Active Suspension Control Based on Estimated Road Class ...downloads.hindawi.com/journals/mpe/2019/3483710.pdfMathematicalProblemsinEngineering 4 6 8 10 12 14 16 18 20 22 Longit ositio

Research ArticleActive Suspension Control Based on Estimated Road Class forOff-Road Vehicle

Mingde Gong 12 HaohaoWang3 and XinWang1

1School of Mechanical Engineering Yanshan University Qinhuangdao 066004 China2Hebei Key Laboratory of Special Delivery Equipment Yanshan University Qinhuangdao 066004 China3School of Mechanical Science and Engineering Jilin University Changchun 130025 China

Correspondence should be addressed to Mingde Gong 307070001qqcom

Received 24 January 2019 Revised 30 May 2019 Accepted 16 June 2019 Published 27 June 2019

Academic Editor Andrzej Swierniak

Copyright copy 2019 Mingde Gong 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

Road input can be provided for a vehicle in advance by using an optical sensor to preview the front terrain and suspensionparameters can be adjusted before a corresponding moment to keep the body as smooth as possible and thus improve ride comfortand handling stability However few studies have described this phenomenon in detail In this study a LiDAR coupled with globalpositioning and inertial navigation systems was used to obtain the digital terrain in front of a vehicle in the form of a 3D pointcloud which was processed by a statistical filter in the Point Cloud Library for the acquisition of accurate data Next the inversedistance weighting interpolationmethod and fractal interpolation were adopted to extract the road height profile from the 3D pointcloud and improve its accuracyThe roughness grade of the road height profile was utilised as the input of active suspensionThenthe active suspension which was based on an LQG controller used the analytic hierarchy process method to select proper weightcoefficients of performance indicators according to the previously calculated road grade Finally the road experiment verified thatreasonable selection of active suspensionrsquos LQG controller weightings based on estimated road profile and road class through fractalinterpolation can improve the ride comfort and handling stability of the vehicle more than passive suspension did

1 Introduction

In the running course of a vehicle the road excitation charac-teristics considerably influence its ride comfort and handlingstability [1 2] Suspension is an important component thatplays a critical function in the transmission of force betweenvehiclersquos tire and body and alleviates the impact of the roadsurface Consequently obtaining the pavement roughnessin front of a vehicle to control its suspension and improveride comfort and handling stability has been attracting theattention of researchers

Existing active suspension regulates its parameters onlywhen the road interferes with the vehicle [1 2] The use ofthe sensor system to detect the road on which a vehicle willbe running and of the built-in controllers to process therelated data can maximise the potential of active suspensionTherefore a suspension system with a preview function isvital to future intelligent chassis

Stereo cameras [3 4] and LiDAR sensors [5ndash7] can beadopted for this function bymeasuring the road height infor-mation in front of a running vehicle with certain accuracyIn [2] the use of a low-cost ultrasonic sensor was examinedto compute road surface height estimation in real time byusing simple algorithms due to the drawbacks of sensors (iestereo vision or LiDAR) such as costliness and computationcomplexity However the ultrasonic sensor does not have thehigh precision of a laser radar In [7] a LiDAR coupled with aglobal positioning system (GPS) and an inertial navigationsystem (INS) was used to measure two off-road surfacesbut details of the measurement and data processing werenot provided In [8] the author proposed the use of theoverall response of the preceding vehicles to generate previewcontroller information for follower vehicles no sensor is usedto measure the terrain online

Although many scholars have proposed the adoptionof preview information for active suspension detailed

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 3483710 17 pageshttpsdoiorg10115520193483710

2 Mathematical Problems in Engineering

3D laser scanner

Scan filter

Vehicle INS system GPS

Real-Time Kinematic (RTK)Kalman filter algorithm

Scan assembly Data fusionPoint cloud data filter

Interpolation of elevation data

Elevation profile

Sequencesignal

Vehicle suspension

control

Point cloud in LiDAR coordinate system

Accurate position and attitude information

Point cloud in geodetic coordinate system

Preview Pavement Elevation Information

Figure 1 Mobile mapping system

descriptions have not been provided Instead researchershave merely introduced concepts or assumed the road heightprofile to be known and obtained by statistical methodsThis study proposes an approach that uses LiDAR cou-pled with INS and GPS to obtain a preview of the 3Dpoint cloud of the terrain in a geodetic coordinate systemand extract the road height profile in front of a vehicleThis work also elaborates the methods and steps of dataprocessing The road roughness grade is identified on thebasis of the aforementioned information Then the Linear-Quadratic-Gaussian (LQG) controllerrsquos weight coefficientsof performance indicators are optimised in different roadroughness grades Finally we select proper weight coeffi-cients that correspond to previously calculated road gradesto improve the vehiclersquos ride comfort and handling stabil-ity

This study is organised as follows Section 2 providesthe mobile mapping system Section 3 presents the dataprocessing of sensors Section 4 uses the methods of thetwo preceding sections to experiment and obtain the 3Dpoint cloud of the test site Section 5 adopts the road heightprofile information utilised in fractal interpolation theoryin Section 6 to improve accuracy This profile informationis extracted from the 3D point cloud by using the inversedistance weighting (IDW) interpolation method Section 7solves the grade of the road height profilersquos roughnessSection 8 establishes the suspension vertical dynamic modeland designs the LQG controller for active suspensionSection 9 calculates the controllerrsquos weight coefficients ofperformance indicators for different road grades using theanalytic hierarchy process (AHP) In Section 10 a road exper-iment verifies that reasonable selection of LQG controllerweightings based on estimated road profile and road classthrough fractal interpolation can improve the ride comfortand handling stability of the vehicle more than passivesuspension does

2 Mobile Mapping System

To control active suspension with preview information adevice thatmaps the front terrain of the running vehiclemustbe developed Laser radar scanning is a rapid 3D measure-mentmethod and has existed for awhile But it is seldomusedfor the realization of active suspension The mobile LiDARsystem is mainly composed of a LiDAR sensor GPS and INS(Figure 1) In the running course the LiDAR is mounted infront of the vehicle and returns the 3D point cloud of the roadon the basis of its own coordinate system One vital challengein accurate mapping is estimating the attitude of LiDARwhich affects the precision of the point cloud To solve thisproblem the INS which is installed near the LiDAR collectsthe attitude angle that is used to compensate for motion dur-ing scan acquisition to improve the precision of data in realtime We need to realise the accurate location of the LiDARto spread the 3D point cloud on the earth surface AlthoughGPS has higher positioning accuracy than other approachesit is vulnerable to a wide variety of interferences such as themultipath effect from radars electromagnetic interferenceand block of signals [9 10] INS is a self-contained system thatconsists of inertial measurement units which can providedynamic measurement for a short period with high frequentupdate It is a complete autonomous navigation system withremarkable concealment strong anti-interference capabilityand immunity to meteorological conditions [11] Howeverits measurement error may accumulate over time due todrifting effects [12] The integration of GPS and INS has beenproposed and widely implemented for vehicle applications toleverage the strengths of the two systems and to offset theirindividual drawbacks because they are more advantageousthan single navigation systems [13 14] In these integratedsystems the Kalman filter is a popular fusion method thathas been used in recent years due to its practicability andsuitability [15]The centimeter-level positioning accuracy can

Mathematical Problems in Engineering 3

L1

L2

L4

L3

XL

ZL

YL

XLLi

OL

OL

Figure 2 Operation mode of ibeo LUX 2010

be obtained in real time by using real-time kinematic (RTK)technology

3 Point Cloud Data Processing

The multilinear LiDAR can launch more one laser lineswhilst rotating around the laser emission centre at a certainfrequency and returns the point cloud of the obstacle in theform of relative distance 119871 119894 and angle 120579 which depend onthe LiDARrsquos coordinate system The point cloud is stackedin front of the vehicle instead of being spread on the earthsurface because the LiDAR coordinate system moves in realtime with the vehicle To resolve this problem in the firststep we should coordinate the transformation of the pointcloud Relative distance 119871 119894 and angle 120579 of the point cloudare transformed into the Cartesian coordinate In the secondstep data fusion algorithm is then implemented GPSINSintegrated navigation system is used to transform the dataobtained from first step into theWGS-84 geodetic coordinatesystem which is stationary

31 Coordinate Transformation of Point Cloud The LiDARrsquosvertical angle resolution is marked as 120572 Figure 2 shows that119874119871 minus119883119871119884119871119885119871 is a LiDAR coordinate system whose origin119874119871coincides with the laser emission centre The angle betweenthe instantaneous laser line and the 119883119871 axis is defined as 120579and 119871 119894 is the distance between119874119871 and the obstacle Accordingto geometric relations the 3D point cloud is represented inthe Cartesian coordinate system as follows

[[[119909119871119910119871119911119871]]]= 119871 119894 sdot [[

[cos120572119894lowast sdot cos 120579cos120572119894lowast sdot sin 120579

sin120572119894lowast]]]

120572119894lowast =

2120572 119894 = 1120572 119894 = 2minus120572 119894 = 3minus2120572 119894 = 4

120579 isin [minus60∘ 50∘]

(1)

32 Data Fusion The coordinate systems of the mobilemapping system include the LiDAR coordinate system 119874119871 minus119883119871119884119871119885119871 INS coordinate system119874119868minus119883119868119884119868119885119868 local horizontalcoordinate system 119874119866 minus 119883119866119884119866119885119866 and WGS-84 geodeticcoordinate system 11987484 minus 119883841198848411988584 (Figure 3)

The coordinate values of the point cloud [119909119871 119910119871 119911119871] basedon the LiDAR coordinate system are transformed to theWGS-84 geodetic coordinate system as follows

[[[119909119882841199101198828411991111988284

]]]

= [[[119909119866119875119878119910119866119875119878119911119866119875119878

]]]

+ 119877119882[[[119877119873[[

[119877119872[[

[119909119871119910119871119911119871]]]+ [[[Δ119909119871119868Δ119910119871119868Δ119911119871119868

]]]minus [[[Δ119909119868119866Δ119910119868119866Δ119911119868119866

]]]]]]]]]

(2)

where 119877119872 is the rotation matrix between LiDAR and INSwhose element is calculated from the installation angle errorof two sensors The error angles around 119909 119910 and 119911 are 120573 120572and 120574 which represent roll pitch and course angle errorsrespectively during the installation process

119877119872 = 119877120574119911 sdot 119877120572119910 sdot 119877120573119909119877120574119911 = [[

[cos 120574 sin 120574 0minus sin 120574 cos 120574 00 0 1

]]]

119877120572119910 = [[[cos120572 0 minus sin1205720 1 0

sin120572 0 cos120572]]]

119877120573119909 = [[[1 0 00 cos120573 sin1205730 minussin120573 cos120573

]]]

(3)


point

O84

X84

Z84

OL

XL

YL

ZL

OI

XI

YI

ZI

Y84

ZG

OG

XG

YG

L

Figure 3 Coordinate systems of mobile mapping system

where 119877119873 is the rotation matrix between INS and the localhorizontal coordinate system and the matrix element iscalculated from the measured values of INS where 120593 120596 and119896 represent pitch roll and course angles respectively

119877119873 = 119877120581 sdot 119877120593 sdot 119877120596119877120581 = [[


]]]

119877120593 = [[[cos120593 0 minus sin1205930 1 0

sin120593 0 cos120593]]]

119877120596 = [[[1 0 00 cos120596 sin1205960 minus sin120596 cos120596

]]]

(4)

where 119877119882 is the rotation matrix between the local horizontalcoordinate system and the WGS-84 geodetic coordinate sys-tem and the matrix element is calculated from the measuredvalues of the loosely coupled GPSINS system where 119861 119871and119867 represent latitude longitude and altitude respectively

119877119882 = [[[minus cos 119871 sin119861 minus sin 119871 minus cos 119871 cos119861minus sin 119871 sin119861 cos 119871 minus sin 119871 cos119861

cos119861 0 minus sin119861]]]

(5)

where [119909119871119868119910119871119868119911119871119868]119879 are the offset values between theLiDAR and the INS coordinate system [119909119868119866119910119868119866119911119868119866]119879

represent the offset values between the INS and the GPScoordinate system

The location of GPS in the WGS-84 geodetic coordinatesystem can be expressed as follows

[[[119909119866119875119878119910119866119875119878119911119866119875119878

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

( 119886radic1 minus 1198902sin2119861 + 119867) sdot cos119861 sdot cos 119871( 119886radic1 minus 1198902sin2119861 + 119867) sdot cos119861 sdot sin 119871[ 119886radic1 minus 1198902sin2119861 sdot (1 minus 1198902) + 119867] sdot sin119861

]]]]]]]]

(6)

where 120572 is a semimajor axis of WGS-84 and 119890 is the firsteccentricity

119886 = 6378137 1198981198902 = 0006694379 (7)

4 Terrain Measurement

In this study integrated navigation system of MEMSis adopted of which core components are the gyro-scope accelerometer and the high-performance BeidouGPSreceiver The position velocity azimuth angle attitude anglethree-axis acceleration and angular velocity can be outputby the Kalman filter integrated navigation algorithm thatcan solve the problem of INS error drift over time andRTK technology It enters the pure inertial navigation modewithout satellite navigation signal and can maintain highnavigation accuracy in a short time

The ibeo LUX 2010 selected for project implementationdetects objects and their distance by means of laser beams


Table 1 Specification of integrated navigation system

Azimuthaccuracy

Attitudeaccuracy

Positioningaccuracy

Speedaccuracy

le 01∘ le 01∘ le 2119888119898 le 01119898119904Interface Data update rate Voltage Power waste119877119878232 le 500119867119911 119863119862 9119881 sim 36119881 le 7119882

Table 2 Specification of LiDAR

Scanningfrequency

Angularresolution

Number of laserlines

Laserwavelength

50119867119911 05∘ 4 905119899119898Laser grade

Verticalangular

resolutionVoltage Interface

119862119897119886119904119904 119868 120572 = 08∘ 119863119862 9119881 sim 27119881 119862119860119873119864119905ℎ119890119903119899119890119905

It scans the surroundings with several rotating laser beamsreceives the echoes with a photo diode receiver processes thedata by means of a time of flight calculation and issues theprocessed data via the interfaces Ethernet andor CAN

The specifications of integrated navigation system andLiDAR are shown in Tables 1 and 2

The calibration of the integrated navigation system andthe LiDAR needs to be carried out The main work is tomeasure the relative position and angular error between thesensors

The sensor position center is given by the manufacturerso the relative position error between sensors is measured bya ruler a triangle ruler and the like

Sensors are mounted on an adjustable platform Inte-grated navigation systemrsquos pitch angle and roll angle areadjusted to 0 according to its output value and the headingangle ismade to coincidewith the longitudinal symmetry axisof the vehicle by using a laser scale Based on the adjustedintegrated navigation system the relative angle error betweensensors is measured by a level meter

The data rate of LiDAR is 50Hz and that of integratednavigation system is 500Hz In the process of data solutiona unified time stamp and Kalman filter prediction model areused to solve the problem of inconsistent working frequencyof multisensor The vehicle computer time is regarded asthe unified time stamp of the system According to Kalmanfilter prediction model the sensor data are converted tothe working frequency of the LiDAR thus forming themultisensor data at the same frequency

The dirt terrain around a university was selected asthe test site (Figure 4) Multithreading technology was usedto ensure simultaneous data acquisition from integratednavigation system and LiDAR The contact surface betweenthe tire and the ground is the reference zero surface Figure 5shows the road map of the selected dirt terrain after dataprocessing in MATLAB

Table 3 shows the point cloud properties of the dirtterrain

Figure 4 Test site

Vert

ical h

eight

(m)

Longitudinal position (m

)

Lateral position (m)

20

minus2minus4 5

10

15

20

Figure 5 A road map of dirt terrain

5 Extraction of Road Height Profile

The 3D point cloud of the dirt terrain acquired with LiDARinevitably suffers from noise contamination and containsoutliers [16ndash18] due to the limitations of sensors [19] theinherent noise of the acquisition device [20] and the lightingor reflective nature of the surface or artefact in the scene [21]

Therefore the raw point cloud should be filtered to obtainaccurate data that are suitable for further processing For thisproject we only consider the road that the vehicle is going topass through and other road elevation information is ignoredto simplify the calculation Thus the use of the road heightprofile extracted from the 3D point cloud of the terrain isbeneficial This paper assumes that the road ahead is emptyIn the follow-up study vehicle trajectory planning should becarried out in a short time according to the steering wheelangle

51 Filtering of Point Cloud The Point Cloud Library (PCL)is a large-scale open project [22] for 2D3D image and pointcloud processing the framework of which contains numer-ous state-of-the-art algorithms including filtering featureestimation surface reconstruction registration model fit-ting and segmentation For example these algorithms can beused to filter outliers from noisy data stitch 3D point cloudstogether segment relevant parts of a scene extract key pointsand compute descriptors to recognise objects in the worldon the basis of their geometric appearance and create andvisualise surfaces from point clouds

TheStatisticalOutlierRemoval filter in PCL is used to filteroutliers Compared with the unused filter the denoised 3Dpoint cloud of the dirt terrain decreased from 346036 to322628 (Figure 6)


Table 3 3D point cloud properties of dirt terrain

Point numbers 346036Longitudinal position range

Minimum40002 (m)

Maximum209994 (m)

Lateral position range minus49935 (m) 29998 (m)Vertical height range minus01477 (m) 01997 (m)


)


2 0 minus2 minus4 5

10

15

20

Vert

ical h

eight

(m)

Figure 6 Denoised road map of the dirt terrain

52 Extraction of Road Height Profile We can predict thetrajectory of a vehicle in advance according to the vehiclersquosdriving direction We assume that the vehicle drives in astraight line and the equation for the hypothetical drivingtrack is shown in the following formula

119897119900119899119892119894119905119906119889119894119899119886119897 119903119886119899119892119890 isin [4119898 21119898]119897119886119905119890119903119886119897 119903119886119899119892119890 = minus2119898 (8)

