (s)

(a) The yaw rate of vehicle

2 4 6 8 10Time (s)

minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)

Lateral acceleration(b) The lateral acceleration of vehicle

Figure 11

2 3 4 5 6 7 8 9 10Time (s)

minus15

minus1

minus05

0

05

1

15

Side

slip

angl

e (de

gs)

IMM-CKFACTUALCKF

EKFUKF

Figure 12 The estimated sideslip angle on flat level road

transferWith the increase of vehiclemass the error increasesgradually

53 Simulation Results of the DLC Manoeuver on Banked10 Road In order to evaluate the correction of corneringstiffness the steering input of DLC manoeuver on banked10 road is generated The simulation is done on ice road(road friction 120583 = 03) and the vehicle velocity is 60kmh

The simulation results with and without the correction ofcornering stiffness are presented in Figure 14 The change ofcornering stiffness is due to the drastic lateral load transferwhich immediately leads to the increase of the sideslip angleAs such the estimation result with correction of corner-ing stiffness shows more robust performance especially inpresence of large sideslip angle regime compared with thesimulation result in the absence of correction about cornering


2 3 4 5 6 7 8 9 10Time (s)

minus3000

minus2000

minus1000

0

1000La

tera

l tire

-roa

d fo

rce o

f fro

nt ri

ght t

ire (N

)

IMM-CKFCKFACTUAL

EKFUKF

(a) Front right lateral tire force

2 3 4 5 6 7 8 9 10Time (s)

CKFActualIMM-CKF

EKFUKF

minus1500

minus1000

minus500

0

500

1000

1500

2000

Late

ral t

ire-r

oad

forc

e of f

ront

left

tires

(N)

(b) Front left lateral tire force

0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

3000

Long

itudi

nal t

ire-r

oad

forc

e of r

ight

fron

t tire

(N)

(c) Front right longitudinal tire force

0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

Long

itudi

nal t

ire-r

oad

forc

e of L

eft fr

ont t

ire (N

)

(d) Front left longitudinal tire force

Figure 13

stiffness Such an obvious difference in the performanceis observed clearly in the period between 85s and 9s inFigure 14(a) The result of correction is presented in Figures14(b) and 14(c)

6 Conclusions

(1) For the sake of the various driving conditions twovehicle models were built for estimation The 7-degree-of-freedom (7DOF) vehiclemodel with lineartire model was built for normal driving conditionand the other is 7DOF vehicle model with nonlinearDugoff tire model for the extreme driving conditionTwomodelswere switched by IMM-CKF tomatch thedifferent driving condition

(2) To eliminate the estimation error caused by lateralacceleration sensor in case of vehicle driving on theslope road the lateral acceleration correction forsideslip angle estimation is considered The overallresults from HIL verify that the proposed methodcan realize accurate estimation about vehicle state

parameters in a wide range of road conditions Mean-while the method is only applied on the mild bankedroad and the application of the proposed approachunder higher banking angle road would be concernedin the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Henan Science and TechnologyProject (192102210063 182102210034)


6 65 7 75 8 85 9Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

Side

slip

Ang

le (

deg)

estimated with correctionestimated without correctionactual

(a) Estimated sideslip angle with and without cornering stiffness correction

6 65 7 75 8 85 9Time (s)

08

1

12

14

16

18

2

22

Fron

t cor

nerin

g sti

ffnes

s (N

rad

)

with correctionwithout correction

times105

(b) Front cornering stiffness

6 65 7 75 8 85 9Time (s)

04

06

08

1

12

14

16

Rear

corn

erin

g sti

ffnes

s


times105

(c) Rear cornering stiffness

Figure 14

References

[1] M Milanese C Novara and I Gerlero ldquoRobust estimation ofvehicle sideslip angle from variables measured by ESC systemrdquoInternationales Stuttgarter Symposium pp 1063ndash1076 2015

[2] X Li X Song and C Chan ldquoReliable vehicle sideslip anglefusion estimation using low-cost sensorsrdquoMeasurement vol 51no 1 pp 241ndash258 2014

[3] H Zhang X Huang J Wang and H R Karimi ldquoRobustenergy-to-peak sideslip angle estimation with applications toground vehiclesrdquoMechatronics vol 30 no 1 pp 338ndash347 2015

[4] B L Boada M J L Boada and V Diaz ldquoVehicle sideslip anglemeasurement based on sensor data fusion using an integratedANFIS and anUnscented Kalman Filter algorithmrdquoMechanicalSystems and Signal Processing vol 72 no 3 pp 723ndash739 2016

[5] N Ding and S Taheri ldquoApplication of recursive least squarealgorithm on estimation of vehicle sideslip angle and roadfrictionrdquo Mathematical Problems in Engineering vol 68 no 6pp 1ndash18 2010

[6] C Song F Xiao S Song S Li and J Li ldquoDesign of anovel nonlinear observer to estimate sideslip angle and tireforces for distributed electric vehiclerdquo Mathematical Problemsin Engineering vol 215 no 1 pp 1ndash11 2015

[7] Z Changzhu C Qijun and Q Jianbin ldquoRobust filteringfor vehicle sideslip angle estimation with sampled-data mea-surementsrdquo Transactions of the Institute of Measurement andControl vol 73 no 1 pp 1ndash12 2016

[8] F Cheli E Sabbioni M Pesce and S Melzi ldquoA methodologyfor vehicle sideslip angle identification comparison with exper-imental datardquo Vehicle System Dynamics vol 45 no 6 pp 549ndash563 2007

[9] P J T Venhovens and K Naab ldquoVehicle dynamics estimationusing Kalman filtersrdquo Vehicle System Dynamics vol 32 no 2-3pp 171ndash184 1999

[10] G Baffet A Charara and D Lechner ldquoEstimation of vehiclesideslip tire force and wheel cornering stiffnessrdquo Control Engi-neering Practice vol 17 no 11 pp 1255ndash1264 2009


[11] J Stephant A Charara and D Meizel ldquoEvaluation of a slidingmode observer for vehicle sideslip anglerdquo Control EngineeringPractice vol 15 no 7 pp 803ndash812 2007

[12] L Li G Jia X Ran J Song and K Wu ldquoA variable structureextended Kalman filter for vehicle sideslip angle estimation ona low friction roadrdquo Vehicle System Dynamics vol 52 no 2 pp280ndash308 2014

[13] Y Chen Y Ji and K Guo ldquoA reduced-order nonlinear slidingmode observer for vehicle slip angle and tyre forcesrdquo VehicleSystem Dynamics vol 52 no 12 pp 1716ndash1728 2014

[14] H F Grip L Imsland T A Johansen T I Fossen J CKalkkuhl andA Suissa ldquoNonlinear vehicle side-slip estimationwith friction adaptationrdquoAutomatica vol 44 no 3 pp 611ndash6222008

[15] Y F Lian Y Zhao L L Hu and Y T Tian ldquoCorneringstiffness and sideslip angle estimation based on simplifiedlateral dynamicmodels for four-in-wheel-motor-driven electricvehicles with lateral tire force informationrdquo International Jour-nal of Automotive Technology vol 16 no 4 pp 669ndash683 2015

[16] B L Boada M J L Boada A Gauchıa E Olmeda and V DıazldquoSideslip angle estimator based on ANFIS for vehicle handlingand stabilityrdquo Journal of Mechanical Science and Technology vol29 no 4 pp 1473ndash1481 2015

