Research ArticleDriver Steering Control and Full Vehicle DynamicsStudy Based on a Nonlinear Three-Directional CoupledHeavy-Duty Vehicle Model
S H Li and J Y Ren
School of Mechanical Engineering Shijiazhuang Tiedao University Shijiazhuang 050043 China
Correspondence should be addressed to S H Li lshsjz163com
Received 14 August 2014 Accepted 27 November 2014 Published 24 December 2014
Academic Editor Hongbin Zhang
Copyright copy 2014 S H Li and J Y RenThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Under complicated driving situations such as cornering brake lane change or barrier avoidance the vertical lateral andlongitudinal dynamics of a vehicle are coupled and interacted obviously This work aims to propose the suitable vehicle and drivermodels for researching full vehicle dynamics in complicated conditions A nonlinear three-directional coupled lumped parameters(TCLP) model of a heavy-duty vehicle considering the nonlinearity of suspension damping and tire stiffness is built firstly Then amodified preview driver model with nonlinear time delay is proposed and connected to the TCLP model to form a driver-vehicleclosed-loop systemThe presented driver-vehicle closed-loop system is evaluated during a double-lane change and compared withtest data traditional handling stability vehicle model linear full vehicle model and other driver models The results show that thenew driver model has better lane keeping performances than the other two driver models In addition the effects of driver modelparameters on lane keeping performances handling stability ride comfort and roll stability are discussed The models and resultsof this paper are useful to enhance understanding the effects of driver behaviour on full vehicle dynamics
1 Introduction
Due to rapid development of highway transportation theresearch in the field of vehicle dynamics and control systemshas attracted many scholarsrsquo attention In order to simulatedifferent driving scenarios and reduce test cost a lot ofdriver steering control models have been proposed Howeveralthough automobile and driver form a couple the aim andreason of the present investigations are often focused eitheron the vehicle or on the driver Plochl and Edelmann [1]and MacAdam [2] respectively gave a detailed review ofdriver models and their application in automobile dynamicsAmong many different driver models the single- or mul-tipoint preview driver models play an important role Guoand Guan [3 4] proposed a single-point preview optimalcurvaturemodel that has been widely used Sharp [5] added alow-pass road excitation filter to a single-point time-invariantoptimal preview control and calculated the optimal controlvariable value based on the state space equations Legouiset al [6] proposed two driver models with fixed gains and
linear or nonlinear time delay Liu et al [7 8] used these twomodels to study the nonlinear lateral dynamics of a 2DOFvehicle model Liu et al [9] established a fuzzy-PID driversteering model for a truck and adjusted the gains by fuzzyrule Besides above single-point preview models some mul-tipoint preview models are also widely used One of the mostwell-known multipoint preview models is MacAdamrsquos drivermodel which is based on the preview control framework forlinear systems and calculates the average position deviationduring the preview time [10] Ungoren and Peng [11] pro-posed a generalized version of the MacAdamrsquos model thatcan simulate steering actions of human drivers with differentdriving styles Chatzikomis and Spentzas [12] presented acombined longitudinal-lateral controller bymodifying Sharprsquomultipoint preview model [13] Pick and Cole [14] extendeda multipoint preview path-following controller to includethe muscle reflex control loop and steering torque feedbackIt should be noted that most substantial progress in drivermodeling still lies in a linear world and focuses on two-axlecab car However the investigation on steering control of the
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 352374 16 pageshttpdxdoiorg1011552014352374
2 Mathematical Problems in Engineering
three-axle heavy vehicles with nonlinear suspension and tireproperty is also necessary and has attracted some scholarsrsquoattention The UMTRIrsquos yaw-roll truck model described in[15] considered nonlinearities of suspension and tire corner-ing force as tabular functions and was further adapted toinclude the road roughness effect [16] but did not set up thedifference equations for the longitudinalmotion andwas onlysuitable to constant-speed driving conditions
In addition the present driver models are often appliedto evaluate lateral and yaw dynamic characteristics based onthe handling stability vehicle model and the results aboutthe effect of driver behavior on vehicle vertical lateral andlongitudinal dynamics are still inadequate In fact the roadsurface offers not only lateral and longitudinal forces to avehicle but also vertical forces to suspension especially undercomplicated driving situations such as lane change corner-ing or barrier avoidance In these cases the vertical rolland pitch dynamics of a vehicle are coupled with the lateraland yaw motions obviously Due to bigger inertia longerwheelbase and higher roll center compared with cab cars theheavy-duty vehicles show poorer stability and greater three-directional coupling effects when entering a turn or lanechange Hence it is quite necessary to build a three-direc-tional coupled vehicle model and research the effect of driversteering on full vehicle dynamics
This work applies a modified preview driver model in fullvehicle dynamics simulation of a heavy vehicle and investi-gates the effect of driver parameters on both lane keeping per-formances and full vehicle dynamic characteristics Firstly wegive a description on how tomodel the three-directional cou-pling effect and nonlinear properties of suspension and tire ina heavy-duty vehicle Then a modified single-point previewdriver control model with nonlinear time delay consideringvehicle speed variation is proposed and a driver-vehicleclosed-loop system is constructed The full vehicle dynamicresponses of the closed-loop system in double-lane changeare obtained by numerical integration The validity of thisclosed-loop system is verified by comparison with the resultsof a field test the traditional steering stability vehicle modelthe linear vehicle model and other driver control modelsFinally the effects of driver model parameters such as vehiclerunning speed time delay preview distance and permitposition error on the path-following behavior and three-directional dynamics of the vehicle are also analyzed
2 Model Building
21 The Nonlinear Three-Directional Coupled Vehicle ModelA nonlinear TCLP model for a three-axial heavy-duty truckwith 23-DOF is presented as shown in Figure 1 The vehicleis front wheel steered and rear wheel driven 119911
119888 120579119888 120601119888 119911119887
120579 120601 stand for the vertical pitch and roll displacementsof driver cab and vehicle body 120595 119909 119910 represent the yawlongitudinal and lateral displacements of the full vehicle119911119906119894 120601119906119894(119894 = 1ndash3) denote the vertical and roll displacements
of three wheel axles 1205791199011and 1205791199012are the pitch angles of the left
or right balancing pole on rear suspension 119872 119872119887 and 119872
119888
denote themasses of full vehicle vehicle body and driver cabrespectively The origin of vehicle coordinate system (119909 119910 119911)
is located in the intersection between the roll axle and thevertical line passing vehicle center of gravity ℎ
119904and 119897119904are the
vertical and longitudinal distance from the sprung mass cen-ter of gravity to the coordinate origin 119889
1199051and 1198891199052are the front
and rear wheel track width 1198891199041and 119889
1199042are the lateral dis-
tance between left and right springs on front or rear suspen-sion
The movements of the heavy-duty vehicle are coupledwith each other greatly Equations (1) give the longitudinallateral and yaw dynamics of the full vehicle Equations (2)give the vertical roll and pitch dynamics of the sprung mass
119872( minus 119910) + 119872119887119887
120579 + 119872
119887ℎ119904
120579 minus 119872
119887119897119904(2
+1205792
)
= (11986511990511990911
+ 11986511990511990912
) cos 120575 minus (11986511990511991011
+ 11986511990511991012
) sin 120575
+
3
sum
119894=2
2
sum
119895=1
119865119905119909119894119895
119872 ( 119910 + ) minus 119872119887119887 minus 119872
119887ℎ119904
= (11986511990511990911
+ 11986511990511990912
) sin 120575 + (11986511990511991011
+ 11986511990511991012
) cos 120575
+
3
sum
119894=2
2
sum
119895=1
119865119905119910119894119895
119868119911 + 2119868
119911119887120579 minus (119868
119887119909119911+119872119887119897119904ℎ119904) minus 119872
(3)For this heavy-duty vehicle two hydraulic dampers are
fixed on the left and right front suspensions and the tandembalanced suspension does not have any shock absorber Inorder to represent the frictional property of leaf spring thedamping forces of tandem balanced suspension are modeledlinearly The suspension forces between middle or rear axleand vehicle body are given by
11986511990421
= 11987011990421(119911119887+ (120579 minus 120579
0) 1198972minus
12057911990111198973
2
minus 1199111199062+
(120593 minus 1205931199062) 1198891199042
2
)
+ 11986211990421(119887+
1205791198972minus
12057911990111198973
2
minus 1199062+
( minus 1199062) 1198891199042
2
)
11986511990422
= 11987011990422(119911119887+ (120579 minus 120579
0) 1198972minus
12057911990121198973
2
minus 1199111199062minus
(120593 minus 1205931199062) 1198891199042
2
)
+ 11986211990422(119887+
1205791198972minus
12057911990121198973
2
minus 1199062minus
( minus 1199062) 1198891199042
2
)
11986511990431
= 11987011990431(119911119887+ (120579 minus 120579
0) 1198972+
12057911990111198973
2
minus 1199111199063+
(120593 minus 1205931199063) 1198891199043
2
)
+ 11986211990431(119887+
1205791198972+
12057911990111198973
2
minus 1199063+
( minus 1199063) 1198891199043
2
)
11986511990432
= 11987011990432(119911119887+ (120579 minus 120579
0) 1198972+
12057911990121198973
2
minus 1199111199063minus
(120593 minus 1205931199063) 1198891199043
2
)
+ 11986211990432(119887+
1205791198972+
12057911990121198973
2
minus 1199063minus
( minus 1199063) 1198891199043
2
)
(4)
Since hydraulic dampers show obvious nonlinearitymany dynamic models for shock absorber have been pro-posed amongwhich the fittedmodel is quite suitable tomod-eling the ascertained shock absorber but needs a large amountof experimental work [17 18] In this work the dynamic prop-erty of the damper on front suspension ismeasured byHT-911testing machine under sinusoidal excitation Since the inher-ence frequency of the vehicle body is from 1Hz to 25Hzfour excitation frequencies are selected as 1Hz 15Hz 2Hzand 25Hz Limited by the machinersquos tonnage the excitationamplitude is chosen as 10mm Figure 2 shows the measuredforce-velocity curves Since the hysteresis loops of dampingforce depend on the excitation frequency and amplitudegreatly the parameters of hysteresis model under randomexcitations are difficult to identify Hence a nonlinear seg-mented model describing the damperrsquos scheme framework isproposed here
119865119889=
11986501sgn (V
119889) V
119889gt Vlim 1
119862 (1 + 120573 sgn (V119889)) V119889
1003816100381610038161003816V119889
1003816100381610038161003816
119899 Vlim 2 le V119889le Vlim 1
11986502sgn (V
119889) V
119889lt Vlim 2
(5)
where V119889 119862 120573 and 119899 are the relative velocity of cylinder
and plunger the damping coefficient the asymmetry ratioand the exponent respectively 119865
01 11986502 Vlim 1 and Vlim 2 are
the damping force and relative velocity when the damperreaching saturation in tension or compression process
Parameters in model (5) fitted to the measured data are119862 = 30893 120573 = 056 119899 = 016 119865
01= 4119N 119865
02= 726N
Vlim 1 = 012ms and Vlim 2 = 008ms The damping forcecurve obtained from the theoretical damper model is shownas the thick solid line in Figure 2 It can be seen that thepresented damping forcemodel is able to describe the schemeframework and saturation property of the damper Thoughmodel (5) neglects the damperrsquos hysteresis characteristics itis simple and accurate enough for numerical simulation
Using model (5) to calculate the damping force the frontsuspension forces are expressed by
11986511990411
= 11987011990411(119911119887minus (120579 minus 120579
0) 1198971minus 1199111199061+
(120593 minus 1205931199061) 1198891199041
2
) + 11986511988911
11986511990412
= 11987011990412(119911119887minus (120579 minus 120579
0) 1198971minus 1199111199061minus
(120593 minus 1205931199061) 1198891199041
2
) + 11986511988912
(6)
where 11986511988911
and 11986511988912
are the left and right damping forceof front suspension The relative velocities of left and right
Mathematical Problems in Engineering 5
(a)
0 005 01 015 02 025
0
1000
2000
3000
4000
25 Hz20 Hz15 Hz
10 HzTheoretical
minus02 minus01minus015 minus005
minus1000
d (ms)
Fc
(N)
(b)
Figure 2 Dynamic test and modeling of the damper
damper are obtained by V1198891= (119887minus
1205791198971minus 1199061+ ( minus
1199061)11988911990412)
and V1198892= (119887minus
1205791198971minus 1199061minus ( minus
1199061)11988911990412)
The following equations give the balancing pole pitchthe cab vertical roll and pitch and the axle vertical and rollmovements
(119894 = 1 sim 3 119895 = 1 sim 2) are the drivingtorque and braking torque of six wheels
23 Driver Model According to Guorsquos preview of optimalcurvature drivermodel [3 4] the optimal front steering angleis expressed by
120575119901=
2119871
1198892[119891 (119905 + 119879) minus 119910 (119905) minus 119879 119910 (119905)] (13)
where 119889 119879 and 119871 are the preview distance preview time andwheelbase respectively 119891(119905 + 119879) is the lateral position of thedesired route at preview point and 119910(119905) is the vehicle lateralposition at current time
The above model is very simple and suitable to simulatelateral dynamics of vehicle running at a constant speed How-ever it neglects the effect of time delay and the desired routefunction 119891(119905) needs to be computed according to vehiclespeed and trajectory before simulation
Legouisrsquo driver model with nonlinear time delay [6ndash8]calculates the front steering angle by
120575119901= minus119870[119910
119873(119905 minus 119879
119903) +
119889
119880
119910119873(119905 minus 119879
119903)] (14)
where119870 119879119903 and119880 are feedback gain time delay and vehicle
running speed respectively 119910119873(119905 minus 119879
119903) and 119910
119873(119905 minus 119879
119903) +
(119889119880) 119910119873(119905 minus 119879
119903) are the lateral position of vehicle gravity
center and preview point in the inertial frames respectivelyThis model introduced time delay and calculated positiondeviation between vehicle and desired route in the inertialframes However the feedback gain in Legouisrsquo driver modelis a constant and cannot be obtained by vehicle parametersThe ideal route function is not included in the model becausethe straight-line driving condition is researched In additionthe preview point is gained from vehicle gravity center andneglects the difference of longitudinal distance and yaw anglebetween vehicle gravity center and driver position
By combining the above two models a modified drivermodel is proposed here as shown in Figure 3The front wheelsteering angle is given by
120575119901(119905) =
2119871
1198892[119877119884119889(119905 minus 119879
119903) minus 119884119889(119905 minus 119879
119903)]
1003816100381610038161003816119877119884119889(119905 minus 119879
where119877119884119889119884119889 119890119888119903 and119889119905 are the required lateral position and
real lateral displacement of the preview point in the ground
Y
O Xd
Required route RYdYd
Fty12Fty22Fty32
x
o120595Vx120596r
Vy120575
dt2
Fty31Fty21
Fty11l3
l2l1
dy
Figure 3 The new driver control model with nonlinear time delay
coordinate system the permit position error and the inte-gration time step respectively For three-axle vehicle theparameter 119871 in (15) is the distance between front wheel andcenter of balance suspension and is expressed by 119871 = 119897
1+ 1198972
This modified model has a feedback gain 21198711198892 that is
defined by wheelbase and preview distance It can be noticedthat a big wheelbase or a small preview distance will lead toa big gain This statement is well understood because driversfeel it difficult to control the vehicle direction in the case of bigwheelbase or small preview distance so they have to increasethe intervention of steering angle
The displacement of the preview point in the groundcoordinate system is
119883119889= 119883 (119905) + 119897
1cos120595 + 119889
119880
( minus 1198971 sin120595)
119884119889= 119884 (119905) + 119897
1sin120595 + 119889
119880
( + 1198971 cos120595)
(16)
where119883(119905) 119884(119905) and 120595 are displacements and heading angleof vehicle gravity center in the ground coordinate system Itshould be noted that the preview point lies in 119889meter aheadof driver seat not vehicle gravity center According to 119883
119889
and the required route function the required lateral position119877119884119889is easily obtained Substituting 119877119884
119889and 119884
119889into (15) and
introducing time delay the front wheel angle can be obtainedIt should be noted that themodified drivermodel depends onthe vehicle longitudinal speed as strongly as Legouisrsquo drivermodel
The displacements and velocities in vehicle coordinatesystem (119909 119910 119911) can be gained from vehicle model andtransferred to the ground coordinate system (119883 119884 119885) by thefollowing relation
= 119881119909cos120595 minus 119881
119910sin120595
= 119881119909sin120595 + 119881
119910cos120595
(17)
Finally the vehicle model tire model and driver modelare coupled into the driver-vehicle closed-loop system Thelongitudinal slip ratios of six wheels are calculated in real-time with vehicle responses as inputThe front wheel steeringangle is obtained by the modified driver model and fed back
Mathematical Problems in Engineering 7
Coupled vehicle model
Wheel rotate equations
Braking toque
Three-directionaltire model
Slip ratio equation
Drivermodel
Output responses
Road surface properties
Requiredroute
Vehicle initial conditions
120596ij Mb
ztij
Ftxij FtyijFtzijMtzij
120575rij 120583ij
Vx
Ssij
RY119889(t minus Tr)
Yd(t minus Tr)
Xd(t minus Tr)
Figure 4 The driver-heavy-vehicle closed-loop model
to the tire model Then the vertical longitudinal and lateraltire forces are calculated by the tire model and input intothe vehicle model to gain vehicle responses and positionsin next time step The simulation process of this driver-vehicle closed-loop system is shown in Figure 4 Due to thetime variability nonlinearity and high-dimensional propertyof this system the closed-loop system equations are solvednumerically by the quick integration method [22] and theRunge-Kutta method of order four
3 Model Evaluation
In order to verify the presented TCLP vehicle model andthe new driver model simulation results of this driver-vehicle closed-loop system are obtained using different vehi-cle models or driver models During simulation the vehicleparameters are chosen for a DFL1250A9 truck manufacturedby Dongfeng Motor Group Company Limited [23ndash25] andthe B-class road roughness is selected referring to [26]
The parameters of double-lane change route for theheavy-duty vehicle are chosen referring to [27 28] and shownin Figure 5
31 Comparison with the Handling Stability Vehicle Model Atraditional two-degree of freedom (2DOF) handling stabilityvehicle model for a three-axle heavy vehicle is set up whichconsiders only the lateral and yaw motion [29] The ordinarydifferential equations of motion of this 2DOF model may beexpressed by
where 119898 119868119911 119881119909 119881119910 and 120596
119903are vehicle mass vehicle inertia
around 119911 axial and longitudinal lateral and yaw rate of thevehicle respectively 120575
119894119865120572119894 and119872
119911119894are steering angle lateral
tire force and self-aligning torque of wheels respectively 119897119909119894
and 119897119910119894are the distance from wheel center to vehicle gravity
center in longitudinal and lateral direction respectivelyThe double-lane change responses of this TCLP model
and the traditional 2DOF model at an entrance speedof 60 kmh are simulated respectively and compared inFigure 6 It can be seen from Figure 6 that the results of thesetwo models are very consistent in magnitude and trendsHence the two vehicle models verify each other The TCLPmodel has a worse path-following ability than the 2DOFmodel and the vehicle running speed of TCLP model fluctu-ates randomlyThe yaw rate and lateral acceleration obtainedfrom TCLP model is bigger than that from 2DOF modelThe reason for these differences between two models is thatthe TCLP vehicle model considers B-class road roughnessand the coupled effect of roll vertical longitudinal and pitchmotion on yaw and lateral motion while the 2DOF modelneglects them