The 2D road height profile whose accuracy shouldcorrespondwith that of the 3Dpoint cloud is discrete becausethe 3D point cloud data is discrete Therefore we shouldcalculate the accuracy and then extract the road height profileaccording to the calculated precision

Since the resolution of point cloud is affected by manyfactors such as the number of lasers in LiDAR and vehiclespeed the average resolution of point cloud is used in thesubsequent calculation

The volume of the denoised 3D point cloud 119881 and thesingle point occupancy volume V can be written as follows

119881 = 119903119886119899119892119890119909 times 119903119886119899119892119890119910 times 119903119886119899119892119890119911 (9)

119903119886119899119892119890119909 = 209994 minus 40003 = 169991119903119886119899119892119890119910 = 29998 + 49935 = 799330119903119886119899119892119890119911 = 01977 + 01477 = 034540

(10)

V = 119881119905ℎ119890 119899119906119898119887119890119903 119900119891 3119863 119901119900119894119899119905 119888119897119900119906119889 (11)

We regard V as the cube boxThus the average resolution119889 of the 3D point cloud can be considered as the lengthof the cube The calculation shows that the value of 119889 isapproximately 5cm

IDW interpolation is a frequently used technique forLiDARdata [23]The core idea of this technique is that nearby

points are more alike than distant ones are The weightsassigned to the points closer to the prediction location aregreater than those far away It is calculated by using thefollowing equation

ℎ = sum119899119894=1 (1119889119894) ℎ119894sum119899119894=1 (1119889119894) (12)

where ℎ119894 is the known height value and 119889119894 is the distance tothe known points

According to the precision of the denoised point cloud apoint to be interpolated is selected every 5 cm from the vehicletrajectory Six nearest neighbours to that point are selected tointerpolate using IDWwith the weight of 1119889119894Thereafter theroad height profile is obtained (Figure 7)The key road pointsof the vehiclersquos trajectory which not only effectively reducethe number of point clouds but also improve the calculationspeed are thus obtained These points are vital for the nextstep

6 Interpolation of Road Height ProfileBased on Fractal Theory

The preview sensor must be able to image sudden obstaclessuch as the high curb stone of standard EU 1340 and DIN483 For this the preview sensor has to scan the road with agrid width of 2cm So an interpolation method close to thenature geometry should be selected to improve the accuracyof the road height profile thereby enabling precise control ofthe active suspension

Fractal interpolation which is based on fractal theoryirregularity roughness and self-similarity of the dataset itselfuses the iterated function system (IFS) to focus on theretention of the global characteristics of a dataset [24 25]Recently several studies have been including fractal com-pression and fractal interpolation methods in many digitisedterrain reconstruction applications [26 27] Fractal inter-polation can describe natural phenomena more realisticallythan can classic interpolation methods [25 28] This theoryutilises its similarity principle to observe detailed structureshidden in a chaotic appearance as a thinking tool for theunderstanding of global appearance from local characteristics[28] Figure 8 shows the difference between traditional andfractal geometries on data simulation Fractal geometry canexpress the details of a simulated object whereas traditionalgeometry cannot [29]The calculation of fractal interpolationis described in Section 10 ldquoMethodologyrdquo

61 Quantitative Analysis of Road Height Profilersquos FractalCharacteristics The quantitative analysis of fractal charac-teristics uses the ldquobox covering methodrdquo to obtain the dual


4 6 8 10 12 14 16 18 20 22Longitudinal position (m)

minus006

minus004

minus002

0

002

004

006

Vert

ical

hei

ght (

m)

Figure 7 Road height profile using IDW algorithm

Data PointSimulate Curve

(a)


(b)

Figure 8 Data interpolation (a) Spline interpolation (b) Fractal interpolation

Table 4 Coverage statistics

119903119894(m) 0005 001 002 004 008119873(119903119894) 71400 18700 5100 1275 426ln (1119903119894) 52983 46052 39120 32189 25257ln (1119873 (119903119894)) 111761 98363 85370 71507 60544

logarithmic coordinate diagram and to analyse its propertiesfurther The calculation steps are as follows(1)The square with a length 119903 that covers the fractal curveis used (Figure 7)(2) For different 119903 counting the number of squares119873(119903)is to cover the fractal curve(3)The dual logarithmic coordinate diagram of ln119873(119903) minusln(1119903) is plotted(4) The correlation coefficient of discrete points in thedual logarithmic coordinate diagram is calculated

Table 4 lists the values of 119903119894119873(119903119894) ln(1119903119894) and ln119873(119903119894)

The data of ln(1119903119894) and ln(1119873(119903119894)) are fitted by using theleast squaresmethodThus a dual logarithmic regression linegraph is obtained (Figure 9)The regression linear equation isas follows

ln119873(119903119894) = 18652 sdot ln( 1119903119894) + 12541 (13)

The correlation coefficient119877of discrete points ln(1119873(119903119894))and ln(1119903119894) is as follows

119877 = 9994 (14)

where 119877 = 9994 indicates that the road height profile hasstrong self-similarity It also reflects the fractal characteristicsof the curve quantitativelyTherefore the theory and methodof fractal interpolation can be used to study Figure 7

62 Fractal Interpolation of Road Height Profile The dataobtained from Section 52 are interpolated by fractal interpo-lation Four data points are inserted between each interval


4

5

6

7

8

9

10

11

12

lnN

(r)

lnN(r)=18652ln(1r)+12541R=09994

25 3 35 4 45 5 552ln(1r)

Figure 9 Double logarithmic map of road height profile based on box covering method

4 6 8 10 12 14 16 18 20Longitudinal position (m)

4 402 404 406 408 41001

0015002

Primary dataInterpolated data

minus006

minus004

minus002

0

002

004

006

Vert

ical

hei

ght (

m)

Figure 10 Road height information after fractal interpolation

which is equivalent to obtaining a road height profile every 1cm Figure 10 shows the road height information after fractalinterpolation

7 Roughness Grade Division of RoadHeight Information

Vehicle suspension is actively controlled using the LQG con-troller to enable the vehicle to obtain improved ride comfortand handling stability on the basis of road height informationTherefore suitable LQG controllerrsquos weight coefficients ofperformance indicators that vary in different road roughnessgrades should be selectedThe International Organisation forStandardisation (ISO 8608 2016) proposed a methodologyfor classifying road profiles on the basis of power spectraldensity [30] which divides road roughness into eight grades(AndashH)

The road height information must be initially dividedThe road surface roughness index unifies the traditionalstatistical parameter and the fractal dimension and is thussuitable for the grade of the road surface [31] In [31] theexpression of road roughness index is as follows

119877lowast = 1198771199021(119863minus1) (15)

where D and 119877119902 are the fractal dimension and root meansquare (RMS) error of the road surface respectivelyThe roadheight information in this study is as follows

119877119902 = 157037 times 10minus3119898119863 = 18652119877lowast = 241180 times 10minus3119898

(16)


ms

Ks1

Kt

Cs

Zs1

q

Zu1

mu

Vehicle body

Passive suspension

Wheel

Tire

(a)

ms

Ks2

Kt

u

Zs2

q

Zu2

mu

(b)

Figure 11 Quarter vehicle model (a) Passive suspension (b) Active suspension

Table 5 Quarter vehicle model parameters

Symbol Quantity Value Unit119898119904 Sprung mass 320 119896119892119898119906 Unsprung mass 40 1198961198921198701199041 Passive suspension stiffness 22000 1198731198981198701199042 Active suspension stiffness 20000 119873119898119870119905 Tire stiffness 200000 119873119898119862119904 Passive suspension damping 1000 119873 sdot 1199041198981198851199041 Sprung massrsquos displacements of passive suspension - 1198981198851199042 Sprung massrsquos displacements of active suspension - 1198981198851199061 Unsprung massrsquos displacements of passive suspension - 1198981198851199062 Unsprung massrsquos displacements of active suspension - 119898119902 Displacements of road excitation - 119898119906 Active force - 119873

according to the data provided in [31] which belongs to 119861grade Next the LQG controller of the square active suspen-sion is designed to solve the controllerrsquos weight coefficientsunder different road roughness grades

8 Optimal Design of LQG Controller

Before designing the LQG controller of the active suspensiona dynamic model of passive and active suspensions of thevehicle which is the basis of follow-up calculation must beestablished

81 Suspension System Dynamics Figure 11 establishes aquarter vehicle model with passive and active suspensionswhich have two degrees of freedom and heave dynamicsUnlike passive suspension active suspension has only oneinput of road excitation and more active controlling forcemoreover a force actuator is installed in place of the dampervariable In this implementation the tire is represented by alinear spring In Table 5 the parameters of the quarter vehicle

model are shown Since the active suspension is a parallelstructure of actuator and spring the passive suspensionreplaces actuator with damper so the suspension stiffness ofboth is set to different values

According to the quarter vehicle model based on New-tonrsquos second law the equations of motion of the system are asfollows [32](1) Passive suspension

1198981199061199061 = 119870119905 (119902 minus 1198851199061) minus 1198701199041 (1198851199061 minus 1198851199041)minus 119862119878 (1199061 minus 1199041)

1198981199041199041 = 1198701199041 (1198851199061 minus 1198851199041) + 119862119878 (1199061 minus 1199041)(17)

(2) Active suspension1198981199061199062 = 119870119905 (119902 minus 1198851199062) minus 1198701199041 (1198851199062 minus 1198851199042) minus 1199061198981199041199042 = 1198701199041 (1198851199062 minus 1198851199042) + 119906 (18)


The state equation is established and the system statevariable119883 is defined as follows

1198831 = [1199041 1199061 1198851199041 1198851199061]1198791198832 = [1199042 1199062 1198851199042 1198851199062]119879

(19)

The suspension system can be expressed in the followingstate equation

1 = 1198601 sdot 1198831 + 1198611 sdot 1199022 = 1198602 sdot 1198832 + 1198612 sdot 119880119880 = [119906119902]

(20)

where

1198601 =[[[[[[[[[

minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

1198701199041119898119904119862119904119898119906

minus 119862119904119898119906

1198701199041119898119906

minus1198701199041 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]]

1198611 =[[[[[[[

011987011990511989811990600

]]]]]]]

1198602 =[[[[[[[[

0 0 minus1198701199042119898119904

11987011990421198981199040 0 1198701199042119898119906

minus1198701199042 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]

1198612 =[[[[[[[[

1119898119904

0minus 1119898119906

1198701199051198981199060 00 0

]]]]]]]]

(21)

The system output 119884 is defined as follows

1198841 = [1199041 1198851199061 minus 1198851199041 119902 minus 1198851199061]1198791198842 = [1199042 1198851199062 minus 1198851199042 119902 minus 1198851199062]119879

(22)

The output equation of the system is as follows

1198841 = 1198621 sdot 119883 + 1198631 sdot 1199021198842 = 1198622 sdot 119883 + 1198632 sdot 119880 (23)

where

1198621 = [[[[minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198631 = [[[001]]]

1198622 = [[[[0 0 minus1198701199042119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198632 = [[[[

1119898119904

00 00 1

]]]]

(24)

82 LQG Controller Themain performance evaluation indi-cators in designing the controller for the quarter active sus-pension are as follows (1) vehicle body acceleration whichevaluates ride comfort and handing stability (2) suspensiondynamic deflection which affects body posture and (3) tiredynamic deflection on behalf of the tire road holding [33]

Here the comprehensive performance evaluation indica-tor denoted by 119869 is defined as follows [34 35]

119869 = lim119879997888rarrinfin

1119879 1199021 sdot [1198851199062 (119905) minus 119902]2 + 1199022sdot [1198851199042 (119905) minus 1198851199062 (119905)]2 + 1199023 sdot 11990422 (119905) 119889119905

(25)

where 1199021 1199022 1199023 are the weight coefficients of tire dynamicdeflection suspension dynamic deflection and vehicle bodyacceleration respectively The weight coefficient 1199023 of thevehicle body acceleration which is used as a base value forthe other coefficients to simplify the calculation is set to 1because it is important for vehicle body acceleration

Then formula (25) can be rewritten as an integral type ofquadratic function by the following

119869 = lim119879997888rarrinfin

1119879 11988321198791198761198832 + 119880119879119877119880 + 21198832

119879119873119880119889119905 (26)

where

119876 =[[[[[[[[[[

0 0 0 00 0 0 00 0 1199022 + 11990231198702

11990421198982119904

minus1199022 minus 1199023119870211990421198982119904

0 0 minus1199022 minus 1199023119870211990421198982119904

1199021 + 1199022 + 1199023119870211990421198982119904

]]]]]]]]]]


119873 =[[[[[[[[[

0 00 0

minus11987011990421198982119904

011987011990421198982

119904

minus1199021

]]]]]]]]]

119877 = [[11990231198982119904

00 1199021

]]

(27)

When the vehicle parameter values and weight coeffi-cients 1199021 1199022 1199023 are determined the optimal control feedbackgain matrix 119870 can be obtained from the following Riccatiequation

1198751198602 + 1198601198792119875 minus (1198751198612 + 119873)119877minus1 (1198611198792119875 + 119873119879) + 119876 = 0 (28)

119870 = 119877minus1 (1198611198792119875 + 119873119879) (29)

Then according to the state variables 1198832 at any time theoptimal control force of the actuator denoted by 119906 can bedescribed as follows

119906 = minus119870 sdot 1198832 (30)

The solving method of the weight coefficients in differentroad grades is discussed in the succeeding section

9 Solution of Weight Coefficients

Weneed to solve eight sets of weight coefficients to enable theactive suspension to adapt to different road grades Accordingto [36] random road excitation can be written as follows

(119905) = minus212058711989900V119902 (119905) + 21205871198990radic119878119902 (1198990) V119882(119905) (31)

where 11989900 = 0011119898minus1 is the spatial lower cut-off frequency119882(119905) stands for Gaussian white noise with zero mean Vrepresents the speed of the vehicle

According to formula (31) under the speed 40 119896119898ℎ weobtain eight grades of road surface roughness as shown inFigure 12

AHP is a multicriteria decision-aiding method that isbased on a solid axiomatic foundation It is a systematic pro-cedure for dealing with complex decision-making problemsin which many competing alternatives exist [37ndash39] Thuswe adopt this method to solve the weight coefficients of theLQG controller The process of calculating LQG controllercoefficients using AHP method is described in Section 10ldquoMethodologyrdquo

Table 6 lists the passive suspension performance statisticsunder different road grades

On the high-grade road surface the ride comfort ismainly optimized and on the poor grade road surface

0 20 40 60 80 100 120 140 160Time t (s)

minus3

minus2

minus1

0

1

2

3

Road

elev

atio

n q

(m)

Class A Class B Class D Class E Class HClass F Class GClass C

Figure 12 Road surface roughness of different grades

Table 6 Performance statistics of passive suspension under differ-ent road grades

Road grades 1205901198611198601 (1198981199042) 1205901198631198791198631 (119898) 1205901198781198821198781 (119898)A 03418 00011 00034B 06836 00023 00069C 13671 00046 00137D 27342 00091 00275E 54684 00182 00550F 109368 00365 01099G 218737 00729 02198H 437473 01458 04396

the handling stability is mainly optimized The subjectivejudgment matrices of different road grades are as follows

119867119860 =[[[[[[

1 2 912 1 319 13 1

]]]]]]

119867119861 =[[[[[[

1 3 913 1 319 13 1

]]]]]]

119867119862 =[[[[[[[

1 34 843 1 418 14 1

]]]]]]]

119867119863 =[[[[[[[

1 34 643 1 416 14 1

]]]]]]]


L◼SS T◼L◼S R◼T◼L◼S

-

Figure 13 Affine transform decomposition diagram

Table 7 LQG controller weighting factor in different road grades

Road grades 1199021 1199022A (qA) 393234009 12595536B (qB) 300093714 11003214C (qC) 865620060 17165706D (qD) 952738155 20794772E (qE) 624220080 47608143F (qF) 714553616 62384316G (qG) 526487606 169337086H (qH) 495443806 179947505

119867119864 = [[[[[

1 1 31 1 113 1 1

]]]]]

119867119865 = [[[[[

1 1 21 1 112 1 1

]]]]]

119867119866 = [[[[

1 1 11 1 151 5 1

]]]]

119867119867 = [[[[

1 1 11 1 161 6 1

]]]]

(32)

Table 7 shows the calculated LQG controllerrsquos weightcoefficients in different road roughness grades

Because the road height profile calculated in Section 7belongs to 119861 grade the weight coefficients in the LQGcontroller are selected as follows

1199021 = 3000937141199022 = 11003214 (33)

10 Methodology

101 Fractal Interpolation

1011 Affine Transformation A two-dimensional affinetransformation119882 is formed as follows

119882[11990910158401199101015840] = [119886 119887119888 119889] [

119909119910] + [

119890119891] (34)

where 119886 119887 119888 119889 119890 119891 are real numbers 119901(119909 119910) is a point on theplane and119882 is a linear affine transformation

The transformation matrix 119860 can be decomposed intorotation 119877 expansion 119871 and distortion 119879

119860 = [119886 119887119888 119889] = 119877 sdot 119879 sdot 119871

119877 = [cos 120579 minus sin 120579sin 120579 cos 120579 ]

119879 = [ 1 tan120572tan120572 1 ]

119871 = [1198971 00 1198972]

(35)

Figure 13 shows a decomposition diagram of an affinetransformation

1012 IFS and Fractal Interpolation A set of data points 119875119894 =(119909119894 119910119894) isin 1198772 | 119894 = 0 1 2 sdot sdot sdot 119873 satisfies 1199090 lt 1199091 lt 1199092 ltsdot sdot sdot lt 119909119873 The attractor of IFS is a continuous function ofinterpolating data An interpolation curve across a series ofdata points 119875119894 and map 119875119900 and 119875119873 to 119875119899minus1 and 119875119899 is formed