[17] W Liu H He and F Sun ldquoVehicle state estimation basedon minimum model error criterion combining with extendedkalman filterrdquo Journal of The Franklin Institute vol 353 no 4pp 834ndash856 2016

[18] M Dawood C Cappelle M E El Najjar M Khalil andD Pomorski ldquoVehicle geo-localization based on IMM-UKFdata fusion using a GPS receiver a video camera and a 3Dcity modelrdquo in Proceedings of the 2011 IEEE Intelligent VehiclesSymposium IVrsquo11 pp 510ndash515 Germany June 2011

[19] X Gao Z Yu J Neubeck and J Wiedemann ldquoSideslip angleestimation based on inputndashoutput linearisation with tirendashroadfriction adaptationrdquo Vehicle System Dynamics vol 48 no 2 pp217ndash234 2010

[20] HDugoff P S Fancher and L Segel ldquoAn analysis of tire proper-ties and their influence on vehicle dynamic performancerdquo SAEpaper Article ID 700377 1970

[21] J-F Liao and B-S Chen ldquoRobust mobile location estimatorwith NLOS mitigation using interacting multiple model algo-rithmrdquo IEEE Transactions on Wireless Communications vol 5no 11 pp 3002ndash3006 2006

[22] S-L Koo and H-S Tan ldquoTire dynamic deflection andits impact on vehicle longitudinal dynamics and controlrdquoIEEEASME Transactions on Mechatronics vol 12 no 6 pp623ndash631 2007

[23] H Zhang Y Shi and J Wang ldquoObserver-based trackingcontroller design for networked predictive control systems withuncertain Markov delaysrdquo International Journal of Control vol86 no 10 pp 1824ndash1836 2013

[24] K Jo K Chu and M Sunwoo ldquoInteracting multiple model-filter-based sensor fusion of GPS with in-vehicle sensors forreal-time vehicle positioningrdquo IEEE Transactions on IntelligentTransportation Systems vol 13 no 1 pp 329ndash343 2012

[25] S S Blackman and R Popoli Design and Analysis of ModernTracking Systems Artech House Noorwood Mass USA 1999

[26] Y Shi H Zhang and J Wang ldquoOn energy-to-peakfiltering fornonuniformly sampled nonlinear systems a Markovian jumpsystem approachrdquo IEEE Transactions on Fuzzy Systems vol 22no 1 pp 212ndash222 2014

[27] Y Fukada ldquoSlip-angle estimation for vehicle stability controlrdquoVehicle System Dynamics vol 32 no 4 pp 375ndash388 1999

[28] H B Pacejka Tyre and Vehicle Dynamics Butterworth Heine-mann London UK 2002

[29] ldquoVeDyna Userrsquos Guide [G]rdquo 1997[30] A Nishio K Tozu H Yamaguchi K Asano and Y Amano

ldquoDevelopment of vehicle stability control system based onvehicle sideslip angle estimationrdquo SAE Technical Papers ArticleID 010137 2001

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Submit your manuscripts atwwwhindawicom


0 10Wheel slip angle (deg)

0

5La

tera

l for

ce (K

N)

Burckhardtasphalt wetBurckhardtasphalt dry

Burckhardt cobblestone wetBurckhardtice

LinearLinear

5

Figure 1 Lateral force curve

the road surface especially on a low friction road Conse-quently the parameters estimated based on linear vehiclemodel are error due to the tirersquos inherent nature Comparedto type 1 type 2 uses the estimation approach based onthe nonlinear vehicle model [10 11] which is more close toreflect the vehicle dynamics nature to predict the vehiclestate under extreme driving conditions A lot of literatureshave been proposed to estimate the parameters based on thenonlinear vehicle model Liang Li and Gang Jia proposed avariable structure extended Kalman filter with the sideslipangle rate feedback to estimate the sideslip angle [12] Theirsimulations and vehicle winter test show that the proposedapproach can provide accurate value of sideslip under lowfriction road conditions Yu andKhGuo proposed a reduced-order sliding mode observer for vehicle state estimation [1314] The precise nonlinear tire model ldquoUniTirerdquo is used toimprove the accuracy of vehicle states parameters estima-tion A novel approach is proposed to estimate parametersby using a mathematical tool which includes a nonlinearMagic tire model [15] BLBoada proposed a novel observerbased on adaptive Neuro-Fuzzy Inference System (ANFIS)combined with Unscented Kalman Filter in order to estimatethe sideslip angle and tire force [16] Wei L proposed anestimation approach based on the Extended Kalman Filter(EKF) combing with the Minimum Model Error (MME)criterion for vehicle state estimation [17] However theestimation based on nonlinear vehicle model is not suitablefor parameters estimations under normal driving conditionsbecause of the excessive computational power in embeddedsystems In sum the time to choose an optimal model forvarious driving conditions is also important besides thechoice of vehicle model The multiple model (MM) methodtakes advantage of several models to express possible vehiclemodel under different driving conditions which can achievemore accurate vehicle state parameters by combined filtersbased on corresponding vehicle model of a limited numberof different models compared with single vehicle modelAmong several MM estimation algorithms the interacting

12

21

22

Fx12 Fx12Fy22Fx22

11

Fy21Fy11 Fx11

Fx21

x

y

gBf

Lr Lf

Figure 2 7-DOF vehicle model

MM (IMM) has attractedwide attention because it can obtainaccurate calculation results with low computational load [18]

In this paper we use the interacting multiple model(IMM) to choose an optimal model for various driving con-ditions In the IMM method two filters are used in parallelto estimate sideslip angle and tire force One filter is CKFfilter based on four-wheel nonlinear vehicle dynamics modelwith linear tire model for normal driving conditions andthe other is based on four-wheel nonlinear vehicle dynamicsmodel with nonlinear Dugoff tire model for extreme drivingconditions The sideslip angle and tire force predicted by thevariable structure IMM-CKF filter aremore accurate than thesingle filter for various driving conditions

This paper is organized as follows In Section 2 thedynamics models of the vehicle linear tire and nonlinearDugoff tiremodel are presented respectively In Section 3 thedetailed structure of IMM-CKF is represented In Section 4the key dynamics variable is given in detail In Section 5 thesimulations are presented to validate the performance of theestimation method

2 Design of Vehicle Dynamics Model

21 Vehicle Model In Figure 2 the 7-DOF vehicle dynamicsmodel is proposed to estimate the vehicle state parametersThe 7-DOF vehicle model includes longitudinal motionlateral motion yaw motion and 4-wheel rotation [13]

The equation of longitudinal motion is expressed asfollows

119898(V119909 minus V119910) = 2sum119894119895=1

119865119909119894119895= (11986511990911 + 11986511990912) cos 120575 + (11986511990921 + 11986511990922)minus (11986511990911 + 11986511990912) sin 120575 = 119898119886119909

(1)

The equation of lateral motion is expressed as follows

119898(V119910 + V119909) = 2sum119894119895=1

119865119910119894119895= (11986511991011 + 11986511991012) cos 120575 + (11986511991021 + 11986511991022)minus (11986511991011 + 11986511991012) sin 120575 = 119898119886119910

(2)




(3)




(5)


(6)




(7)





[[[[[[

12057211120572121205722112057222

]]]]]]= [[[[[[

12057512057500

]]]]]]minus [[[[[[

12058511120585121205852112058522

]]]]]]

(8)