32 Comparison with the Linear Vehicle Model When thenonlinearity of front suspension damper and tire force isneglected as shown in Figure 7 the vehicle responses becomesmaller and the tracking performance is better Thus thelinear vehicle model may predict more conservative results
8 Mathematical Problems in Engineering
20m
20m40m40m
35m35m
15m
30m
35
m
35
m
35
m
375
m
Figure 5 The lane change route for the heavy-duty vehicle
0 5 10
162
164
166
168
17
0 5 10 15
0
5
Coupled model2DOF model
Required route
minus5
50 100 150 200
0
2
4
Y(m
)
X (m)
u(m
s)
t (s)
0 5 10 15
0
02
04
Coupled model2DOF model
Required route
minus04
minus02
t (s) t (s)
wr (
rad
s)
ay(m
s2)
Figure 6 Double-lane change responses of TCLP model and 2DOF model at 60 kmh
In the meantime it is found that the difference between thelinear and nonlinear vehicle models in the maneuver of lanechange is much greater than that in the maneuver of drivingstraight The lateral load transfer yaw motion and lateralmotion while lane change destabilizes the vehicle and conse-quently aggravates other vehicle motions due to couple effectIn addition by simulating responses at different entrancespeeds we found that the difference between the linearand nonlinear vehicle models becomes greater with the riseof vehicle speed Therefore the nonlinear vehicle modelshould be used so as to simulate vehicle dynamics accurately
especially in lane change maneuver or high speed drivingcondition
33 Comparison with Test Data During double-lane changetest an empty loaded DFL1250A9 truck was selected to runat 30 kmh speed A three-axis piezoelectric accelerometer(frequency range from 1Hz to 500Hz) and a gyro (measur-ing range plusmn50 degs) were placed at the center of gravity ofwhole vehicle and a cable displacement sensor was placedat front wheel to measure front wheel steering angle asshown in Figure 8 The measured signals were amplified by
Mathematical Problems in Engineering 9
50 100 150 200 250 300
0
1
2
3
4
minus1
Y(m
)
X (m)0 5 10 15 20
162
164
166
168
u(m
s)
t (s)
5 10 15 20
0
2
4
minus4
minus2
t (s)
ay
(ms
2 )
5 10 15 20
0
01
02
03
minus02
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
001
minus002
minus001
t (s)
120573(r
ad)
7 8 9
0
10
20
minus20
minus10
t (s)
azb
(ms
2 )
5 10 15 20
0
01
02
minus02
minus01
NonlinearLinear
Requied route
t (s)
Φ(r
ad)
5 10 15 20
0
005
minus01
minus005
NonlinearLinear
Requied route
t (s)
120579(r
ad)
Figure 7 The double-lane change responses of the linear and nonlinear TCLP model at 60 kmh
10 Mathematical Problems in Engineering
Figure 8 Vehicle and sensors in field test
Table 1 Vehicle responses obtained from three driver models
Vehicle speed Driver model RMS value of vehicle responses119906 119908119903 119886119910 119886119911119887 119891119894 119909119894119905119886
a multichannel charge amplifier (YE5853A) and convertedto digital signals by an intelligent acquisition and processinganalyzer (INV360DF) The sampling frequency was 200Hz
Figure 9 gives the experimental time history of frontwheel steering angle Using this measured front wheel steer-ing angle as input the yaw rate and lateral acceleration are cal-culated from TCLP model 2DOF model and linear modelrespectively These simulation results are compared with
the test results as shown in Figure 10 It is obvious thatthe simulation results of TCLP model are most consistentwith the results of field test Due to neglecting of the three-directional coupling effects of vehiclemotions or nonlinearityof suspension and tire the time histories of yaw rate obtainedfrom 2DOF model and linear model are too smooth and thelateral accelerations of 2DOF model and linear model aresmaller than the test results Thus the verification of thisproposed TCLP model is verified
34 Comparison with Other Driver Models Using the pro-posed modified driver model the optimal curvature modelby Guo and nonlinear time delay driver model by Legouisthe trajectories of vehicle gravity center during double-lanechange maneuver at four entrance speeds of 20 40 60 and80 kmh are simulated respectively as shown in Figure 11Table 1 lists the root mean square (RMS) value of vehicleresponses in six directions at different speeds applying thesethree driver models As shown in Figure 11 it is obvious thatthe tracking performance of the modified driver model issuperior to the other two driver models at different entrancespeeds It is also found from Figure 11 that though an over-shoot of the newmodel at a vehicle speed of 80 kmh exists in119883 = [150 180]m the results of the other two models divergewhen the vehicle arrives at 119883 = 200m while the resultof the new model is stable From Table 1 we can see thatthe real vehicle speed of the modified driver model is the
Mathematical Problems in Engineering 11
Simulation results of TCLP model
Test results
Simulation results of 2DOF model
Simulation results of linear model
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
02
04
minus04
minus02
t (s)
ay(m
s2 )
5 10 15 20
0
05
1
minus05
t (s)
ay(m
s2 )
Figure 10 Comparison of simulation and test results
12 Mathematical Problems in Engineering
50 100 150 200 250 300
0
1
2
3
4
5
minus1
Y(m
)
X (m)
(a) 20 kmh
50 100 150 200 250
0
1
2
3
4
Y(m
)
X (m)
(b) 40 kmh
50 100 150 200 250 300
0
1
2
3
4
5
NewLiu
GuoRoute
minus1
Y(m
)
X (m)
(c) 60 kmh
50 100 150 200
0
2
4
NewLiu
GuoRoute
minus2
Y(m
)
X (m)
(d) 80 kmh
Figure 11 Comparison of modified driver model with other driver models at different speeds
closest to the assumed speed and the modified driver modelhas smaller responses except for vertical vibration and betterstability than the other models Though the modified drivermodel obtains greater vertical vehicle acceleration and worseride comfort than Legouisrsquos model at speeds lower than60 kmh the new driver model can give the vehicle betterpath-following ability and handling stability
4 Effects of Driver Model Parameters onFull Vehicle Dynamics and Stability
The important parameters in driver model include vehiclerunning speed time delay preview distance and permitposition error which decide the steering control effect andvehicle dynamics During the same driving maneuver theskilled drivers may use higher vehicle speed longer previewdistance shorter time delay and larger permit position error
than the new drivers The match of these four driverparameters also influences the vehicle dynamics RecentlyPauwelussen [30] researched the dependencies of driversteering control parameters on vehicle lateral properties andpath keeping and gives some useful conclusions This workfocuses on the effects of driver model parameters on bothpath keeping and full vehicle dynamics such as vehiclehandling stability ride comfort and roll stability
Since the vehicle lateral displacement is randomly causedby road roughness excitations and the lateral position devi-ation is always changing during lane change maneuvers astatistic index is introduced to evaluate the vehiclersquos path-following ability globally which is formulated by
Route error = radicsum (119877119884
119889(119905) minus 119884
119889(119905))2
119873
(20)
Mathematical Problems in Engineering 13
20 40 60 80 100 120 140 160 180 200
0
1
2
3
4
5
20 kmh30 kmh40 kmh
50 kmh60 kmhRoute
Y(m
)
X (m)
minus1
0 50 100 150 200
0
1
2
3
4
0 s002 s005 s
01 s02 sRoute
Y(m
)
X (m)
minus1
Figure 12 Routes of vehicle under different speed or time delay
where 119873 is the total number of data points 119877119884119889and 119884
119889are
the required lateral position and real lateral displacement ofthe preview point in the ground coordinate system
When a vehicle is undergoing a lane change its steeringstability and roll stability are vital problems The path-following error lateral acceleration and yaw rate can be usedto evaluate the steering stability The roll angle can be used toevaluate the roll stability Being proportional to the rolloverindex LTR (lateral-load transfer rate) the lateral accelerationis also able to reflect the vehicle roll stability [31] In additionthe vertical acceleration and pitch angle are simulated so asto research the whole-body dynamics in the vehicle Usingthese evaluation indexes the effects of parameters on steeringstability roll stability and riding comfort are discussed indetail
Figure 12 shows the vehicle trajectories under differentinitial vehicle speeds or driver time delays The effects ofvehicle speed and driver time delay on full vehicle dynamicsand stability are shown in Figure 13 As we can see fromFigures 12 and 13 a higher speed or a longer time delay willlead to a greater path-following error However low speed(20 kmh) and longtime delay (035 s) may enlarge the path-following error With the rise of vehicle speed the verticalacceleration lateral acceleration yaw rate and roll angleincrease while the pitch angle decreases Rise of time delaycauses the increase of lateral acceleration yaw rate and rollangle but hardly changes the vertical acceleration and pitchangle As for the effects of vehicle speed and driver time delayon full vehicle dynamics the following may be concluded
(1) A high vehicle speed during double-lane changeyields bad path keeping performance ride comfortsteering stability and roll stability When the vehiclespeed is higher than 60 kmh the ride comfort of thisvehicle gets worse greatly
(2) A long time delay is harmful to path keeping per-formance steering stability and roll stability and hasalmost no effect on the ride comfort
Thus a low vehicle speed ranging from 20 kmh to 60 kmh and a small time delay less than 035 s are recommended forimproving this truckrsquos stability when going through the lanechange
In case of varying the preview distance or permit positionerror the variation in vehicle trajectories at the speed of40 kmh is shown in Figure 14The effects of preview distanceor permit position error on the path-following ability andfull vehicle dynamics are shown in Figure 15 As shown inFigures 14 and 15 a large preview distance deteriorates thepath-following ability but has small effect on other dynamicresponses On the other hand a too small preview distanceleads to a hop of lateral acceleration yaw rate and roll anglewhich conflicts with closed-loop stability and is harmfulto steering control This conclusion is consistent with thatof Gillespie and MacAdam [15] and Pauwelussen [30] Theinfluence of permit position error on path-following ability issmall at short preview distance but great at long preview dis-tance As shown in Figure 14 a too small or too large permitposition error (001m or 08m)will worsen the vehiclersquos path-following ability In addition large permit position error mayincrease vertical acceleration lateral acceleration yaw rateand roll angle and reduce the handling stability and rollstability and ride comfort The effect of preview distance andpermit position error on pitch angle is small and shows fluc-tuation Thus a preview distance ranging from 10m to 30mand a small permit position error but more than 001m arerecommended for improving this truckrsquos stability when mov-ing through the lane change A smaller permit position errorshould be matched with a shorter preview distance
14 Mathematical Problems in Engineering
002
04
2040
60800
1
2
Rout
e err
or (m
)
u0 (kmh) 0
02
04
2040
60807
8
9
u0 (kmh)
002
04
204060800
02
04
wr (
rad
s)
u0 (kmh)
0
02
04
204060800
005
01
120579(r
ad)
u0 (kmh)002
04
2040
60800
01
02
Φ(r
ad)
u0 (kmh)
002
04
2040
60800
2
4
u0(kmh)
ay(m
s2 )
Tr(s) Tr
(s)
Tr(s)
Tr(s)
Tr(s) Tr
(s)
azb
(ms2 )
Figure 13 The effects of vehicle speed and driver time delay
0 50 100 150 200
0
1
2
3
4
5
001 m02 m04 m
06 m08 mRoute
Y(m
)
X (m)
minus10 50 100 150 200
0
2
4
6
8
10
12
2 m10 m20 m
30 m40 mRoute
Y(m
)
X (m)
minus2
minus4
Figure 14 Routes of vehicle under different preview distance and permit position error
Mathematical Problems in Engineering 15
020
4060
005
1150
1
2
Rout
e err
or (m
)
er (m) Ld (m)
020
4060
01
25
10
15
er (m) Ld (m)
020
4060
01
20
5
er (m) Ld (m) 020
4060
01
20
02
04
er (m) Ld (m)wr (
rad
s)
2040
601
20
02
04
00er (m) Ld (m)
Φ(r
ad)
020
40 60
01
20
005
01
er (m) Ld (m)
120579(r
ad)
ay(m
s2 )
azb
(ms2 )
Figure 15 The effects of preview distance and permit position error
5 Conclusions
In this paper a nonlinear three-directional coupled lumpedparameters (TCLP) model of heavy-duty vehicle consideringthe nonlinearity of suspension damping and tire stiffness isbuilt and connected with a proposedmodified preview drivermodel with nonlinear time delayThe proposed driver modelis simple and has a feedback gain 2119871119889
2 that is decided bywheelbase and previewdistanceThe validity of this presenteddriver-vehicle closed-loop system is verified by comparisonwith the test data the traditional steering stability vehiclemodel the linear vehicle model and other driver controlmodelsThe results show that the new drivermodel has betterlane keeping performances than the other two driver modelsThe coupling effects of vehicle motions and the nonlinear-ity of suspension damping and tire stiffness could not beneglected in turning or high speed driving situation
The effects of driver parameters on path-following abilityand full vehicle dynamics are also discussed It is foundthat the high vehicle speed large time delay long previewdistance or big permit position error will deteriorate path-following performances and handling stability Of course atoo short preview distance or a too small permit position
error is also harmful to path-following performances andhandling stability The ride comfort mainly depends on vehi-cle speed preview distance and permit position error Thekey factors influencing roll characteristics are vehicle speedand time delay
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The National Natural Science Foundation of China(11472180) theNatural Science Foundation ofHebei Province(E2012210025) and the New Century Talent Foundation ofMinistry of Education (NCET-13-0913) support this work
References
[1] M Plochl and J Edelmann ldquoDriver models in automobiledynamics applicationrdquoVehicle SystemDynamics vol 45 no 7-8pp 699ndash741 2007
16 Mathematical Problems in Engineering
[2] C C MacAdam ldquoUnderstanding and modeling the humandriverrdquo Vehicle System Dynamics vol 40 no 1ndash3 pp 101ndash1342003
[3] K Guo and H Guan ldquoModeling of drivervehicle directionalcontrol systemrdquo Vehicle System Dynamics vol 22 no 3-4 pp141ndash184 1993
[4] K GuoVehicle Handling DynamicsTheory Jiangsu Science andTechnology Publishing House Nanjing China 2011
[5] R S Sharp ldquoDriver steering control and a new perspective oncar handling qualitiesrdquo Proceedings of the Institution ofMechani-cal Engineers Part C Journal of Mechanical Engineering Sciencevol 219 no 10 pp 1041ndash1051 2005
[6] T Legouis A Laneville P Bourassa and G Payre ldquoCharac-terization of dynamic vehicle stability using two models of thehuman pilot behaviorrdquo Vehicle System Dynamics vol 15 no 1pp 1ndash18 1986
[7] Z Liu G Payre and P Bourassa ldquoNonlinear oscillations andchaotic motions in a road vehicle system with driver steeringcontrolrdquo Nonlinear Dynamics vol 9 no 3 pp 281ndash304 1996
[8] Z Liu G Payre and P Bourassa ldquoStability and oscillations ina time-delayed vehicle system with driver controlrdquo NonlinearDynamics vol 35 no 2 pp 159ndash173 2004
[9] X D Liu W Feng and N Kang ldquoResearch of simulationmethod of tractor-semitrailer carrying liquid considering liquidsloshingrdquo in Proceedings of the 10th National Conference onVibrationTheory and Application pp 581ndash590 Nanjing ChinaOctober 2011
[10] C C MacAdam ldquoApplication of an optimal preview control forsimulation of closed-loop automobile drivingrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 11 no 6 pp 393ndash399 1981
[11] A Y Ungoren and H Peng ldquoAn adaptive lateral preview drivermodelrdquo Vehicle System Dynamics vol 43 no 4 pp 245ndash2592005
[12] C I Chatzikomis and K N Spentzas ldquoA path-following drivermodel with longitudinal and lateral control of vehiclersquos motionrdquoForschung im Ingenieurwesen vol 73 no 4 pp 257ndash266 2009
[13] R S Sharp D Casanova and P Symonds ldquoA mathematicalmodel for driver steering control with design tuning andperformance resultsrdquo Vehicle System Dynamics vol 33 no 5pp 289ndash326 2000
[14] A J Pick and D J Cole ldquoNeuromuscular dynamics in thedriver-vehicle systemrdquo Vehicle System Dynamics vol 44 sup-plement 1 pp 624ndash631 2006
[15] T D Gillespie and C C MacAdam Constant Velocity YawRollProgram Userrsquos Manual The University of Michigan Trans-portation Research Institute Ann Arbor Mich USA 1982
[16] R I S Raj Influence of road roughness and directional maneuveron the dynamic performance of heavy vehicles [MS thesis]Department of Mechanical Engineering Concordia UniversityMontreal Canada 1998
[17] KWordenDHickeyMHaroon andD E Adams ldquoNonlinearsystem identification of automotive dampers a time and fre-quency-domain analysisrdquo Mechanical Systems and Signal Pro-cessing vol 23 no 1 pp 104ndash126 2009
[18] S H Li Y J Lu and L Y Li ldquoDynamical test and modelingfor hydraulic shock absorber on heavy vehicle under harmonicand random loadingsrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 13 pp 1903ndash1910 2012
[19] X W Ji and Y M Gao ldquoThe dynamic stiffness and dampingcharacteristics of the tirerdquo Automotive Engineering vol 5 pp315ndash321 1994
[20] G Gim and P E Nikravesh ldquoA three dimensional tire modelfor steady-state simulations of vehiclesrdquo SAE vol 102 no 2 pp150ndash159 1993
[21] J D Zhuang Principles of Automobile Tire Beijing Institute ofTechnology Press Beijing China 1996
[22] W-M Zhai ldquoTwo simple fast integration methods for large-scale dynamic problems in engineeringrdquo International Journalfor Numerical Methods in Engineering vol 39 no 24 pp 4199ndash4214 1996
[23] Dongfeng Motor Group Company Dongfeng DFL1250A8DFL1250A9 Truck Chassis Refit Manual Dongfeng MotorGroup Company 2008
[24] S Li S Yang L Chen and Y Lu ldquoEffects of parameters ondynamic responses for a heavy vehicle-pavement- foundationcoupled systemrdquo International Journal of Heavy Vehicle Systemsvol 19 no 2 pp 207ndash224 2012
[25] S Yang S Li andY Lu ldquoInvestigation on dynamical interactionbetween a heavy vehicle and road pavementrdquo Vehicle SystemDynamics vol 48 no 8 pp 923ndash944 2010
[26] GBT 7031-2005ISO 86081995 Mechanical Vibration-RoadSurface Profiles-Reporting of Measured Data 2005
[27] M Li X Pu and Z Changfu ldquoModeling and simulationof heavy-duty semi-trailer in extreme condition based onADAMSrdquo Automobile Technology vol 2 pp 22ndash25 2009
[28] ldquoPassenger carsmdashtest track for a severe lane-change manoeu-vremdashpart 1 double lane-changerdquo ISO 3888-11999
[29] S H Li S P Yang and N Chen ldquoDirectional control of adriver-heavy-vehicle closed-loop systemrdquo Advanced Engineer-ing Forum vol 2-3 pp 33ndash38 2012
[30] J Pauwelussen ldquoDependencies of driver steering controlparametersrdquo Vehicle System Dynamics vol 50 no 6 pp 939ndash959 2012
[31] H Imine and V Dolcemascolo ldquoRollover risk prediction ofheavy vehicle in interaction with infrastructurerdquo InternationalJournal ofHeavyVehicle Systems vol 14 no 3 pp 294ndash307 2007
three-axle heavy vehicles with nonlinear suspension and tireproperty is also necessary and has attracted some scholarsrsquoattention The UMTRIrsquos yaw-roll truck model described in[15] considered nonlinearities of suspension and tire corner-ing force as tabular functions and was further adapted toinclude the road roughness effect [16] but did not set up thedifference equations for the longitudinalmotion andwas onlysuitable to constant-speed driving conditions
In addition the present driver models are often appliedto evaluate lateral and yaw dynamic characteristics based onthe handling stability vehicle model and the results aboutthe effect of driver behavior on vehicle vertical lateral andlongitudinal dynamics are still inadequate In fact the roadsurface offers not only lateral and longitudinal forces to avehicle but also vertical forces to suspension especially undercomplicated driving situations such as lane change corner-ing or barrier avoidance In these cases the vertical rolland pitch dynamics of a vehicle are coupled with the lateraland yaw motions obviously Due to bigger inertia longerwheelbase and higher roll center compared with cab cars theheavy-duty vehicles show poorer stability and greater three-directional coupling effects when entering a turn or lanechange Hence it is quite necessary to build a three-direc-tional coupled vehicle model and research the effect of driversteering on full vehicle dynamics
This work applies a modified preview driver model in fullvehicle dynamics simulation of a heavy vehicle and investi-gates the effect of driver parameters on both lane keeping per-formances and full vehicle dynamic characteristics Firstly wegive a description on how tomodel the three-directional cou-pling effect and nonlinear properties of suspension and tire ina heavy-duty vehicle Then a modified single-point previewdriver control model with nonlinear time delay consideringvehicle speed variation is proposed and a driver-vehicleclosed-loop system is constructed The full vehicle dynamicresponses of the closed-loop system in double-lane changeare obtained by numerical integration The validity of thisclosed-loop system is verified by comparison with the resultsof a field test the traditional steering stability vehicle modelthe linear vehicle model and other driver control modelsFinally the effects of driver model parameters such as vehiclerunning speed time delay preview distance and permitposition error on the path-following behavior and three-directional dynamics of the vehicle are also analyzed