Considering IFS 1198772119882119899 119899 = 0 1 2 sdot sdot sdot 119873 119882119899 is anaffine transformation with the following form

119882119899 [119909119910] = [119886119899 0119888119899 119889119899] sdot [

119909119910] + [

119890119899119891119899] (36)


Table 8 Comparison values of importance between indicators

119894119895 Equally Moderately Strongly Very Extremelyℎ119894119895 1 3 5 7 9

The aforementioned equation satisfies the following

119882119899 [11990901199100] = [119909119899minus1119910119899minus1]

119882119899 [119909119873119910119873] = [119909119899119910119899]

(37)

where 119887119899 = 0 guarantees that the interpolation function ofeach cell does not overlap Assume that 119889119899 is constant thenwe can solve the parameters as follows [38]

119886119899 = (119909119899 minus 119909119899minus1)(119909119873 minus 1199090)119890119899 = (119909119873119909119899minus1 minus 1199090119909119899)(119909119873 minus 1199090)119888119899 = (119910119899 minus 119910119899minus1 minus 119889119899 (119910119873 minus 1199100))(119909119873 minus 1199090)119891119899 = (119909119873119910119899minus1 minus 1199090119910119899 minus 119889119899 (1199091198731199100 minus 1199090119910119873))(119909119873 minus 1199090)

(38)

For a set of data points 119875119894 = (119909119894 119910119894) isin 1198772 | 119894 = 0 1 2 sdot sdot sdot 119873 119873 numbers of affine transformations exist The fractalinterpolation function of the affine transformation of 119899119905ℎ isas follows

119909119899119894 = 119886119899119909119894 + 119890119899119910119899119894 = 119888119899119909119894 + 119889119899119910119894 + 119891119899

119894 = 0 1 sdot sdot sdot 119873 119899 = 1 2 sdot sdot sdot 119873(39)

102 Analytic Hierarchy Process

1021 Calculating Objective Weight Coefficient According tothe parameters of a passive suspension its dynamic simula-tion model is operated under eight road grades to obtain theperformance statistics 1205901198611198601 1205901198631198791198631 1205901198781198821198781 which are the RMSof vehiclersquos vertical acceleration tire dynamic deflection andsuspension dynamic deflection respectively The objectiveweight coefficient of vertical acceleration is supposed to be 1According to [38] the equation of calculating other objectiveweight coefficients is as follows

12059021198611198601 sdot 1 = 1205731 sdot 12059021198631198791198631 = 1205732 sdot 12059021198781198821198781 (40)

1022 Determining Subjective Weight Coefficients (1)A sub-jective judgement matrix is created as followsℎ119894119895 is the comparison value of the importance of theindicators 119894 and 119895 and a fundamental 1-to-9 scale can be usedto rank the judgments (Table 8)

The subjective judgement matrix 119867 is built throughpairwise comparison of each decision factor

119867 =

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

1 ℎ12 sdot sdot sdot sdot sdot sdot ℎ11198991ℎ12 1 ℎ23 sdot sdot sdot ℎ2119899sdot sdot sdot 1ℎ23 sdot sdot sdot sdot sdot sdot sdot sdot sdot d d

1ℎ1119899

1ℎ2119899 sdot sdot sdot sdot sdot sdot 1

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

(41)

(2) Matrix 119867 is calculated by multiplying the vector ofevery row as follows

119872 = [11987211198722 sdot sdot sdot 119872119899]119879119872119894 = 119899prod

119895=1

ℎ119894119895(119894 119895 = 1 2 sdot sdot sdot 119899)

(42)

(3) 119899radic119872 is calculated as follows

119882 = [11988211198822 sdot sdot sdot 119882119899]119879119882119894 = 119899radic119872119894

(119894 = 1 2 sdot sdot sdot 119899)(43)

(4) Positive vector119882 is calculated as follows

119882 = 119882sum119899119894=1119882119894

(44)

(5) The maximum eigenvalue of matrix 119867 is calculatedand checked for consistency as follows

If every element in matrix 119867 satisfies the equation ℎ119894119895 =1ℎ119895119894 and ℎ119894119895 = ℎ119894119896 ∙ ℎ119896119895 then the matrix is the consistencymatrixThe subjective judgementmatrix is often not perfectlyconsistent due to the randomness of peoplersquos judgmentThesejudgment errors can be detected by the consistency ratio 119862119877

120582max = 119899sum119894=1

(119867119882)119894119899119882119894

(119894 = 1 2 sdot sdot sdot 119899) (45)

119862119877 = 120582max minus 119899119877119868 (119899 minus 1) (46)

where 119877119868 is a random consistency index of matrix119867When 119899 is 10 119877119868 is equal to 14851 The solution is correct

when 119862119877 is less than 1 If 119862119877 is more than 1 then theevaluation matrix119867 needs to be revised on the basis of [39](6) Subjective weight coefficients

The subjective weight coefficient of the vehiclersquos verticalacceleration is 1 The subjective weight coefficients of the tiredynamic deflection 1205741 and suspension dynamic deflection 1205742are determined according to the following formula

1198821 = 11988221205741 = 11988231205742 (47)


Figure 14 Prototype vehicle with active suspension

Figure 15 Single active suspension actuator

1023 Calculating LQGControllerrsquosWeight Coefficients 1199021 and1199022 The general weights of the evaluating indexes related toride comfort can be obtained on the basis of the objectiveweights and subjective weight coefficients

1199021 = 1205731 sdot 12057411199022 = 1205732 sdot 1205742 (48)

11 Experiment

When the vehiclersquos speed is constant the more point clouddata are collected and the higher data fusion rate is requiredTherefore in order to meet the needs of high-speed dataprocessing of active suspension firstly only collect andstore the topographic data in front of the vehicle secondlycollect the data by using the suitable number of lines ofLiDAR thirdly the algorithm adopted in this paper not onlyeffectively collects enough point cloud data on the groundbut also satisfies the accuracy of the solution after data fusionand obtains the elevation information of the road surface foreffective control of suspension

Road experiments of the prototype vehicle are performedto verify the accuracy and efficiency of the active suspensioncontrol algorithm that is based on LiDAR-previewed terraininformation The main body of the prototype vehicle isassembled and constructed using the FAW Besturn car chas-sis (Figure 14) It includes the mechanical hydraulic servoand the electric control parts The single active suspensionactuator unit is shown in Figure 15 and the left side is theservo control actuator cylinder Figure 4 shows the test site

0 2 4 6 8 10Time (s)

minus3

minus2

minus1

0

1

2

3

Vert

ical

acce

lera

tion

(ms

2 )

qFqGqEqH

qDqAqCqB

Figure 16 Vertical acceleration of active suspension

0 2 4 6 8 10Time (s)

minus008minus006minus004minus002

0002004006008

Tire

dyn

amic

defl

ectio

n (m

)

qCqFqEqG

qDqAqHqB

Figure 17 Tire dynamic deflection of active suspension

0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)

qAqHqDqF

qEqCqGqB

Figure 18 Active suspension dynamic deflection

Figures 16ndash22 show the experiment data In order toproperly validate selection of control weightings based onestimated road profile and road class the performance eval-uation indicators of active control for different weight factorsq1 and q2 that are under different road class are reportedin Figures 16ndash18 Figures 19ndash21 depict the performanceevaluation indicator comparison between active suspension


Table 9 RMS values of active and passive suspension performance indicators

Active suspension Passive suspension119902119860 119902119861 119902119862 119902119863 119902119864 119902119865 119902119866 119902119867Vertical acceleration (1198981199042) 08581 08277 08342 08909 09173 09231 09206 09028 10277Tire deflection (119898) 00246 00242 00258 00247 00251 00256 00248 00245 00270Suspension deflection (119898) 00344 00335 00337 00342 00338 00339 00336 00343 00368

0 2 4 6 8 10Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

Vert

ical

acce

lera

tion

(ms

2 )

Passive suspensionActive suspension (qB)

Figure 19 Vertical acceleration

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Tire

dyn

amic

defl

ectio

n (m

)


Figure 20 Tire dynamic deflection

0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)


Figure 21 Suspension dynamic deflection

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus2000minus1500minus1000

minus5000

5001000150020002500

Activ

e con

trol f

orce

(N)

Figure 22 Active control force

08

085

09

095

1

105

qF qG qEqH

qD

qA354

7098329771009

1033

1946

qC qB078

Passivesuspension

Active suspension

Figure 23 Bar-chart of suspensionrsquos vertical acceleration

with LQG controller whose controller coefficient is selectedas qB and passive suspension Figure 22 shows changes in thecontrol force of the active suspension actuators

Table 9 shows the RMS of performance statistics of LQG-active suspension system based on different weight coeffi-cients q1 and q2 andpassive suspension systemDraw the datain Table 9 as a bar chart as shown in Figures 23ndash25 As canbe seen from Figure 23 the RMS of vertical acceleration ofLQG active suspension controller based on grade B pavementcoefficients qB is about 1946 lower than that of passivesuspension and about 1033 1009 977 832 709354 and 078 lower than the LQG active suspensioncontroller based on other pavement grade coefficients qF qGqE qH qD qA and qC

In Figure 24 the RMS of tire dynamic deflection of LQGactive suspension controller based on grade B pavementcoefficients qB is about 159 lower than that of passivesuspension and about 1033 1009 977 832 709354 and 078 lower than the LQG active suspensioncontroller based on other pavement grade coefficients qF qGqE qH qD qA and qC


002300235

002400245

002500255

002600265

002700275

0028

qC qFqE

qG qD qA qHqB

620547

358122162202248

Passive suspension Active suspension

159

Figure 24 Bar-chart of suspensionrsquos tire dynamic deflection

0032

0033

0034

0035

0036

0037

0038

897

262 233 146 118 089 059

qA qH qDqF qE qC qG qB

029

Passivesuspension Active suspension

Figure 25 Bar-chart of suspensionrsquos dynamic deflection

In Figure 25 the RMS of suspension dynamic deflectionof LQG active suspension controller based on grade B pave-ment coefficients qB is about 897 lower than that of passivesuspension and about 262 233 146 118 089059 and 029 lower than the LQG active suspensioncontroller based on other pavement grade coefficients qAqH qD qF qE qC and qG

The experimental data show that reasonable selection ofLQG controller weightings based on estimated road profileand road class through fractal interpolation can improvevehicle ride comfort and handling stability effectively morethan passive suspension does

12 Conclusions

This study focuses on extracting a road height profile fromthe 3D point cloud data of the terrain in front of a vehicleas input to the suspension control model Three innovationsare mainly presented Firstly a LiDAR coupled with INSand GPS is used to measure the 3D point cloud of theterrain in front of the vehicle Although many researchershave proposed the use of preview information for activesuspension they did not provide detailed descriptions andinstead only presented concepts Secondly the road heightprofile is extracted using the IDW interpolation methodThe main difference is that this method has been usedfor 3D digital terrain obtained by airborne LiDAR systemspreviously and is rarely used in a vehiclersquos LiDAR systemto obtain preview information of a road Thirdly fractalinterpolation is used to improve the accuracy of road heightinformation This interpolation method is close to natural

geometry Thus the interpolated data in this work are moreprecise than traditionally interpolated data

Road experiment verifies that reasonable selection ofLQG controller weightings based on estimated road profileand road class through fractal interpolation can improve theride comfort and handling stability of the vehicle more thanpassive suspension does

Data Availability

All data generated or analysed during this study are includedin this published article

Conflicts of Interest

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

Authorsrsquo Contributions

Mingde Gong conceived the idea designed the experimentsandwrote the paperHaohaoWanghelpedwith the algorithmand analysed the experimental data Xin Wang collated andanalysed the experimental data

Acknowledgments

This work was supported by the National Key Research andDevelopment Program of China (Grant no 2016YFC0802900)

References

[1] C Gohrle A Schindler A Wagner and O Sawodny ldquoDesignand vehicle implementation of preview active suspension con-trollersrdquo IEEE Transactions on Control Systems Technology vol22 no 3 pp 1135ndash1142 2014

[2] MH Kim and S B Choi ldquoEstimation of road surface height forpreview system using ultrasonic sensorrdquo in Proceedings of theInternational Conference on Networking Sensing and Controlpp 1ndash4 IEEE 2016

[3] F Oniga and S Nedevschi ldquoProcessing dense stereo data usingelevationmaps road surface traffic isle and obstacle detectionrdquoIEEE Transactions on Vehicular Technology vol 59 no 3 pp1172ndash1182 2010

[4] A Jaakkola J Hyyppa H Hyyppa and A Kukko ldquoRetrievalalgorithms for road surface modelling using laser-based mobilemappingrdquo Sensors vol 8 no 9 pp 5238ndash5249 2008

[5] J Laurent M Talbot and M Doucet ldquoRoad surface inspectionusing laser scanners adapted for the high precision measure-ments of large flat surfacesrdquo in Proceedings of the InternationalConference on Recent Advances in 3-D Digital Imaging andModeling pp 303ndash310 IEEE 1997

[6] J J Dawkins D M Bevly and R L Jackson ldquoEvaluation offractal terrain model for vehicle dynamic simulationsrdquo Journalof Terramechanics vol 49 no 6 pp 299ndash307 2012

[7] M Rahman and G Rideout ldquoUsing the lead vehicle as previewsensor in convoy vehicle active suspension controlrdquo VehicleSystem Dynamics vol 50 no 12 pp 1923ndash1948 2012


[8] B Fu L Liu and J Bao ldquoGPSINSspeed log integrated naviga-tion systembased onMAKF and priori velocity informationrdquo inProceedings of the IEEE International Conference on Informationand Automation pp 54ndash58 IEEE 2014

[9] I Skog and P Handel ldquoIn-car positioning and navigationtechnologiesa surveyrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 10 no 1 pp 4ndash21 2009

[10] A Noureldin T B Karamat M D Eberts and A El-ShafieldquoPerformance enhancement of MEMS-based INSGPS integra-tion for low-cost navigation applicationsrdquo IEEE Transactions onVehicular Technology vol 58 no 3 pp 1077ndash1096 2009

[11] M S Grewal L RWeill andA P Andrews ldquoGlobal postioningsystems inertial navigation and integrationrdquo Wiley Interdisci-plinary Reviews Computational Statistics vol 3 no 4 pp 383-384 2007

[12] H Zhao Z Xiong L Shi F Yu and J Liu ldquoA robust filteringalgorithm for integrated navigation system of aerospace vehiclein launch inertial coordinaterdquo Aerospace Science and Technol-ogy vol 58 pp 629ndash640 2016

[13] X Ning M Gui Y Xu X Bai and J Fang ldquoINSVNSCNSintegrated navigation method for planetary roversrdquo AerospaceScience and Technology vol 48 no 1 pp 102ndash114 2015

[14] M Enkhtur S Y Cho and K H Kim ldquoModified unscentedkalman filter for a multirate INSGPS integrated navigationsystemrdquo ETRI Journal vol 35 no 5 pp 943ndash946 2013

[15] J Landa D Prochazka and J Strsquoastny ldquoPoint cloud processingfor smart systemsrdquo Acta Universitatis Agriculturae et Silvicul-turae Mendelianae Brunensis vol 61 no 7 pp 2415ndash2421 2013

[16] X Han J S Jin M Wang W Jiang L Gao and L Xiao ldquoAreview of algorithms for filtering the 3D point cloudrdquo SignalProcessing Image Communication vol 57 pp 103ndash112 2017

[17] H Xie K T McDonnell and H Qin ldquoSurface reconstructionof noisy and defective data setsrdquo in Proceedings of the IEEEVisualization 2004 pp 259ndash266 IEEE USA 2004

[18] J Park H Kim Y W Tai M S Brown and I KweonldquoHigh quality depth map upsampling for 3D-TOF camerasrdquoin Proceedings of the 2011 IEEE International Conference onComputer Vision ICCV 2011 pp 1623ndash1630 IEEE Spain 2011

[19] E A L Narvaez and N E L Narvaez ldquoPoint cloud denoisingusing robust principal component analysisrdquo in Proceedings ofthe First International Conference on Computer Graphics Theoryand Applications Setubal Portugal 2006

[20] F Zaman Y PWong and B Y Ng ldquoDensity-based denoising ofpoint cloudrdquo in Proceedings of the 9th International Conferenceon Robotic Vision Signal Processing and Power Applications pp287ndash295 Springer Singapore 2017

[21] R B Rusu and S Cousins ldquo3D is here Point Cloud Library(PCL)rdquo in Proceedings of the IEEE International Conference onRobotics and Automation (ICRA rsquo11) pp 1ndash4 IEEE ShanghaiChina 2011

[22] I Ashraf S Hur and Y Park ldquoAn investigation of interpolationtechniques to generate 2D intensity image from LIDAR datardquoIEEE Access vol 5 no 99 pp 8250ndash8260 2017

[23] M F Barnsley ldquoFractal functions and interpolationrdquo Construc-tive Approximation vol 2 no 4 pp 303ndash329 1986

[24] Y M Huang and C-J Chen ldquo3D Fractal reconstruction ofterrain profile data based on digital elevation modelrdquo ChaosSolitons amp Fractals vol 40 no 4 pp 1741ndash1749 2009

[25] K Falconer Fractal Geometry Mathematical Foundations andApplications K Falconer Ed Wiley 2nd edition 2003

[26] R Małysz ldquoConvergence of trajectories in fractal interpolationof stochastic processesrdquo Chaos Solitons amp Fractals vol 27 no 5pp 1328ndash1338 2006

[27] C-J Chen T-Y Lee Y M Huang and F-J Lai ldquoExtraction ofcharacteristic points and its fractal reconstruction for terrainprofile datardquo Chaos Solitons amp Fractals vol 39 no 4 pp 1732ndash1743 2009

[28] J Chen and T Lee ldquoFractal reality of random data compressionfor equal-interval seriesrdquo Fractals-complex Geometry Patterns ampScaling in Nature amp Society vol 8 no 2 pp 205ndash214 2000