[[[[[[

12058511120585121205852112058522

]]]]]]= tanminus1

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

asymp

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

(9)







119865119910119894119895 = 119862119910119894119895 tan1205721198941198951 minus 120582119894119895 119891 (119878) (12)

119865119909119894119895 = 1198621199091198941198951205821198941198951 minus 120582119894119895 119891 (119878) (13)



Yaw-rate sensors

Accelerationsensor

Wheel-speed sensors




IMM algorithm

11 12

21 22

ax ay

ℎ


119878 = 120583119865119911119894119895 (1 minus 120576V119909radic1205821198941198952 + tan2120572119894119895)

2radic11986211989411989521205821198941198952 + 1198621198941198952tan2120572119894119895 (1 minus 1205821198941198952) (14)

119891 (119878) = 1 119878 gt 1119878 (2 minus 119878) 119878 lt 1

120582119894119895 = 119877119908120596119894119895 minus V119909max (119877119908120596119894119895 V119909)

(15)



119910119894119895 = V119892120590119894119895 (minus119865119910119894119895 + 119865119910119894119895) (16)

119909119894119895 = V119892120590119894119895 (minus119865119909119894119895 + 119865119909119894119895) (17)


3 IMM-CKF








119901(119894)119896minus1|119896minus1 = 119903sum119894=1



) (119909(119895)119896minus1|119896minus1


)119879]sdot 120583(119894119895)119896minus1|119896minus1

(19)


Previous Step

Initialconditions



tire

Modeprobability

Update



1kminus

11kp minus

2| 1kminus

21kP minus

1|2k

2|1k

( ) ( )1

1i j ik ij k

Cπ minus=

1|1k

2|2k

21kx minus

11kx minus

11kp minus

21kp minus

1| 1k kminus

1| 1k kp minus

2| 1k kminus

2| 1k kp minus

1k1kp2k2kp

Estimation Fusion

kkp


x

x

x x

xx

x1k

2k



= 119875(119898(119894)119896minus1

| 119898(119894)119896119885119896minus1) = (1119862)120587119894119895120583(119894)119896minus1 119894 119895 =1 2 119903120583(119894119895)



120587119894119895 = [099 001002 098] (20)





(119905) = 119891 (119909 (119905) 119906 (119905)) + 119882 (119905)119910 (119905) = ℎ (119909 (119905) 119906 (119905)) + 119881 (119905) (21)



119906 = [120575ℎ 12059611 12059612 12059621 12059622] (22)


119910 = [119886119909 119886119910 12059611 12059612 12059621 12059622] (23)


119909 = [ V119892 120573 11986511991011 11986511991012 11986511991021 11986511991022 11986511990911 11986511990912 11986511990921 11986511991022] (24)







(25)


ℎ1 = 1119898 [(1199098 + 1199099) cos 1199061 + 1199096 + 1199097minus (1199098 + 1199099) sin 1199061]

ℎ2 = 1119898 [(1199094 + 1199095) cos 1199061 + 11990910 + 11990911minus (1199094 + 1199095) sin 1199061]

ℎ3 = 1199091ℎ4 = 1199062ℎ5 = 1199063ℎ6 = 1199064ℎ7 = 1199065

(26)



(27)



(i) Time Update


119898sum119894=1


119898sum119894=1


(28)



119898sum119894=1

119885119894119896|119896minus1119875119911119911119896|119896minus1 = 1119898

119898sum119894=1


119898sum119894=1


(29)




(30)

120583(119894)119896 = 119875 (119898(119894)119896 | 119911119896) = 1119888Λ(119894)119896 119888119895 (31)

119888 = 119903sum119894=1

Λ(119894)119896 119888119895 (32)


where 119878(119894)119896

= 119867(119894)119896119875(119894)119896|119896minus1

(119867(119894)119896)119879 + 119877(119894)

119896 119878(119894)119896



119909119896|119896 = 119903sum119894=1

119909(119894)119896 120583(119894)119896 (33)

119875119896|119896 = 119903sum119894=1

[119875(119894)119896|119896 + (119909119896|119896 minus 119909(119894)119896|119896) (119909119896|119896 minus 119909(119894)119896|119896)119879] 120583(119894)119896 (34)




119886119910 = 119886119910119898 minus 119892120572119903119900119897119897 (35)



120572119903119900119897119897 = 119898119886119910ℎ119903119900119897119897119870119865 + 119870119877 minus 119898119892ℎ119903119900119897119897 (36)


120583 = 1003816100381610038161003816100381610038161003816100381611988611991011989210038161003816100381610038161003816100381610038161003816 + 120583119890 (37)



120593119897119891 = radic12058321198652119911119894119895 minus 1198652119909119894119895120583119865119911119894119895 (38)


119862120572 (120583 119865119911119894119895 119865119909119894119895)= 120593119897119891 119862120572 (120583 119865119911119894119895 119865119909119894119895 = 0) minus 12120583119865119911119894119895+ 12 (120583119865119911119894119895 minus 119865119909119894119895)

(39)







Yawrate


11 12

21 22

ax

ay Fy11 Fy12 Fy21

Fy22 Fx12 Fx21 Fy22


2 4 6 8 10Time (s)

minus60

minus40

minus20

0

20

40

60

Stee

ring

angl

e (de

g)

Steering angle


minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)


2 4 6 8 10Time (s)


Figure 7






0 2 4 6 8 10 12Time (s)

minus3

minus2

minus1

0

1

2

3

Side

slip

angl

e (de

gs)



10Time (s)

minus3

minus2

minus1

0

1

2

3

4

Side

slip

angl

e (de

gs)


2 4 6 8








CKFEKFUKF of7DOF


IMM-CKFscope


minus15

minus10

minus5

0

5

10

15

Yaw

rate

(deg

s)

Yaw rate

2 4 6 8 10Time (s)


2 4 6 8 10Time (s)

minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)


Figure 11

2 3 4 5 6 7 8 9 10Time (s)

minus15

minus1

minus05

0

05

1

15

Side

slip

angl

e (de

gs)

IMM-CKFACTUALCKF

EKFUKF






2 3 4 5 6 7 8 9 10Time (s)

minus3000

minus2000

minus1000

0

1000La

tera

l tire

-roa

d fo

rce o

f fro

nt ri

ght t

ire (N

)

IMM-CKFCKFACTUAL

EKFUKF


2 3 4 5 6 7 8 9 10Time (s)

CKFActualIMM-CKF

EKFUKF

minus1500

minus1000

minus500

0

500

1000

1500

2000

Late

ral t

ire-r

oad

forc

e of f

ront

left

tires

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

3000

Long

itudi

nal t

ire-r

oad

forc

e of r

ight

fron

t tire

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

Long

itudi

nal t

ire-r

oad

forc

e of L

eft fr

ont t

ire (N

)


Figure 13


6 Conclusions




Data Availability




Acknowledgments



6 65 7 75 8 85 9Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

Side

slip

Ang

le (

deg)



6 65 7 75 8 85 9Time (s)

08

1

12

14

16

18

2

22

Fron

t cor

nerin

g sti

ffnes

s (N

rad

)


times105


6 65 7 75 8 85 9Time (s)

04

06

08

1

12

14

16

Rear

corn

erin

g sti

ffnes

s


times105


Figure 14

References







































Journal of












Journal of







Volume 2018






Volume 2018











(3)




(5)


(6)




(7)