2 Model Building
21 The Nonlinear Three-Directional Coupled Vehicle ModelA nonlinear TCLP model for a three-axial heavy-duty truckwith 23-DOF is presented as shown in Figure 1 The vehicleis front wheel steered and rear wheel driven 119911
119888 120579119888 120601119888 119911119887
120579 120601 stand for the vertical pitch and roll displacementsof driver cab and vehicle body 120595 119909 119910 represent the yawlongitudinal and lateral displacements of the full vehicle119911119906119894 120601119906119894(119894 = 1ndash3) denote the vertical and roll displacements
of three wheel axles 1205791199011and 1205791199012are the pitch angles of the left
or right balancing pole on rear suspension 119872 119872119887 and 119872
119888
denote themasses of full vehicle vehicle body and driver cabrespectively The origin of vehicle coordinate system (119909 119910 119911)
is located in the intersection between the roll axle and thevertical line passing vehicle center of gravity ℎ
119904and 119897119904are the
vertical and longitudinal distance from the sprung mass cen-ter of gravity to the coordinate origin 119889
1199051and 1198891199052are the front
and rear wheel track width 1198891199041and 119889
1199042are the lateral dis-
tance between left and right springs on front or rear suspen-sion
The movements of the heavy-duty vehicle are coupledwith each other greatly Equations (1) give the longitudinallateral and yaw dynamics of the full vehicle Equations (2)give the vertical roll and pitch dynamics of the sprung mass
119872( minus 119910) + 119872119887119887
120579 + 119872
119887ℎ119904
120579 minus 119872
119887119897119904(2
+1205792
)
= (11986511990511990911
+ 11986511990511990912
) cos 120575 minus (11986511990511991011
+ 11986511990511991012
) sin 120575
+
3
sum
119894=2
2
sum
119895=1
119865119905119909119894119895
119872 ( 119910 + ) minus 119872119887119887 minus 119872
119887ℎ119904
= (11986511990511990911
+ 11986511990511990912
) sin 120575 + (11986511990511991011
+ 11986511990511991012
) cos 120575
+
3
sum
119894=2
2
sum
119895=1
119865119905119910119894119895
119868119911 + 2119868
119911119887120579 minus (119868
119887119909119911+119872119887119897119904ℎ119904) minus 119872
(3)For this heavy-duty vehicle two hydraulic dampers are
fixed on the left and right front suspensions and the tandembalanced suspension does not have any shock absorber Inorder to represent the frictional property of leaf spring thedamping forces of tandem balanced suspension are modeledlinearly The suspension forces between middle or rear axleand vehicle body are given by
11986511990421
= 11987011990421(119911119887+ (120579 minus 120579
0) 1198972minus
12057911990111198973
2
minus 1199111199062+
(120593 minus 1205931199062) 1198891199042
2
)
+ 11986211990421(119887+
1205791198972minus
12057911990111198973
2
minus 1199062+
( minus 1199062) 1198891199042
2
)
11986511990422
= 11987011990422(119911119887+ (120579 minus 120579
0) 1198972minus
12057911990121198973
2
minus 1199111199062minus
(120593 minus 1205931199062) 1198891199042
2
)
+ 11986211990422(119887+
1205791198972minus
12057911990121198973
2
minus 1199062minus
( minus 1199062) 1198891199042
2
)
11986511990431
= 11987011990431(119911119887+ (120579 minus 120579
0) 1198972+
12057911990111198973
2
minus 1199111199063+
(120593 minus 1205931199063) 1198891199043
2
)
+ 11986211990431(119887+
1205791198972+
12057911990111198973
2
minus 1199063+
( minus 1199063) 1198891199043
2
)
11986511990432
= 11987011990432(119911119887+ (120579 minus 120579
0) 1198972+
12057911990121198973
2
minus 1199111199063minus
(120593 minus 1205931199063) 1198891199043
2
)
+ 11986211990432(119887+
1205791198972+
12057911990121198973
2
minus 1199063minus
( minus 1199063) 1198891199043
2
)
(4)
Since hydraulic dampers show obvious nonlinearitymany dynamic models for shock absorber have been pro-posed amongwhich the fittedmodel is quite suitable tomod-eling the ascertained shock absorber but needs a large amountof experimental work [17 18] In this work the dynamic prop-erty of the damper on front suspension ismeasured byHT-911testing machine under sinusoidal excitation Since the inher-ence frequency of the vehicle body is from 1Hz to 25Hzfour excitation frequencies are selected as 1Hz 15Hz 2Hzand 25Hz Limited by the machinersquos tonnage the excitationamplitude is chosen as 10mm Figure 2 shows the measuredforce-velocity curves Since the hysteresis loops of dampingforce depend on the excitation frequency and amplitudegreatly the parameters of hysteresis model under randomexcitations are difficult to identify Hence a nonlinear seg-mented model describing the damperrsquos scheme framework isproposed here
119865119889=
11986501sgn (V
119889) V
119889gt Vlim 1
119862 (1 + 120573 sgn (V119889)) V119889
1003816100381610038161003816V119889
1003816100381610038161003816
119899 Vlim 2 le V119889le Vlim 1
11986502sgn (V
119889) V
119889lt Vlim 2
(5)
where V119889 119862 120573 and 119899 are the relative velocity of cylinder
and plunger the damping coefficient the asymmetry ratioand the exponent respectively 119865
01 11986502 Vlim 1 and Vlim 2 are
the damping force and relative velocity when the damperreaching saturation in tension or compression process
Parameters in model (5) fitted to the measured data are119862 = 30893 120573 = 056 119899 = 016 119865
01= 4119N 119865
02= 726N
Vlim 1 = 012ms and Vlim 2 = 008ms The damping forcecurve obtained from the theoretical damper model is shownas the thick solid line in Figure 2 It can be seen that thepresented damping forcemodel is able to describe the schemeframework and saturation property of the damper Thoughmodel (5) neglects the damperrsquos hysteresis characteristics itis simple and accurate enough for numerical simulation
Using model (5) to calculate the damping force the frontsuspension forces are expressed by
11986511990411
= 11987011990411(119911119887minus (120579 minus 120579
0) 1198971minus 1199111199061+
(120593 minus 1205931199061) 1198891199041
2
) + 11986511988911
11986511990412
= 11987011990412(119911119887minus (120579 minus 120579
0) 1198971minus 1199111199061minus
(120593 minus 1205931199061) 1198891199041
2
) + 11986511988912
(6)
where 11986511988911
and 11986511988912
are the left and right damping forceof front suspension The relative velocities of left and right
Mathematical Problems in Engineering 5
(a)
0 005 01 015 02 025
0
1000
2000
3000
4000
25 Hz20 Hz15 Hz
10 HzTheoretical
minus02 minus01minus015 minus005
minus1000
d (ms)
Fc
(N)
(b)
Figure 2 Dynamic test and modeling of the damper
damper are obtained by V1198891= (119887minus
1205791198971minus 1199061+ ( minus
1199061)11988911990412)
and V1198892= (119887minus
1205791198971minus 1199061minus ( minus
1199061)11988911990412)
The following equations give the balancing pole pitchthe cab vertical roll and pitch and the axle vertical and rollmovements
(119894 = 1 sim 3 119895 = 1 sim 2) are the drivingtorque and braking torque of six wheels
23 Driver Model According to Guorsquos preview of optimalcurvature drivermodel [3 4] the optimal front steering angleis expressed by
120575119901=
2119871
1198892[119891 (119905 + 119879) minus 119910 (119905) minus 119879 119910 (119905)] (13)
where 119889 119879 and 119871 are the preview distance preview time andwheelbase respectively 119891(119905 + 119879) is the lateral position of thedesired route at preview point and 119910(119905) is the vehicle lateralposition at current time
The above model is very simple and suitable to simulatelateral dynamics of vehicle running at a constant speed How-ever it neglects the effect of time delay and the desired routefunction 119891(119905) needs to be computed according to vehiclespeed and trajectory before simulation
Legouisrsquo driver model with nonlinear time delay [6ndash8]calculates the front steering angle by
120575119901= minus119870[119910
119873(119905 minus 119879
119903) +
119889
119880
119910119873(119905 minus 119879
119903)] (14)
where119870 119879119903 and119880 are feedback gain time delay and vehicle
running speed respectively 119910119873(119905 minus 119879
119903) and 119910
119873(119905 minus 119879
119903) +
(119889119880) 119910119873(119905 minus 119879
119903) are the lateral position of vehicle gravity
center and preview point in the inertial frames respectivelyThis model introduced time delay and calculated positiondeviation between vehicle and desired route in the inertialframes However the feedback gain in Legouisrsquo driver modelis a constant and cannot be obtained by vehicle parametersThe ideal route function is not included in the model becausethe straight-line driving condition is researched In additionthe preview point is gained from vehicle gravity center andneglects the difference of longitudinal distance and yaw anglebetween vehicle gravity center and driver position
By combining the above two models a modified drivermodel is proposed here as shown in Figure 3The front wheelsteering angle is given by
120575119901(119905) =
2119871
1198892[119877119884119889(119905 minus 119879
119903) minus 119884119889(119905 minus 119879
119903)]
1003816100381610038161003816119877119884119889(119905 minus 119879
where119877119884119889119884119889 119890119888119903 and119889119905 are the required lateral position and
real lateral displacement of the preview point in the ground
Y
O Xd
Required route RYdYd
Fty12Fty22Fty32
x
o120595Vx120596r
Vy120575
dt2
Fty31Fty21
Fty11l3
l2l1
dy
Figure 3 The new driver control model with nonlinear time delay
coordinate system the permit position error and the inte-gration time step respectively For three-axle vehicle theparameter 119871 in (15) is the distance between front wheel andcenter of balance suspension and is expressed by 119871 = 119897
1+ 1198972
This modified model has a feedback gain 21198711198892 that is
defined by wheelbase and preview distance It can be noticedthat a big wheelbase or a small preview distance will lead toa big gain This statement is well understood because driversfeel it difficult to control the vehicle direction in the case of bigwheelbase or small preview distance so they have to increasethe intervention of steering angle
The displacement of the preview point in the groundcoordinate system is
119883119889= 119883 (119905) + 119897
1cos120595 + 119889
119880
( minus 1198971 sin120595)
119884119889= 119884 (119905) + 119897
1sin120595 + 119889
119880
( + 1198971 cos120595)
(16)
where119883(119905) 119884(119905) and 120595 are displacements and heading angleof vehicle gravity center in the ground coordinate system Itshould be noted that the preview point lies in 119889meter aheadof driver seat not vehicle gravity center According to 119883
119889
and the required route function the required lateral position119877119884119889is easily obtained Substituting 119877119884
119889and 119884
119889into (15) and
introducing time delay the front wheel angle can be obtainedIt should be noted that themodified drivermodel depends onthe vehicle longitudinal speed as strongly as Legouisrsquo drivermodel
The displacements and velocities in vehicle coordinatesystem (119909 119910 119911) can be gained from vehicle model andtransferred to the ground coordinate system (119883 119884 119885) by thefollowing relation
= 119881119909cos120595 minus 119881
119910sin120595
= 119881119909sin120595 + 119881
119910cos120595
(17)
Finally the vehicle model tire model and driver modelare coupled into the driver-vehicle closed-loop system Thelongitudinal slip ratios of six wheels are calculated in real-time with vehicle responses as inputThe front wheel steeringangle is obtained by the modified driver model and fed back
Mathematical Problems in Engineering 7
Coupled vehicle model
Wheel rotate equations
Braking toque
Three-directionaltire model
Slip ratio equation
Drivermodel
Output responses
Road surface properties
Requiredroute
Vehicle initial conditions
120596ij Mb
ztij
Ftxij FtyijFtzijMtzij
120575rij 120583ij
Vx
Ssij
RY119889(t minus Tr)
Yd(t minus Tr)
Xd(t minus Tr)
Figure 4 The driver-heavy-vehicle closed-loop model
to the tire model Then the vertical longitudinal and lateraltire forces are calculated by the tire model and input intothe vehicle model to gain vehicle responses and positionsin next time step The simulation process of this driver-vehicle closed-loop system is shown in Figure 4 Due to thetime variability nonlinearity and high-dimensional propertyof this system the closed-loop system equations are solvednumerically by the quick integration method [22] and theRunge-Kutta method of order four
3 Model Evaluation
In order to verify the presented TCLP vehicle model andthe new driver model simulation results of this driver-vehicle closed-loop system are obtained using different vehi-cle models or driver models During simulation the vehicleparameters are chosen for a DFL1250A9 truck manufacturedby Dongfeng Motor Group Company Limited [23ndash25] andthe B-class road roughness is selected referring to [26]
The parameters of double-lane change route for theheavy-duty vehicle are chosen referring to [27 28] and shownin Figure 5
31 Comparison with the Handling Stability Vehicle Model Atraditional two-degree of freedom (2DOF) handling stabilityvehicle model for a three-axle heavy vehicle is set up whichconsiders only the lateral and yaw motion [29] The ordinarydifferential equations of motion of this 2DOF model may beexpressed by
where 119898 119868119911 119881119909 119881119910 and 120596
119903are vehicle mass vehicle inertia
around 119911 axial and longitudinal lateral and yaw rate of thevehicle respectively 120575
119894119865120572119894 and119872
119911119894are steering angle lateral
tire force and self-aligning torque of wheels respectively 119897119909119894
and 119897119910119894are the distance from wheel center to vehicle gravity
center in longitudinal and lateral direction respectivelyThe double-lane change responses of this TCLP model
and the traditional 2DOF model at an entrance speedof 60 kmh are simulated respectively and compared inFigure 6 It can be seen from Figure 6 that the results of thesetwo models are very consistent in magnitude and trendsHence the two vehicle models verify each other The TCLPmodel has a worse path-following ability than the 2DOFmodel and the vehicle running speed of TCLP model fluctu-ates randomlyThe yaw rate and lateral acceleration obtainedfrom TCLP model is bigger than that from 2DOF modelThe reason for these differences between two models is thatthe TCLP vehicle model considers B-class road roughnessand the coupled effect of roll vertical longitudinal and pitchmotion on yaw and lateral motion while the 2DOF modelneglects them
32 Comparison with the Linear Vehicle Model When thenonlinearity of front suspension damper and tire force isneglected as shown in Figure 7 the vehicle responses becomesmaller and the tracking performance is better Thus thelinear vehicle model may predict more conservative results
8 Mathematical Problems in Engineering
20m
20m40m40m
35m35m
15m
30m
35
m
35
m
35
m
375
m
Figure 5 The lane change route for the heavy-duty vehicle
0 5 10
162
164
166
168
17
0 5 10 15
0
5
Coupled model2DOF model
Required route
minus5
50 100 150 200
0
2
4
Y(m
)
X (m)
u(m
s)
t (s)
0 5 10 15
0
02
04
Coupled model2DOF model
Required route
minus04
minus02
t (s) t (s)
wr (
rad
s)
ay(m
s2)
Figure 6 Double-lane change responses of TCLP model and 2DOF model at 60 kmh
In the meantime it is found that the difference between thelinear and nonlinear vehicle models in the maneuver of lanechange is much greater than that in the maneuver of drivingstraight The lateral load transfer yaw motion and lateralmotion while lane change destabilizes the vehicle and conse-quently aggravates other vehicle motions due to couple effectIn addition by simulating responses at different entrancespeeds we found that the difference between the linearand nonlinear vehicle models becomes greater with the riseof vehicle speed Therefore the nonlinear vehicle modelshould be used so as to simulate vehicle dynamics accurately
especially in lane change maneuver or high speed drivingcondition
33 Comparison with Test Data During double-lane changetest an empty loaded DFL1250A9 truck was selected to runat 30 kmh speed A three-axis piezoelectric accelerometer(frequency range from 1Hz to 500Hz) and a gyro (measur-ing range plusmn50 degs) were placed at the center of gravity ofwhole vehicle and a cable displacement sensor was placedat front wheel to measure front wheel steering angle asshown in Figure 8 The measured signals were amplified by
Mathematical Problems in Engineering 9
50 100 150 200 250 300
0
1
2
3
4
minus1
Y(m
)
X (m)0 5 10 15 20
162
164
166
168
u(m
s)
t (s)
5 10 15 20
0
2
4
minus4
minus2
t (s)
ay
(ms
2 )
5 10 15 20
0
01
02
03
minus02
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
001
minus002
minus001
t (s)
120573(r
ad)
7 8 9
0
10
20
minus20
minus10
t (s)
azb
(ms
2 )
5 10 15 20
0
01
02
minus02
minus01
NonlinearLinear
Requied route
t (s)
Φ(r
ad)
5 10 15 20
0
005
minus01
minus005
NonlinearLinear
Requied route
t (s)
120579(r
ad)
Figure 7 The double-lane change responses of the linear and nonlinear TCLP model at 60 kmh
10 Mathematical Problems in Engineering
Figure 8 Vehicle and sensors in field test
Table 1 Vehicle responses obtained from three driver models
Vehicle speed Driver model RMS value of vehicle responses119906 119908119903 119886119910 119886119911119887 119891119894 119909119894119905119886
a multichannel charge amplifier (YE5853A) and convertedto digital signals by an intelligent acquisition and processinganalyzer (INV360DF) The sampling frequency was 200Hz
Figure 9 gives the experimental time history of frontwheel steering angle Using this measured front wheel steer-ing angle as input the yaw rate and lateral acceleration are cal-culated from TCLP model 2DOF model and linear modelrespectively These simulation results are compared with
the test results as shown in Figure 10 It is obvious thatthe simulation results of TCLP model are most consistentwith the results of field test Due to neglecting of the three-directional coupling effects of vehiclemotions or nonlinearityof suspension and tire the time histories of yaw rate obtainedfrom 2DOF model and linear model are too smooth and thelateral accelerations of 2DOF model and linear model aresmaller than the test results Thus the verification of thisproposed TCLP model is verified
34 Comparison with Other Driver Models Using the pro-posed modified driver model the optimal curvature modelby Guo and nonlinear time delay driver model by Legouisthe trajectories of vehicle gravity center during double-lanechange maneuver at four entrance speeds of 20 40 60 and80 kmh are simulated respectively as shown in Figure 11Table 1 lists the root mean square (RMS) value of vehicleresponses in six directions at different speeds applying thesethree driver models As shown in Figure 11 it is obvious thatthe tracking performance of the modified driver model issuperior to the other two driver models at different entrancespeeds It is also found from Figure 11 that though an over-shoot of the newmodel at a vehicle speed of 80 kmh exists in119883 = [150 180]m the results of the other two models divergewhen the vehicle arrives at 119883 = 200m while the resultof the new model is stable From Table 1 we can see thatthe real vehicle speed of the modified driver model is the
Mathematical Problems in Engineering 11
Simulation results of TCLP model
Test results
Simulation results of 2DOF model
Simulation results of linear model
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
02
04
minus04
minus02
t (s)
ay(m
s2 )
5 10 15 20
0
05
1
minus05
t (s)
ay(m
s2 )
Figure 10 Comparison of simulation and test results
12 Mathematical Problems in Engineering
50 100 150 200 250 300
0
1
2
3
4
5
minus1
Y(m
)
X (m)
(a) 20 kmh
50 100 150 200 250
0
1
2
3
4
Y(m
)
X (m)
(b) 40 kmh
50 100 150 200 250 300
0
1
2
3
4
5
NewLiu
GuoRoute
minus1
Y(m
)
X (m)
(c) 60 kmh
50 100 150 200
0
2
4
NewLiu
GuoRoute
minus2
Y(m
)
X (m)
(d) 80 kmh
Figure 11 Comparison of modified driver model with other driver models at different speeds
closest to the assumed speed and the modified driver modelhas smaller responses except for vertical vibration and betterstability than the other models Though the modified drivermodel obtains greater vertical vehicle acceleration and worseride comfort than Legouisrsquos model at speeds lower than60 kmh the new driver model can give the vehicle betterpath-following ability and handling stability
4 Effects of Driver Model Parameters onFull Vehicle Dynamics and Stability
The important parameters in driver model include vehiclerunning speed time delay preview distance and permitposition error which decide the steering control effect andvehicle dynamics During the same driving maneuver theskilled drivers may use higher vehicle speed longer previewdistance shorter time delay and larger permit position error