[29] M Barnsley Fractals Everywhere Academic Press BostonMass USA 2nd edition 1993

[30] Z Lanying L Zhixiong and H Zhanfeng ldquoFractal parametersof the grade of road surface roughnessrdquoComputer and Commu-nications vol 26 no 6 pp 158ndash161 2008

[31] C Li M Liang Y Wang and Y Dong ldquoVibration suppressionusing two terminal flywheel Part II Application to vehiclepassive suspensionrdquo Journal of Vibration and Control vol 18no 9 pp 1353ndash1365 2012

[32] H Pang YChen J Chen andX Liu ldquoDesign of LQGcontrollerfor active suspension without considering road input signalsrdquoShock and Vibration vol 2017 13 pages 2017

[33] H Habibullah H R Pota I R Petersen and M S RanaldquoTracking of triangular reference signals using LQG controllersfor lateral positioning of an AFM scanner stagerdquo IEEEASMETransactions on Mechatronics vol 19 no 4 pp 1105ndash1114 2014

[34] K S Grewal R Dixon and J Pearson ldquoLQG controller designapplied to a pneumatic stewart-gough platformrdquo InternationalJournal of Automation and Computing vol 9 no 1 pp 45ndash532012

[35] DMiaomiao Z Dingxuan B Yang andW Lili ldquoTerminal slid-ing mode control for full vehicle active suspe-nsion systemsrdquoJournal of Mechanical Science and Technology vol 32 no 6 pp2851ndash2866 2018

[36] A Them ldquoThe design of LQG controller for active suspensionbased on analytic hierarchy processrdquoMathematical Problems inEngineering vol 2010 19 pages 2010

[37] T L Saaty The Analytic Hierarchy Process McGraw-Hill NewYork NY USA 1980

[38] B Goenaga L Fuentes and O Mora ldquoEvaluation of themethodologies used to generate random pavement profilesbased on the power spectral density An approach based on theinternational roughness indexrdquo Ingenieria e Investigacion vol37 no 1 pp 49ndash57 2017

[39] S H Yan and X Tian ldquoMethod of comparison matrix consis-tency adjustment based on AHPrdquo Armament Automation vol27 no 4 article no 14 pp 8-9 2008

Hindawiwwwhindawicom Volume 2018

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International Journal of


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International Journal of Mathematics and Mathematical Sciences


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Hindawiwwwhindawicom Volume 2018Volume 2018

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Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


3D laser scanner

Scan filter

Vehicle INS system GPS

Real-Time Kinematic (RTK)Kalman filter algorithm

Scan assembly Data fusionPoint cloud data filter

Interpolation of elevation data

Elevation profile

Sequencesignal

Vehicle suspension

control

Point cloud in LiDAR coordinate system

Accurate position and attitude information

Point cloud in geodetic coordinate system

Preview Pavement Elevation Information

Figure 1 Mobile mapping system

descriptions have not been provided Instead researchershave merely introduced concepts or assumed the road heightprofile to be known and obtained by statistical methodsThis study proposes an approach that uses LiDAR cou-pled with INS and GPS to obtain a preview of the 3Dpoint cloud of the terrain in a geodetic coordinate systemand extract the road height profile in front of a vehicleThis work also elaborates the methods and steps of dataprocessing The road roughness grade is identified on thebasis of the aforementioned information Then the Linear-Quadratic-Gaussian (LQG) controllerrsquos weight coefficientsof performance indicators are optimised in different roadroughness grades Finally we select proper weight coeffi-cients that correspond to previously calculated road gradesto improve the vehiclersquos ride comfort and handling stabil-ity

This study is organised as follows Section 2 providesthe mobile mapping system Section 3 presents the dataprocessing of sensors Section 4 uses the methods of thetwo preceding sections to experiment and obtain the 3Dpoint cloud of the test site Section 5 adopts the road heightprofile information utilised in fractal interpolation theoryin Section 6 to improve accuracy This profile informationis extracted from the 3D point cloud by using the inversedistance weighting (IDW) interpolation method Section 7solves the grade of the road height profilersquos roughnessSection 8 establishes the suspension vertical dynamic modeland designs the LQG controller for active suspensionSection 9 calculates the controllerrsquos weight coefficients ofperformance indicators for different road grades using theanalytic hierarchy process (AHP) In Section 10 a road exper-iment verifies that reasonable selection of LQG controllerweightings based on estimated road profile and road classthrough fractal interpolation can improve the ride comfortand handling stability of the vehicle more than passivesuspension does

2 Mobile Mapping System

To control active suspension with preview information adevice thatmaps the front terrain of the running vehiclemustbe developed Laser radar scanning is a rapid 3D measure-mentmethod and has existed for awhile But it is seldomusedfor the realization of active suspension The mobile LiDARsystem is mainly composed of a LiDAR sensor GPS and INS(Figure 1) In the running course the LiDAR is mounted infront of the vehicle and returns the 3D point cloud of the roadon the basis of its own coordinate system One vital challengein accurate mapping is estimating the attitude of LiDARwhich affects the precision of the point cloud To solve thisproblem the INS which is installed near the LiDAR collectsthe attitude angle that is used to compensate for motion dur-ing scan acquisition to improve the precision of data in realtime We need to realise the accurate location of the LiDARto spread the 3D point cloud on the earth surface AlthoughGPS has higher positioning accuracy than other approachesit is vulnerable to a wide variety of interferences such as themultipath effect from radars electromagnetic interferenceand block of signals [9 10] INS is a self-contained system thatconsists of inertial measurement units which can providedynamic measurement for a short period with high frequentupdate It is a complete autonomous navigation system withremarkable concealment strong anti-interference capabilityand immunity to meteorological conditions [11] Howeverits measurement error may accumulate over time due todrifting effects [12] The integration of GPS and INS has beenproposed and widely implemented for vehicle applications toleverage the strengths of the two systems and to offset theirindividual drawbacks because they are more advantageousthan single navigation systems [13 14] In these integratedsystems the Kalman filter is a popular fusion method thathas been used in recent years due to its practicability andsuitability [15]The centimeter-level positioning accuracy can


L1

L2

L4

L3

XL

ZL

YL

XLLi

OL

OL






[[[119909119871119910119871119911119871]]]= 119871 119894 sdot [[


sin120572119894lowast]]]

120572119894lowast =

2120572 119894 = 1120572 119894 = 2minus120572 119894 = 3minus2120572 119894 = 4

120579 isin [minus60∘ 50∘]

(1)



[[[119909119882841199101198828411991111988284

]]]

= [[[119909119866119875119878119910119866119875119878119911119866119875119878

]]]

+ 119877119882[[[119877119873[[

[119877119872[[

[119909119871119910119871119911119871]]]+ [[[Δ119909119871119868Δ119910119871119868Δ119911119871119868

]]]minus [[[Δ119909119868119866Δ119910119868119866Δ119911119868119866

]]]]]]]]]

(2)


119877119872 = 119877120574119911 sdot 119877120572119910 sdot 119877120573119909119877120574119911 = [[


]]]

119877120572119910 = [[[cos120572 0 minus sin1205720 1 0

sin120572 0 cos120572]]]


]]]

(3)


point

O84

X84

Z84

OL

XL

YL

ZL

OI

XI

YI

ZI

Y84

ZG

OG

XG

YG

L



119877119873 = 119877120581 sdot 119877120593 sdot 119877120596119877120581 = [[


]]]

119877120593 = [[[cos120593 0 minus sin1205930 1 0

sin120593 0 cos120593]]]


]]]

(4)



cos119861 0 minus sin119861]]]

(5)




[[[119909119866119875119878119910119866119875119878119911119866119875119878

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


]]]]]]]]

(6)


119886 = 6378137 1198981198902 = 0006694379 (7)






Azimuthaccuracy

Attitudeaccuracy

Positioningaccuracy

Speedaccuracy



Scanningfrequency

Angularresolution


Laserwavelength

50119867119911 05∘ 4 905119899119898Laser grade

Verticalangular


119862119897119886119904119904 119868 120572 = 08∘ 119863119862 9119881 sim 27119881 119862119860119873119864119905ℎ119890119903119899119890119905









Figure 4 Test site

Vert

ical h

eight

(m)


)


20

minus2minus4 5

10

15

20










Minimum40002 (m)

Maximum209994 (m)



)


2 0 minus2 minus4 5

10

15

20

Vert

ical h

eight

(m)



119897119900119899119892119894119905119906119889119894119899119886119897 119903119886119899119892119890 isin [4119898 21119898]119897119886119905119890119903119886119897 119903119886119899119892119890 = minus2119898 (8)




119881 = 119903119886119899119892119890119909 times 119903119886119899119892119890119910 times 119903119886119899119892119890119911 (9)

119903119886119899119892119890119909 = 209994 minus 40003 = 169991119903119886119899119892119890119910 = 29998 + 49935 = 799330119903119886119899119892119890119911 = 01977 + 01477 = 034540

(10)

V = 119881119905ℎ119890 119899119906119898119887119890119903 119900119891 3119863 119901119900119894119899119905 119888119897119900119906119889 (11)




ℎ = sum119899119894=1 (1119889119894) ℎ119894sum119899119894=1 (1119889119894) (12)









minus006

minus004

minus002

0

002

004

006

Vert

ical

hei

ght (

m)



(a)


(b)



119903119894(m) 0005 001 002 004 008119873(119903119894) 71400 18700 5100 1275 426ln (1119903119894) 52983 46052 39120 32189 25257ln (1119873 (119903119894)) 111761 98363 85370 71507 60544




ln119873(119903119894) = 18652 sdot ln( 1119903119894) + 12541 (13)


119877 = 9994 (14)




4

5

6

7

8

9

10

11

12

lnN

(r)

lnN(r)=18652ln(1r)+12541R=09994

25 3 35 4 45 5 552ln(1r)



4 402 404 406 408 41001

0015002


minus006

minus004

minus002

0

002

004

006

Vert

ical

hei

ght (

m)






119877lowast = 1198771199021(119863minus1) (15)



(16)


ms

Ks1

Kt

Cs

Zs1

q

Zu1

mu

Vehicle body

Passive suspension

Wheel

Tire

(a)

ms

Ks2

Kt

u

Zs2

q

Zu2

mu

(b)











1198981199041199041 = 1198701199041 (1198851199061 minus 1198851199041) + 119862119878 (1199061 minus 1199041)(17)




1198831 = [1199041 1199061 1198851199041 1198851199061]1198791198832 = [1199042 1199062 1198851199042 1198851199062]119879

(19)



(20)

where

1198601 =[[[[[[[[[

minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

1198701199041119898119904119862119904119898119906

minus 119862119904119898119906

1198701199041119898119906

minus1198701199041 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]]

1198611 =[[[[[[[

011987011990511989811990600

]]]]]]]

1198602 =[[[[[[[[

0 0 minus1198701199042119898119904

11987011990421198981199040 0 1198701199042119898119906

minus1198701199042 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]

1198612 =[[[[[[[[

1119898119904

0minus 1119898119906

1198701199051198981199060 00 0

]]]]]]]]

(21)



(22)



where

1198621 = [[[[minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198631 = [[[001]]]

1198622 = [[[[0 0 minus1198701199042119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198632 = [[[[

1119898119904

00 00 1

]]]]

(24)



119869 = lim119879997888rarrinfin


(25)



119869 = lim119879997888rarrinfin

1119879 11988321198791198761198832 + 119880119879119877119880 + 21198832

119879119873119880119889119905 (26)

where

119876 =[[[[[[[[[[

0 0 0 00 0 0 00 0 1199022 + 11990231198702

11990421198982119904

minus1199022 minus 1199023119870211990421198982119904

0 0 minus1199022 minus 1199023119870211990421198982119904

1199021 + 1199022 + 1199023119870211990421198982119904

]]]]]]]]]]


119873 =[[[[[[[[[

0 00 0

minus11987011990421198982119904

011987011990421198982

119904

minus1199021

]]]]]]]]]

119877 = [[11990231198982119904

00 1199021

]]

(27)


1198751198602 + 1198601198792119875 minus (1198751198612 + 119873)119877minus1 (1198611198792119875 + 119873119879) + 119876 = 0 (28)

119870 = 119877minus1 (1198611198792119875 + 119873119879) (29)


119906 = minus119870 sdot 1198832 (30)




(119905) = minus212058711989900V119902 (119905) + 21205871198990radic119878119902 (1198990) V119882(119905) (31)






0 20 40 60 80 100 120 140 160Time t (s)

minus3

minus2

minus1

0

1

2

3

Road

elev

atio

n q

(m)






119867119860 =[[[[[[

1 2 912 1 319 13 1

]]]]]]

119867119861 =[[[[[[

1 3 913 1 319 13 1

]]]]]]

119867119862 =[[[[[[[

1 34 843 1 418 14 1

]]]]]]]

119867119863 =[[[[[[[

1 34 643 1 416 14 1

]]]]]]]



-




119867119864 = [[[[[

1 1 31 1 113 1 1

]]]]]

119867119865 = [[[[[

1 1 21 1 112 1 1

]]]]]

119867119866 = [[[[

1 1 11 1 151 5 1

]]]]

119867119867 = [[[[

1 1 11 1 161 6 1

]]]]

(32)



1199021 = 3000937141199022 = 11003214 (33)

10 Methodology



119882[11990910158401199101015840] = [119886 119887119888 119889] [

119909119910] + [

119890119891] (34)



119860 = [119886 119887119888 119889] = 119877 sdot 119879 sdot 119871


119879 = [ 1 tan120572tan120572 1 ]

119871 = [1198971 00 1198972]

(35)




119882119899 [119909119910] = [119886119899 0119888119899 119889119899] sdot [

119909119910] + [

119890119899119891119899] (36)





119882119899 [11990901199100] = [119909119899minus1119910119899minus1]

119882119899 [119909119873119910119873] = [119909119899119910119899]

(37)



(38)


119909119899119894 = 119886119899119909119894 + 119890119899119910119899119894 = 119888119899119909119894 + 119889119899119910119894 + 119891119899




12059021198611198601 sdot 1 = 1205731 sdot 12059021198631198791198631 = 1205732 sdot 12059021198781198821198781 (40)



119867 =

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


1ℎ1119899


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

(41)



119895=1

ℎ119894119895(119894 119895 = 1 2 sdot sdot sdot 119899)

(42)



(119894 = 1 2 sdot sdot sdot 119899)(43)


119882 = 119882sum119899119894=1119882119894

(44)



120582max = 119899sum119894=1

(119867119882)119894119899119882119894

(119894 = 1 2 sdot sdot sdot 119899) (45)

119862119877 = 120582max minus 119899119877119868 (119899 minus 1) (46)




1198821 = 11988221205741 = 11988231205742 (47)





1199021 = 1205731 sdot 12057411199022 = 1205732 sdot 1205742 (48)

11 Experiment



0 2 4 6 8 10Time (s)

minus3

minus2

minus1

0

1

2

3

Vert

ical

acce

lera

tion

(ms

2 )

qFqGqEqH

qDqAqCqB


0 2 4 6 8 10Time (s)


0002004006008

Tire

dyn

amic

defl

ectio

n (m

)

qCqFqEqG

qDqAqHqB


0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)

qAqHqDqF

qEqCqGqB






0 2 4 6 8 10Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

Vert

ical

acce

lera

tion

(ms

2 )



0 2 4 6 8 10Time (s)


0002004006008

01

Tire

dyn

amic

defl

ectio

n (m

)



0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)



0 1 2 3 4 5 6 7 8 9 10Time (s)


minus5000

5001000150020002500

Activ

e con

trol f

orce

(N)


08

085

09

095

1

105

qF qG qEqH

qD

qA354

7098329771009

1033

1946

qC qB078

Passivesuspension

Active suspension






002300235

002400245

002500255

002600265

002700275

0028

qC qFqE

qG qD qA qHqB

620547

358122162202248


159


0032

0033

0034

0035

0036

0037

0038

897

262 233 146 118 089 059


029





12 Conclusions




Data Availability






Acknowledgments


References
















































Journal of












Journal of







Volume 2018






Volume 2018









L1

L2

L4

L3

XL

ZL

YL

XLLi

OL

OL






[[[119909119871119910119871119911119871]]]= 119871 119894 sdot [[


sin120572119894lowast]]]

120572119894lowast =

2120572 119894 = 1120572 119894 = 2minus120572 119894 = 3minus2120572 119894 = 4

120579 isin [minus60∘ 50∘]

(1)



[[[119909119882841199101198828411991111988284

]]]

= [[[119909119866119875119878119910119866119875119878119911119866119875119878

]]]

+ 119877119882[[[119877119873[[

[119877119872[[

[119909119871119910119871119911119871]]]+ [[[Δ119909119871119868Δ119910119871119868Δ119911119871119868

]]]minus [[[Δ119909119868119866Δ119910119868119866Δ119911119868119866

]]]]]]]]]

(2)


119877119872 = 119877120574119911 sdot 119877120572119910 sdot 119877120573119909119877120574119911 = [[


]]]

119877120572119910 = [[[cos120572 0 minus sin1205720 1 0

sin120572 0 cos120572]]]


]]]

(3)


point

O84

X84

Z84

OL

XL

YL

ZL

OI

XI

YI

ZI

Y84

ZG

OG

XG

YG

L



119877119873 = 119877120581 sdot 119877120593 sdot 119877120596119877120581 = [[


]]]

119877120593 = [[[cos120593 0 minus sin1205930 1 0

sin120593 0 cos120593]]]


]]]

(4)



cos119861 0 minus sin119861]]]

(5)




[[[119909119866119875119878119910119866119875119878119911119866119875119878

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


]]]]]]]]

(6)


119886 = 6378137 1198981198902 = 0006694379 (7)






Azimuthaccuracy

Attitudeaccuracy

Positioningaccuracy

Speedaccuracy



Scanningfrequency

Angularresolution


Laserwavelength

50119867119911 05∘ 4 905119899119898Laser grade

Verticalangular


119862119897119886119904119904 119868 120572 = 08∘ 119863119862 9119881 sim 27119881 119862119860119873119864119905ℎ119890119903119899119890119905