[[[[[[

12057211120572121205722112057222

]]]]]]= [[[[[[

12057512057500

]]]]]]minus [[[[[[

12058511120585121205852112058522

]]]]]]

(8)


[[[[[[

12058511120585121205852112058522

]]]]]]= tanminus1

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

asymp

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

(9)







119865119910119894119895 = 119862119910119894119895 tan1205721198941198951 minus 120582119894119895 119891 (119878) (12)

119865119909119894119895 = 1198621199091198941198951205821198941198951 minus 120582119894119895 119891 (119878) (13)



Yaw-rate sensors

Accelerationsensor

Wheel-speed sensors




IMM algorithm

11 12

21 22

ax ay

ℎ


119878 = 120583119865119911119894119895 (1 minus 120576V119909radic1205821198941198952 + tan2120572119894119895)

2radic11986211989411989521205821198941198952 + 1198621198941198952tan2120572119894119895 (1 minus 1205821198941198952) (14)

119891 (119878) = 1 119878 gt 1119878 (2 minus 119878) 119878 lt 1

120582119894119895 = 119877119908120596119894119895 minus V119909max (119877119908120596119894119895 V119909)

(15)



119910119894119895 = V119892120590119894119895 (minus119865119910119894119895 + 119865119910119894119895) (16)

119909119894119895 = V119892120590119894119895 (minus119865119909119894119895 + 119865119909119894119895) (17)


3 IMM-CKF








119901(119894)119896minus1|119896minus1 = 119903sum119894=1



) (119909(119895)119896minus1|119896minus1


)119879]sdot 120583(119894119895)119896minus1|119896minus1

(19)


Previous Step

Initialconditions



tire

Modeprobability

Update



1kminus

11kp minus

2| 1kminus

21kP minus

1|2k

2|1k

( ) ( )1

1i j ik ij k

Cπ minus=

1|1k

2|2k

21kx minus

11kx minus

11kp minus

21kp minus

1| 1k kminus

1| 1k kp minus

2| 1k kminus

2| 1k kp minus

1k1kp2k2kp

Estimation Fusion

kkp


x

x

x x

xx

x1k

2k



= 119875(119898(119894)119896minus1

| 119898(119894)119896119885119896minus1) = (1119862)120587119894119895120583(119894)119896minus1 119894 119895 =1 2 119903120583(119894119895)



120587119894119895 = [099 001002 098] (20)





(119905) = 119891 (119909 (119905) 119906 (119905)) + 119882 (119905)119910 (119905) = ℎ (119909 (119905) 119906 (119905)) + 119881 (119905) (21)



119906 = [120575ℎ 12059611 12059612 12059621 12059622] (22)


119910 = [119886119909 119886119910 12059611 12059612 12059621 12059622] (23)


119909 = [ V119892 120573 11986511991011 11986511991012 11986511991021 11986511991022 11986511990911 11986511990912 11986511990921 11986511991022] (24)







(25)


ℎ1 = 1119898 [(1199098 + 1199099) cos 1199061 + 1199096 + 1199097minus (1199098 + 1199099) sin 1199061]

ℎ2 = 1119898 [(1199094 + 1199095) cos 1199061 + 11990910 + 11990911minus (1199094 + 1199095) sin 1199061]

ℎ3 = 1199091ℎ4 = 1199062ℎ5 = 1199063ℎ6 = 1199064ℎ7 = 1199065

(26)



(27)



(i) Time Update


119898sum119894=1


119898sum119894=1


(28)



119898sum119894=1

119885119894119896|119896minus1119875119911119911119896|119896minus1 = 1119898

119898sum119894=1


119898sum119894=1


(29)




(30)

120583(119894)119896 = 119875 (119898(119894)119896 | 119911119896) = 1119888Λ(119894)119896 119888119895 (31)

119888 = 119903sum119894=1

Λ(119894)119896 119888119895 (32)


where 119878(119894)119896

= 119867(119894)119896119875(119894)119896|119896minus1

(119867(119894)119896)119879 + 119877(119894)

119896 119878(119894)119896



119909119896|119896 = 119903sum119894=1

119909(119894)119896 120583(119894)119896 (33)

119875119896|119896 = 119903sum119894=1

[119875(119894)119896|119896 + (119909119896|119896 minus 119909(119894)119896|119896) (119909119896|119896 minus 119909(119894)119896|119896)119879] 120583(119894)119896 (34)




119886119910 = 119886119910119898 minus 119892120572119903119900119897119897 (35)



120572119903119900119897119897 = 119898119886119910ℎ119903119900119897119897119870119865 + 119870119877 minus 119898119892ℎ119903119900119897119897 (36)


120583 = 1003816100381610038161003816100381610038161003816100381611988611991011989210038161003816100381610038161003816100381610038161003816 + 120583119890 (37)



120593119897119891 = radic12058321198652119911119894119895 minus 1198652119909119894119895120583119865119911119894119895 (38)


119862120572 (120583 119865119911119894119895 119865119909119894119895)= 120593119897119891 119862120572 (120583 119865119911119894119895 119865119909119894119895 = 0) minus 12120583119865119911119894119895+ 12 (120583119865119911119894119895 minus 119865119909119894119895)

(39)







Yawrate


11 12

21 22

ax

ay Fy11 Fy12 Fy21

Fy22 Fx12 Fx21 Fy22


2 4 6 8 10Time (s)

minus60

minus40

minus20

0

20

40

60

Stee

ring

angl

e (de

g)

Steering angle


minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)


2 4 6 8 10Time (s)


Figure 7






0 2 4 6 8 10 12Time (s)

minus3

minus2

minus1

0

1

2

3

Side

slip

angl

e (de

gs)



10Time (s)

minus3

minus2

minus1

0

1

2

3

4

Side

slip

angl

e (de

gs)


2 4 6 8








CKFEKFUKF of7DOF


IMM-CKFscope


minus15

minus10

minus5

0

5

10

15

Yaw

rate

(deg

s)

Yaw rate

2 4 6 8 10Time (s)


2 4 6 8 10Time (s)

minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)


Figure 11

2 3 4 5 6 7 8 9 10Time (s)

minus15

minus1

minus05

0

05

1

15

Side

slip

angl

e (de

gs)

IMM-CKFACTUALCKF

EKFUKF






2 3 4 5 6 7 8 9 10Time (s)

minus3000

minus2000

minus1000

0

1000La

tera

l tire

-roa

d fo

rce o

f fro

nt ri

ght t

ire (N

)

IMM-CKFCKFACTUAL

EKFUKF


2 3 4 5 6 7 8 9 10Time (s)

CKFActualIMM-CKF

EKFUKF

minus1500

minus1000

minus500

0

500

1000

1500

2000

Late

ral t

ire-r

oad

forc

e of f

ront

left

tires

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

3000

Long

itudi

nal t

ire-r

oad

forc

e of r

ight

fron

t tire

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

Long

itudi

nal t

ire-r

oad

forc

e of L

eft fr

ont t

ire (N

)


Figure 13


6 Conclusions




Data Availability




Acknowledgments



6 65 7 75 8 85 9Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

Side

slip

Ang

le (

deg)



6 65 7 75 8 85 9Time (s)

08

1

12

14

16

18

2

22

Fron

t cor

nerin

g sti

ffnes

s (N

rad

)


times105


6 65 7 75 8 85 9Time (s)