than the new drivers The match of these four driverparameters also influences the vehicle dynamics RecentlyPauwelussen [30] researched the dependencies of driversteering control parameters on vehicle lateral properties andpath keeping and gives some useful conclusions This workfocuses on the effects of driver model parameters on bothpath keeping and full vehicle dynamics such as vehiclehandling stability ride comfort and roll stability
Since the vehicle lateral displacement is randomly causedby road roughness excitations and the lateral position devi-ation is always changing during lane change maneuvers astatistic index is introduced to evaluate the vehiclersquos path-following ability globally which is formulated by
Route error = radicsum (119877119884
119889(119905) minus 119884
119889(119905))2
119873
(20)
Mathematical Problems in Engineering 13
20 40 60 80 100 120 140 160 180 200
0
1
2
3
4
5
20 kmh30 kmh40 kmh
50 kmh60 kmhRoute
Y(m
)
X (m)
minus1
0 50 100 150 200
0
1
2
3
4
0 s002 s005 s
01 s02 sRoute
Y(m
)
X (m)
minus1
Figure 12 Routes of vehicle under different speed or time delay
where 119873 is the total number of data points 119877119884119889and 119884
119889are
the required lateral position and real lateral displacement ofthe preview point in the ground coordinate system
When a vehicle is undergoing a lane change its steeringstability and roll stability are vital problems The path-following error lateral acceleration and yaw rate can be usedto evaluate the steering stability The roll angle can be used toevaluate the roll stability Being proportional to the rolloverindex LTR (lateral-load transfer rate) the lateral accelerationis also able to reflect the vehicle roll stability [31] In additionthe vertical acceleration and pitch angle are simulated so asto research the whole-body dynamics in the vehicle Usingthese evaluation indexes the effects of parameters on steeringstability roll stability and riding comfort are discussed indetail
Figure 12 shows the vehicle trajectories under differentinitial vehicle speeds or driver time delays The effects ofvehicle speed and driver time delay on full vehicle dynamicsand stability are shown in Figure 13 As we can see fromFigures 12 and 13 a higher speed or a longer time delay willlead to a greater path-following error However low speed(20 kmh) and longtime delay (035 s) may enlarge the path-following error With the rise of vehicle speed the verticalacceleration lateral acceleration yaw rate and roll angleincrease while the pitch angle decreases Rise of time delaycauses the increase of lateral acceleration yaw rate and rollangle but hardly changes the vertical acceleration and pitchangle As for the effects of vehicle speed and driver time delayon full vehicle dynamics the following may be concluded
(1) A high vehicle speed during double-lane changeyields bad path keeping performance ride comfortsteering stability and roll stability When the vehiclespeed is higher than 60 kmh the ride comfort of thisvehicle gets worse greatly
(2) A long time delay is harmful to path keeping per-formance steering stability and roll stability and hasalmost no effect on the ride comfort
Thus a low vehicle speed ranging from 20 kmh to 60 kmh and a small time delay less than 035 s are recommended forimproving this truckrsquos stability when going through the lanechange
In case of varying the preview distance or permit positionerror the variation in vehicle trajectories at the speed of40 kmh is shown in Figure 14The effects of preview distanceor permit position error on the path-following ability andfull vehicle dynamics are shown in Figure 15 As shown inFigures 14 and 15 a large preview distance deteriorates thepath-following ability but has small effect on other dynamicresponses On the other hand a too small preview distanceleads to a hop of lateral acceleration yaw rate and roll anglewhich conflicts with closed-loop stability and is harmfulto steering control This conclusion is consistent with thatof Gillespie and MacAdam [15] and Pauwelussen [30] Theinfluence of permit position error on path-following ability issmall at short preview distance but great at long preview dis-tance As shown in Figure 14 a too small or too large permitposition error (001m or 08m)will worsen the vehiclersquos path-following ability In addition large permit position error mayincrease vertical acceleration lateral acceleration yaw rateand roll angle and reduce the handling stability and rollstability and ride comfort The effect of preview distance andpermit position error on pitch angle is small and shows fluc-tuation Thus a preview distance ranging from 10m to 30mand a small permit position error but more than 001m arerecommended for improving this truckrsquos stability when mov-ing through the lane change A smaller permit position errorshould be matched with a shorter preview distance
14 Mathematical Problems in Engineering
002
04
2040
60800
1
2
Rout
e err
or (m
)
u0 (kmh) 0
02
04
2040
60807
8
9
u0 (kmh)
002
04
204060800
02
04
wr (
rad
s)
u0 (kmh)
0
02
04
204060800
005
01
120579(r
ad)
u0 (kmh)002
04
2040
60800
01
02
Φ(r
ad)
u0 (kmh)
002
04
2040
60800
2
4
u0(kmh)
ay(m
s2 )
Tr(s) Tr
(s)
Tr(s)
Tr(s)
Tr(s) Tr
(s)
azb
(ms2 )
Figure 13 The effects of vehicle speed and driver time delay
0 50 100 150 200
0
1
2
3
4
5
001 m02 m04 m
06 m08 mRoute
Y(m
)
X (m)
minus10 50 100 150 200
0
2
4
6
8
10
12
2 m10 m20 m
30 m40 mRoute
Y(m
)
X (m)
minus2
minus4
Figure 14 Routes of vehicle under different preview distance and permit position error
Mathematical Problems in Engineering 15
020
4060
005
1150
1
2
Rout
e err
or (m
)
er (m) Ld (m)
020
4060
01
25
10
15
er (m) Ld (m)
020
4060
01
20
5
er (m) Ld (m) 020
4060
01
20
02
04
er (m) Ld (m)wr (
rad
s)
2040
601
20
02
04
00er (m) Ld (m)
Φ(r
ad)
020
40 60
01
20
005
01
er (m) Ld (m)
120579(r
ad)
ay(m
s2 )
azb
(ms2 )
Figure 15 The effects of preview distance and permit position error
5 Conclusions
In this paper a nonlinear three-directional coupled lumpedparameters (TCLP) model of heavy-duty vehicle consideringthe nonlinearity of suspension damping and tire stiffness isbuilt and connected with a proposedmodified preview drivermodel with nonlinear time delayThe proposed driver modelis simple and has a feedback gain 2119871119889
2 that is decided bywheelbase and previewdistanceThe validity of this presenteddriver-vehicle closed-loop system is verified by comparisonwith the test data the traditional steering stability vehiclemodel the linear vehicle model and other driver controlmodelsThe results show that the new drivermodel has betterlane keeping performances than the other two driver modelsThe coupling effects of vehicle motions and the nonlinear-ity of suspension damping and tire stiffness could not beneglected in turning or high speed driving situation
The effects of driver parameters on path-following abilityand full vehicle dynamics are also discussed It is foundthat the high vehicle speed large time delay long previewdistance or big permit position error will deteriorate path-following performances and handling stability Of course atoo short preview distance or a too small permit position
error is also harmful to path-following performances andhandling stability The ride comfort mainly depends on vehi-cle speed preview distance and permit position error Thekey factors influencing roll characteristics are vehicle speedand time delay
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The National Natural Science Foundation of China(11472180) theNatural Science Foundation ofHebei Province(E2012210025) and the New Century Talent Foundation ofMinistry of Education (NCET-13-0913) support this work
References
[1] M Plochl and J Edelmann ldquoDriver models in automobiledynamics applicationrdquoVehicle SystemDynamics vol 45 no 7-8pp 699ndash741 2007
16 Mathematical Problems in Engineering
[2] C C MacAdam ldquoUnderstanding and modeling the humandriverrdquo Vehicle System Dynamics vol 40 no 1ndash3 pp 101ndash1342003
[3] K Guo and H Guan ldquoModeling of drivervehicle directionalcontrol systemrdquo Vehicle System Dynamics vol 22 no 3-4 pp141ndash184 1993
[4] K GuoVehicle Handling DynamicsTheory Jiangsu Science andTechnology Publishing House Nanjing China 2011
[5] R S Sharp ldquoDriver steering control and a new perspective oncar handling qualitiesrdquo Proceedings of the Institution ofMechani-cal Engineers Part C Journal of Mechanical Engineering Sciencevol 219 no 10 pp 1041ndash1051 2005
[6] T Legouis A Laneville P Bourassa and G Payre ldquoCharac-terization of dynamic vehicle stability using two models of thehuman pilot behaviorrdquo Vehicle System Dynamics vol 15 no 1pp 1ndash18 1986
[7] Z Liu G Payre and P Bourassa ldquoNonlinear oscillations andchaotic motions in a road vehicle system with driver steeringcontrolrdquo Nonlinear Dynamics vol 9 no 3 pp 281ndash304 1996
[8] Z Liu G Payre and P Bourassa ldquoStability and oscillations ina time-delayed vehicle system with driver controlrdquo NonlinearDynamics vol 35 no 2 pp 159ndash173 2004
[9] X D Liu W Feng and N Kang ldquoResearch of simulationmethod of tractor-semitrailer carrying liquid considering liquidsloshingrdquo in Proceedings of the 10th National Conference onVibrationTheory and Application pp 581ndash590 Nanjing ChinaOctober 2011
[10] C C MacAdam ldquoApplication of an optimal preview control forsimulation of closed-loop automobile drivingrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 11 no 6 pp 393ndash399 1981
[11] A Y Ungoren and H Peng ldquoAn adaptive lateral preview drivermodelrdquo Vehicle System Dynamics vol 43 no 4 pp 245ndash2592005
[12] C I Chatzikomis and K N Spentzas ldquoA path-following drivermodel with longitudinal and lateral control of vehiclersquos motionrdquoForschung im Ingenieurwesen vol 73 no 4 pp 257ndash266 2009
[13] R S Sharp D Casanova and P Symonds ldquoA mathematicalmodel for driver steering control with design tuning andperformance resultsrdquo Vehicle System Dynamics vol 33 no 5pp 289ndash326 2000
[14] A J Pick and D J Cole ldquoNeuromuscular dynamics in thedriver-vehicle systemrdquo Vehicle System Dynamics vol 44 sup-plement 1 pp 624ndash631 2006
[15] T D Gillespie and C C MacAdam Constant Velocity YawRollProgram Userrsquos Manual The University of Michigan Trans-portation Research Institute Ann Arbor Mich USA 1982
[16] R I S Raj Influence of road roughness and directional maneuveron the dynamic performance of heavy vehicles [MS thesis]Department of Mechanical Engineering Concordia UniversityMontreal Canada 1998
[17] KWordenDHickeyMHaroon andD E Adams ldquoNonlinearsystem identification of automotive dampers a time and fre-quency-domain analysisrdquo Mechanical Systems and Signal Pro-cessing vol 23 no 1 pp 104ndash126 2009
[18] S H Li Y J Lu and L Y Li ldquoDynamical test and modelingfor hydraulic shock absorber on heavy vehicle under harmonicand random loadingsrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 13 pp 1903ndash1910 2012
[19] X W Ji and Y M Gao ldquoThe dynamic stiffness and dampingcharacteristics of the tirerdquo Automotive Engineering vol 5 pp315ndash321 1994
[20] G Gim and P E Nikravesh ldquoA three dimensional tire modelfor steady-state simulations of vehiclesrdquo SAE vol 102 no 2 pp150ndash159 1993
[21] J D Zhuang Principles of Automobile Tire Beijing Institute ofTechnology Press Beijing China 1996
[22] W-M Zhai ldquoTwo simple fast integration methods for large-scale dynamic problems in engineeringrdquo International Journalfor Numerical Methods in Engineering vol 39 no 24 pp 4199ndash4214 1996
[23] Dongfeng Motor Group Company Dongfeng DFL1250A8DFL1250A9 Truck Chassis Refit Manual Dongfeng MotorGroup Company 2008
[24] S Li S Yang L Chen and Y Lu ldquoEffects of parameters ondynamic responses for a heavy vehicle-pavement- foundationcoupled systemrdquo International Journal of Heavy Vehicle Systemsvol 19 no 2 pp 207ndash224 2012
[25] S Yang S Li andY Lu ldquoInvestigation on dynamical interactionbetween a heavy vehicle and road pavementrdquo Vehicle SystemDynamics vol 48 no 8 pp 923ndash944 2010
[26] GBT 7031-2005ISO 86081995 Mechanical Vibration-RoadSurface Profiles-Reporting of Measured Data 2005
[27] M Li X Pu and Z Changfu ldquoModeling and simulationof heavy-duty semi-trailer in extreme condition based onADAMSrdquo Automobile Technology vol 2 pp 22ndash25 2009
[28] ldquoPassenger carsmdashtest track for a severe lane-change manoeu-vremdashpart 1 double lane-changerdquo ISO 3888-11999
[29] S H Li S P Yang and N Chen ldquoDirectional control of adriver-heavy-vehicle closed-loop systemrdquo Advanced Engineer-ing Forum vol 2-3 pp 33ndash38 2012
[30] J Pauwelussen ldquoDependencies of driver steering controlparametersrdquo Vehicle System Dynamics vol 50 no 6 pp 939ndash959 2012
[31] H Imine and V Dolcemascolo ldquoRollover risk prediction ofheavy vehicle in interaction with infrastructurerdquo InternationalJournal ofHeavyVehicle Systems vol 14 no 3 pp 294ndash307 2007
(3)For this heavy-duty vehicle two hydraulic dampers are
fixed on the left and right front suspensions and the tandembalanced suspension does not have any shock absorber Inorder to represent the frictional property of leaf spring thedamping forces of tandem balanced suspension are modeledlinearly The suspension forces between middle or rear axleand vehicle body are given by
11986511990421
= 11987011990421(119911119887+ (120579 minus 120579
0) 1198972minus
12057911990111198973
2
minus 1199111199062+
(120593 minus 1205931199062) 1198891199042
2
)
+ 11986211990421(119887+
1205791198972minus
12057911990111198973
2
minus 1199062+
( minus 1199062) 1198891199042
2
)
11986511990422
= 11987011990422(119911119887+ (120579 minus 120579
0) 1198972minus
12057911990121198973
2
minus 1199111199062minus
(120593 minus 1205931199062) 1198891199042
2
)
+ 11986211990422(119887+
1205791198972minus
12057911990121198973
2
minus 1199062minus
( minus 1199062) 1198891199042
2
)
11986511990431
= 11987011990431(119911119887+ (120579 minus 120579
0) 1198972+
12057911990111198973
2
minus 1199111199063+
(120593 minus 1205931199063) 1198891199043
2
)
+ 11986211990431(119887+
1205791198972+
12057911990111198973
2
minus 1199063+
( minus 1199063) 1198891199043
2
)
11986511990432
= 11987011990432(119911119887+ (120579 minus 120579
0) 1198972+
12057911990121198973
2
minus 1199111199063minus
(120593 minus 1205931199063) 1198891199043
2
)
+ 11986211990432(119887+
1205791198972+
12057911990121198973
2
minus 1199063minus
( minus 1199063) 1198891199043
2
)
(4)
Since hydraulic dampers show obvious nonlinearitymany dynamic models for shock absorber have been pro-posed amongwhich the fittedmodel is quite suitable tomod-eling the ascertained shock absorber but needs a large amountof experimental work [17 18] In this work the dynamic prop-erty of the damper on front suspension ismeasured byHT-911testing machine under sinusoidal excitation Since the inher-ence frequency of the vehicle body is from 1Hz to 25Hzfour excitation frequencies are selected as 1Hz 15Hz 2Hzand 25Hz Limited by the machinersquos tonnage the excitationamplitude is chosen as 10mm Figure 2 shows the measuredforce-velocity curves Since the hysteresis loops of dampingforce depend on the excitation frequency and amplitudegreatly the parameters of hysteresis model under randomexcitations are difficult to identify Hence a nonlinear seg-mented model describing the damperrsquos scheme framework isproposed here
119865119889=
11986501sgn (V
119889) V
119889gt Vlim 1
119862 (1 + 120573 sgn (V119889)) V119889
1003816100381610038161003816V119889
1003816100381610038161003816
119899 Vlim 2 le V119889le Vlim 1
11986502sgn (V
119889) V
119889lt Vlim 2
(5)
where V119889 119862 120573 and 119899 are the relative velocity of cylinder
and plunger the damping coefficient the asymmetry ratioand the exponent respectively 119865
01 11986502 Vlim 1 and Vlim 2 are
the damping force and relative velocity when the damperreaching saturation in tension or compression process
Parameters in model (5) fitted to the measured data are119862 = 30893 120573 = 056 119899 = 016 119865
01= 4119N 119865
02= 726N
Vlim 1 = 012ms and Vlim 2 = 008ms The damping forcecurve obtained from the theoretical damper model is shownas the thick solid line in Figure 2 It can be seen that thepresented damping forcemodel is able to describe the schemeframework and saturation property of the damper Thoughmodel (5) neglects the damperrsquos hysteresis characteristics itis simple and accurate enough for numerical simulation
Using model (5) to calculate the damping force the frontsuspension forces are expressed by
11986511990411
= 11987011990411(119911119887minus (120579 minus 120579
0) 1198971minus 1199111199061+
(120593 minus 1205931199061) 1198891199041
2
) + 11986511988911
11986511990412
= 11987011990412(119911119887minus (120579 minus 120579
0) 1198971minus 1199111199061minus
(120593 minus 1205931199061) 1198891199041
2
) + 11986511988912
(6)
where 11986511988911
and 11986511988912
are the left and right damping forceof front suspension The relative velocities of left and right
Mathematical Problems in Engineering 5
(a)
0 005 01 015 02 025
0
1000
2000
3000
4000
25 Hz20 Hz15 Hz
10 HzTheoretical
minus02 minus01minus015 minus005
minus1000
d (ms)
Fc
(N)
(b)
Figure 2 Dynamic test and modeling of the damper
damper are obtained by V1198891= (119887minus
1205791198971minus 1199061+ ( minus
1199061)11988911990412)
and V1198892= (119887minus
1205791198971minus 1199061minus ( minus
1199061)11988911990412)
The following equations give the balancing pole pitchthe cab vertical roll and pitch and the axle vertical and rollmovements
(119894 = 1 sim 3 119895 = 1 sim 2) are the drivingtorque and braking torque of six wheels
23 Driver Model According to Guorsquos preview of optimalcurvature drivermodel [3 4] the optimal front steering angleis expressed by
120575119901=
2119871
1198892[119891 (119905 + 119879) minus 119910 (119905) minus 119879 119910 (119905)] (13)
where 119889 119879 and 119871 are the preview distance preview time andwheelbase respectively 119891(119905 + 119879) is the lateral position of thedesired route at preview point and 119910(119905) is the vehicle lateralposition at current time
The above model is very simple and suitable to simulatelateral dynamics of vehicle running at a constant speed How-ever it neglects the effect of time delay and the desired routefunction 119891(119905) needs to be computed according to vehiclespeed and trajectory before simulation
Legouisrsquo driver model with nonlinear time delay [6ndash8]calculates the front steering angle by
120575119901= minus119870[119910
119873(119905 minus 119879
119903) +
119889
119880
119910119873(119905 minus 119879
119903)] (14)
where119870 119879119903 and119880 are feedback gain time delay and vehicle
running speed respectively 119910119873(119905 minus 119879
119903) and 119910
119873(119905 minus 119879
119903) +
(119889119880) 119910119873(119905 minus 119879
119903) are the lateral position of vehicle gravity
center and preview point in the inertial frames respectivelyThis model introduced time delay and calculated positiondeviation between vehicle and desired route in the inertialframes However the feedback gain in Legouisrsquo driver modelis a constant and cannot be obtained by vehicle parametersThe ideal route function is not included in the model becausethe straight-line driving condition is researched In additionthe preview point is gained from vehicle gravity center andneglects the difference of longitudinal distance and yaw anglebetween vehicle gravity center and driver position
By combining the above two models a modified drivermodel is proposed here as shown in Figure 3The front wheelsteering angle is given by
120575119901(119905) =
2119871
1198892[119877119884119889(119905 minus 119879
119903) minus 119884119889(119905 minus 119879
119903)]
1003816100381610038161003816119877119884119889(119905 minus 119879
where119877119884119889119884119889 119890119888119903 and119889119905 are the required lateral position and
real lateral displacement of the preview point in the ground
Y
O Xd
Required route RYdYd
Fty12Fty22Fty32
x
o120595Vx120596r
Vy120575
dt2
Fty31Fty21
Fty11l3
l2l1
dy
Figure 3 The new driver control model with nonlinear time delay
coordinate system the permit position error and the inte-gration time step respectively For three-axle vehicle theparameter 119871 in (15) is the distance between front wheel andcenter of balance suspension and is expressed by 119871 = 119897
1+ 1198972
This modified model has a feedback gain 21198711198892 that is
defined by wheelbase and preview distance It can be noticedthat a big wheelbase or a small preview distance will lead toa big gain This statement is well understood because driversfeel it difficult to control the vehicle direction in the case of bigwheelbase or small preview distance so they have to increasethe intervention of steering angle
The displacement of the preview point in the groundcoordinate system is
119883119889= 119883 (119905) + 119897
1cos120595 + 119889
119880
( minus 1198971 sin120595)
119884119889= 119884 (119905) + 119897
1sin120595 + 119889
119880
( + 1198971 cos120595)
(16)
where119883(119905) 119884(119905) and 120595 are displacements and heading angleof vehicle gravity center in the ground coordinate system Itshould be noted that the preview point lies in 119889meter aheadof driver seat not vehicle gravity center According to 119883
119889