Figure 4 Test site

Vert

ical h

eight

(m)


)


20

minus2minus4 5

10

15

20










Minimum40002 (m)

Maximum209994 (m)



)


2 0 minus2 minus4 5

10

15

20

Vert

ical h

eight

(m)



119897119900119899119892119894119905119906119889119894119899119886119897 119903119886119899119892119890 isin [4119898 21119898]119897119886119905119890119903119886119897 119903119886119899119892119890 = minus2119898 (8)




119881 = 119903119886119899119892119890119909 times 119903119886119899119892119890119910 times 119903119886119899119892119890119911 (9)

119903119886119899119892119890119909 = 209994 minus 40003 = 169991119903119886119899119892119890119910 = 29998 + 49935 = 799330119903119886119899119892119890119911 = 01977 + 01477 = 034540

(10)

V = 119881119905ℎ119890 119899119906119898119887119890119903 119900119891 3119863 119901119900119894119899119905 119888119897119900119906119889 (11)




ℎ = sum119899119894=1 (1119889119894) ℎ119894sum119899119894=1 (1119889119894) (12)









minus006

minus004

minus002

0

002

004

006

Vert

ical

hei

ght (

m)



(a)


(b)



119903119894(m) 0005 001 002 004 008119873(119903119894) 71400 18700 5100 1275 426ln (1119903119894) 52983 46052 39120 32189 25257ln (1119873 (119903119894)) 111761 98363 85370 71507 60544




ln119873(119903119894) = 18652 sdot ln( 1119903119894) + 12541 (13)


119877 = 9994 (14)




4

5

6

7

8

9

10

11

12

lnN

(r)

lnN(r)=18652ln(1r)+12541R=09994

25 3 35 4 45 5 552ln(1r)



4 402 404 406 408 41001

0015002


minus006

minus004

minus002

0

002

004

006

Vert

ical

hei

ght (

m)






119877lowast = 1198771199021(119863minus1) (15)



(16)


ms

Ks1

Kt

Cs

Zs1

q

Zu1

mu

Vehicle body

Passive suspension

Wheel

Tire

(a)

ms

Ks2

Kt

u

Zs2

q

Zu2

mu

(b)











1198981199041199041 = 1198701199041 (1198851199061 minus 1198851199041) + 119862119878 (1199061 minus 1199041)(17)




1198831 = [1199041 1199061 1198851199041 1198851199061]1198791198832 = [1199042 1199062 1198851199042 1198851199062]119879

(19)



(20)

where

1198601 =[[[[[[[[[

minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

1198701199041119898119904119862119904119898119906

minus 119862119904119898119906

1198701199041119898119906

minus1198701199041 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]]

1198611 =[[[[[[[

011987011990511989811990600

]]]]]]]

1198602 =[[[[[[[[

0 0 minus1198701199042119898119904

11987011990421198981199040 0 1198701199042119898119906

minus1198701199042 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]

1198612 =[[[[[[[[

1119898119904

0minus 1119898119906

1198701199051198981199060 00 0

]]]]]]]]

(21)



(22)



where

1198621 = [[[[minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198631 = [[[001]]]

1198622 = [[[[0 0 minus1198701199042119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198632 = [[[[

1119898119904

00 00 1

]]]]

(24)



119869 = lim119879997888rarrinfin


(25)



119869 = lim119879997888rarrinfin

1119879 11988321198791198761198832 + 119880119879119877119880 + 21198832

119879119873119880119889119905 (26)

where

119876 =[[[[[[[[[[

0 0 0 00 0 0 00 0 1199022 + 11990231198702

11990421198982119904

minus1199022 minus 1199023119870211990421198982119904

0 0 minus1199022 minus 1199023119870211990421198982119904

1199021 + 1199022 + 1199023119870211990421198982119904

]]]]]]]]]]


119873 =[[[[[[[[[

0 00 0

minus11987011990421198982119904

011987011990421198982

119904

minus1199021

]]]]]]]]]

119877 = [[11990231198982119904

00 1199021

]]

(27)


1198751198602 + 1198601198792119875 minus (1198751198612 + 119873)119877minus1 (1198611198792119875 + 119873119879) + 119876 = 0 (28)

119870 = 119877minus1 (1198611198792119875 + 119873119879) (29)


119906 = minus119870 sdot 1198832 (30)




(119905) = minus212058711989900V119902 (119905) + 21205871198990radic119878119902 (1198990) V119882(119905) (31)






0 20 40 60 80 100 120 140 160Time t (s)

minus3

minus2

minus1

0

1

2

3

Road

elev

atio

n q

(m)






119867119860 =[[[[[[

1 2 912 1 319 13 1

]]]]]]

119867119861 =[[[[[[

1 3 913 1 319 13 1

]]]]]]

119867119862 =[[[[[[[

1 34 843 1 418 14 1

]]]]]]]

119867119863 =[[[[[[[

1 34 643 1 416 14 1

]]]]]]]



-




119867119864 = [[[[[

1 1 31 1 113 1 1

]]]]]

119867119865 = [[[[[

1 1 21 1 112 1 1

]]]]]

119867119866 = [[[[

1 1 11 1 151 5 1

]]]]

119867119867 = [[[[

1 1 11 1 161 6 1

]]]]

(32)



1199021 = 3000937141199022 = 11003214 (33)

10 Methodology



119882[11990910158401199101015840] = [119886 119887119888 119889] [

119909119910] + [

119890119891] (34)



119860 = [119886 119887119888 119889] = 119877 sdot 119879 sdot 119871


119879 = [ 1 tan120572tan120572 1 ]

119871 = [1198971 00 1198972]

(35)




119882119899 [119909119910] = [119886119899 0119888119899 119889119899] sdot [

119909119910] + [

119890119899119891119899] (36)





119882119899 [11990901199100] = [119909119899minus1119910119899minus1]

119882119899 [119909119873119910119873] = [119909119899119910119899]

(37)



(38)


119909119899119894 = 119886119899119909119894 + 119890119899119910119899119894 = 119888119899119909119894 + 119889119899119910119894 + 119891119899




12059021198611198601 sdot 1 = 1205731 sdot 12059021198631198791198631 = 1205732 sdot 12059021198781198821198781 (40)



119867 =

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


1ℎ1119899


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

(41)



119895=1

ℎ119894119895(119894 119895 = 1 2 sdot sdot sdot 119899)

(42)



(119894 = 1 2 sdot sdot sdot 119899)(43)


119882 = 119882sum119899119894=1119882119894

(44)



120582max = 119899sum119894=1

(119867119882)119894119899119882119894

(119894 = 1 2 sdot sdot sdot 119899) (45)

119862119877 = 120582max minus 119899119877119868 (119899 minus 1) (46)




1198821 = 11988221205741 = 11988231205742 (47)





1199021 = 1205731 sdot 12057411199022 = 1205732 sdot 1205742 (48)

11 Experiment



0 2 4 6 8 10Time (s)

minus3

minus2

minus1

0

1

2

3

Vert

ical

acce

lera

tion

(ms

2 )

qFqGqEqH

qDqAqCqB


0 2 4 6 8 10Time (s)


0002004006008

Tire

dyn

amic

defl

ectio

n (m

)

qCqFqEqG

qDqAqHqB


0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)

qAqHqDqF

qEqCqGqB






0 2 4 6 8 10Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

Vert

ical

acce

lera

tion

(ms

2 )



0 2 4 6 8 10Time (s)


0002004006008

01

Tire

dyn

amic

defl

ectio

n (m

)



0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)



0 1 2 3 4 5 6 7 8 9 10Time (s)


minus5000

5001000150020002500

Activ

e con

trol f

orce

(N)


08

085

09

095

1

105

qF qG qEqH

qD

qA354

7098329771009

1033

1946

qC qB078

Passivesuspension

Active suspension






002300235

002400245

002500255

002600265

002700275

0028

qC qFqE

qG qD qA qHqB

620547

358122162202248


159


0032

0033

0034

0035

0036

0037

0038

897

262 233 146 118 089 059


029





12 Conclusions




Data Availability






Acknowledgments


References
















































Journal of












Journal of







Volume 2018






Volume 2018









point

O84

X84

Z84

OL

XL

YL

ZL

OI

XI

YI

ZI

Y84

ZG

OG

XG

YG

L



119877119873 = 119877120581 sdot 119877120593 sdot 119877120596119877120581 = [[


]]]

119877120593 = [[[cos120593 0 minus sin1205930 1 0

sin120593 0 cos120593]]]


]]]

(4)



cos119861 0 minus sin119861]]]

(5)




[[[119909119866119875119878119910119866119875119878119911119866119875119878

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


]]]]]]]]

(6)


119886 = 6378137 1198981198902 = 0006694379 (7)






Azimuthaccuracy

Attitudeaccuracy

Positioningaccuracy

Speedaccuracy



Scanningfrequency

Angularresolution


Laserwavelength

50119867119911 05∘ 4 905119899119898Laser grade

Verticalangular


119862119897119886119904119904 119868 120572 = 08∘ 119863119862 9119881 sim 27119881 119862119860119873119864119905ℎ119890119903119899119890119905









Figure 4 Test site

Vert

ical h

eight

(m)


)


20

minus2minus4 5

10

15

20










Minimum40002 (m)

Maximum209994 (m)



)


2 0 minus2 minus4 5

10

15

20

Vert

ical h

eight

(m)



119897119900119899119892119894119905119906119889119894119899119886119897 119903119886119899119892119890 isin [4119898 21119898]119897119886119905119890119903119886119897 119903119886119899119892119890 = minus2119898 (8)




119881 = 119903119886119899119892119890119909 times 119903119886119899119892119890119910 times 119903119886119899119892119890119911 (9)

119903119886119899119892119890119909 = 209994 minus 40003 = 169991119903119886119899119892119890119910 = 29998 + 49935 = 799330119903119886119899119892119890119911 = 01977 + 01477 = 034540

(10)

V = 119881119905ℎ119890 119899119906119898119887119890119903 119900119891 3119863 119901119900119894119899119905 119888119897119900119906119889 (11)




ℎ = sum119899119894=1 (1119889119894) ℎ119894sum119899119894=1 (1119889119894) (12)









minus006

minus004

minus002

0

002

004

006

Vert

ical

hei

ght (

m)



(a)


(b)



119903119894(m) 0005 001 002 004 008119873(119903119894) 71400 18700 5100 1275 426ln (1119903119894) 52983 46052 39120 32189 25257ln (1119873 (119903119894)) 111761 98363 85370 71507 60544




ln119873(119903119894) = 18652 sdot ln( 1119903119894) + 12541 (13)


119877 = 9994 (14)




4

5

6

7

8

9

10

11

12

lnN

(r)

lnN(r)=18652ln(1r)+12541R=09994

25 3 35 4 45 5 552ln(1r)



4 402 404 406 408 41001

0015002


minus006

minus004

minus002

0

002

004

006

Vert

ical

hei

ght (

m)






119877lowast = 1198771199021(119863minus1) (15)



(16)


ms

Ks1

Kt

Cs

Zs1

q

Zu1

mu

Vehicle body

Passive suspension

Wheel

Tire

(a)

ms

Ks2

Kt

u

Zs2

q

Zu2

mu

(b)











1198981199041199041 = 1198701199041 (1198851199061 minus 1198851199041) + 119862119878 (1199061 minus 1199041)(17)




1198831 = [1199041 1199061 1198851199041 1198851199061]1198791198832 = [1199042 1199062 1198851199042 1198851199062]119879

(19)



(20)

where

1198601 =[[[[[[[[[

minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

1198701199041119898119904119862119904119898119906

minus 119862119904119898119906

1198701199041119898119906

minus1198701199041 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]]

1198611 =[[[[[[[

011987011990511989811990600

]]]]]]]

1198602 =[[[[[[[[

0 0 minus1198701199042119898119904

11987011990421198981199040 0 1198701199042119898119906

minus1198701199042 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]

1198612 =[[[[[[[[

1119898119904

0minus 1119898119906

1198701199051198981199060 00 0

]]]]]]]]

(21)



(22)



where

1198621 = [[[[minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198631 = [[[001]]]

1198622 = [[[[0 0 minus1198701199042119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198632 = [[[[

1119898119904

00 00 1

]]]]

(24)



119869 = lim119879997888rarrinfin


(25)



119869 = lim119879997888rarrinfin

1119879 11988321198791198761198832 + 119880119879119877119880 + 21198832

119879119873119880119889119905 (26)

where

119876 =[[[[[[[[[[

0 0 0 00 0 0 00 0 1199022 + 11990231198702

11990421198982119904

minus1199022 minus 1199023119870211990421198982119904

0 0 minus1199022 minus 1199023119870211990421198982119904

1199021 + 1199022 + 1199023119870211990421198982119904

]]]]]]]]]]


119873 =[[[[[[[[[

0 00 0

minus11987011990421198982119904

011987011990421198982

119904

minus1199021

]]]]]]]]]

119877 = [[11990231198982119904

00 1199021

]]

(27)


1198751198602 + 1198601198792119875 minus (1198751198612 + 119873)119877minus1 (1198611198792119875 + 119873119879) + 119876 = 0 (28)

119870 = 119877minus1 (1198611198792119875 + 119873119879) (29)


119906 = minus119870 sdot 1198832 (30)




(119905) = minus212058711989900V119902 (119905) + 21205871198990radic119878119902 (1198990) V119882(119905) (31)






0 20 40 60 80 100 120 140 160Time t (s)

minus3

minus2

minus1

0

1

2

3

Road

elev

atio

n q

(m)






119867119860 =[[[[[[

1 2 912 1 319 13 1

]]]]]]

119867119861 =[[[[[[

1 3 913 1 319 13 1

]]]]]]

119867119862 =[[[[[[[

1 34 843 1 418 14 1

]]]]]]]

119867119863 =[[[[[[[

1 34 643 1 416 14 1

]]]]]]]



-




119867119864 = [[[[[

1 1 31 1 113 1 1

]]]]]

119867119865 = [[[[[

1 1 21 1 112 1 1

]]]]]

119867119866 = [[[[

1 1 11 1 151 5 1

]]]]

119867119867 = [[[[

1 1 11 1 161 6 1

]]]]

(32)



1199021 = 3000937141199022 = 11003214 (33)

10 Methodology



119882[11990910158401199101015840] = [119886 119887119888 119889] [

119909119910] + [

119890119891] (34)



119860 = [119886 119887119888 119889] = 119877 sdot 119879 sdot 119871


119879 = [ 1 tan120572tan120572 1 ]

119871 = [1198971 00 1198972]

(35)




119882119899 [119909119910] = [119886119899 0119888119899 119889119899] sdot [

119909119910] + [

119890119899119891119899] (36)





119882119899 [11990901199100] = [119909119899minus1119910119899minus1]

119882119899 [119909119873119910119873] = [119909119899119910119899]

(37)



(38)


119909119899119894 = 119886119899119909119894 + 119890119899119910119899119894 = 119888119899119909119894 + 119889119899119910119894 + 119891119899




12059021198611198601 sdot 1 = 1205731 sdot 12059021198631198791198631 = 1205732 sdot 12059021198781198821198781 (40)



119867 =

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


1ℎ1119899


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

(41)



119895=1

ℎ119894119895(119894 119895 = 1 2 sdot sdot sdot 119899)

(42)



(119894 = 1 2 sdot sdot sdot 119899)(43)


119882 = 119882sum119899119894=1119882119894

(44)



120582max = 119899sum119894=1

(119867119882)119894119899119882119894

(119894 = 1 2 sdot sdot sdot 119899) (45)

119862119877 = 120582max minus 119899119877119868 (119899 minus 1) (46)




1198821 = 11988221205741 = 11988231205742 (47)





1199021 = 1205731 sdot 12057411199022 = 1205732 sdot 1205742 (48)

11 Experiment



0 2 4 6 8 10Time (s)

minus3

minus2

minus1

0

1

2

3

Vert

ical

acce

lera

tion

(ms

2 )

qFqGqEqH

qDqAqCqB


0 2 4 6 8 10Time (s)


0002004006008

Tire

dyn

amic

defl

ectio

n (m

)

qCqFqEqG

qDqAqHqB


0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)

qAqHqDqF

qEqCqGqB






0 2 4 6 8 10Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

Vert

ical

acce

lera

tion

(ms

2 )



0 2 4 6 8 10Time (s)


0002004006008

01

Tire

dyn

amic

defl

ectio

n (m

)



0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)



0 1 2 3 4 5 6 7 8 9 10Time (s)


minus5000

5001000150020002500

Activ

e con

trol f

orce

(N)


08

085

09

095

1

105

qF qG qEqH

qD

qA354

7098329771009

1033

1946

qC qB078

Passivesuspension

Active suspension






002300235

002400245

002500255

002600265

002700275

0028

qC qFqE

qG qD qA qHqB

620547

358122162202248


159


0032

0033

0034

0035

0036

0037

0038

897

262 233 146 118 089 059


029





12 Conclusions




Data Availability






Acknowledgments


References
















































Journal of












Journal of







Volume 2018






Volume 2018










Azimuthaccuracy

Attitudeaccuracy

Positioningaccuracy

Speedaccuracy



Scanningfrequency

Angularresolution


Laserwavelength

50119867119911 05∘ 4 905119899119898Laser grade

Verticalangular


119862119897119886119904119904 119868 120572 = 08∘ 119863119862 9119881 sim 27119881 119862119860119873119864119905ℎ119890119903119899119890119905









Figure 4 Test site

Vert

ical h

eight

(m)


)


20

minus2minus4 5

10

15

20










Minimum40002 (m)

Maximum209994 (m)



)


2 0 minus2 minus4 5

10

15

20

Vert

ical h

eight

(m)