04

06

08

1

12

14

16

Rear

corn

erin

g sti

ffnes

s


times105


Figure 14

References







































Journal of












Journal of







Volume 2018






Volume 2018










Yaw-rate sensors

Accelerationsensor

Wheel-speed sensors




IMM algorithm

11 12

21 22

ax ay

ℎ


119878 = 120583119865119911119894119895 (1 minus 120576V119909radic1205821198941198952 + tan2120572119894119895)

2radic11986211989411989521205821198941198952 + 1198621198941198952tan2120572119894119895 (1 minus 1205821198941198952) (14)

119891 (119878) = 1 119878 gt 1119878 (2 minus 119878) 119878 lt 1

120582119894119895 = 119877119908120596119894119895 minus V119909max (119877119908120596119894119895 V119909)

(15)



119910119894119895 = V119892120590119894119895 (minus119865119910119894119895 + 119865119910119894119895) (16)

119909119894119895 = V119892120590119894119895 (minus119865119909119894119895 + 119865119909119894119895) (17)


3 IMM-CKF








119901(119894)119896minus1|119896minus1 = 119903sum119894=1



) (119909(119895)119896minus1|119896minus1


)119879]sdot 120583(119894119895)119896minus1|119896minus1

(19)


Previous Step

Initialconditions



tire

Modeprobability

Update



1kminus

11kp minus

2| 1kminus

21kP minus

1|2k

2|1k

( ) ( )1

1i j ik ij k

Cπ minus=

1|1k

2|2k

21kx minus

11kx minus

11kp minus

21kp minus

1| 1k kminus

1| 1k kp minus

2| 1k kminus

2| 1k kp minus

1k1kp2k2kp

Estimation Fusion

kkp


x

x

x x

xx

x1k

2k



= 119875(119898(119894)119896minus1

| 119898(119894)119896119885119896minus1) = (1119862)120587119894119895120583(119894)119896minus1 119894 119895 =1 2 119903120583(119894119895)



120587119894119895 = [099 001002 098] (20)





(119905) = 119891 (119909 (119905) 119906 (119905)) + 119882 (119905)119910 (119905) = ℎ (119909 (119905) 119906 (119905)) + 119881 (119905) (21)



119906 = [120575ℎ 12059611 12059612 12059621 12059622] (22)


119910 = [119886119909 119886119910 12059611 12059612 12059621 12059622] (23)


119909 = [ V119892 120573 11986511991011 11986511991012 11986511991021 11986511991022 11986511990911 11986511990912 11986511990921 11986511991022] (24)







(25)


ℎ1 = 1119898 [(1199098 + 1199099) cos 1199061 + 1199096 + 1199097minus (1199098 + 1199099) sin 1199061]

ℎ2 = 1119898 [(1199094 + 1199095) cos 1199061 + 11990910 + 11990911minus (1199094 + 1199095) sin 1199061]

ℎ3 = 1199091ℎ4 = 1199062ℎ5 = 1199063ℎ6 = 1199064ℎ7 = 1199065

(26)



(27)



(i) Time Update


119898sum119894=1


119898sum119894=1


(28)



119898sum119894=1

119885119894119896|119896minus1119875119911119911119896|119896minus1 = 1119898

119898sum119894=1


119898sum119894=1


(29)




(30)

120583(119894)119896 = 119875 (119898(119894)119896 | 119911119896) = 1119888Λ(119894)119896 119888119895 (31)

119888 = 119903sum119894=1

Λ(119894)119896 119888119895 (32)


where 119878(119894)119896

= 119867(119894)119896119875(119894)119896|119896minus1

(119867(119894)119896)119879 + 119877(119894)

119896 119878(119894)119896



119909119896|119896 = 119903sum119894=1

119909(119894)119896 120583(119894)119896 (33)

119875119896|119896 = 119903sum119894=1

[119875(119894)119896|119896 + (119909119896|119896 minus 119909(119894)119896|119896) (119909119896|119896 minus 119909(119894)119896|119896)119879] 120583(119894)119896 (34)




119886119910 = 119886119910119898 minus 119892120572119903119900119897119897 (35)



120572119903119900119897119897 = 119898119886119910ℎ119903119900119897119897119870119865 + 119870119877 minus 119898119892ℎ119903119900119897119897 (36)


120583 = 1003816100381610038161003816100381610038161003816100381611988611991011989210038161003816100381610038161003816100381610038161003816 + 120583119890 (37)



120593119897119891 = radic12058321198652119911119894119895 minus 1198652119909119894119895120583119865119911119894119895 (38)


119862120572 (120583 119865119911119894119895 119865119909119894119895)= 120593119897119891 119862120572 (120583 119865119911119894119895 119865119909119894119895 = 0) minus 12120583119865119911119894119895+ 12 (120583119865119911119894119895 minus 119865119909119894119895)

(39)







Yawrate


11 12

21 22

ax

ay Fy11 Fy12 Fy21

Fy22 Fx12 Fx21 Fy22


2 4 6 8 10Time (s)

minus60

minus40

minus20

0

20

40

60

Stee

ring

angl

e (de

g)

Steering angle


minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)


2 4 6 8 10Time (s)


Figure 7






0 2 4 6 8 10 12Time (s)

minus3

minus2

minus1

0

1

2

3

Side

slip

angl

e (de

gs)



10Time (s)

minus3

minus2

minus1

0

1

2

3

4

Side

slip

angl

e (de

gs)


2 4 6 8








CKFEKFUKF of7DOF


IMM-CKFscope


minus15

minus10

minus5

0

5

10

15

Yaw

rate

(deg

s)

Yaw rate

2 4 6 8 10Time (s)


2 4 6 8 10Time (s)

minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)


Figure 11

2 3 4 5 6 7 8 9 10Time (s)

minus15

minus1

minus05

0

05

1

15

Side

slip

angl

e (de

gs)

IMM-CKFACTUALCKF

EKFUKF






2 3 4 5 6 7 8 9 10Time (s)

minus3000

minus2000

minus1000

0

1000La

tera

l tire

-roa

d fo

rce o

f fro

nt ri

ght t

ire (N

)

IMM-CKFCKFACTUAL

EKFUKF


2 3 4 5 6 7 8 9 10Time (s)

CKFActualIMM-CKF

EKFUKF

minus1500

minus1000

minus500

0

500

1000

1500

2000

Late

ral t

ire-r

oad

forc

e of f

ront

left

tires

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

3000

Long

itudi

nal t

ire-r

oad

forc

e of r

ight

fron

t tire

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

Long

itudi

nal t

ire-r

oad

forc

e of L

eft fr

ont t

ire (N

)


Figure 13


6 Conclusions




Data Availability




Acknowledgments



6 65 7 75 8 85 9Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

Side

slip

Ang

le (

deg)



6 65 7 75 8 85 9Time (s)

08

1

12

14

16

18

2

22

Fron

t cor

nerin

g sti

ffnes

s (N

rad

)


times105


6 65 7 75 8 85 9Time (s)