and the required route function the required lateral position119877119884119889is easily obtained Substituting 119877119884
119889and 119884
119889into (15) and
introducing time delay the front wheel angle can be obtainedIt should be noted that themodified drivermodel depends onthe vehicle longitudinal speed as strongly as Legouisrsquo drivermodel
The displacements and velocities in vehicle coordinatesystem (119909 119910 119911) can be gained from vehicle model andtransferred to the ground coordinate system (119883 119884 119885) by thefollowing relation
= 119881119909cos120595 minus 119881
119910sin120595
= 119881119909sin120595 + 119881
119910cos120595
(17)
Finally the vehicle model tire model and driver modelare coupled into the driver-vehicle closed-loop system Thelongitudinal slip ratios of six wheels are calculated in real-time with vehicle responses as inputThe front wheel steeringangle is obtained by the modified driver model and fed back
Mathematical Problems in Engineering 7
Coupled vehicle model
Wheel rotate equations
Braking toque
Three-directionaltire model
Slip ratio equation
Drivermodel
Output responses
Road surface properties
Requiredroute
Vehicle initial conditions
120596ij Mb
ztij
Ftxij FtyijFtzijMtzij
120575rij 120583ij
Vx
Ssij
RY119889(t minus Tr)
Yd(t minus Tr)
Xd(t minus Tr)
Figure 4 The driver-heavy-vehicle closed-loop model
to the tire model Then the vertical longitudinal and lateraltire forces are calculated by the tire model and input intothe vehicle model to gain vehicle responses and positionsin next time step The simulation process of this driver-vehicle closed-loop system is shown in Figure 4 Due to thetime variability nonlinearity and high-dimensional propertyof this system the closed-loop system equations are solvednumerically by the quick integration method [22] and theRunge-Kutta method of order four
3 Model Evaluation
In order to verify the presented TCLP vehicle model andthe new driver model simulation results of this driver-vehicle closed-loop system are obtained using different vehi-cle models or driver models During simulation the vehicleparameters are chosen for a DFL1250A9 truck manufacturedby Dongfeng Motor Group Company Limited [23ndash25] andthe B-class road roughness is selected referring to [26]
The parameters of double-lane change route for theheavy-duty vehicle are chosen referring to [27 28] and shownin Figure 5
31 Comparison with the Handling Stability Vehicle Model Atraditional two-degree of freedom (2DOF) handling stabilityvehicle model for a three-axle heavy vehicle is set up whichconsiders only the lateral and yaw motion [29] The ordinarydifferential equations of motion of this 2DOF model may beexpressed by
where 119898 119868119911 119881119909 119881119910 and 120596
119903are vehicle mass vehicle inertia
around 119911 axial and longitudinal lateral and yaw rate of thevehicle respectively 120575
119894119865120572119894 and119872
119911119894are steering angle lateral
tire force and self-aligning torque of wheels respectively 119897119909119894
and 119897119910119894are the distance from wheel center to vehicle gravity
center in longitudinal and lateral direction respectivelyThe double-lane change responses of this TCLP model
and the traditional 2DOF model at an entrance speedof 60 kmh are simulated respectively and compared inFigure 6 It can be seen from Figure 6 that the results of thesetwo models are very consistent in magnitude and trendsHence the two vehicle models verify each other The TCLPmodel has a worse path-following ability than the 2DOFmodel and the vehicle running speed of TCLP model fluctu-ates randomlyThe yaw rate and lateral acceleration obtainedfrom TCLP model is bigger than that from 2DOF modelThe reason for these differences between two models is thatthe TCLP vehicle model considers B-class road roughnessand the coupled effect of roll vertical longitudinal and pitchmotion on yaw and lateral motion while the 2DOF modelneglects them
32 Comparison with the Linear Vehicle Model When thenonlinearity of front suspension damper and tire force isneglected as shown in Figure 7 the vehicle responses becomesmaller and the tracking performance is better Thus thelinear vehicle model may predict more conservative results
8 Mathematical Problems in Engineering
20m
20m40m40m
35m35m
15m
30m
35
m
35
m
35
m
375
m
Figure 5 The lane change route for the heavy-duty vehicle
0 5 10
162
164
166
168
17
0 5 10 15
0
5
Coupled model2DOF model
Required route
minus5
50 100 150 200
0
2
4
Y(m
)
X (m)
u(m
s)
t (s)
0 5 10 15
0
02
04
Coupled model2DOF model
Required route
minus04
minus02
t (s) t (s)
wr (
rad
s)
ay(m
s2)
Figure 6 Double-lane change responses of TCLP model and 2DOF model at 60 kmh
In the meantime it is found that the difference between thelinear and nonlinear vehicle models in the maneuver of lanechange is much greater than that in the maneuver of drivingstraight The lateral load transfer yaw motion and lateralmotion while lane change destabilizes the vehicle and conse-quently aggravates other vehicle motions due to couple effectIn addition by simulating responses at different entrancespeeds we found that the difference between the linearand nonlinear vehicle models becomes greater with the riseof vehicle speed Therefore the nonlinear vehicle modelshould be used so as to simulate vehicle dynamics accurately
especially in lane change maneuver or high speed drivingcondition
33 Comparison with Test Data During double-lane changetest an empty loaded DFL1250A9 truck was selected to runat 30 kmh speed A three-axis piezoelectric accelerometer(frequency range from 1Hz to 500Hz) and a gyro (measur-ing range plusmn50 degs) were placed at the center of gravity ofwhole vehicle and a cable displacement sensor was placedat front wheel to measure front wheel steering angle asshown in Figure 8 The measured signals were amplified by
Mathematical Problems in Engineering 9
50 100 150 200 250 300
0
1
2
3
4
minus1
Y(m
)
X (m)0 5 10 15 20
162
164
166
168
u(m
s)
t (s)
5 10 15 20
0
2
4
minus4
minus2
t (s)
ay
(ms
2 )
5 10 15 20
0
01
02
03
minus02
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
001
minus002
minus001
t (s)
120573(r
ad)
7 8 9
0
10
20
minus20
minus10
t (s)
azb
(ms
2 )
5 10 15 20
0
01
02
minus02
minus01
NonlinearLinear
Requied route
t (s)
Φ(r
ad)
5 10 15 20
0
005
minus01
minus005
NonlinearLinear
Requied route
t (s)
120579(r
ad)
Figure 7 The double-lane change responses of the linear and nonlinear TCLP model at 60 kmh
10 Mathematical Problems in Engineering
Figure 8 Vehicle and sensors in field test
Table 1 Vehicle responses obtained from three driver models
Vehicle speed Driver model RMS value of vehicle responses119906 119908119903 119886119910 119886119911119887 119891119894 119909119894119905119886
a multichannel charge amplifier (YE5853A) and convertedto digital signals by an intelligent acquisition and processinganalyzer (INV360DF) The sampling frequency was 200Hz
Figure 9 gives the experimental time history of frontwheel steering angle Using this measured front wheel steer-ing angle as input the yaw rate and lateral acceleration are cal-culated from TCLP model 2DOF model and linear modelrespectively These simulation results are compared with
the test results as shown in Figure 10 It is obvious thatthe simulation results of TCLP model are most consistentwith the results of field test Due to neglecting of the three-directional coupling effects of vehiclemotions or nonlinearityof suspension and tire the time histories of yaw rate obtainedfrom 2DOF model and linear model are too smooth and thelateral accelerations of 2DOF model and linear model aresmaller than the test results Thus the verification of thisproposed TCLP model is verified
34 Comparison with Other Driver Models Using the pro-posed modified driver model the optimal curvature modelby Guo and nonlinear time delay driver model by Legouisthe trajectories of vehicle gravity center during double-lanechange maneuver at four entrance speeds of 20 40 60 and80 kmh are simulated respectively as shown in Figure 11Table 1 lists the root mean square (RMS) value of vehicleresponses in six directions at different speeds applying thesethree driver models As shown in Figure 11 it is obvious thatthe tracking performance of the modified driver model issuperior to the other two driver models at different entrancespeeds It is also found from Figure 11 that though an over-shoot of the newmodel at a vehicle speed of 80 kmh exists in119883 = [150 180]m the results of the other two models divergewhen the vehicle arrives at 119883 = 200m while the resultof the new model is stable From Table 1 we can see thatthe real vehicle speed of the modified driver model is the
Mathematical Problems in Engineering 11
Simulation results of TCLP model
Test results
Simulation results of 2DOF model
Simulation results of linear model
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
02
04
minus04
minus02
t (s)
ay(m
s2 )
5 10 15 20
0
05
1
minus05
t (s)
ay(m
s2 )
Figure 10 Comparison of simulation and test results
12 Mathematical Problems in Engineering
50 100 150 200 250 300
0
1
2
3
4
5
minus1
Y(m
)
X (m)
(a) 20 kmh
50 100 150 200 250
0
1
2
3
4
Y(m
)
X (m)
(b) 40 kmh
50 100 150 200 250 300
0
1
2
3
4
5
NewLiu
GuoRoute
minus1
Y(m
)
X (m)
(c) 60 kmh
50 100 150 200
0
2
4
NewLiu
GuoRoute
minus2
Y(m
)
X (m)
(d) 80 kmh
Figure 11 Comparison of modified driver model with other driver models at different speeds
closest to the assumed speed and the modified driver modelhas smaller responses except for vertical vibration and betterstability than the other models Though the modified drivermodel obtains greater vertical vehicle acceleration and worseride comfort than Legouisrsquos model at speeds lower than60 kmh the new driver model can give the vehicle betterpath-following ability and handling stability
4 Effects of Driver Model Parameters onFull Vehicle Dynamics and Stability
The important parameters in driver model include vehiclerunning speed time delay preview distance and permitposition error which decide the steering control effect andvehicle dynamics During the same driving maneuver theskilled drivers may use higher vehicle speed longer previewdistance shorter time delay and larger permit position error
than the new drivers The match of these four driverparameters also influences the vehicle dynamics RecentlyPauwelussen [30] researched the dependencies of driversteering control parameters on vehicle lateral properties andpath keeping and gives some useful conclusions This workfocuses on the effects of driver model parameters on bothpath keeping and full vehicle dynamics such as vehiclehandling stability ride comfort and roll stability
Since the vehicle lateral displacement is randomly causedby road roughness excitations and the lateral position devi-ation is always changing during lane change maneuvers astatistic index is introduced to evaluate the vehiclersquos path-following ability globally which is formulated by
Route error = radicsum (119877119884
119889(119905) minus 119884
119889(119905))2
119873
(20)
Mathematical Problems in Engineering 13
20 40 60 80 100 120 140 160 180 200
0
1
2
3
4
5
20 kmh30 kmh40 kmh
50 kmh60 kmhRoute
Y(m
)
X (m)
minus1
0 50 100 150 200
0
1
2
3
4
0 s002 s005 s
01 s02 sRoute
Y(m
)
X (m)
minus1
Figure 12 Routes of vehicle under different speed or time delay
where 119873 is the total number of data points 119877119884119889and 119884
119889are
the required lateral position and real lateral displacement ofthe preview point in the ground coordinate system
When a vehicle is undergoing a lane change its steeringstability and roll stability are vital problems The path-following error lateral acceleration and yaw rate can be usedto evaluate the steering stability The roll angle can be used toevaluate the roll stability Being proportional to the rolloverindex LTR (lateral-load transfer rate) the lateral accelerationis also able to reflect the vehicle roll stability [31] In additionthe vertical acceleration and pitch angle are simulated so asto research the whole-body dynamics in the vehicle Usingthese evaluation indexes the effects of parameters on steeringstability roll stability and riding comfort are discussed indetail
Figure 12 shows the vehicle trajectories under differentinitial vehicle speeds or driver time delays The effects ofvehicle speed and driver time delay on full vehicle dynamicsand stability are shown in Figure 13 As we can see fromFigures 12 and 13 a higher speed or a longer time delay willlead to a greater path-following error However low speed(20 kmh) and longtime delay (035 s) may enlarge the path-following error With the rise of vehicle speed the verticalacceleration lateral acceleration yaw rate and roll angleincrease while the pitch angle decreases Rise of time delaycauses the increase of lateral acceleration yaw rate and rollangle but hardly changes the vertical acceleration and pitchangle As for the effects of vehicle speed and driver time delayon full vehicle dynamics the following may be concluded
(1) A high vehicle speed during double-lane changeyields bad path keeping performance ride comfortsteering stability and roll stability When the vehiclespeed is higher than 60 kmh the ride comfort of thisvehicle gets worse greatly
(2) A long time delay is harmful to path keeping per-formance steering stability and roll stability and hasalmost no effect on the ride comfort
Thus a low vehicle speed ranging from 20 kmh to 60 kmh and a small time delay less than 035 s are recommended forimproving this truckrsquos stability when going through the lanechange
In case of varying the preview distance or permit positionerror the variation in vehicle trajectories at the speed of40 kmh is shown in Figure 14The effects of preview distanceor permit position error on the path-following ability andfull vehicle dynamics are shown in Figure 15 As shown inFigures 14 and 15 a large preview distance deteriorates thepath-following ability but has small effect on other dynamicresponses On the other hand a too small preview distanceleads to a hop of lateral acceleration yaw rate and roll anglewhich conflicts with closed-loop stability and is harmfulto steering control This conclusion is consistent with thatof Gillespie and MacAdam [15] and Pauwelussen [30] Theinfluence of permit position error on path-following ability issmall at short preview distance but great at long preview dis-tance As shown in Figure 14 a too small or too large permitposition error (001m or 08m)will worsen the vehiclersquos path-following ability In addition large permit position error mayincrease vertical acceleration lateral acceleration yaw rateand roll angle and reduce the handling stability and rollstability and ride comfort The effect of preview distance andpermit position error on pitch angle is small and shows fluc-tuation Thus a preview distance ranging from 10m to 30mand a small permit position error but more than 001m arerecommended for improving this truckrsquos stability when mov-ing through the lane change A smaller permit position errorshould be matched with a shorter preview distance
14 Mathematical Problems in Engineering
002
04
2040
60800
1
2
Rout
e err
or (m
)
u0 (kmh) 0
02
04
2040
60807
8
9
u0 (kmh)
002
04
204060800
02
04
wr (
rad
s)
u0 (kmh)
0
02
04
204060800
005
01
120579(r
ad)
u0 (kmh)002
04
2040
60800
01
02
Φ(r
ad)
u0 (kmh)
002
04
2040
60800
2
4
u0(kmh)
ay(m
s2 )
Tr(s) Tr
(s)
Tr(s)
Tr(s)
Tr(s) Tr
(s)
azb
(ms2 )
Figure 13 The effects of vehicle speed and driver time delay
0 50 100 150 200
0
1
2
3
4
5
001 m02 m04 m
06 m08 mRoute
Y(m
)
X (m)
minus10 50 100 150 200
0
2
4
6
8
10
12
2 m10 m20 m
30 m40 mRoute
Y(m
)
X (m)
minus2
minus4
Figure 14 Routes of vehicle under different preview distance and permit position error
Mathematical Problems in Engineering 15
020
4060
005
1150
1
2
Rout
e err
or (m
)
er (m) Ld (m)
020
4060
01
25
10
15
er (m) Ld (m)
020
4060
01
20
5
er (m) Ld (m) 020
4060
01
20
02
04
er (m) Ld (m)wr (
rad
s)
2040
601
20
02
04
00er (m) Ld (m)
Φ(r
ad)
020
40 60
01
20
005
01
er (m) Ld (m)
120579(r
ad)
ay(m
s2 )
azb
(ms2 )
Figure 15 The effects of preview distance and permit position error
5 Conclusions
In this paper a nonlinear three-directional coupled lumpedparameters (TCLP) model of heavy-duty vehicle consideringthe nonlinearity of suspension damping and tire stiffness isbuilt and connected with a proposedmodified preview drivermodel with nonlinear time delayThe proposed driver modelis simple and has a feedback gain 2119871119889
2 that is decided bywheelbase and previewdistanceThe validity of this presenteddriver-vehicle closed-loop system is verified by comparisonwith the test data the traditional steering stability vehiclemodel the linear vehicle model and other driver controlmodelsThe results show that the new drivermodel has betterlane keeping performances than the other two driver modelsThe coupling effects of vehicle motions and the nonlinear-ity of suspension damping and tire stiffness could not beneglected in turning or high speed driving situation
The effects of driver parameters on path-following abilityand full vehicle dynamics are also discussed It is foundthat the high vehicle speed large time delay long previewdistance or big permit position error will deteriorate path-following performances and handling stability Of course atoo short preview distance or a too small permit position
error is also harmful to path-following performances andhandling stability The ride comfort mainly depends on vehi-cle speed preview distance and permit position error Thekey factors influencing roll characteristics are vehicle speedand time delay
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The National Natural Science Foundation of China(11472180) theNatural Science Foundation ofHebei Province(E2012210025) and the New Century Talent Foundation ofMinistry of Education (NCET-13-0913) support this work
References
[1] M Plochl and J Edelmann ldquoDriver models in automobiledynamics applicationrdquoVehicle SystemDynamics vol 45 no 7-8pp 699ndash741 2007
16 Mathematical Problems in Engineering
[2] C C MacAdam ldquoUnderstanding and modeling the humandriverrdquo Vehicle System Dynamics vol 40 no 1ndash3 pp 101ndash1342003
[3] K Guo and H Guan ldquoModeling of drivervehicle directionalcontrol systemrdquo Vehicle System Dynamics vol 22 no 3-4 pp141ndash184 1993
[4] K GuoVehicle Handling DynamicsTheory Jiangsu Science andTechnology Publishing House Nanjing China 2011
[5] R S Sharp ldquoDriver steering control and a new perspective oncar handling qualitiesrdquo Proceedings of the Institution ofMechani-cal Engineers Part C Journal of Mechanical Engineering Sciencevol 219 no 10 pp 1041ndash1051 2005
[6] T Legouis A Laneville P Bourassa and G Payre ldquoCharac-terization of dynamic vehicle stability using two models of thehuman pilot behaviorrdquo Vehicle System Dynamics vol 15 no 1pp 1ndash18 1986
[7] Z Liu G Payre and P Bourassa ldquoNonlinear oscillations andchaotic motions in a road vehicle system with driver steeringcontrolrdquo Nonlinear Dynamics vol 9 no 3 pp 281ndash304 1996
[8] Z Liu G Payre and P Bourassa ldquoStability and oscillations ina time-delayed vehicle system with driver controlrdquo NonlinearDynamics vol 35 no 2 pp 159ndash173 2004
[9] X D Liu W Feng and N Kang ldquoResearch of simulationmethod of tractor-semitrailer carrying liquid considering liquidsloshingrdquo in Proceedings of the 10th National Conference onVibrationTheory and Application pp 581ndash590 Nanjing ChinaOctober 2011
[10] C C MacAdam ldquoApplication of an optimal preview control forsimulation of closed-loop automobile drivingrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 11 no 6 pp 393ndash399 1981
[11] A Y Ungoren and H Peng ldquoAn adaptive lateral preview drivermodelrdquo Vehicle System Dynamics vol 43 no 4 pp 245ndash2592005
[12] C I Chatzikomis and K N Spentzas ldquoA path-following drivermodel with longitudinal and lateral control of vehiclersquos motionrdquoForschung im Ingenieurwesen vol 73 no 4 pp 257ndash266 2009
[13] R S Sharp D Casanova and P Symonds ldquoA mathematicalmodel for driver steering control with design tuning andperformance resultsrdquo Vehicle System Dynamics vol 33 no 5pp 289ndash326 2000
[14] A J Pick and D J Cole ldquoNeuromuscular dynamics in thedriver-vehicle systemrdquo Vehicle System Dynamics vol 44 sup-plement 1 pp 624ndash631 2006
[15] T D Gillespie and C C MacAdam Constant Velocity YawRollProgram Userrsquos Manual The University of Michigan Trans-portation Research Institute Ann Arbor Mich USA 1982
[16] R I S Raj Influence of road roughness and directional maneuveron the dynamic performance of heavy vehicles [MS thesis]Department of Mechanical Engineering Concordia UniversityMontreal Canada 1998
[17] KWordenDHickeyMHaroon andD E Adams ldquoNonlinearsystem identification of automotive dampers a time and fre-quency-domain analysisrdquo Mechanical Systems and Signal Pro-cessing vol 23 no 1 pp 104ndash126 2009
[18] S H Li Y J Lu and L Y Li ldquoDynamical test and modelingfor hydraulic shock absorber on heavy vehicle under harmonicand random loadingsrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 13 pp 1903ndash1910 2012
[19] X W Ji and Y M Gao ldquoThe dynamic stiffness and dampingcharacteristics of the tirerdquo Automotive Engineering vol 5 pp315ndash321 1994