119897119900119899119892119894119905119906119889119894119899119886119897 119903119886119899119892119890 isin [4119898 21119898]119897119886119905119890119903119886119897 119903119886119899119892119890 = minus2119898 (8)




119881 = 119903119886119899119892119890119909 times 119903119886119899119892119890119910 times 119903119886119899119892119890119911 (9)

119903119886119899119892119890119909 = 209994 minus 40003 = 169991119903119886119899119892119890119910 = 29998 + 49935 = 799330119903119886119899119892119890119911 = 01977 + 01477 = 034540

(10)

V = 119881119905ℎ119890 119899119906119898119887119890119903 119900119891 3119863 119901119900119894119899119905 119888119897119900119906119889 (11)




ℎ = sum119899119894=1 (1119889119894) ℎ119894sum119899119894=1 (1119889119894) (12)









minus006

minus004

minus002

0

002

004

006

Vert

ical

hei

ght (

m)



(a)


(b)



119903119894(m) 0005 001 002 004 008119873(119903119894) 71400 18700 5100 1275 426ln (1119903119894) 52983 46052 39120 32189 25257ln (1119873 (119903119894)) 111761 98363 85370 71507 60544




ln119873(119903119894) = 18652 sdot ln( 1119903119894) + 12541 (13)


119877 = 9994 (14)




4

5

6

7

8

9

10

11

12

lnN

(r)

lnN(r)=18652ln(1r)+12541R=09994

25 3 35 4 45 5 552ln(1r)



4 402 404 406 408 41001

0015002


minus006

minus004

minus002

0

002

004

006

Vert

ical

hei

ght (

m)






119877lowast = 1198771199021(119863minus1) (15)



(16)


ms

Ks1

Kt

Cs

Zs1

q

Zu1

mu

Vehicle body

Passive suspension

Wheel

Tire

(a)

ms

Ks2

Kt

u

Zs2

q

Zu2

mu

(b)











1198981199041199041 = 1198701199041 (1198851199061 minus 1198851199041) + 119862119878 (1199061 minus 1199041)(17)




1198831 = [1199041 1199061 1198851199041 1198851199061]1198791198832 = [1199042 1199062 1198851199042 1198851199062]119879

(19)



(20)

where

1198601 =[[[[[[[[[

minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

1198701199041119898119904119862119904119898119906

minus 119862119904119898119906

1198701199041119898119906

minus1198701199041 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]]

1198611 =[[[[[[[

011987011990511989811990600

]]]]]]]

1198602 =[[[[[[[[

0 0 minus1198701199042119898119904

11987011990421198981199040 0 1198701199042119898119906

minus1198701199042 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]

1198612 =[[[[[[[[

1119898119904

0minus 1119898119906

1198701199051198981199060 00 0

]]]]]]]]

(21)



(22)



where

1198621 = [[[[minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198631 = [[[001]]]

1198622 = [[[[0 0 minus1198701199042119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198632 = [[[[

1119898119904

00 00 1

]]]]

(24)



119869 = lim119879997888rarrinfin


(25)



119869 = lim119879997888rarrinfin

1119879 11988321198791198761198832 + 119880119879119877119880 + 21198832

119879119873119880119889119905 (26)

where

119876 =[[[[[[[[[[

0 0 0 00 0 0 00 0 1199022 + 11990231198702

11990421198982119904

minus1199022 minus 1199023119870211990421198982119904

0 0 minus1199022 minus 1199023119870211990421198982119904

1199021 + 1199022 + 1199023119870211990421198982119904

]]]]]]]]]]


119873 =[[[[[[[[[

0 00 0

minus11987011990421198982119904

011987011990421198982

119904

minus1199021

]]]]]]]]]

119877 = [[11990231198982119904

00 1199021

]]

(27)


1198751198602 + 1198601198792119875 minus (1198751198612 + 119873)119877minus1 (1198611198792119875 + 119873119879) + 119876 = 0 (28)

119870 = 119877minus1 (1198611198792119875 + 119873119879) (29)


119906 = minus119870 sdot 1198832 (30)




(119905) = minus212058711989900V119902 (119905) + 21205871198990radic119878119902 (1198990) V119882(119905) (31)






0 20 40 60 80 100 120 140 160Time t (s)

minus3

minus2

minus1

0

1

2

3

Road

elev

atio

n q

(m)






119867119860 =[[[[[[

1 2 912 1 319 13 1

]]]]]]

119867119861 =[[[[[[

1 3 913 1 319 13 1

]]]]]]

119867119862 =[[[[[[[

1 34 843 1 418 14 1

]]]]]]]

119867119863 =[[[[[[[

1 34 643 1 416 14 1

]]]]]]]



-




119867119864 = [[[[[

1 1 31 1 113 1 1

]]]]]

119867119865 = [[[[[

1 1 21 1 112 1 1

]]]]]

119867119866 = [[[[

1 1 11 1 151 5 1

]]]]

119867119867 = [[[[

1 1 11 1 161 6 1

]]]]

(32)



1199021 = 3000937141199022 = 11003214 (33)

10 Methodology



119882[11990910158401199101015840] = [119886 119887119888 119889] [

119909119910] + [

119890119891] (34)



119860 = [119886 119887119888 119889] = 119877 sdot 119879 sdot 119871


119879 = [ 1 tan120572tan120572 1 ]

119871 = [1198971 00 1198972]

(35)




119882119899 [119909119910] = [119886119899 0119888119899 119889119899] sdot [

119909119910] + [

119890119899119891119899] (36)





119882119899 [11990901199100] = [119909119899minus1119910119899minus1]

119882119899 [119909119873119910119873] = [119909119899119910119899]

(37)



(38)


119909119899119894 = 119886119899119909119894 + 119890119899119910119899119894 = 119888119899119909119894 + 119889119899119910119894 + 119891119899




12059021198611198601 sdot 1 = 1205731 sdot 12059021198631198791198631 = 1205732 sdot 12059021198781198821198781 (40)



119867 =

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


1ℎ1119899


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

(41)



119895=1

ℎ119894119895(119894 119895 = 1 2 sdot sdot sdot 119899)

(42)



(119894 = 1 2 sdot sdot sdot 119899)(43)


119882 = 119882sum119899119894=1119882119894

(44)



120582max = 119899sum119894=1

(119867119882)119894119899119882119894

(119894 = 1 2 sdot sdot sdot 119899) (45)

119862119877 = 120582max minus 119899119877119868 (119899 minus 1) (46)




1198821 = 11988221205741 = 11988231205742 (47)





1199021 = 1205731 sdot 12057411199022 = 1205732 sdot 1205742 (48)

11 Experiment



0 2 4 6 8 10Time (s)

minus3

minus2

minus1

0

1

2

3

Vert

ical

acce

lera

tion

(ms

2 )

qFqGqEqH

qDqAqCqB


0 2 4 6 8 10Time (s)


0002004006008

Tire

dyn

amic

defl

ectio

n (m

)

qCqFqEqG

qDqAqHqB


0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)

qAqHqDqF

qEqCqGqB






0 2 4 6 8 10Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

Vert

ical

acce

lera

tion

(ms

2 )



0 2 4 6 8 10Time (s)


0002004006008

01

Tire

dyn

amic

defl

ectio

n (m

)



0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)



0 1 2 3 4 5 6 7 8 9 10Time (s)


minus5000

5001000150020002500

Activ

e con

trol f

orce

(N)


08

085

09

095

1

105

qF qG qEqH

qD

qA354

7098329771009

1033

1946

qC qB078

Passivesuspension

Active suspension






002300235

002400245

002500255

002600265

002700275

0028

qC qFqE

qG qD qA qHqB

620547

358122162202248


159


0032

0033

0034

0035

0036

0037

0038

897

262 233 146 118 089 059


029





12 Conclusions




Data Availability






Acknowledgments


References
















































Journal of












Journal of







Volume 2018






Volume 2018











Minimum40002 (m)

Maximum209994 (m)



)


2 0 minus2 minus4 5

10

15

20

Vert

ical h

eight

(m)



119897119900119899119892119894119905119906119889119894119899119886119897 119903119886119899119892119890 isin [4119898 21119898]119897119886119905119890119903119886119897 119903119886119899119892119890 = minus2119898 (8)




119881 = 119903119886119899119892119890119909 times 119903119886119899119892119890119910 times 119903119886119899119892119890119911 (9)

119903119886119899119892119890119909 = 209994 minus 40003 = 169991119903119886119899119892119890119910 = 29998 + 49935 = 799330119903119886119899119892119890119911 = 01977 + 01477 = 034540

(10)

V = 119881119905ℎ119890 119899119906119898119887119890119903 119900119891 3119863 119901119900119894119899119905 119888119897119900119906119889 (11)




ℎ = sum119899119894=1 (1119889119894) ℎ119894sum119899119894=1 (1119889119894) (12)









minus006

minus004

minus002

0

002

004

006

Vert

ical

hei

ght (

m)



(a)


(b)



119903119894(m) 0005 001 002 004 008119873(119903119894) 71400 18700 5100 1275 426ln (1119903119894) 52983 46052 39120 32189 25257ln (1119873 (119903119894)) 111761 98363 85370 71507 60544




ln119873(119903119894) = 18652 sdot ln( 1119903119894) + 12541 (13)


119877 = 9994 (14)




4

5

6

7

8

9

10

11

12

lnN

(r)

lnN(r)=18652ln(1r)+12541R=09994

25 3 35 4 45 5 552ln(1r)



4 402 404 406 408 41001

0015002


minus006

minus004

minus002

0

002

004

006

Vert

ical

hei

ght (

m)






119877lowast = 1198771199021(119863minus1) (15)



(16)


ms

Ks1

Kt

Cs

Zs1

q

Zu1

mu

Vehicle body

Passive suspension

Wheel

Tire

(a)

ms

Ks2

Kt

u

Zs2

q

Zu2

mu

(b)











1198981199041199041 = 1198701199041 (1198851199061 minus 1198851199041) + 119862119878 (1199061 minus 1199041)(17)




1198831 = [1199041 1199061 1198851199041 1198851199061]1198791198832 = [1199042 1199062 1198851199042 1198851199062]119879

(19)



(20)

where

1198601 =[[[[[[[[[

minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

1198701199041119898119904119862119904119898119906

minus 119862119904119898119906

1198701199041119898119906

minus1198701199041 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]]

1198611 =[[[[[[[

011987011990511989811990600

]]]]]]]

1198602 =[[[[[[[[

0 0 minus1198701199042119898119904

11987011990421198981199040 0 1198701199042119898119906

minus1198701199042 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]

1198612 =[[[[[[[[

1119898119904

0minus 1119898119906

1198701199051198981199060 00 0

]]]]]]]]

(21)



(22)



where

1198621 = [[[[minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198631 = [[[001]]]

1198622 = [[[[0 0 minus1198701199042119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198632 = [[[[

1119898119904

00 00 1

]]]]

(24)



119869 = lim119879997888rarrinfin


(25)



119869 = lim119879997888rarrinfin

1119879 11988321198791198761198832 + 119880119879119877119880 + 21198832

119879119873119880119889119905 (26)

where

119876 =[[[[[[[[[[

0 0 0 00 0 0 00 0 1199022 + 11990231198702

11990421198982119904

minus1199022 minus 1199023119870211990421198982119904

0 0 minus1199022 minus 1199023119870211990421198982119904

1199021 + 1199022 + 1199023119870211990421198982119904

]]]]]]]]]]


119873 =[[[[[[[[[

0 00 0

minus11987011990421198982119904

011987011990421198982

119904

minus1199021

]]]]]]]]]

119877 = [[11990231198982119904

00 1199021

]]

(27)


1198751198602 + 1198601198792119875 minus (1198751198612 + 119873)119877minus1 (1198611198792119875 + 119873119879) + 119876 = 0 (28)

119870 = 119877minus1 (1198611198792119875 + 119873119879) (29)


119906 = minus119870 sdot 1198832 (30)




(119905) = minus212058711989900V119902 (119905) + 21205871198990radic119878119902 (1198990) V119882(119905) (31)






0 20 40 60 80 100 120 140 160Time t (s)

minus3

minus2

minus1

0

1

2

3

Road

elev

atio

n q

(m)






119867119860 =[[[[[[

1 2 912 1 319 13 1

]]]]]]

119867119861 =[[[[[[

1 3 913 1 319 13 1

]]]]]]

119867119862 =[[[[[[[

1 34 843 1 418 14 1

]]]]]]]

119867119863 =[[[[[[[

1 34 643 1 416 14 1

]]]]]]]



-




119867119864 = [[[[[

1 1 31 1 113 1 1

]]]]]

119867119865 = [[[[[

1 1 21 1 112 1 1

]]]]]

119867119866 = [[[[

1 1 11 1 151 5 1

]]]]

119867119867 = [[[[

1 1 11 1 161 6 1

]]]]

(32)



1199021 = 3000937141199022 = 11003214 (33)

10 Methodology



119882[11990910158401199101015840] = [119886 119887119888 119889] [

119909119910] + [

119890119891] (34)



119860 = [119886 119887119888 119889] = 119877 sdot 119879 sdot 119871


119879 = [ 1 tan120572tan120572 1 ]

119871 = [1198971 00 1198972]

(35)




119882119899 [119909119910] = [119886119899 0119888119899 119889119899] sdot [

119909119910] + [

119890119899119891119899] (36)





119882119899 [11990901199100] = [119909119899minus1119910119899minus1]

119882119899 [119909119873119910119873] = [119909119899119910119899]

(37)



(38)


119909119899119894 = 119886119899119909119894 + 119890119899119910119899119894 = 119888119899119909119894 + 119889119899119910119894 + 119891119899




12059021198611198601 sdot 1 = 1205731 sdot 12059021198631198791198631 = 1205732 sdot 12059021198781198821198781 (40)



119867 =

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


1ℎ1119899


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

(41)



119895=1

ℎ119894119895(119894 119895 = 1 2 sdot sdot sdot 119899)

(42)



(119894 = 1 2 sdot sdot sdot 119899)(43)


119882 = 119882sum119899119894=1119882119894

(44)



120582max = 119899sum119894=1

(119867119882)119894119899119882119894

(119894 = 1 2 sdot sdot sdot 119899) (45)

119862119877 = 120582max minus 119899119877119868 (119899 minus 1) (46)




1198821 = 11988221205741 = 11988231205742 (47)





1199021 = 1205731 sdot 12057411199022 = 1205732 sdot 1205742 (48)

11 Experiment



0 2 4 6 8 10Time (s)

minus3

minus2

minus1

0

1

2

3

Vert

ical

acce

lera

tion

(ms

2 )

qFqGqEqH

qDqAqCqB


0 2 4 6 8 10Time (s)


0002004006008

Tire

dyn

amic

defl

ectio

n (m

)

qCqFqEqG

qDqAqHqB


0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)

qAqHqDqF

qEqCqGqB






0 2 4 6 8 10Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

Vert

ical

acce

lera

tion

(ms

2 )



0 2 4 6 8 10Time (s)


0002004006008

01

Tire

dyn

amic

defl

ectio

n (m

)



0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)



0 1 2 3 4 5 6 7 8 9 10Time (s)


minus5000

5001000150020002500

Activ

e con

trol f

orce

(N)


08

085

09

095

1

105

qF qG qEqH

qD

qA354

7098329771009

1033

1946

qC qB078

Passivesuspension

Active suspension






002300235

002400245

002500255

002600265

002700275

0028

qC qFqE

qG qD qA qHqB

620547

358122162202248


159


0032

0033

0034

0035

0036

0037

0038

897

262 233 146 118 089 059


029





12 Conclusions




Data Availability






Acknowledgments


References
















































Journal of












Journal of







Volume 2018






Volume 2018










minus006

minus004

minus002

0

002

004

006

Vert

ical

hei

ght (

m)



(a)


(b)



119903119894(m) 0005 001 002 004 008119873(119903119894) 71400 18700 5100 1275 426ln (1119903119894) 52983 46052 39120 32189 25257ln (1119873 (119903119894)) 111761 98363 85370 71507 60544




ln119873(119903119894) = 18652 sdot ln( 1119903119894) + 12541 (13)


119877 = 9994 (14)




4

5

6

7

8

9

10

11

12

lnN

(r)

lnN(r)=18652ln(1r)+12541R=09994

25 3 35 4 45 5 552ln(1r)



4 402 404 406 408 41001

0015002


minus006

minus004

minus002

0

002

004

006

Vert

ical

hei

ght (

m)






119877lowast = 1198771199021(119863minus1) (15)



(16)


ms

Ks1

Kt

Cs

Zs1

q

Zu1

mu

Vehicle body

Passive suspension

Wheel

Tire

(a)

ms

Ks2

Kt

u

Zs2

q

Zu2

mu

(b)











1198981199041199041 = 1198701199041 (1198851199061 minus 1198851199041) + 119862119878 (1199061 minus 1199041)(17)




1198831 = [1199041 1199061 1198851199041 1198851199061]1198791198832 = [1199042 1199062 1198851199042 1198851199062]119879

(19)



(20)

where

1198601 =[[[[[[[[[

minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

1198701199041119898119904119862119904119898119906

minus 119862119904119898119906

1198701199041119898119906

minus1198701199041 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]]

1198611 =[[[[[[[

011987011990511989811990600

]]]]]]]

1198602 =[[[[[[[[

0 0 minus1198701199042119898119904

11987011990421198981199040 0 1198701199042119898119906

minus1198701199042 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]

1198612 =[[[[[[[[

1119898119904

0minus 1119898119906

1198701199051198981199060 00 0

]]]]]]]]

(21)



(22)



where

1198621 = [[[[minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198631 = [[[001]]]

1198622 = [[[[0 0 minus1198701199042119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198632 = [[[[

1119898119904

00 00 1

]]]]

(24)



119869 = lim119879997888rarrinfin


(25)