04

06

08

1

12

14

16

Rear

corn

erin

g sti

ffnes

s


times105


Figure 14

References







































Journal of












Journal of







Volume 2018






Volume 2018









Previous Step

Initialconditions



tire

Modeprobability

Update



1kminus

11kp minus

2| 1kminus

21kP minus

1|2k

2|1k

( ) ( )1

1i j ik ij k

Cπ minus=

1|1k

2|2k

21kx minus

11kx minus

11kp minus

21kp minus

1| 1k kminus

1| 1k kp minus

2| 1k kminus

2| 1k kp minus

1k1kp2k2kp

Estimation Fusion

kkp


x

x

x x

xx

x1k

2k



= 119875(119898(119894)119896minus1

| 119898(119894)119896119885119896minus1) = (1119862)120587119894119895120583(119894)119896minus1 119894 119895 =1 2 119903120583(119894119895)



120587119894119895 = [099 001002 098] (20)





(119905) = 119891 (119909 (119905) 119906 (119905)) + 119882 (119905)119910 (119905) = ℎ (119909 (119905) 119906 (119905)) + 119881 (119905) (21)



119906 = [120575ℎ 12059611 12059612 12059621 12059622] (22)


119910 = [119886119909 119886119910 12059611 12059612 12059621 12059622] (23)


119909 = [ V119892 120573 11986511991011 11986511991012 11986511991021 11986511991022 11986511990911 11986511990912 11986511990921 11986511991022] (24)







(25)


ℎ1 = 1119898 [(1199098 + 1199099) cos 1199061 + 1199096 + 1199097minus (1199098 + 1199099) sin 1199061]

ℎ2 = 1119898 [(1199094 + 1199095) cos 1199061 + 11990910 + 11990911minus (1199094 + 1199095) sin 1199061]

ℎ3 = 1199091ℎ4 = 1199062ℎ5 = 1199063ℎ6 = 1199064ℎ7 = 1199065

(26)



(27)



(i) Time Update


119898sum119894=1


119898sum119894=1


(28)



119898sum119894=1

119885119894119896|119896minus1119875119911119911119896|119896minus1 = 1119898

119898sum119894=1


119898sum119894=1


(29)




(30)

120583(119894)119896 = 119875 (119898(119894)119896 | 119911119896) = 1119888Λ(119894)119896 119888119895 (31)

119888 = 119903sum119894=1

Λ(119894)119896 119888119895 (32)


where 119878(119894)119896

= 119867(119894)119896119875(119894)119896|119896minus1

(119867(119894)119896)119879 + 119877(119894)

119896 119878(119894)119896



119909119896|119896 = 119903sum119894=1

119909(119894)119896 120583(119894)119896 (33)

119875119896|119896 = 119903sum119894=1

[119875(119894)119896|119896 + (119909119896|119896 minus 119909(119894)119896|119896) (119909119896|119896 minus 119909(119894)119896|119896)119879] 120583(119894)119896 (34)




119886119910 = 119886119910119898 minus 119892120572119903119900119897119897 (35)



120572119903119900119897119897 = 119898119886119910ℎ119903119900119897119897119870119865 + 119870119877 minus 119898119892ℎ119903119900119897119897 (36)


120583 = 1003816100381610038161003816100381610038161003816100381611988611991011989210038161003816100381610038161003816100381610038161003816 + 120583119890 (37)



120593119897119891 = radic12058321198652119911119894119895 minus 1198652119909119894119895120583119865119911119894119895 (38)


119862120572 (120583 119865119911119894119895 119865119909119894119895)= 120593119897119891 119862120572 (120583 119865119911119894119895 119865119909119894119895 = 0) minus 12120583119865119911119894119895+ 12 (120583119865119911119894119895 minus 119865119909119894119895)

(39)







Yawrate


11 12

21 22

ax

ay Fy11 Fy12 Fy21

Fy22 Fx12 Fx21 Fy22


2 4 6 8 10Time (s)

minus60

minus40

minus20

0

20

40

60

Stee

ring

angl

e (de

g)

Steering angle


minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)


2 4 6 8 10Time (s)


Figure 7






0 2 4 6 8 10 12Time (s)

minus3

minus2

minus1

0

1

2

3

Side

slip

angl

e (de

gs)



10Time (s)

minus3

minus2

minus1

0

1

2

3

4

Side

slip

angl

e (de

gs)


2 4 6 8








CKFEKFUKF of7DOF


IMM-CKFscope


minus15

minus10

minus5

0

5

10

15

Yaw

rate

(deg

s)

Yaw rate

2 4 6 8 10Time (s)


2 4 6 8 10Time (s)

minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)


Figure 11

2 3 4 5 6 7 8 9 10Time (s)

minus15

minus1

minus05

0

05

1

15

Side

slip

angl

e (de

gs)

IMM-CKFACTUALCKF

EKFUKF






2 3 4 5 6 7 8 9 10Time (s)

minus3000

minus2000

minus1000

0

1000La

tera

l tire

-roa

d fo

rce o

f fro

nt ri

ght t

ire (N

)

IMM-CKFCKFACTUAL

EKFUKF


2 3 4 5 6 7 8 9 10Time (s)

CKFActualIMM-CKF

EKFUKF

minus1500

minus1000

minus500

0

500

1000

1500

2000

Late

ral t

ire-r

oad

forc

e of f

ront

left

tires

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

3000

Long

itudi

nal t

ire-r

oad

forc

e of r

ight

fron

t tire

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

Long

itudi

nal t

ire-r

oad

forc

e of L

eft fr

ont t

ire (N

)


Figure 13


6 Conclusions




Data Availability




Acknowledgments



6 65 7 75 8 85 9Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

Side

slip

Ang

le (

deg)



6 65 7 75 8 85 9Time (s)

08

1

12

14

16

18

2

22

Fron

t cor

nerin

g sti

ffnes

s (N

rad

)


times105


6 65 7 75 8 85 9Time (s)

04

06

08

1

12

14

16

Rear

corn

erin

g sti

ffnes

s


times105


Figure 14

References







































Journal of












Journal of







Volume 2018






Volume 2018












(25)


ℎ1 = 1119898 [(1199098 + 1199099) cos 1199061 + 1199096 + 1199097minus (1199098 + 1199099) sin 1199061]

ℎ2 = 1119898 [(1199094 + 1199095) cos 1199061 + 11990910 + 11990911minus (1199094 + 1199095) sin 1199061]

ℎ3 = 1199091ℎ4 = 1199062ℎ5 = 1199063ℎ6 = 1199064ℎ7 = 1199065

(26)



(27)



(i) Time Update


119898sum119894=1


119898sum119894=1


(28)



119898sum119894=1

119885119894119896|119896minus1119875119911119911119896|119896minus1 = 1119898

119898sum119894=1


119898sum119894=1


(29)




(30)

120583(119894)119896 = 119875 (119898(119894)119896 | 119911119896) = 1119888Λ(119894)119896 119888119895 (31)

119888 = 119903sum119894=1

Λ(119894)119896 119888119895 (32)


where 119878(119894)119896

= 119867(119894)119896119875(119894)119896|119896minus1

(119867(119894)119896)119879 + 119877(119894)

119896 119878(119894)119896



119909119896|119896 = 119903sum119894=1

119909(119894)119896 120583(119894)119896 (33)

119875119896|119896 = 119903sum119894=1

[119875(119894)119896|119896 + (119909119896|119896 minus 119909(119894)119896|119896) (119909119896|119896 minus 119909(119894)119896|119896)119879] 120583(119894)119896 (34)




119886119910 = 119886119910119898 minus 119892120572119903119900119897119897 (35)



120572119903119900119897119897 = 119898119886119910ℎ119903119900119897119897119870119865 + 119870119877 minus 119898119892ℎ119903119900119897119897 (36)