[20] G Gim and P E Nikravesh ldquoA three dimensional tire modelfor steady-state simulations of vehiclesrdquo SAE vol 102 no 2 pp150ndash159 1993
[21] J D Zhuang Principles of Automobile Tire Beijing Institute ofTechnology Press Beijing China 1996
[22] W-M Zhai ldquoTwo simple fast integration methods for large-scale dynamic problems in engineeringrdquo International Journalfor Numerical Methods in Engineering vol 39 no 24 pp 4199ndash4214 1996
[23] Dongfeng Motor Group Company Dongfeng DFL1250A8DFL1250A9 Truck Chassis Refit Manual Dongfeng MotorGroup Company 2008
[24] S Li S Yang L Chen and Y Lu ldquoEffects of parameters ondynamic responses for a heavy vehicle-pavement- foundationcoupled systemrdquo International Journal of Heavy Vehicle Systemsvol 19 no 2 pp 207ndash224 2012
[25] S Yang S Li andY Lu ldquoInvestigation on dynamical interactionbetween a heavy vehicle and road pavementrdquo Vehicle SystemDynamics vol 48 no 8 pp 923ndash944 2010
[26] GBT 7031-2005ISO 86081995 Mechanical Vibration-RoadSurface Profiles-Reporting of Measured Data 2005
[27] M Li X Pu and Z Changfu ldquoModeling and simulationof heavy-duty semi-trailer in extreme condition based onADAMSrdquo Automobile Technology vol 2 pp 22ndash25 2009
[28] ldquoPassenger carsmdashtest track for a severe lane-change manoeu-vremdashpart 1 double lane-changerdquo ISO 3888-11999
[29] S H Li S P Yang and N Chen ldquoDirectional control of adriver-heavy-vehicle closed-loop systemrdquo Advanced Engineer-ing Forum vol 2-3 pp 33ndash38 2012
[30] J Pauwelussen ldquoDependencies of driver steering controlparametersrdquo Vehicle System Dynamics vol 50 no 6 pp 939ndash959 2012
[31] H Imine and V Dolcemascolo ldquoRollover risk prediction ofheavy vehicle in interaction with infrastructurerdquo InternationalJournal ofHeavyVehicle Systems vol 14 no 3 pp 294ndash307 2007
(3)For this heavy-duty vehicle two hydraulic dampers are
fixed on the left and right front suspensions and the tandembalanced suspension does not have any shock absorber Inorder to represent the frictional property of leaf spring thedamping forces of tandem balanced suspension are modeledlinearly The suspension forces between middle or rear axleand vehicle body are given by
11986511990421
= 11987011990421(119911119887+ (120579 minus 120579
0) 1198972minus
12057911990111198973
2
minus 1199111199062+
(120593 minus 1205931199062) 1198891199042
2
)
+ 11986211990421(119887+
1205791198972minus
12057911990111198973
2
minus 1199062+
( minus 1199062) 1198891199042
2
)
11986511990422
= 11987011990422(119911119887+ (120579 minus 120579
0) 1198972minus
12057911990121198973
2
minus 1199111199062minus
(120593 minus 1205931199062) 1198891199042
2
)
+ 11986211990422(119887+
1205791198972minus
12057911990121198973
2
minus 1199062minus
( minus 1199062) 1198891199042
2
)
11986511990431
= 11987011990431(119911119887+ (120579 minus 120579
0) 1198972+
12057911990111198973
2
minus 1199111199063+
(120593 minus 1205931199063) 1198891199043
2
)
+ 11986211990431(119887+
1205791198972+
12057911990111198973
2
minus 1199063+
( minus 1199063) 1198891199043
2
)
11986511990432
= 11987011990432(119911119887+ (120579 minus 120579
0) 1198972+
12057911990121198973
2
minus 1199111199063minus
(120593 minus 1205931199063) 1198891199043
2
)
+ 11986211990432(119887+
1205791198972+
12057911990121198973
2
minus 1199063minus
( minus 1199063) 1198891199043
2
)
(4)
Since hydraulic dampers show obvious nonlinearitymany dynamic models for shock absorber have been pro-posed amongwhich the fittedmodel is quite suitable tomod-eling the ascertained shock absorber but needs a large amountof experimental work [17 18] In this work the dynamic prop-erty of the damper on front suspension ismeasured byHT-911testing machine under sinusoidal excitation Since the inher-ence frequency of the vehicle body is from 1Hz to 25Hzfour excitation frequencies are selected as 1Hz 15Hz 2Hzand 25Hz Limited by the machinersquos tonnage the excitationamplitude is chosen as 10mm Figure 2 shows the measuredforce-velocity curves Since the hysteresis loops of dampingforce depend on the excitation frequency and amplitudegreatly the parameters of hysteresis model under randomexcitations are difficult to identify Hence a nonlinear seg-mented model describing the damperrsquos scheme framework isproposed here
119865119889=
11986501sgn (V
119889) V
119889gt Vlim 1
119862 (1 + 120573 sgn (V119889)) V119889
1003816100381610038161003816V119889
1003816100381610038161003816
119899 Vlim 2 le V119889le Vlim 1
11986502sgn (V
119889) V
119889lt Vlim 2
(5)
where V119889 119862 120573 and 119899 are the relative velocity of cylinder
and plunger the damping coefficient the asymmetry ratioand the exponent respectively 119865
01 11986502 Vlim 1 and Vlim 2 are
the damping force and relative velocity when the damperreaching saturation in tension or compression process
Parameters in model (5) fitted to the measured data are119862 = 30893 120573 = 056 119899 = 016 119865
01= 4119N 119865
02= 726N
Vlim 1 = 012ms and Vlim 2 = 008ms The damping forcecurve obtained from the theoretical damper model is shownas the thick solid line in Figure 2 It can be seen that thepresented damping forcemodel is able to describe the schemeframework and saturation property of the damper Thoughmodel (5) neglects the damperrsquos hysteresis characteristics itis simple and accurate enough for numerical simulation
Using model (5) to calculate the damping force the frontsuspension forces are expressed by
11986511990411
= 11987011990411(119911119887minus (120579 minus 120579
0) 1198971minus 1199111199061+
(120593 minus 1205931199061) 1198891199041
2
) + 11986511988911
11986511990412
= 11987011990412(119911119887minus (120579 minus 120579
0) 1198971minus 1199111199061minus
(120593 minus 1205931199061) 1198891199041
2
) + 11986511988912
(6)
where 11986511988911
and 11986511988912
are the left and right damping forceof front suspension The relative velocities of left and right
Mathematical Problems in Engineering 5
(a)
0 005 01 015 02 025
0
1000
2000
3000
4000
25 Hz20 Hz15 Hz
10 HzTheoretical
minus02 minus01minus015 minus005
minus1000
d (ms)
Fc
(N)
(b)
Figure 2 Dynamic test and modeling of the damper
damper are obtained by V1198891= (119887minus
1205791198971minus 1199061+ ( minus
1199061)11988911990412)
and V1198892= (119887minus
1205791198971minus 1199061minus ( minus
1199061)11988911990412)
The following equations give the balancing pole pitchthe cab vertical roll and pitch and the axle vertical and rollmovements
(119894 = 1 sim 3 119895 = 1 sim 2) are the drivingtorque and braking torque of six wheels
23 Driver Model According to Guorsquos preview of optimalcurvature drivermodel [3 4] the optimal front steering angleis expressed by
120575119901=
2119871
1198892[119891 (119905 + 119879) minus 119910 (119905) minus 119879 119910 (119905)] (13)
where 119889 119879 and 119871 are the preview distance preview time andwheelbase respectively 119891(119905 + 119879) is the lateral position of thedesired route at preview point and 119910(119905) is the vehicle lateralposition at current time
The above model is very simple and suitable to simulatelateral dynamics of vehicle running at a constant speed How-ever it neglects the effect of time delay and the desired routefunction 119891(119905) needs to be computed according to vehiclespeed and trajectory before simulation
Legouisrsquo driver model with nonlinear time delay [6ndash8]calculates the front steering angle by
120575119901= minus119870[119910
119873(119905 minus 119879
119903) +
119889
119880
119910119873(119905 minus 119879
119903)] (14)
where119870 119879119903 and119880 are feedback gain time delay and vehicle
running speed respectively 119910119873(119905 minus 119879
119903) and 119910
119873(119905 minus 119879
119903) +
(119889119880) 119910119873(119905 minus 119879
119903) are the lateral position of vehicle gravity
center and preview point in the inertial frames respectivelyThis model introduced time delay and calculated positiondeviation between vehicle and desired route in the inertialframes However the feedback gain in Legouisrsquo driver modelis a constant and cannot be obtained by vehicle parametersThe ideal route function is not included in the model becausethe straight-line driving condition is researched In additionthe preview point is gained from vehicle gravity center andneglects the difference of longitudinal distance and yaw anglebetween vehicle gravity center and driver position
By combining the above two models a modified drivermodel is proposed here as shown in Figure 3The front wheelsteering angle is given by
120575119901(119905) =
2119871
1198892[119877119884119889(119905 minus 119879
119903) minus 119884119889(119905 minus 119879
119903)]
1003816100381610038161003816119877119884119889(119905 minus 119879
where119877119884119889119884119889 119890119888119903 and119889119905 are the required lateral position and
real lateral displacement of the preview point in the ground
Y
O Xd
Required route RYdYd
Fty12Fty22Fty32
x
o120595Vx120596r
Vy120575
dt2
Fty31Fty21
Fty11l3
l2l1
dy
Figure 3 The new driver control model with nonlinear time delay
coordinate system the permit position error and the inte-gration time step respectively For three-axle vehicle theparameter 119871 in (15) is the distance between front wheel andcenter of balance suspension and is expressed by 119871 = 119897
1+ 1198972
This modified model has a feedback gain 21198711198892 that is
defined by wheelbase and preview distance It can be noticedthat a big wheelbase or a small preview distance will lead toa big gain This statement is well understood because driversfeel it difficult to control the vehicle direction in the case of bigwheelbase or small preview distance so they have to increasethe intervention of steering angle
The displacement of the preview point in the groundcoordinate system is
119883119889= 119883 (119905) + 119897
1cos120595 + 119889
119880
( minus 1198971 sin120595)
119884119889= 119884 (119905) + 119897
1sin120595 + 119889
119880
( + 1198971 cos120595)
(16)
where119883(119905) 119884(119905) and 120595 are displacements and heading angleof vehicle gravity center in the ground coordinate system Itshould be noted that the preview point lies in 119889meter aheadof driver seat not vehicle gravity center According to 119883
119889
and the required route function the required lateral position119877119884119889is easily obtained Substituting 119877119884
119889and 119884
119889into (15) and
introducing time delay the front wheel angle can be obtainedIt should be noted that themodified drivermodel depends onthe vehicle longitudinal speed as strongly as Legouisrsquo drivermodel
The displacements and velocities in vehicle coordinatesystem (119909 119910 119911) can be gained from vehicle model andtransferred to the ground coordinate system (119883 119884 119885) by thefollowing relation
= 119881119909cos120595 minus 119881
119910sin120595
= 119881119909sin120595 + 119881
119910cos120595
(17)
Finally the vehicle model tire model and driver modelare coupled into the driver-vehicle closed-loop system Thelongitudinal slip ratios of six wheels are calculated in real-time with vehicle responses as inputThe front wheel steeringangle is obtained by the modified driver model and fed back
Mathematical Problems in Engineering 7
Coupled vehicle model
Wheel rotate equations
Braking toque
Three-directionaltire model
Slip ratio equation
Drivermodel
Output responses
Road surface properties
Requiredroute
Vehicle initial conditions
120596ij Mb
ztij
Ftxij FtyijFtzijMtzij
120575rij 120583ij
Vx
Ssij
RY119889(t minus Tr)
Yd(t minus Tr)
Xd(t minus Tr)
Figure 4 The driver-heavy-vehicle closed-loop model
to the tire model Then the vertical longitudinal and lateraltire forces are calculated by the tire model and input intothe vehicle model to gain vehicle responses and positionsin next time step The simulation process of this driver-vehicle closed-loop system is shown in Figure 4 Due to thetime variability nonlinearity and high-dimensional propertyof this system the closed-loop system equations are solvednumerically by the quick integration method [22] and theRunge-Kutta method of order four
3 Model Evaluation
In order to verify the presented TCLP vehicle model andthe new driver model simulation results of this driver-vehicle closed-loop system are obtained using different vehi-cle models or driver models During simulation the vehicleparameters are chosen for a DFL1250A9 truck manufacturedby Dongfeng Motor Group Company Limited [23ndash25] andthe B-class road roughness is selected referring to [26]
The parameters of double-lane change route for theheavy-duty vehicle are chosen referring to [27 28] and shownin Figure 5
31 Comparison with the Handling Stability Vehicle Model Atraditional two-degree of freedom (2DOF) handling stabilityvehicle model for a three-axle heavy vehicle is set up whichconsiders only the lateral and yaw motion [29] The ordinarydifferential equations of motion of this 2DOF model may beexpressed by
where 119898 119868119911 119881119909 119881119910 and 120596
119903are vehicle mass vehicle inertia
around 119911 axial and longitudinal lateral and yaw rate of thevehicle respectively 120575
119894119865120572119894 and119872
119911119894are steering angle lateral
tire force and self-aligning torque of wheels respectively 119897119909119894
and 119897119910119894are the distance from wheel center to vehicle gravity
center in longitudinal and lateral direction respectivelyThe double-lane change responses of this TCLP model
and the traditional 2DOF model at an entrance speedof 60 kmh are simulated respectively and compared inFigure 6 It can be seen from Figure 6 that the results of thesetwo models are very consistent in magnitude and trendsHence the two vehicle models verify each other The TCLPmodel has a worse path-following ability than the 2DOFmodel and the vehicle running speed of TCLP model fluctu-ates randomlyThe yaw rate and lateral acceleration obtainedfrom TCLP model is bigger than that from 2DOF modelThe reason for these differences between two models is thatthe TCLP vehicle model considers B-class road roughnessand the coupled effect of roll vertical longitudinal and pitchmotion on yaw and lateral motion while the 2DOF modelneglects them
32 Comparison with the Linear Vehicle Model When thenonlinearity of front suspension damper and tire force isneglected as shown in Figure 7 the vehicle responses becomesmaller and the tracking performance is better Thus thelinear vehicle model may predict more conservative results
8 Mathematical Problems in Engineering
20m
20m40m40m
35m35m
15m
30m
35
m
35
m
35
m
375
m
Figure 5 The lane change route for the heavy-duty vehicle
0 5 10
162
164
166
168
17
0 5 10 15
0
5
Coupled model2DOF model
Required route
minus5
50 100 150 200
0
2
4
Y(m
)
X (m)
u(m
s)
t (s)
0 5 10 15
0
02
04
Coupled model2DOF model
Required route
minus04
minus02
t (s) t (s)
wr (
rad
s)
ay(m
s2)
Figure 6 Double-lane change responses of TCLP model and 2DOF model at 60 kmh
In the meantime it is found that the difference between thelinear and nonlinear vehicle models in the maneuver of lanechange is much greater than that in the maneuver of drivingstraight The lateral load transfer yaw motion and lateralmotion while lane change destabilizes the vehicle and conse-quently aggravates other vehicle motions due to couple effectIn addition by simulating responses at different entrancespeeds we found that the difference between the linearand nonlinear vehicle models becomes greater with the riseof vehicle speed Therefore the nonlinear vehicle modelshould be used so as to simulate vehicle dynamics accurately
especially in lane change maneuver or high speed drivingcondition
33 Comparison with Test Data During double-lane changetest an empty loaded DFL1250A9 truck was selected to runat 30 kmh speed A three-axis piezoelectric accelerometer(frequency range from 1Hz to 500Hz) and a gyro (measur-ing range plusmn50 degs) were placed at the center of gravity ofwhole vehicle and a cable displacement sensor was placedat front wheel to measure front wheel steering angle asshown in Figure 8 The measured signals were amplified by
Mathematical Problems in Engineering 9
50 100 150 200 250 300
0
1
2
3
4
minus1
Y(m
)
X (m)0 5 10 15 20
162
164
166
168
u(m
s)
t (s)
5 10 15 20
0
2
4
minus4
minus2
t (s)
ay
(ms
2 )
5 10 15 20
0
01
02
03
minus02
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
001
minus002
minus001
t (s)
120573(r
ad)
7 8 9
0
10
20
minus20
minus10
t (s)
azb
(ms
2 )
5 10 15 20
0
01
02
minus02
minus01
NonlinearLinear
Requied route
t (s)
Φ(r
ad)
5 10 15 20
0
005
minus01
minus005
NonlinearLinear
Requied route
t (s)
120579(r
ad)
Figure 7 The double-lane change responses of the linear and nonlinear TCLP model at 60 kmh
10 Mathematical Problems in Engineering
Figure 8 Vehicle and sensors in field test
Table 1 Vehicle responses obtained from three driver models
Vehicle speed Driver model RMS value of vehicle responses119906 119908119903 119886119910 119886119911119887 119891119894 119909119894119905119886
a multichannel charge amplifier (YE5853A) and convertedto digital signals by an intelligent acquisition and processinganalyzer (INV360DF) The sampling frequency was 200Hz
Figure 9 gives the experimental time history of frontwheel steering angle Using this measured front wheel steer-ing angle as input the yaw rate and lateral acceleration are cal-culated from TCLP model 2DOF model and linear modelrespectively These simulation results are compared with
the test results as shown in Figure 10 It is obvious thatthe simulation results of TCLP model are most consistentwith the results of field test Due to neglecting of the three-directional coupling effects of vehiclemotions or nonlinearityof suspension and tire the time histories of yaw rate obtainedfrom 2DOF model and linear model are too smooth and thelateral accelerations of 2DOF model and linear model aresmaller than the test results Thus the verification of thisproposed TCLP model is verified
34 Comparison with Other Driver Models Using the pro-posed modified driver model the optimal curvature modelby Guo and nonlinear time delay driver model by Legouisthe trajectories of vehicle gravity center during double-lanechange maneuver at four entrance speeds of 20 40 60 and80 kmh are simulated respectively as shown in Figure 11Table 1 lists the root mean square (RMS) value of vehicleresponses in six directions at different speeds applying thesethree driver models As shown in Figure 11 it is obvious thatthe tracking performance of the modified driver model issuperior to the other two driver models at different entrancespeeds It is also found from Figure 11 that though an over-shoot of the newmodel at a vehicle speed of 80 kmh exists in119883 = [150 180]m the results of the other two models divergewhen the vehicle arrives at 119883 = 200m while the resultof the new model is stable From Table 1 we can see thatthe real vehicle speed of the modified driver model is the
Mathematical Problems in Engineering 11
Simulation results of TCLP model
Test results
Simulation results of 2DOF model
Simulation results of linear model
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
02
04
minus04
minus02
t (s)
ay(m
s2 )
5 10 15 20
0
05
1
minus05
t (s)
ay(m
s2 )
Figure 10 Comparison of simulation and test results
12 Mathematical Problems in Engineering
50 100 150 200 250 300
0
1
2
3
4
5
minus1
Y(m
)
X (m)
(a) 20 kmh
50 100 150 200 250
0
1
2
3
4
Y(m
)
X (m)
(b) 40 kmh
50 100 150 200 250 300
0
1
2
3
4
5
NewLiu
GuoRoute
minus1
Y(m
)
X (m)
(c) 60 kmh
50 100 150 200
0
2
4
NewLiu
GuoRoute
minus2
Y(m
)
X (m)
(d) 80 kmh
Figure 11 Comparison of modified driver model with other driver models at different speeds
closest to the assumed speed and the modified driver modelhas smaller responses except for vertical vibration and betterstability than the other models Though the modified drivermodel obtains greater vertical vehicle acceleration and worseride comfort than Legouisrsquos model at speeds lower than60 kmh the new driver model can give the vehicle betterpath-following ability and handling stability
4 Effects of Driver Model Parameters onFull Vehicle Dynamics and Stability
The important parameters in driver model include vehiclerunning speed time delay preview distance and permitposition error which decide the steering control effect andvehicle dynamics During the same driving maneuver theskilled drivers may use higher vehicle speed longer previewdistance shorter time delay and larger permit position error
than the new drivers The match of these four driverparameters also influences the vehicle dynamics RecentlyPauwelussen [30] researched the dependencies of driversteering control parameters on vehicle lateral properties andpath keeping and gives some useful conclusions This workfocuses on the effects of driver model parameters on bothpath keeping and full vehicle dynamics such as vehiclehandling stability ride comfort and roll stability
Since the vehicle lateral displacement is randomly causedby road roughness excitations and the lateral position devi-ation is always changing during lane change maneuvers astatistic index is introduced to evaluate the vehiclersquos path-following ability globally which is formulated by
Route error = radicsum (119877119884
119889(119905) minus 119884
119889(119905))2
119873
(20)