119869 = lim119879997888rarrinfin

1119879 11988321198791198761198832 + 119880119879119877119880 + 21198832

119879119873119880119889119905 (26)

where

119876 =[[[[[[[[[[

0 0 0 00 0 0 00 0 1199022 + 11990231198702

11990421198982119904

minus1199022 minus 1199023119870211990421198982119904

0 0 minus1199022 minus 1199023119870211990421198982119904

1199021 + 1199022 + 1199023119870211990421198982119904

]]]]]]]]]]


119873 =[[[[[[[[[

0 00 0

minus11987011990421198982119904

011987011990421198982

119904

minus1199021

]]]]]]]]]

119877 = [[11990231198982119904

00 1199021

]]

(27)


1198751198602 + 1198601198792119875 minus (1198751198612 + 119873)119877minus1 (1198611198792119875 + 119873119879) + 119876 = 0 (28)

119870 = 119877minus1 (1198611198792119875 + 119873119879) (29)


119906 = minus119870 sdot 1198832 (30)




(119905) = minus212058711989900V119902 (119905) + 21205871198990radic119878119902 (1198990) V119882(119905) (31)






0 20 40 60 80 100 120 140 160Time t (s)

minus3

minus2

minus1

0

1

2

3

Road

elev

atio

n q

(m)






119867119860 =[[[[[[

1 2 912 1 319 13 1

]]]]]]

119867119861 =[[[[[[

1 3 913 1 319 13 1

]]]]]]

119867119862 =[[[[[[[

1 34 843 1 418 14 1

]]]]]]]

119867119863 =[[[[[[[

1 34 643 1 416 14 1

]]]]]]]



-




119867119864 = [[[[[

1 1 31 1 113 1 1

]]]]]

119867119865 = [[[[[

1 1 21 1 112 1 1

]]]]]

119867119866 = [[[[

1 1 11 1 151 5 1

]]]]

119867119867 = [[[[

1 1 11 1 161 6 1

]]]]

(32)



1199021 = 3000937141199022 = 11003214 (33)

10 Methodology



119882[11990910158401199101015840] = [119886 119887119888 119889] [

119909119910] + [

119890119891] (34)



119860 = [119886 119887119888 119889] = 119877 sdot 119879 sdot 119871


119879 = [ 1 tan120572tan120572 1 ]

119871 = [1198971 00 1198972]

(35)




119882119899 [119909119910] = [119886119899 0119888119899 119889119899] sdot [

119909119910] + [

119890119899119891119899] (36)





119882119899 [11990901199100] = [119909119899minus1119910119899minus1]

119882119899 [119909119873119910119873] = [119909119899119910119899]

(37)



(38)


119909119899119894 = 119886119899119909119894 + 119890119899119910119899119894 = 119888119899119909119894 + 119889119899119910119894 + 119891119899




12059021198611198601 sdot 1 = 1205731 sdot 12059021198631198791198631 = 1205732 sdot 12059021198781198821198781 (40)



119867 =

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


1ℎ1119899


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

(41)



119895=1

ℎ119894119895(119894 119895 = 1 2 sdot sdot sdot 119899)

(42)



(119894 = 1 2 sdot sdot sdot 119899)(43)


119882 = 119882sum119899119894=1119882119894

(44)



120582max = 119899sum119894=1

(119867119882)119894119899119882119894

(119894 = 1 2 sdot sdot sdot 119899) (45)

119862119877 = 120582max minus 119899119877119868 (119899 minus 1) (46)




1198821 = 11988221205741 = 11988231205742 (47)





1199021 = 1205731 sdot 12057411199022 = 1205732 sdot 1205742 (48)

11 Experiment



0 2 4 6 8 10Time (s)

minus3

minus2

minus1

0

1

2

3

Vert

ical

acce

lera

tion

(ms

2 )

qFqGqEqH

qDqAqCqB


0 2 4 6 8 10Time (s)


0002004006008

Tire

dyn

amic

defl

ectio

n (m

)

qCqFqEqG

qDqAqHqB


0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)

qAqHqDqF

qEqCqGqB






0 2 4 6 8 10Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

Vert

ical

acce

lera

tion

(ms

2 )



0 2 4 6 8 10Time (s)


0002004006008

01

Tire

dyn

amic

defl

ectio

n (m

)



0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)



0 1 2 3 4 5 6 7 8 9 10Time (s)


minus5000

5001000150020002500

Activ

e con

trol f

orce

(N)


08

085

09

095

1

105

qF qG qEqH

qD

qA354

7098329771009

1033

1946

qC qB078

Passivesuspension

Active suspension






002300235

002400245

002500255

002600265

002700275

0028

qC qFqE

qG qD qA qHqB

620547

358122162202248


159


0032

0033

0034

0035

0036

0037

0038

897

262 233 146 118 089 059


029





12 Conclusions




Data Availability






Acknowledgments


References
















































Journal of












Journal of







Volume 2018






Volume 2018









4

5

6

7

8

9

10

11

12

lnN

(r)

lnN(r)=18652ln(1r)+12541R=09994

25 3 35 4 45 5 552ln(1r)



4 402 404 406 408 41001

0015002


minus006

minus004

minus002

0

002

004

006

Vert

ical

hei

ght (

m)






119877lowast = 1198771199021(119863minus1) (15)



(16)


ms

Ks1

Kt

Cs

Zs1

q

Zu1

mu

Vehicle body

Passive suspension

Wheel

Tire

(a)

ms

Ks2

Kt

u

Zs2

q

Zu2

mu

(b)











1198981199041199041 = 1198701199041 (1198851199061 minus 1198851199041) + 119862119878 (1199061 minus 1199041)(17)




1198831 = [1199041 1199061 1198851199041 1198851199061]1198791198832 = [1199042 1199062 1198851199042 1198851199062]119879

(19)



(20)

where

1198601 =[[[[[[[[[

minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

1198701199041119898119904119862119904119898119906

minus 119862119904119898119906

1198701199041119898119906

minus1198701199041 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]]

1198611 =[[[[[[[

011987011990511989811990600

]]]]]]]

1198602 =[[[[[[[[

0 0 minus1198701199042119898119904

11987011990421198981199040 0 1198701199042119898119906

minus1198701199042 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]

1198612 =[[[[[[[[

1119898119904

0minus 1119898119906

1198701199051198981199060 00 0

]]]]]]]]

(21)



(22)



where

1198621 = [[[[minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198631 = [[[001]]]

1198622 = [[[[0 0 minus1198701199042119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198632 = [[[[

1119898119904

00 00 1

]]]]

(24)



119869 = lim119879997888rarrinfin


(25)



119869 = lim119879997888rarrinfin

1119879 11988321198791198761198832 + 119880119879119877119880 + 21198832

119879119873119880119889119905 (26)

where

119876 =[[[[[[[[[[

0 0 0 00 0 0 00 0 1199022 + 11990231198702

11990421198982119904

minus1199022 minus 1199023119870211990421198982119904

0 0 minus1199022 minus 1199023119870211990421198982119904

1199021 + 1199022 + 1199023119870211990421198982119904

]]]]]]]]]]


119873 =[[[[[[[[[

0 00 0

minus11987011990421198982119904

011987011990421198982

119904

minus1199021

]]]]]]]]]

119877 = [[11990231198982119904

00 1199021

]]

(27)


1198751198602 + 1198601198792119875 minus (1198751198612 + 119873)119877minus1 (1198611198792119875 + 119873119879) + 119876 = 0 (28)

119870 = 119877minus1 (1198611198792119875 + 119873119879) (29)


119906 = minus119870 sdot 1198832 (30)




(119905) = minus212058711989900V119902 (119905) + 21205871198990radic119878119902 (1198990) V119882(119905) (31)






0 20 40 60 80 100 120 140 160Time t (s)

minus3

minus2

minus1

0

1

2

3

Road

elev

atio

n q

(m)






119867119860 =[[[[[[

1 2 912 1 319 13 1

]]]]]]

119867119861 =[[[[[[

1 3 913 1 319 13 1

]]]]]]

119867119862 =[[[[[[[

1 34 843 1 418 14 1

]]]]]]]

119867119863 =[[[[[[[

1 34 643 1 416 14 1

]]]]]]]



-




119867119864 = [[[[[

1 1 31 1 113 1 1

]]]]]

119867119865 = [[[[[

1 1 21 1 112 1 1

]]]]]

119867119866 = [[[[

1 1 11 1 151 5 1

]]]]

119867119867 = [[[[

1 1 11 1 161 6 1

]]]]

(32)



1199021 = 3000937141199022 = 11003214 (33)

10 Methodology



119882[11990910158401199101015840] = [119886 119887119888 119889] [

119909119910] + [

119890119891] (34)



119860 = [119886 119887119888 119889] = 119877 sdot 119879 sdot 119871


119879 = [ 1 tan120572tan120572 1 ]

119871 = [1198971 00 1198972]

(35)




119882119899 [119909119910] = [119886119899 0119888119899 119889119899] sdot [

119909119910] + [

119890119899119891119899] (36)





119882119899 [11990901199100] = [119909119899minus1119910119899minus1]

119882119899 [119909119873119910119873] = [119909119899119910119899]

(37)



(38)


119909119899119894 = 119886119899119909119894 + 119890119899119910119899119894 = 119888119899119909119894 + 119889119899119910119894 + 119891119899




12059021198611198601 sdot 1 = 1205731 sdot 12059021198631198791198631 = 1205732 sdot 12059021198781198821198781 (40)



119867 =

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


1ℎ1119899


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

(41)



119895=1

ℎ119894119895(119894 119895 = 1 2 sdot sdot sdot 119899)

(42)



(119894 = 1 2 sdot sdot sdot 119899)(43)


119882 = 119882sum119899119894=1119882119894

(44)



120582max = 119899sum119894=1

(119867119882)119894119899119882119894

(119894 = 1 2 sdot sdot sdot 119899) (45)

119862119877 = 120582max minus 119899119877119868 (119899 minus 1) (46)




1198821 = 11988221205741 = 11988231205742 (47)





1199021 = 1205731 sdot 12057411199022 = 1205732 sdot 1205742 (48)

11 Experiment



0 2 4 6 8 10Time (s)

minus3

minus2

minus1

0

1

2

3

Vert

ical

acce

lera

tion

(ms

2 )

qFqGqEqH

qDqAqCqB


0 2 4 6 8 10Time (s)


0002004006008

Tire

dyn

amic

defl

ectio

n (m

)

qCqFqEqG

qDqAqHqB


0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)

qAqHqDqF

qEqCqGqB






0 2 4 6 8 10Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

Vert

ical

acce

lera

tion

(ms

2 )



0 2 4 6 8 10Time (s)


0002004006008

01

Tire

dyn

amic

defl

ectio

n (m

)



0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)



0 1 2 3 4 5 6 7 8 9 10Time (s)


minus5000

5001000150020002500

Activ

e con

trol f

orce

(N)


08

085

09

095

1

105

qF qG qEqH

qD

qA354

7098329771009

1033

1946

qC qB078

Passivesuspension

Active suspension






002300235

002400245

002500255

002600265

002700275

0028

qC qFqE

qG qD qA qHqB

620547

358122162202248


159


0032

0033

0034

0035

0036

0037

0038

897

262 233 146 118 089 059


029





12 Conclusions




Data Availability






Acknowledgments


References
















































Journal of












Journal of







Volume 2018






Volume 2018









ms

Ks1

Kt

Cs

Zs1

q

Zu1

mu

Vehicle body

Passive suspension

Wheel

Tire

(a)

ms

Ks2

Kt

u

Zs2

q

Zu2

mu

(b)











1198981199041199041 = 1198701199041 (1198851199061 minus 1198851199041) + 119862119878 (1199061 minus 1199041)(17)




1198831 = [1199041 1199061 1198851199041 1198851199061]1198791198832 = [1199042 1199062 1198851199042 1198851199062]119879

(19)



(20)

where

1198601 =[[[[[[[[[

minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

1198701199041119898119904119862119904119898119906

minus 119862119904119898119906

1198701199041119898119906

minus1198701199041 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]]

1198611 =[[[[[[[

011987011990511989811990600

]]]]]]]

1198602 =[[[[[[[[

0 0 minus1198701199042119898119904

11987011990421198981199040 0 1198701199042119898119906

minus1198701199042 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]

1198612 =[[[[[[[[

1119898119904

0minus 1119898119906

1198701199051198981199060 00 0

]]]]]]]]

(21)



(22)



where

1198621 = [[[[minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198631 = [[[001]]]

1198622 = [[[[0 0 minus1198701199042119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198632 = [[[[

1119898119904

00 00 1

]]]]

(24)



119869 = lim119879997888rarrinfin


(25)



119869 = lim119879997888rarrinfin

1119879 11988321198791198761198832 + 119880119879119877119880 + 21198832

119879119873119880119889119905 (26)

where

119876 =[[[[[[[[[[

0 0 0 00 0 0 00 0 1199022 + 11990231198702

11990421198982119904

minus1199022 minus 1199023119870211990421198982119904

0 0 minus1199022 minus 1199023119870211990421198982119904

1199021 + 1199022 + 1199023119870211990421198982119904

]]]]]]]]]]


119873 =[[[[[[[[[

0 00 0

minus11987011990421198982119904

011987011990421198982

119904

minus1199021

]]]]]]]]]

119877 = [[11990231198982119904

00 1199021

]]

(27)


1198751198602 + 1198601198792119875 minus (1198751198612 + 119873)119877minus1 (1198611198792119875 + 119873119879) + 119876 = 0 (28)

119870 = 119877minus1 (1198611198792119875 + 119873119879) (29)


119906 = minus119870 sdot 1198832 (30)




(119905) = minus212058711989900V119902 (119905) + 21205871198990radic119878119902 (1198990) V119882(119905) (31)






0 20 40 60 80 100 120 140 160Time t (s)

minus3

minus2

minus1

0

1

2

3

Road

elev

atio

n q

(m)






119867119860 =[[[[[[

1 2 912 1 319 13 1

]]]]]]

119867119861 =[[[[[[

1 3 913 1 319 13 1

]]]]]]

119867119862 =[[[[[[[

1 34 843 1 418 14 1

]]]]]]]

119867119863 =[[[[[[[

1 34 643 1 416 14 1

]]]]]]]



-




119867119864 = [[[[[

1 1 31 1 113 1 1

]]]]]

119867119865 = [[[[[

1 1 21 1 112 1 1

]]]]]

119867119866 = [[[[

1 1 11 1 151 5 1

]]]]

119867119867 = [[[[

1 1 11 1 161 6 1

]]]]

(32)



1199021 = 3000937141199022 = 11003214 (33)

10 Methodology



119882[11990910158401199101015840] = [119886 119887119888 119889] [

119909119910] + [

119890119891] (34)



119860 = [119886 119887119888 119889] = 119877 sdot 119879 sdot 119871


119879 = [ 1 tan120572tan120572 1 ]

119871 = [1198971 00 1198972]

(35)




119882119899 [119909119910] = [119886119899 0119888119899 119889119899] sdot [

119909119910] + [

119890119899119891119899] (36)





119882119899 [11990901199100] = [119909119899minus1119910119899minus1]

119882119899 [119909119873119910119873] = [119909119899119910119899]

(37)



(38)


119909119899119894 = 119886119899119909119894 + 119890119899119910119899119894 = 119888119899119909119894 + 119889119899119910119894 + 119891119899




12059021198611198601 sdot 1 = 1205731 sdot 12059021198631198791198631 = 1205732 sdot 12059021198781198821198781 (40)



119867 =

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


1ℎ1119899


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

(41)



119895=1

ℎ119894119895(119894 119895 = 1 2 sdot sdot sdot 119899)

(42)



(119894 = 1 2 sdot sdot sdot 119899)(43)


119882 = 119882sum119899119894=1119882119894

(44)



120582max = 119899sum119894=1

(119867119882)119894119899119882119894

(119894 = 1 2 sdot sdot sdot 119899) (45)

119862119877 = 120582max minus 119899119877119868 (119899 minus 1) (46)




1198821 = 11988221205741 = 11988231205742 (47)





1199021 = 1205731 sdot 12057411199022 = 1205732 sdot 1205742 (48)

11 Experiment



0 2 4 6 8 10Time (s)

minus3

minus2

minus1

0

1

2

3

Vert

ical

acce

lera

tion

(ms

2 )

qFqGqEqH

qDqAqCqB


0 2 4 6 8 10Time (s)


0002004006008

Tire

dyn

amic

defl

ectio

n (m

)

qCqFqEqG

qDqAqHqB


0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)

qAqHqDqF

qEqCqGqB






0 2 4 6 8 10Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

Vert

ical

acce

lera

tion

(ms

2 )



0 2 4 6 8 10Time (s)


0002004006008

01

Tire

dyn

amic

defl

ectio

n (m

)



0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)



0 1 2 3 4 5 6 7 8 9 10Time (s)


minus5000

5001000150020002500

Activ

e con

trol f

orce

(N)


08

085

09

095

1

105

qF qG qEqH

qD

qA354

7098329771009

1033

1946

qC qB078

Passivesuspension

Active suspension






002300235

002400245

002500255

002600265

002700275

0028

qC qFqE

qG qD qA qHqB

620547

358122162202248


159


0032

0033

0034

0035

0036

0037

0038

897

262 233 146 118 089 059


029





12 Conclusions




Data Availability






Acknowledgments


References
















































Journal of












Journal of







Volume 2018






Volume 2018










1198831 = [1199041 1199061 1198851199041 1198851199061]1198791198832 = [1199042 1199062 1198851199042 1198851199062]119879

(19)



(20)

where

1198601 =[[[[[[[[[

minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

1198701199041119898119904119862119904119898119906

minus 119862119904119898119906

1198701199041119898119906

minus1198701199041 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]]

1198611 =[[[[[[[

011987011990511989811990600

]]]]]]]

1198602 =[[[[[[[[

0 0 minus1198701199042119898119904

11987011990421198981199040 0 1198701199042119898119906

minus1198701199042 + 1198701199051198981199061 0 0 00 1 0 0

]]]]]]]]