120583 = 1003816100381610038161003816100381610038161003816100381611988611991011989210038161003816100381610038161003816100381610038161003816 + 120583119890 (37)



120593119897119891 = radic12058321198652119911119894119895 minus 1198652119909119894119895120583119865119911119894119895 (38)


119862120572 (120583 119865119911119894119895 119865119909119894119895)= 120593119897119891 119862120572 (120583 119865119911119894119895 119865119909119894119895 = 0) minus 12120583119865119911119894119895+ 12 (120583119865119911119894119895 minus 119865119909119894119895)

(39)







Yawrate


11 12

21 22

ax

ay Fy11 Fy12 Fy21

Fy22 Fx12 Fx21 Fy22


2 4 6 8 10Time (s)

minus60

minus40

minus20

0

20

40

60

Stee

ring

angl

e (de

g)

Steering angle


minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)


2 4 6 8 10Time (s)


Figure 7






0 2 4 6 8 10 12Time (s)

minus3

minus2

minus1

0

1

2

3

Side

slip

angl

e (de

gs)



10Time (s)

minus3

minus2

minus1

0

1

2

3

4

Side

slip

angl

e (de

gs)


2 4 6 8








CKFEKFUKF of7DOF


IMM-CKFscope


minus15

minus10

minus5

0

5

10

15

Yaw

rate

(deg

s)

Yaw rate

2 4 6 8 10Time (s)


2 4 6 8 10Time (s)

minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)


Figure 11

2 3 4 5 6 7 8 9 10Time (s)

minus15

minus1

minus05

0

05

1

15

Side

slip

angl

e (de

gs)

IMM-CKFACTUALCKF

EKFUKF






2 3 4 5 6 7 8 9 10Time (s)

minus3000

minus2000

minus1000

0

1000La

tera

l tire

-roa

d fo

rce o

f fro

nt ri

ght t

ire (N

)

IMM-CKFCKFACTUAL

EKFUKF


2 3 4 5 6 7 8 9 10Time (s)

CKFActualIMM-CKF

EKFUKF

minus1500

minus1000

minus500

0

500

1000

1500

2000

Late

ral t

ire-r

oad

forc

e of f

ront

left

tires

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

3000

Long

itudi

nal t

ire-r

oad

forc

e of r

ight

fron

t tire

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

Long

itudi

nal t

ire-r

oad

forc

e of L

eft fr

ont t

ire (N

)


Figure 13


6 Conclusions




Data Availability




Acknowledgments



6 65 7 75 8 85 9Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

Side

slip

Ang

le (

deg)



6 65 7 75 8 85 9Time (s)

08

1

12

14

16

18

2

22

Fron

t cor

nerin

g sti

ffnes

s (N

rad

)


times105


6 65 7 75 8 85 9Time (s)

04

06

08

1

12

14

16

Rear

corn

erin

g sti

ffnes

s


times105


Figure 14

References







































Journal of












Journal of







Volume 2018






Volume 2018









where 119878(119894)119896

= 119867(119894)119896119875(119894)119896|119896minus1

(119867(119894)119896)119879 + 119877(119894)

119896 119878(119894)119896



119909119896|119896 = 119903sum119894=1

119909(119894)119896 120583(119894)119896 (33)

119875119896|119896 = 119903sum119894=1

[119875(119894)119896|119896 + (119909119896|119896 minus 119909(119894)119896|119896) (119909119896|119896 minus 119909(119894)119896|119896)119879] 120583(119894)119896 (34)




119886119910 = 119886119910119898 minus 119892120572119903119900119897119897 (35)



120572119903119900119897119897 = 119898119886119910ℎ119903119900119897119897119870119865 + 119870119877 minus 119898119892ℎ119903119900119897119897 (36)


120583 = 1003816100381610038161003816100381610038161003816100381611988611991011989210038161003816100381610038161003816100381610038161003816 + 120583119890 (37)



120593119897119891 = radic12058321198652119911119894119895 minus 1198652119909119894119895120583119865119911119894119895 (38)


119862120572 (120583 119865119911119894119895 119865119909119894119895)= 120593119897119891 119862120572 (120583 119865119911119894119895 119865119909119894119895 = 0) minus 12120583119865119911119894119895+ 12 (120583119865119911119894119895 minus 119865119909119894119895)

(39)







Yawrate


11 12

21 22

ax

ay Fy11 Fy12 Fy21

Fy22 Fx12 Fx21 Fy22


2 4 6 8 10Time (s)

minus60

minus40

minus20

0

20

40

60

Stee

ring

angl

e (de

g)

Steering angle


minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)


2 4 6 8 10Time (s)


Figure 7






0 2 4 6 8 10 12Time (s)

minus3

minus2

minus1

0

1

2

3

Side

slip

angl

e (de

gs)



10Time (s)

minus3

minus2

minus1

0

1

2

3

4

Side

slip

angl

e (de

gs)


2 4 6 8








CKFEKFUKF of7DOF


IMM-CKFscope


minus15

minus10

minus5

0

5

10

15

Yaw

rate

(deg

s)

Yaw rate

2 4 6 8 10Time (s)


2 4 6 8 10Time (s)

minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)


Figure 11

2 3 4 5 6 7 8 9 10Time (s)

minus15

minus1

minus05

0

05

1

15

Side

slip

angl

e (de

gs)

IMM-CKFACTUALCKF

EKFUKF






2 3 4 5 6 7 8 9 10Time (s)

minus3000

minus2000

minus1000

0

1000La

tera

l tire

-roa

d fo

rce o

f fro

nt ri

ght t

ire (N

)

IMM-CKFCKFACTUAL

EKFUKF


2 3 4 5 6 7 8 9 10Time (s)

CKFActualIMM-CKF

EKFUKF

minus1500

minus1000

minus500

0

500

1000

1500

2000

Late

ral t

ire-r

oad

forc

e of f

ront

left

tires

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

3000

Long

itudi

nal t

ire-r

oad

forc

e of r

ight

fron

t tire

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

Long

itudi

nal t

ire-r

oad

forc

e of L

eft fr

ont t

ire (N

)


Figure 13


6 Conclusions




Data Availability




Acknowledgments



6 65 7 75 8 85 9Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

Side

slip

Ang

le (

deg)



6 65 7 75 8 85 9Time (s)

08

1

12

14

16

18

2

22

Fron

t cor

nerin

g sti

ffnes

s (N

rad

)


times105


6 65 7 75 8 85 9Time (s)

04

06

08

1

12

14

16

Rear

corn

erin

g sti

ffnes

s


times105


Figure 14

References







































Journal of












Journal of







Volume 2018






Volume 2018











Yawrate


11 12

21 22

ax

ay Fy11 Fy12 Fy21

Fy22 Fx12 Fx21 Fy22


2 4 6 8 10Time (s)

minus60

minus40

minus20

0

20

40

60

Stee

ring

angl

e (de

g)

Steering angle


minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)


2 4 6 8 10Time (s)


Figure 7






0 2 4 6 8 10 12Time (s)

minus3

minus2

minus1

0

1

2

3

Side

slip

angl

e (de

gs)



10Time (s)

minus3

minus2

minus1

0

1

2

3

4

Side

slip

angl

e (de

gs)


2 4 6 8








CKFEKFUKF of7DOF


IMM-CKFscope


minus15

minus10

minus5

0

5

10

15

Yaw

rate

(deg

s)