Mathematical Problems in Engineering 13
20 40 60 80 100 120 140 160 180 200
0
1
2
3
4
5
20 kmh30 kmh40 kmh
50 kmh60 kmhRoute
Y(m
)
X (m)
minus1
0 50 100 150 200
0
1
2
3
4
0 s002 s005 s
01 s02 sRoute
Y(m
)
X (m)
minus1
Figure 12 Routes of vehicle under different speed or time delay
where 119873 is the total number of data points 119877119884119889and 119884
119889are
the required lateral position and real lateral displacement ofthe preview point in the ground coordinate system
When a vehicle is undergoing a lane change its steeringstability and roll stability are vital problems The path-following error lateral acceleration and yaw rate can be usedto evaluate the steering stability The roll angle can be used toevaluate the roll stability Being proportional to the rolloverindex LTR (lateral-load transfer rate) the lateral accelerationis also able to reflect the vehicle roll stability [31] In additionthe vertical acceleration and pitch angle are simulated so asto research the whole-body dynamics in the vehicle Usingthese evaluation indexes the effects of parameters on steeringstability roll stability and riding comfort are discussed indetail
Figure 12 shows the vehicle trajectories under differentinitial vehicle speeds or driver time delays The effects ofvehicle speed and driver time delay on full vehicle dynamicsand stability are shown in Figure 13 As we can see fromFigures 12 and 13 a higher speed or a longer time delay willlead to a greater path-following error However low speed(20 kmh) and longtime delay (035 s) may enlarge the path-following error With the rise of vehicle speed the verticalacceleration lateral acceleration yaw rate and roll angleincrease while the pitch angle decreases Rise of time delaycauses the increase of lateral acceleration yaw rate and rollangle but hardly changes the vertical acceleration and pitchangle As for the effects of vehicle speed and driver time delayon full vehicle dynamics the following may be concluded
(1) A high vehicle speed during double-lane changeyields bad path keeping performance ride comfortsteering stability and roll stability When the vehiclespeed is higher than 60 kmh the ride comfort of thisvehicle gets worse greatly
(2) A long time delay is harmful to path keeping per-formance steering stability and roll stability and hasalmost no effect on the ride comfort
Thus a low vehicle speed ranging from 20 kmh to 60 kmh and a small time delay less than 035 s are recommended forimproving this truckrsquos stability when going through the lanechange
In case of varying the preview distance or permit positionerror the variation in vehicle trajectories at the speed of40 kmh is shown in Figure 14The effects of preview distanceor permit position error on the path-following ability andfull vehicle dynamics are shown in Figure 15 As shown inFigures 14 and 15 a large preview distance deteriorates thepath-following ability but has small effect on other dynamicresponses On the other hand a too small preview distanceleads to a hop of lateral acceleration yaw rate and roll anglewhich conflicts with closed-loop stability and is harmfulto steering control This conclusion is consistent with thatof Gillespie and MacAdam [15] and Pauwelussen [30] Theinfluence of permit position error on path-following ability issmall at short preview distance but great at long preview dis-tance As shown in Figure 14 a too small or too large permitposition error (001m or 08m)will worsen the vehiclersquos path-following ability In addition large permit position error mayincrease vertical acceleration lateral acceleration yaw rateand roll angle and reduce the handling stability and rollstability and ride comfort The effect of preview distance andpermit position error on pitch angle is small and shows fluc-tuation Thus a preview distance ranging from 10m to 30mand a small permit position error but more than 001m arerecommended for improving this truckrsquos stability when mov-ing through the lane change A smaller permit position errorshould be matched with a shorter preview distance
14 Mathematical Problems in Engineering
002
04
2040
60800
1
2
Rout
e err
or (m
)
u0 (kmh) 0
02
04
2040
60807
8
9
u0 (kmh)
002
04
204060800
02
04
wr (
rad
s)
u0 (kmh)
0
02
04
204060800
005
01
120579(r
ad)
u0 (kmh)002
04
2040
60800
01
02
Φ(r
ad)
u0 (kmh)
002
04
2040
60800
2
4
u0(kmh)
ay(m
s2 )
Tr(s) Tr
(s)
Tr(s)
Tr(s)
Tr(s) Tr
(s)
azb
(ms2 )
Figure 13 The effects of vehicle speed and driver time delay
0 50 100 150 200
0
1
2
3
4
5
001 m02 m04 m
06 m08 mRoute
Y(m
)
X (m)
minus10 50 100 150 200
0
2
4
6
8
10
12
2 m10 m20 m
30 m40 mRoute
Y(m
)
X (m)
minus2
minus4
Figure 14 Routes of vehicle under different preview distance and permit position error
Mathematical Problems in Engineering 15
020
4060
005
1150
1
2
Rout
e err
or (m
)
er (m) Ld (m)
020
4060
01
25
10
15
er (m) Ld (m)
020
4060
01
20
5
er (m) Ld (m) 020
4060
01
20
02
04
er (m) Ld (m)wr (
rad
s)
2040
601
20
02
04
00er (m) Ld (m)
Φ(r
ad)
020
40 60
01
20
005
01
er (m) Ld (m)
120579(r
ad)
ay(m
s2 )
azb
(ms2 )
Figure 15 The effects of preview distance and permit position error
5 Conclusions
In this paper a nonlinear three-directional coupled lumpedparameters (TCLP) model of heavy-duty vehicle consideringthe nonlinearity of suspension damping and tire stiffness isbuilt and connected with a proposedmodified preview drivermodel with nonlinear time delayThe proposed driver modelis simple and has a feedback gain 2119871119889
2 that is decided bywheelbase and previewdistanceThe validity of this presenteddriver-vehicle closed-loop system is verified by comparisonwith the test data the traditional steering stability vehiclemodel the linear vehicle model and other driver controlmodelsThe results show that the new drivermodel has betterlane keeping performances than the other two driver modelsThe coupling effects of vehicle motions and the nonlinear-ity of suspension damping and tire stiffness could not beneglected in turning or high speed driving situation
The effects of driver parameters on path-following abilityand full vehicle dynamics are also discussed It is foundthat the high vehicle speed large time delay long previewdistance or big permit position error will deteriorate path-following performances and handling stability Of course atoo short preview distance or a too small permit position
error is also harmful to path-following performances andhandling stability The ride comfort mainly depends on vehi-cle speed preview distance and permit position error Thekey factors influencing roll characteristics are vehicle speedand time delay
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The National Natural Science Foundation of China(11472180) theNatural Science Foundation ofHebei Province(E2012210025) and the New Century Talent Foundation ofMinistry of Education (NCET-13-0913) support this work
References
[1] M Plochl and J Edelmann ldquoDriver models in automobiledynamics applicationrdquoVehicle SystemDynamics vol 45 no 7-8pp 699ndash741 2007
16 Mathematical Problems in Engineering
[2] C C MacAdam ldquoUnderstanding and modeling the humandriverrdquo Vehicle System Dynamics vol 40 no 1ndash3 pp 101ndash1342003
[3] K Guo and H Guan ldquoModeling of drivervehicle directionalcontrol systemrdquo Vehicle System Dynamics vol 22 no 3-4 pp141ndash184 1993
[4] K GuoVehicle Handling DynamicsTheory Jiangsu Science andTechnology Publishing House Nanjing China 2011
[5] R S Sharp ldquoDriver steering control and a new perspective oncar handling qualitiesrdquo Proceedings of the Institution ofMechani-cal Engineers Part C Journal of Mechanical Engineering Sciencevol 219 no 10 pp 1041ndash1051 2005
[6] T Legouis A Laneville P Bourassa and G Payre ldquoCharac-terization of dynamic vehicle stability using two models of thehuman pilot behaviorrdquo Vehicle System Dynamics vol 15 no 1pp 1ndash18 1986
[7] Z Liu G Payre and P Bourassa ldquoNonlinear oscillations andchaotic motions in a road vehicle system with driver steeringcontrolrdquo Nonlinear Dynamics vol 9 no 3 pp 281ndash304 1996
[8] Z Liu G Payre and P Bourassa ldquoStability and oscillations ina time-delayed vehicle system with driver controlrdquo NonlinearDynamics vol 35 no 2 pp 159ndash173 2004
[9] X D Liu W Feng and N Kang ldquoResearch of simulationmethod of tractor-semitrailer carrying liquid considering liquidsloshingrdquo in Proceedings of the 10th National Conference onVibrationTheory and Application pp 581ndash590 Nanjing ChinaOctober 2011
[10] C C MacAdam ldquoApplication of an optimal preview control forsimulation of closed-loop automobile drivingrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 11 no 6 pp 393ndash399 1981
[11] A Y Ungoren and H Peng ldquoAn adaptive lateral preview drivermodelrdquo Vehicle System Dynamics vol 43 no 4 pp 245ndash2592005
[12] C I Chatzikomis and K N Spentzas ldquoA path-following drivermodel with longitudinal and lateral control of vehiclersquos motionrdquoForschung im Ingenieurwesen vol 73 no 4 pp 257ndash266 2009
[13] R S Sharp D Casanova and P Symonds ldquoA mathematicalmodel for driver steering control with design tuning andperformance resultsrdquo Vehicle System Dynamics vol 33 no 5pp 289ndash326 2000
[14] A J Pick and D J Cole ldquoNeuromuscular dynamics in thedriver-vehicle systemrdquo Vehicle System Dynamics vol 44 sup-plement 1 pp 624ndash631 2006
[15] T D Gillespie and C C MacAdam Constant Velocity YawRollProgram Userrsquos Manual The University of Michigan Trans-portation Research Institute Ann Arbor Mich USA 1982
[16] R I S Raj Influence of road roughness and directional maneuveron the dynamic performance of heavy vehicles [MS thesis]Department of Mechanical Engineering Concordia UniversityMontreal Canada 1998
[17] KWordenDHickeyMHaroon andD E Adams ldquoNonlinearsystem identification of automotive dampers a time and fre-quency-domain analysisrdquo Mechanical Systems and Signal Pro-cessing vol 23 no 1 pp 104ndash126 2009
[18] S H Li Y J Lu and L Y Li ldquoDynamical test and modelingfor hydraulic shock absorber on heavy vehicle under harmonicand random loadingsrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 13 pp 1903ndash1910 2012
[19] X W Ji and Y M Gao ldquoThe dynamic stiffness and dampingcharacteristics of the tirerdquo Automotive Engineering vol 5 pp315ndash321 1994
[20] G Gim and P E Nikravesh ldquoA three dimensional tire modelfor steady-state simulations of vehiclesrdquo SAE vol 102 no 2 pp150ndash159 1993
[21] J D Zhuang Principles of Automobile Tire Beijing Institute ofTechnology Press Beijing China 1996
[22] W-M Zhai ldquoTwo simple fast integration methods for large-scale dynamic problems in engineeringrdquo International Journalfor Numerical Methods in Engineering vol 39 no 24 pp 4199ndash4214 1996
[23] Dongfeng Motor Group Company Dongfeng DFL1250A8DFL1250A9 Truck Chassis Refit Manual Dongfeng MotorGroup Company 2008
[24] S Li S Yang L Chen and Y Lu ldquoEffects of parameters ondynamic responses for a heavy vehicle-pavement- foundationcoupled systemrdquo International Journal of Heavy Vehicle Systemsvol 19 no 2 pp 207ndash224 2012
[25] S Yang S Li andY Lu ldquoInvestigation on dynamical interactionbetween a heavy vehicle and road pavementrdquo Vehicle SystemDynamics vol 48 no 8 pp 923ndash944 2010
[26] GBT 7031-2005ISO 86081995 Mechanical Vibration-RoadSurface Profiles-Reporting of Measured Data 2005
[27] M Li X Pu and Z Changfu ldquoModeling and simulationof heavy-duty semi-trailer in extreme condition based onADAMSrdquo Automobile Technology vol 2 pp 22ndash25 2009
[28] ldquoPassenger carsmdashtest track for a severe lane-change manoeu-vremdashpart 1 double lane-changerdquo ISO 3888-11999
[29] S H Li S P Yang and N Chen ldquoDirectional control of adriver-heavy-vehicle closed-loop systemrdquo Advanced Engineer-ing Forum vol 2-3 pp 33ndash38 2012
[30] J Pauwelussen ldquoDependencies of driver steering controlparametersrdquo Vehicle System Dynamics vol 50 no 6 pp 939ndash959 2012
[31] H Imine and V Dolcemascolo ldquoRollover risk prediction ofheavy vehicle in interaction with infrastructurerdquo InternationalJournal ofHeavyVehicle Systems vol 14 no 3 pp 294ndash307 2007
(119894 = 1 sim 3 119895 = 1 sim 2) are the drivingtorque and braking torque of six wheels
23 Driver Model According to Guorsquos preview of optimalcurvature drivermodel [3 4] the optimal front steering angleis expressed by
120575119901=
2119871
1198892[119891 (119905 + 119879) minus 119910 (119905) minus 119879 119910 (119905)] (13)
where 119889 119879 and 119871 are the preview distance preview time andwheelbase respectively 119891(119905 + 119879) is the lateral position of thedesired route at preview point and 119910(119905) is the vehicle lateralposition at current time
The above model is very simple and suitable to simulatelateral dynamics of vehicle running at a constant speed How-ever it neglects the effect of time delay and the desired routefunction 119891(119905) needs to be computed according to vehiclespeed and trajectory before simulation
Legouisrsquo driver model with nonlinear time delay [6ndash8]calculates the front steering angle by
120575119901= minus119870[119910
119873(119905 minus 119879
119903) +
119889
119880
119910119873(119905 minus 119879
119903)] (14)
where119870 119879119903 and119880 are feedback gain time delay and vehicle
running speed respectively 119910119873(119905 minus 119879
119903) and 119910
119873(119905 minus 119879
119903) +
(119889119880) 119910119873(119905 minus 119879
119903) are the lateral position of vehicle gravity
center and preview point in the inertial frames respectivelyThis model introduced time delay and calculated positiondeviation between vehicle and desired route in the inertialframes However the feedback gain in Legouisrsquo driver modelis a constant and cannot be obtained by vehicle parametersThe ideal route function is not included in the model becausethe straight-line driving condition is researched In additionthe preview point is gained from vehicle gravity center andneglects the difference of longitudinal distance and yaw anglebetween vehicle gravity center and driver position
By combining the above two models a modified drivermodel is proposed here as shown in Figure 3The front wheelsteering angle is given by
120575119901(119905) =
2119871
1198892[119877119884119889(119905 minus 119879
119903) minus 119884119889(119905 minus 119879
119903)]
1003816100381610038161003816119877119884119889(119905 minus 119879
where119877119884119889119884119889 119890119888119903 and119889119905 are the required lateral position and
real lateral displacement of the preview point in the ground
Y
O Xd
Required route RYdYd
Fty12Fty22Fty32
x
o120595Vx120596r
Vy120575
dt2
Fty31Fty21
Fty11l3
l2l1
dy
Figure 3 The new driver control model with nonlinear time delay
coordinate system the permit position error and the inte-gration time step respectively For three-axle vehicle theparameter 119871 in (15) is the distance between front wheel andcenter of balance suspension and is expressed by 119871 = 119897
1+ 1198972
This modified model has a feedback gain 21198711198892 that is
defined by wheelbase and preview distance It can be noticedthat a big wheelbase or a small preview distance will lead toa big gain This statement is well understood because driversfeel it difficult to control the vehicle direction in the case of bigwheelbase or small preview distance so they have to increasethe intervention of steering angle
The displacement of the preview point in the groundcoordinate system is
119883119889= 119883 (119905) + 119897
1cos120595 + 119889
119880
( minus 1198971 sin120595)
119884119889= 119884 (119905) + 119897
1sin120595 + 119889
119880
( + 1198971 cos120595)
(16)
where119883(119905) 119884(119905) and 120595 are displacements and heading angleof vehicle gravity center in the ground coordinate system Itshould be noted that the preview point lies in 119889meter aheadof driver seat not vehicle gravity center According to 119883
119889
and the required route function the required lateral position119877119884119889is easily obtained Substituting 119877119884
119889and 119884
119889into (15) and
introducing time delay the front wheel angle can be obtainedIt should be noted that themodified drivermodel depends onthe vehicle longitudinal speed as strongly as Legouisrsquo drivermodel
The displacements and velocities in vehicle coordinatesystem (119909 119910 119911) can be gained from vehicle model andtransferred to the ground coordinate system (119883 119884 119885) by thefollowing relation
= 119881119909cos120595 minus 119881
119910sin120595
= 119881119909sin120595 + 119881
119910cos120595
(17)
Finally the vehicle model tire model and driver modelare coupled into the driver-vehicle closed-loop system Thelongitudinal slip ratios of six wheels are calculated in real-time with vehicle responses as inputThe front wheel steeringangle is obtained by the modified driver model and fed back
Mathematical Problems in Engineering 7
Coupled vehicle model
Wheel rotate equations
Braking toque
Three-directionaltire model
Slip ratio equation
Drivermodel
Output responses
Road surface properties
Requiredroute
Vehicle initial conditions
120596ij Mb
ztij
Ftxij FtyijFtzijMtzij
120575rij 120583ij
Vx
Ssij
RY119889(t minus Tr)
Yd(t minus Tr)
Xd(t minus Tr)
Figure 4 The driver-heavy-vehicle closed-loop model
to the tire model Then the vertical longitudinal and lateraltire forces are calculated by the tire model and input intothe vehicle model to gain vehicle responses and positionsin next time step The simulation process of this driver-vehicle closed-loop system is shown in Figure 4 Due to thetime variability nonlinearity and high-dimensional propertyof this system the closed-loop system equations are solvednumerically by the quick integration method [22] and theRunge-Kutta method of order four
3 Model Evaluation
In order to verify the presented TCLP vehicle model andthe new driver model simulation results of this driver-vehicle closed-loop system are obtained using different vehi-cle models or driver models During simulation the vehicleparameters are chosen for a DFL1250A9 truck manufacturedby Dongfeng Motor Group Company Limited [23ndash25] andthe B-class road roughness is selected referring to [26]
The parameters of double-lane change route for theheavy-duty vehicle are chosen referring to [27 28] and shownin Figure 5
31 Comparison with the Handling Stability Vehicle Model Atraditional two-degree of freedom (2DOF) handling stabilityvehicle model for a three-axle heavy vehicle is set up whichconsiders only the lateral and yaw motion [29] The ordinarydifferential equations of motion of this 2DOF model may beexpressed by
where 119898 119868119911 119881119909 119881119910 and 120596
119903are vehicle mass vehicle inertia
around 119911 axial and longitudinal lateral and yaw rate of thevehicle respectively 120575
119894119865120572119894 and119872
119911119894are steering angle lateral
tire force and self-aligning torque of wheels respectively 119897119909119894
and 119897119910119894are the distance from wheel center to vehicle gravity
center in longitudinal and lateral direction respectivelyThe double-lane change responses of this TCLP model
and the traditional 2DOF model at an entrance speedof 60 kmh are simulated respectively and compared inFigure 6 It can be seen from Figure 6 that the results of thesetwo models are very consistent in magnitude and trendsHence the two vehicle models verify each other The TCLPmodel has a worse path-following ability than the 2DOFmodel and the vehicle running speed of TCLP model fluctu-ates randomlyThe yaw rate and lateral acceleration obtainedfrom TCLP model is bigger than that from 2DOF modelThe reason for these differences between two models is thatthe TCLP vehicle model considers B-class road roughnessand the coupled effect of roll vertical longitudinal and pitchmotion on yaw and lateral motion while the 2DOF modelneglects them
32 Comparison with the Linear Vehicle Model When thenonlinearity of front suspension damper and tire force isneglected as shown in Figure 7 the vehicle responses becomesmaller and the tracking performance is better Thus thelinear vehicle model may predict more conservative results
8 Mathematical Problems in Engineering
20m
20m40m40m
35m35m
15m
30m
35
m
35
m
35
m
375
m
Figure 5 The lane change route for the heavy-duty vehicle
0 5 10
162
164
166
168
17
0 5 10 15
0
5
Coupled model2DOF model
Required route
minus5
50 100 150 200
0
2
4
Y(m
)
X (m)
u(m
s)
t (s)
0 5 10 15
0
02
04
Coupled model2DOF model
Required route
minus04
minus02
t (s) t (s)
wr (
rad
s)
ay(m
s2)
Figure 6 Double-lane change responses of TCLP model and 2DOF model at 60 kmh
In the meantime it is found that the difference between thelinear and nonlinear vehicle models in the maneuver of lanechange is much greater than that in the maneuver of drivingstraight The lateral load transfer yaw motion and lateralmotion while lane change destabilizes the vehicle and conse-quently aggravates other vehicle motions due to couple effectIn addition by simulating responses at different entrancespeeds we found that the difference between the linearand nonlinear vehicle models becomes greater with the riseof vehicle speed Therefore the nonlinear vehicle modelshould be used so as to simulate vehicle dynamics accurately
especially in lane change maneuver or high speed drivingcondition