1198612 =[[[[[[[[

1119898119904

0minus 1119898119906

1198701199051198981199060 00 0

]]]]]]]]

(21)



(22)



where

1198621 = [[[[minus119862119904119898119904

119862119904119898119904

minus1198701199041119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198631 = [[[001]]]

1198622 = [[[[0 0 minus1198701199042119898119904

11987011990411198981199040 0 minus1 10 0 0 minus1

]]]]

1198632 = [[[[

1119898119904

00 00 1

]]]]

(24)



119869 = lim119879997888rarrinfin


(25)



119869 = lim119879997888rarrinfin

1119879 11988321198791198761198832 + 119880119879119877119880 + 21198832

119879119873119880119889119905 (26)

where

119876 =[[[[[[[[[[

0 0 0 00 0 0 00 0 1199022 + 11990231198702

11990421198982119904

minus1199022 minus 1199023119870211990421198982119904

0 0 minus1199022 minus 1199023119870211990421198982119904

1199021 + 1199022 + 1199023119870211990421198982119904

]]]]]]]]]]


119873 =[[[[[[[[[

0 00 0

minus11987011990421198982119904

011987011990421198982

119904

minus1199021

]]]]]]]]]

119877 = [[11990231198982119904

00 1199021

]]

(27)


1198751198602 + 1198601198792119875 minus (1198751198612 + 119873)119877minus1 (1198611198792119875 + 119873119879) + 119876 = 0 (28)

119870 = 119877minus1 (1198611198792119875 + 119873119879) (29)


119906 = minus119870 sdot 1198832 (30)




(119905) = minus212058711989900V119902 (119905) + 21205871198990radic119878119902 (1198990) V119882(119905) (31)






0 20 40 60 80 100 120 140 160Time t (s)

minus3

minus2

minus1

0

1

2

3

Road

elev

atio

n q

(m)






119867119860 =[[[[[[

1 2 912 1 319 13 1

]]]]]]

119867119861 =[[[[[[

1 3 913 1 319 13 1

]]]]]]

119867119862 =[[[[[[[

1 34 843 1 418 14 1

]]]]]]]

119867119863 =[[[[[[[

1 34 643 1 416 14 1

]]]]]]]



-




119867119864 = [[[[[

1 1 31 1 113 1 1

]]]]]

119867119865 = [[[[[

1 1 21 1 112 1 1

]]]]]

119867119866 = [[[[

1 1 11 1 151 5 1

]]]]

119867119867 = [[[[

1 1 11 1 161 6 1

]]]]

(32)



1199021 = 3000937141199022 = 11003214 (33)

10 Methodology



119882[11990910158401199101015840] = [119886 119887119888 119889] [

119909119910] + [

119890119891] (34)



119860 = [119886 119887119888 119889] = 119877 sdot 119879 sdot 119871


119879 = [ 1 tan120572tan120572 1 ]

119871 = [1198971 00 1198972]

(35)




119882119899 [119909119910] = [119886119899 0119888119899 119889119899] sdot [

119909119910] + [

119890119899119891119899] (36)





119882119899 [11990901199100] = [119909119899minus1119910119899minus1]

119882119899 [119909119873119910119873] = [119909119899119910119899]

(37)



(38)


119909119899119894 = 119886119899119909119894 + 119890119899119910119899119894 = 119888119899119909119894 + 119889119899119910119894 + 119891119899




12059021198611198601 sdot 1 = 1205731 sdot 12059021198631198791198631 = 1205732 sdot 12059021198781198821198781 (40)



119867 =

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


1ℎ1119899


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

(41)



119895=1

ℎ119894119895(119894 119895 = 1 2 sdot sdot sdot 119899)

(42)



(119894 = 1 2 sdot sdot sdot 119899)(43)


119882 = 119882sum119899119894=1119882119894

(44)



120582max = 119899sum119894=1

(119867119882)119894119899119882119894

(119894 = 1 2 sdot sdot sdot 119899) (45)

119862119877 = 120582max minus 119899119877119868 (119899 minus 1) (46)




1198821 = 11988221205741 = 11988231205742 (47)





1199021 = 1205731 sdot 12057411199022 = 1205732 sdot 1205742 (48)

11 Experiment



0 2 4 6 8 10Time (s)

minus3

minus2

minus1

0

1

2

3

Vert

ical

acce

lera

tion

(ms

2 )

qFqGqEqH

qDqAqCqB


0 2 4 6 8 10Time (s)


0002004006008

Tire

dyn

amic

defl

ectio

n (m

)

qCqFqEqG

qDqAqHqB


0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)

qAqHqDqF

qEqCqGqB






0 2 4 6 8 10Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

Vert

ical

acce

lera

tion

(ms

2 )



0 2 4 6 8 10Time (s)


0002004006008

01

Tire

dyn

amic

defl

ectio

n (m

)



0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)



0 1 2 3 4 5 6 7 8 9 10Time (s)


minus5000

5001000150020002500

Activ

e con

trol f

orce

(N)


08

085

09

095

1

105

qF qG qEqH

qD

qA354

7098329771009

1033

1946

qC qB078

Passivesuspension

Active suspension






002300235

002400245

002500255

002600265

002700275

0028

qC qFqE

qG qD qA qHqB

620547

358122162202248


159


0032

0033

0034

0035

0036

0037

0038

897

262 233 146 118 089 059


029





12 Conclusions




Data Availability






Acknowledgments


References
















































Journal of












Journal of







Volume 2018






Volume 2018









119873 =[[[[[[[[[

0 00 0

minus11987011990421198982119904

011987011990421198982

119904

minus1199021

]]]]]]]]]

119877 = [[11990231198982119904

00 1199021

]]

(27)


1198751198602 + 1198601198792119875 minus (1198751198612 + 119873)119877minus1 (1198611198792119875 + 119873119879) + 119876 = 0 (28)

119870 = 119877minus1 (1198611198792119875 + 119873119879) (29)


119906 = minus119870 sdot 1198832 (30)




(119905) = minus212058711989900V119902 (119905) + 21205871198990radic119878119902 (1198990) V119882(119905) (31)






0 20 40 60 80 100 120 140 160Time t (s)

minus3

minus2

minus1

0

1

2

3

Road

elev

atio

n q

(m)






119867119860 =[[[[[[

1 2 912 1 319 13 1

]]]]]]

119867119861 =[[[[[[

1 3 913 1 319 13 1

]]]]]]

119867119862 =[[[[[[[

1 34 843 1 418 14 1

]]]]]]]

119867119863 =[[[[[[[

1 34 643 1 416 14 1

]]]]]]]



-




119867119864 = [[[[[

1 1 31 1 113 1 1

]]]]]

119867119865 = [[[[[

1 1 21 1 112 1 1

]]]]]

119867119866 = [[[[

1 1 11 1 151 5 1

]]]]

119867119867 = [[[[

1 1 11 1 161 6 1

]]]]

(32)



1199021 = 3000937141199022 = 11003214 (33)

10 Methodology



119882[11990910158401199101015840] = [119886 119887119888 119889] [

119909119910] + [

119890119891] (34)



119860 = [119886 119887119888 119889] = 119877 sdot 119879 sdot 119871


119879 = [ 1 tan120572tan120572 1 ]

119871 = [1198971 00 1198972]

(35)




119882119899 [119909119910] = [119886119899 0119888119899 119889119899] sdot [

119909119910] + [

119890119899119891119899] (36)





119882119899 [11990901199100] = [119909119899minus1119910119899minus1]

119882119899 [119909119873119910119873] = [119909119899119910119899]

(37)



(38)


119909119899119894 = 119886119899119909119894 + 119890119899119910119899119894 = 119888119899119909119894 + 119889119899119910119894 + 119891119899




12059021198611198601 sdot 1 = 1205731 sdot 12059021198631198791198631 = 1205732 sdot 12059021198781198821198781 (40)



119867 =

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


1ℎ1119899


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

(41)



119895=1

ℎ119894119895(119894 119895 = 1 2 sdot sdot sdot 119899)

(42)



(119894 = 1 2 sdot sdot sdot 119899)(43)


119882 = 119882sum119899119894=1119882119894

(44)



120582max = 119899sum119894=1

(119867119882)119894119899119882119894

(119894 = 1 2 sdot sdot sdot 119899) (45)

119862119877 = 120582max minus 119899119877119868 (119899 minus 1) (46)




1198821 = 11988221205741 = 11988231205742 (47)





1199021 = 1205731 sdot 12057411199022 = 1205732 sdot 1205742 (48)

11 Experiment



0 2 4 6 8 10Time (s)

minus3

minus2

minus1

0

1

2

3

Vert

ical

acce

lera

tion

(ms

2 )

qFqGqEqH

qDqAqCqB


0 2 4 6 8 10Time (s)


0002004006008

Tire

dyn

amic

defl

ectio

n (m

)

qCqFqEqG

qDqAqHqB


0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)

qAqHqDqF

qEqCqGqB






0 2 4 6 8 10Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

Vert

ical

acce

lera

tion

(ms

2 )



0 2 4 6 8 10Time (s)


0002004006008

01

Tire

dyn

amic

defl

ectio

n (m

)



0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)



0 1 2 3 4 5 6 7 8 9 10Time (s)


minus5000

5001000150020002500

Activ

e con

trol f

orce

(N)


08

085

09

095

1

105

qF qG qEqH

qD

qA354

7098329771009

1033

1946

qC qB078

Passivesuspension

Active suspension






002300235

002400245

002500255

002600265

002700275

0028

qC qFqE

qG qD qA qHqB

620547

358122162202248


159


0032

0033

0034

0035

0036

0037

0038

897

262 233 146 118 089 059


029





12 Conclusions




Data Availability






Acknowledgments


References
















































Journal of












Journal of







Volume 2018






Volume 2018










-




119867119864 = [[[[[

1 1 31 1 113 1 1

]]]]]

119867119865 = [[[[[

1 1 21 1 112 1 1

]]]]]

119867119866 = [[[[

1 1 11 1 151 5 1

]]]]

119867119867 = [[[[

1 1 11 1 161 6 1

]]]]

(32)



1199021 = 3000937141199022 = 11003214 (33)

10 Methodology



119882[11990910158401199101015840] = [119886 119887119888 119889] [

119909119910] + [

119890119891] (34)



119860 = [119886 119887119888 119889] = 119877 sdot 119879 sdot 119871


119879 = [ 1 tan120572tan120572 1 ]

119871 = [1198971 00 1198972]

(35)




119882119899 [119909119910] = [119886119899 0119888119899 119889119899] sdot [

119909119910] + [

119890119899119891119899] (36)





119882119899 [11990901199100] = [119909119899minus1119910119899minus1]

119882119899 [119909119873119910119873] = [119909119899119910119899]

(37)



(38)


119909119899119894 = 119886119899119909119894 + 119890119899119910119899119894 = 119888119899119909119894 + 119889119899119910119894 + 119891119899




12059021198611198601 sdot 1 = 1205731 sdot 12059021198631198791198631 = 1205732 sdot 12059021198781198821198781 (40)



119867 =

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


1ℎ1119899


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

(41)



119895=1

ℎ119894119895(119894 119895 = 1 2 sdot sdot sdot 119899)

(42)



(119894 = 1 2 sdot sdot sdot 119899)(43)


119882 = 119882sum119899119894=1119882119894

(44)



120582max = 119899sum119894=1

(119867119882)119894119899119882119894

(119894 = 1 2 sdot sdot sdot 119899) (45)

119862119877 = 120582max minus 119899119877119868 (119899 minus 1) (46)




1198821 = 11988221205741 = 11988231205742 (47)





1199021 = 1205731 sdot 12057411199022 = 1205732 sdot 1205742 (48)

11 Experiment



0 2 4 6 8 10Time (s)

minus3

minus2

minus1

0

1

2

3

Vert

ical

acce

lera

tion

(ms

2 )

qFqGqEqH

qDqAqCqB


0 2 4 6 8 10Time (s)


0002004006008

Tire

dyn

amic

defl

ectio

n (m

)

qCqFqEqG

qDqAqHqB


0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)

qAqHqDqF

qEqCqGqB






0 2 4 6 8 10Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

Vert

ical

acce

lera

tion

(ms

2 )



0 2 4 6 8 10Time (s)


0002004006008

01

Tire

dyn

amic

defl

ectio

n (m

)



0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)



0 1 2 3 4 5 6 7 8 9 10Time (s)


minus5000

5001000150020002500

Activ

e con

trol f

orce

(N)


08

085

09

095

1

105

qF qG qEqH

qD

qA354

7098329771009

1033

1946

qC qB078

Passivesuspension

Active suspension






002300235

002400245

002500255

002600265

002700275

0028

qC qFqE

qG qD qA qHqB

620547

358122162202248


159


0032

0033

0034

0035

0036

0037

0038

897

262 233 146 118 089 059


029





12 Conclusions




Data Availability






Acknowledgments


References
















































Journal of












Journal of







Volume 2018






Volume 2018












119882119899 [11990901199100] = [119909119899minus1119910119899minus1]

119882119899 [119909119873119910119873] = [119909119899119910119899]

(37)



(38)


119909119899119894 = 119886119899119909119894 + 119890119899119910119899119894 = 119888119899119909119894 + 119889119899119910119894 + 119891119899




12059021198611198601 sdot 1 = 1205731 sdot 12059021198631198791198631 = 1205732 sdot 12059021198781198821198781 (40)



119867 =

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


1ℎ1119899


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

(41)



119895=1

ℎ119894119895(119894 119895 = 1 2 sdot sdot sdot 119899)

(42)



(119894 = 1 2 sdot sdot sdot 119899)(43)


119882 = 119882sum119899119894=1119882119894

(44)



120582max = 119899sum119894=1

(119867119882)119894119899119882119894

(119894 = 1 2 sdot sdot sdot 119899) (45)

119862119877 = 120582max minus 119899119877119868 (119899 minus 1) (46)




1198821 = 11988221205741 = 11988231205742 (47)





1199021 = 1205731 sdot 12057411199022 = 1205732 sdot 1205742 (48)

11 Experiment



0 2 4 6 8 10Time (s)

minus3

minus2

minus1

0

1

2

3

Vert

ical

acce

lera

tion

(ms

2 )

qFqGqEqH

qDqAqCqB


0 2 4 6 8 10Time (s)


0002004006008

Tire

dyn

amic

defl

ectio

n (m

)

qCqFqEqG

qDqAqHqB


0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)

qAqHqDqF

qEqCqGqB






0 2 4 6 8 10Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

Vert

ical

acce

lera

tion

(ms

2 )



0 2 4 6 8 10Time (s)


0002004006008

01

Tire

dyn

amic

defl

ectio

n (m

)



0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)



0 1 2 3 4 5 6 7 8 9 10Time (s)


minus5000

5001000150020002500

Activ

e con

trol f

orce

(N)


08

085

09

095

1

105

qF qG qEqH

qD

qA354

7098329771009

1033

1946

qC qB078

Passivesuspension

Active suspension






002300235

002400245

002500255

002600265

002700275

0028

qC qFqE

qG qD qA qHqB

620547

358122162202248


159


0032

0033

0034

0035

0036

0037

0038

897

262 233 146 118 089 059


029





12 Conclusions




Data Availability






Acknowledgments


References
















































Journal of












Journal of







Volume 2018






Volume 2018












1199021 = 1205731 sdot 12057411199022 = 1205732 sdot 1205742 (48)

11 Experiment



0 2 4 6 8 10Time (s)

minus3

minus2

minus1

0

1

2

3

Vert

ical

acce

lera

tion

(ms

2 )

qFqGqEqH

qDqAqCqB


0 2 4 6 8 10Time (s)


0002004006008

Tire

dyn

amic

defl

ectio

n (m

)

qCqFqEqG

qDqAqHqB


0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)

qAqHqDqF

qEqCqGqB






0 2 4 6 8 10Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

Vert

ical

acce

lera

tion

(ms

2 )



0 2 4 6 8 10Time (s)


0002004006008

01

Tire

dyn

amic

defl

ectio

n (m

)



0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)



0 1 2 3 4 5 6 7 8 9 10Time (s)


minus5000

5001000150020002500

Activ

e con

trol f

orce

(N)


08

085

09

095

1

105

qF qG qEqH

qD

qA354

7098329771009

1033

1946

qC qB078

Passivesuspension

Active suspension






002300235

002400245

002500255

002600265

002700275

0028

qC qFqE

qG qD qA qHqB

620547

358122162202248


159


0032

0033

0034

0035

0036

0037

0038

897

262 233 146 118 089 059


029





12 Conclusions




Data Availability






Acknowledgments


References
















































Journal of












Journal of







Volume 2018






Volume 2018











0 2 4 6 8 10Time (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

Vert

ical

acce

lera

tion

(ms

2 )



0 2 4 6 8 10Time (s)


0002004006008

01

Tire

dyn

amic

defl

ectio

n (m

)



0 2 4 6 8 10Time (s)

minus01

minus005

0

005

01

Susp

ensio

n dy

nam

ic d

eflec

tion

(m)



0 1 2 3 4 5 6 7 8 9 10Time (s)


minus5000

5001000150020002500

Activ

e con

trol f

orce

(N)


08

085

09

095

1

105

qF qG qEqH

qD

qA354

7098329771009

1033

1946

qC qB078

Passivesuspension

Active suspension






002300235

002400245

002500255

002600265

002700275

0028

qC qFqE

qG qD qA qHqB

620547

358122162202248


159


0032

0033

0034

0035

0036

0037

0038

897

262 233 146 118 089 059


029





12 Conclusions




Data Availability






Acknowledgments


References
















































Journal of












Journal of







Volume 2018






Volume 2018









002300235

002400245

002500255

002600265

002700275

0028

qC qFqE

qG qD qA qHqB

620547

358122162202248


159


0032

0033

0034

0035

0036

0037

0038

897

262 233 146 118 089 059


029





12 Conclusions




Data Availability






Acknowledgments


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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