Yaw rate

2 4 6 8 10Time (s)


2 4 6 8 10Time (s)

minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)


Figure 11

2 3 4 5 6 7 8 9 10Time (s)

minus15

minus1

minus05

0

05

1

15

Side

slip

angl

e (de

gs)

IMM-CKFACTUALCKF

EKFUKF






2 3 4 5 6 7 8 9 10Time (s)

minus3000

minus2000

minus1000

0

1000La

tera

l tire

-roa

d fo

rce o

f fro

nt ri

ght t

ire (N

)

IMM-CKFCKFACTUAL

EKFUKF


2 3 4 5 6 7 8 9 10Time (s)

CKFActualIMM-CKF

EKFUKF

minus1500

minus1000

minus500

0

500

1000

1500

2000

Late

ral t

ire-r

oad

forc

e of f

ront

left

tires

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

3000

Long

itudi

nal t

ire-r

oad

forc

e of r

ight

fron

t tire

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

Long

itudi

nal t

ire-r

oad

forc

e of L

eft fr

ont t

ire (N

)


Figure 13


6 Conclusions




Data Availability




Acknowledgments



6 65 7 75 8 85 9Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

Side

slip

Ang

le (

deg)



6 65 7 75 8 85 9Time (s)

08

1

12

14

16

18

2

22

Fron

t cor

nerin

g sti

ffnes

s (N

rad

)


times105


6 65 7 75 8 85 9Time (s)

04

06

08

1

12

14

16

Rear

corn

erin

g sti

ffnes

s


times105


Figure 14

References







































Journal of












Journal of







Volume 2018






Volume 2018









0 2 4 6 8 10 12Time (s)

minus3

minus2

minus1

0

1

2

3

Side

slip

angl

e (de

gs)



10Time (s)

minus3

minus2

minus1

0

1

2

3

4

Side

slip

angl

e (de

gs)


2 4 6 8








CKFEKFUKF of7DOF


IMM-CKFscope


minus15

minus10

minus5

0

5

10

15

Yaw

rate

(deg

s)

Yaw rate

2 4 6 8 10Time (s)


2 4 6 8 10Time (s)

minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)


Figure 11

2 3 4 5 6 7 8 9 10Time (s)

minus15

minus1

minus05

0

05

1

15

Side

slip

angl

e (de

gs)

IMM-CKFACTUALCKF

EKFUKF






2 3 4 5 6 7 8 9 10Time (s)

minus3000

minus2000

minus1000

0

1000La

tera

l tire

-roa

d fo

rce o

f fro

nt ri

ght t

ire (N

)

IMM-CKFCKFACTUAL

EKFUKF


2 3 4 5 6 7 8 9 10Time (s)

CKFActualIMM-CKF

EKFUKF

minus1500

minus1000

minus500

0

500

1000

1500

2000

Late

ral t

ire-r

oad

forc

e of f

ront

left

tires

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

3000

Long

itudi

nal t

ire-r

oad

forc

e of r

ight

fron

t tire

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

Long

itudi

nal t

ire-r

oad

forc

e of L

eft fr

ont t

ire (N

)


Figure 13


6 Conclusions




Data Availability




Acknowledgments



6 65 7 75 8 85 9Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

Side

slip

Ang

le (

deg)



6 65 7 75 8 85 9Time (s)

08

1

12

14

16

18

2

22

Fron

t cor

nerin

g sti

ffnes

s (N

rad

)


times105


6 65 7 75 8 85 9Time (s)

04

06

08

1

12

14

16

Rear

corn

erin

g sti

ffnes

s


times105


Figure 14

References







































Journal of












Journal of







Volume 2018






Volume 2018









CKFEKFUKF of7DOF


IMM-CKFscope


minus15

minus10

minus5

0

5

10

15

Yaw

rate

(deg

s)

Yaw rate

2 4 6 8 10Time (s)


2 4 6 8 10Time (s)

minus4

minus2

0

2

4

Late

ral a

ccel

erat

ion

(ms

2)


Figure 11

2 3 4 5 6 7 8 9 10Time (s)

minus15

minus1

minus05

0

05

1

15

Side

slip

angl

e (de

gs)

IMM-CKFACTUALCKF

EKFUKF






2 3 4 5 6 7 8 9 10Time (s)

minus3000

minus2000

minus1000

0

1000La

tera

l tire

-roa

d fo

rce o

f fro

nt ri

ght t

ire (N

)

IMM-CKFCKFACTUAL

EKFUKF


2 3 4 5 6 7 8 9 10Time (s)

CKFActualIMM-CKF

EKFUKF

minus1500

minus1000

minus500

0

500

1000

1500

2000

Late

ral t

ire-r

oad

forc

e of f

ront

left

tires

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

3000

Long

itudi

nal t

ire-r

oad

forc

e of r

ight

fron

t tire

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

Long

itudi

nal t

ire-r

oad

forc

e of L

eft fr

ont t

ire (N

)


Figure 13


6 Conclusions




Data Availability




Acknowledgments



6 65 7 75 8 85 9Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

Side

slip

Ang

le (

deg)



6 65 7 75 8 85 9Time (s)

08

1

12

14

16

18

2

22

Fron

t cor

nerin

g sti

ffnes

s (N

rad

)


times105


6 65 7 75 8 85 9Time (s)

04

06

08

1

12

14

16

Rear

corn

erin

g sti

ffnes

s


times105


Figure 14

References







































Journal of












Journal of







Volume 2018






Volume 2018









2 3 4 5 6 7 8 9 10Time (s)

minus3000

minus2000

minus1000

0

1000La

tera

l tire

-roa

d fo

rce o

f fro

nt ri

ght t

ire (N

)

IMM-CKFCKFACTUAL

EKFUKF


2 3 4 5 6 7 8 9 10Time (s)

CKFActualIMM-CKF

EKFUKF

minus1500

minus1000

minus500

0

500

1000

1500

2000

Late

ral t

ire-r

oad

forc

e of f

ront

left

tires

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

3000

Long

itudi

nal t

ire-r

oad

forc

e of r

ight

fron

t tire

(N)


0 2 4 6 8 10Time (s)

ActualIMM-CKFCKF

EKFUKF

0

500

1000

1500

2000

2500

Long

itudi

nal t

ire-r

oad

forc

e of L

eft fr

ont t

ire (N

)


Figure 13


6 Conclusions




Data Availability




Acknowledgments



6 65 7 75 8 85 9Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

Side

slip

Ang

le (

deg)



6 65 7 75 8 85 9Time (s)

08

1

12

14

16

18

2

22

Fron

t cor

nerin

g sti

ffnes

s (N

rad

)


times105


6 65 7 75 8 85 9Time (s)

04

06

08

1

12

14

16

Rear

corn

erin

g sti

ffnes

s


times105


Figure 14

References







































Journal of












Journal of







Volume 2018






Volume 2018









6 65 7 75 8 85 9Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

Side

slip

Ang

le (

deg)



6 65 7 75 8 85 9Time (s)

08

1

12

14

16

18

2

22

Fron

t cor

nerin

g sti

ffnes

s (N

rad

)


times105


6 65 7 75 8 85 9Time (s)

04

06

08

1

12

14

16

Rear

corn

erin

g sti

ffnes

s


times105


Figure 14

References







































Journal of












Journal of







Volume 2018






Volume 2018




































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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Design of an Interacting Multiple Model-Cubature Kalman