33 Comparison with Test Data During double-lane changetest an empty loaded DFL1250A9 truck was selected to runat 30 kmh speed A three-axis piezoelectric accelerometer(frequency range from 1Hz to 500Hz) and a gyro (measur-ing range plusmn50 degs) were placed at the center of gravity ofwhole vehicle and a cable displacement sensor was placedat front wheel to measure front wheel steering angle asshown in Figure 8 The measured signals were amplified by
Mathematical Problems in Engineering 9
50 100 150 200 250 300
0
1
2
3
4
minus1
Y(m
)
X (m)0 5 10 15 20
162
164
166
168
u(m
s)
t (s)
5 10 15 20
0
2
4
minus4
minus2
t (s)
ay
(ms
2 )
5 10 15 20
0
01
02
03
minus02
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
001
minus002
minus001
t (s)
120573(r
ad)
7 8 9
0
10
20
minus20
minus10
t (s)
azb
(ms
2 )
5 10 15 20
0
01
02
minus02
minus01
NonlinearLinear
Requied route
t (s)
Φ(r
ad)
5 10 15 20
0
005
minus01
minus005
NonlinearLinear
Requied route
t (s)
120579(r
ad)
Figure 7 The double-lane change responses of the linear and nonlinear TCLP model at 60 kmh
10 Mathematical Problems in Engineering
Figure 8 Vehicle and sensors in field test
Table 1 Vehicle responses obtained from three driver models
Vehicle speed Driver model RMS value of vehicle responses119906 119908119903 119886119910 119886119911119887 119891119894 119909119894119905119886
a multichannel charge amplifier (YE5853A) and convertedto digital signals by an intelligent acquisition and processinganalyzer (INV360DF) The sampling frequency was 200Hz
Figure 9 gives the experimental time history of frontwheel steering angle Using this measured front wheel steer-ing angle as input the yaw rate and lateral acceleration are cal-culated from TCLP model 2DOF model and linear modelrespectively These simulation results are compared with
the test results as shown in Figure 10 It is obvious thatthe simulation results of TCLP model are most consistentwith the results of field test Due to neglecting of the three-directional coupling effects of vehiclemotions or nonlinearityof suspension and tire the time histories of yaw rate obtainedfrom 2DOF model and linear model are too smooth and thelateral accelerations of 2DOF model and linear model aresmaller than the test results Thus the verification of thisproposed TCLP model is verified
34 Comparison with Other Driver Models Using the pro-posed modified driver model the optimal curvature modelby Guo and nonlinear time delay driver model by Legouisthe trajectories of vehicle gravity center during double-lanechange maneuver at four entrance speeds of 20 40 60 and80 kmh are simulated respectively as shown in Figure 11Table 1 lists the root mean square (RMS) value of vehicleresponses in six directions at different speeds applying thesethree driver models As shown in Figure 11 it is obvious thatthe tracking performance of the modified driver model issuperior to the other two driver models at different entrancespeeds It is also found from Figure 11 that though an over-shoot of the newmodel at a vehicle speed of 80 kmh exists in119883 = [150 180]m the results of the other two models divergewhen the vehicle arrives at 119883 = 200m while the resultof the new model is stable From Table 1 we can see thatthe real vehicle speed of the modified driver model is the
Mathematical Problems in Engineering 11
Simulation results of TCLP model
Test results
Simulation results of 2DOF model
Simulation results of linear model
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
02
04
minus04
minus02
t (s)
ay(m
s2 )
5 10 15 20
0
05
1
minus05
t (s)
ay(m
s2 )
Figure 10 Comparison of simulation and test results
12 Mathematical Problems in Engineering
50 100 150 200 250 300
0
1
2
3
4
5
minus1
Y(m
)
X (m)
(a) 20 kmh
50 100 150 200 250
0
1
2
3
4
Y(m
)
X (m)
(b) 40 kmh
50 100 150 200 250 300
0
1
2
3
4
5
NewLiu
GuoRoute
minus1
Y(m
)
X (m)
(c) 60 kmh
50 100 150 200
0
2
4
NewLiu
GuoRoute
minus2
Y(m
)
X (m)
(d) 80 kmh
Figure 11 Comparison of modified driver model with other driver models at different speeds
closest to the assumed speed and the modified driver modelhas smaller responses except for vertical vibration and betterstability than the other models Though the modified drivermodel obtains greater vertical vehicle acceleration and worseride comfort than Legouisrsquos model at speeds lower than60 kmh the new driver model can give the vehicle betterpath-following ability and handling stability
4 Effects of Driver Model Parameters onFull Vehicle Dynamics and Stability
The important parameters in driver model include vehiclerunning speed time delay preview distance and permitposition error which decide the steering control effect andvehicle dynamics During the same driving maneuver theskilled drivers may use higher vehicle speed longer previewdistance shorter time delay and larger permit position error
than the new drivers The match of these four driverparameters also influences the vehicle dynamics RecentlyPauwelussen [30] researched the dependencies of driversteering control parameters on vehicle lateral properties andpath keeping and gives some useful conclusions This workfocuses on the effects of driver model parameters on bothpath keeping and full vehicle dynamics such as vehiclehandling stability ride comfort and roll stability
Since the vehicle lateral displacement is randomly causedby road roughness excitations and the lateral position devi-ation is always changing during lane change maneuvers astatistic index is introduced to evaluate the vehiclersquos path-following ability globally which is formulated by
Route error = radicsum (119877119884
119889(119905) minus 119884
119889(119905))2
119873
(20)
Mathematical Problems in Engineering 13
20 40 60 80 100 120 140 160 180 200
0
1
2
3
4
5
20 kmh30 kmh40 kmh
50 kmh60 kmhRoute
Y(m
)
X (m)
minus1
0 50 100 150 200
0
1
2
3
4
0 s002 s005 s
01 s02 sRoute
Y(m
)
X (m)
minus1
Figure 12 Routes of vehicle under different speed or time delay
where 119873 is the total number of data points 119877119884119889and 119884
119889are
the required lateral position and real lateral displacement ofthe preview point in the ground coordinate system
When a vehicle is undergoing a lane change its steeringstability and roll stability are vital problems The path-following error lateral acceleration and yaw rate can be usedto evaluate the steering stability The roll angle can be used toevaluate the roll stability Being proportional to the rolloverindex LTR (lateral-load transfer rate) the lateral accelerationis also able to reflect the vehicle roll stability [31] In additionthe vertical acceleration and pitch angle are simulated so asto research the whole-body dynamics in the vehicle Usingthese evaluation indexes the effects of parameters on steeringstability roll stability and riding comfort are discussed indetail
Figure 12 shows the vehicle trajectories under differentinitial vehicle speeds or driver time delays The effects ofvehicle speed and driver time delay on full vehicle dynamicsand stability are shown in Figure 13 As we can see fromFigures 12 and 13 a higher speed or a longer time delay willlead to a greater path-following error However low speed(20 kmh) and longtime delay (035 s) may enlarge the path-following error With the rise of vehicle speed the verticalacceleration lateral acceleration yaw rate and roll angleincrease while the pitch angle decreases Rise of time delaycauses the increase of lateral acceleration yaw rate and rollangle but hardly changes the vertical acceleration and pitchangle As for the effects of vehicle speed and driver time delayon full vehicle dynamics the following may be concluded
(1) A high vehicle speed during double-lane changeyields bad path keeping performance ride comfortsteering stability and roll stability When the vehiclespeed is higher than 60 kmh the ride comfort of thisvehicle gets worse greatly
(2) A long time delay is harmful to path keeping per-formance steering stability and roll stability and hasalmost no effect on the ride comfort
Thus a low vehicle speed ranging from 20 kmh to 60 kmh and a small time delay less than 035 s are recommended forimproving this truckrsquos stability when going through the lanechange
In case of varying the preview distance or permit positionerror the variation in vehicle trajectories at the speed of40 kmh is shown in Figure 14The effects of preview distanceor permit position error on the path-following ability andfull vehicle dynamics are shown in Figure 15 As shown inFigures 14 and 15 a large preview distance deteriorates thepath-following ability but has small effect on other dynamicresponses On the other hand a too small preview distanceleads to a hop of lateral acceleration yaw rate and roll anglewhich conflicts with closed-loop stability and is harmfulto steering control This conclusion is consistent with thatof Gillespie and MacAdam [15] and Pauwelussen [30] Theinfluence of permit position error on path-following ability issmall at short preview distance but great at long preview dis-tance As shown in Figure 14 a too small or too large permitposition error (001m or 08m)will worsen the vehiclersquos path-following ability In addition large permit position error mayincrease vertical acceleration lateral acceleration yaw rateand roll angle and reduce the handling stability and rollstability and ride comfort The effect of preview distance andpermit position error on pitch angle is small and shows fluc-tuation Thus a preview distance ranging from 10m to 30mand a small permit position error but more than 001m arerecommended for improving this truckrsquos stability when mov-ing through the lane change A smaller permit position errorshould be matched with a shorter preview distance
14 Mathematical Problems in Engineering
002
04
2040
60800
1
2
Rout
e err
or (m
)
u0 (kmh) 0
02
04
2040
60807
8
9
u0 (kmh)
002
04
204060800
02
04
wr (
rad
s)
u0 (kmh)
0
02
04
204060800
005
01
120579(r
ad)
u0 (kmh)002
04
2040
60800
01
02
Φ(r
ad)
u0 (kmh)
002
04
2040
60800
2
4
u0(kmh)
ay(m
s2 )
Tr(s) Tr
(s)
Tr(s)
Tr(s)
Tr(s) Tr
(s)
azb
(ms2 )
Figure 13 The effects of vehicle speed and driver time delay
0 50 100 150 200
0
1
2
3
4
5
001 m02 m04 m
06 m08 mRoute
Y(m
)
X (m)
minus10 50 100 150 200
0
2
4
6
8
10
12
2 m10 m20 m
30 m40 mRoute
Y(m
)
X (m)
minus2
minus4
Figure 14 Routes of vehicle under different preview distance and permit position error
Mathematical Problems in Engineering 15
020
4060
005
1150
1
2
Rout
e err
or (m
)
er (m) Ld (m)
020
4060
01
25
10
15
er (m) Ld (m)
020
4060
01
20
5
er (m) Ld (m) 020
4060
01
20
02
04
er (m) Ld (m)wr (
rad
s)
2040
601
20
02
04
00er (m) Ld (m)
Φ(r
ad)
020
40 60
01
20
005
01
er (m) Ld (m)
120579(r
ad)
ay(m
s2 )
azb
(ms2 )
Figure 15 The effects of preview distance and permit position error
5 Conclusions
In this paper a nonlinear three-directional coupled lumpedparameters (TCLP) model of heavy-duty vehicle consideringthe nonlinearity of suspension damping and tire stiffness isbuilt and connected with a proposedmodified preview drivermodel with nonlinear time delayThe proposed driver modelis simple and has a feedback gain 2119871119889
2 that is decided bywheelbase and previewdistanceThe validity of this presenteddriver-vehicle closed-loop system is verified by comparisonwith the test data the traditional steering stability vehiclemodel the linear vehicle model and other driver controlmodelsThe results show that the new drivermodel has betterlane keeping performances than the other two driver modelsThe coupling effects of vehicle motions and the nonlinear-ity of suspension damping and tire stiffness could not beneglected in turning or high speed driving situation
The effects of driver parameters on path-following abilityand full vehicle dynamics are also discussed It is foundthat the high vehicle speed large time delay long previewdistance or big permit position error will deteriorate path-following performances and handling stability Of course atoo short preview distance or a too small permit position
error is also harmful to path-following performances andhandling stability The ride comfort mainly depends on vehi-cle speed preview distance and permit position error Thekey factors influencing roll characteristics are vehicle speedand time delay
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The National Natural Science Foundation of China(11472180) theNatural Science Foundation ofHebei Province(E2012210025) and the New Century Talent Foundation ofMinistry of Education (NCET-13-0913) support this work
References
[1] M Plochl and J Edelmann ldquoDriver models in automobiledynamics applicationrdquoVehicle SystemDynamics vol 45 no 7-8pp 699ndash741 2007
16 Mathematical Problems in Engineering
[2] C C MacAdam ldquoUnderstanding and modeling the humandriverrdquo Vehicle System Dynamics vol 40 no 1ndash3 pp 101ndash1342003
[3] K Guo and H Guan ldquoModeling of drivervehicle directionalcontrol systemrdquo Vehicle System Dynamics vol 22 no 3-4 pp141ndash184 1993
[4] K GuoVehicle Handling DynamicsTheory Jiangsu Science andTechnology Publishing House Nanjing China 2011
[5] R S Sharp ldquoDriver steering control and a new perspective oncar handling qualitiesrdquo Proceedings of the Institution ofMechani-cal Engineers Part C Journal of Mechanical Engineering Sciencevol 219 no 10 pp 1041ndash1051 2005
[6] T Legouis A Laneville P Bourassa and G Payre ldquoCharac-terization of dynamic vehicle stability using two models of thehuman pilot behaviorrdquo Vehicle System Dynamics vol 15 no 1pp 1ndash18 1986
[7] Z Liu G Payre and P Bourassa ldquoNonlinear oscillations andchaotic motions in a road vehicle system with driver steeringcontrolrdquo Nonlinear Dynamics vol 9 no 3 pp 281ndash304 1996
[8] Z Liu G Payre and P Bourassa ldquoStability and oscillations ina time-delayed vehicle system with driver controlrdquo NonlinearDynamics vol 35 no 2 pp 159ndash173 2004
[9] X D Liu W Feng and N Kang ldquoResearch of simulationmethod of tractor-semitrailer carrying liquid considering liquidsloshingrdquo in Proceedings of the 10th National Conference onVibrationTheory and Application pp 581ndash590 Nanjing ChinaOctober 2011
[10] C C MacAdam ldquoApplication of an optimal preview control forsimulation of closed-loop automobile drivingrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 11 no 6 pp 393ndash399 1981
[11] A Y Ungoren and H Peng ldquoAn adaptive lateral preview drivermodelrdquo Vehicle System Dynamics vol 43 no 4 pp 245ndash2592005
[12] C I Chatzikomis and K N Spentzas ldquoA path-following drivermodel with longitudinal and lateral control of vehiclersquos motionrdquoForschung im Ingenieurwesen vol 73 no 4 pp 257ndash266 2009
[13] R S Sharp D Casanova and P Symonds ldquoA mathematicalmodel for driver steering control with design tuning andperformance resultsrdquo Vehicle System Dynamics vol 33 no 5pp 289ndash326 2000
[14] A J Pick and D J Cole ldquoNeuromuscular dynamics in thedriver-vehicle systemrdquo Vehicle System Dynamics vol 44 sup-plement 1 pp 624ndash631 2006
[15] T D Gillespie and C C MacAdam Constant Velocity YawRollProgram Userrsquos Manual The University of Michigan Trans-portation Research Institute Ann Arbor Mich USA 1982
[16] R I S Raj Influence of road roughness and directional maneuveron the dynamic performance of heavy vehicles [MS thesis]Department of Mechanical Engineering Concordia UniversityMontreal Canada 1998
[17] KWordenDHickeyMHaroon andD E Adams ldquoNonlinearsystem identification of automotive dampers a time and fre-quency-domain analysisrdquo Mechanical Systems and Signal Pro-cessing vol 23 no 1 pp 104ndash126 2009
[18] S H Li Y J Lu and L Y Li ldquoDynamical test and modelingfor hydraulic shock absorber on heavy vehicle under harmonicand random loadingsrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 13 pp 1903ndash1910 2012
[19] X W Ji and Y M Gao ldquoThe dynamic stiffness and dampingcharacteristics of the tirerdquo Automotive Engineering vol 5 pp315ndash321 1994
[20] G Gim and P E Nikravesh ldquoA three dimensional tire modelfor steady-state simulations of vehiclesrdquo SAE vol 102 no 2 pp150ndash159 1993
[21] J D Zhuang Principles of Automobile Tire Beijing Institute ofTechnology Press Beijing China 1996
[22] W-M Zhai ldquoTwo simple fast integration methods for large-scale dynamic problems in engineeringrdquo International Journalfor Numerical Methods in Engineering vol 39 no 24 pp 4199ndash4214 1996
[23] Dongfeng Motor Group Company Dongfeng DFL1250A8DFL1250A9 Truck Chassis Refit Manual Dongfeng MotorGroup Company 2008
[24] S Li S Yang L Chen and Y Lu ldquoEffects of parameters ondynamic responses for a heavy vehicle-pavement- foundationcoupled systemrdquo International Journal of Heavy Vehicle Systemsvol 19 no 2 pp 207ndash224 2012
[25] S Yang S Li andY Lu ldquoInvestigation on dynamical interactionbetween a heavy vehicle and road pavementrdquo Vehicle SystemDynamics vol 48 no 8 pp 923ndash944 2010
[26] GBT 7031-2005ISO 86081995 Mechanical Vibration-RoadSurface Profiles-Reporting of Measured Data 2005
[27] M Li X Pu and Z Changfu ldquoModeling and simulationof heavy-duty semi-trailer in extreme condition based onADAMSrdquo Automobile Technology vol 2 pp 22ndash25 2009
[28] ldquoPassenger carsmdashtest track for a severe lane-change manoeu-vremdashpart 1 double lane-changerdquo ISO 3888-11999
[29] S H Li S P Yang and N Chen ldquoDirectional control of adriver-heavy-vehicle closed-loop systemrdquo Advanced Engineer-ing Forum vol 2-3 pp 33ndash38 2012
[30] J Pauwelussen ldquoDependencies of driver steering controlparametersrdquo Vehicle System Dynamics vol 50 no 6 pp 939ndash959 2012
[31] H Imine and V Dolcemascolo ldquoRollover risk prediction ofheavy vehicle in interaction with infrastructurerdquo InternationalJournal ofHeavyVehicle Systems vol 14 no 3 pp 294ndash307 2007
(119894 = 1 sim 3 119895 = 1 sim 2) are the drivingtorque and braking torque of six wheels
23 Driver Model According to Guorsquos preview of optimalcurvature drivermodel [3 4] the optimal front steering angleis expressed by
120575119901=
2119871
1198892[119891 (119905 + 119879) minus 119910 (119905) minus 119879 119910 (119905)] (13)
where 119889 119879 and 119871 are the preview distance preview time andwheelbase respectively 119891(119905 + 119879) is the lateral position of thedesired route at preview point and 119910(119905) is the vehicle lateralposition at current time
The above model is very simple and suitable to simulatelateral dynamics of vehicle running at a constant speed How-ever it neglects the effect of time delay and the desired routefunction 119891(119905) needs to be computed according to vehiclespeed and trajectory before simulation
Legouisrsquo driver model with nonlinear time delay [6ndash8]calculates the front steering angle by
120575119901= minus119870[119910
119873(119905 minus 119879
119903) +
119889
119880
119910119873(119905 minus 119879
119903)] (14)
where119870 119879119903 and119880 are feedback gain time delay and vehicle
running speed respectively 119910119873(119905 minus 119879
119903) and 119910
119873(119905 minus 119879
119903) +
(119889119880) 119910119873(119905 minus 119879
119903) are the lateral position of vehicle gravity
center and preview point in the inertial frames respectivelyThis model introduced time delay and calculated positiondeviation between vehicle and desired route in the inertialframes However the feedback gain in Legouisrsquo driver modelis a constant and cannot be obtained by vehicle parametersThe ideal route function is not included in the model becausethe straight-line driving condition is researched In additionthe preview point is gained from vehicle gravity center andneglects the difference of longitudinal distance and yaw anglebetween vehicle gravity center and driver position
By combining the above two models a modified drivermodel is proposed here as shown in Figure 3The front wheelsteering angle is given by
120575119901(119905) =
2119871
1198892[119877119884119889(119905 minus 119879
119903) minus 119884119889(119905 minus 119879
119903)]
1003816100381610038161003816119877119884119889(119905 minus 119879
where119877119884119889119884119889 119890119888119903 and119889119905 are the required lateral position and
real lateral displacement of the preview point in the ground
Y
O Xd
Required route RYdYd
Fty12Fty22Fty32
x
o120595Vx120596r
Vy120575
dt2
Fty31Fty21
Fty11l3
l2l1
dy
Figure 3 The new driver control model with nonlinear time delay
coordinate system the permit position error and the inte-gration time step respectively For three-axle vehicle theparameter 119871 in (15) is the distance between front wheel andcenter of balance suspension and is expressed by 119871 = 119897
1+ 1198972
This modified model has a feedback gain 21198711198892 that is
defined by wheelbase and preview distance It can be noticedthat a big wheelbase or a small preview distance will lead toa big gain This statement is well understood because driversfeel it difficult to control the vehicle direction in the case of bigwheelbase or small preview distance so they have to increasethe intervention of steering angle
The displacement of the preview point in the groundcoordinate system is
119883119889= 119883 (119905) + 119897
1cos120595 + 119889
119880
( minus 1198971 sin120595)
119884119889= 119884 (119905) + 119897
1sin120595 + 119889
119880
( + 1198971 cos120595)
(16)
where119883(119905) 119884(119905) and 120595 are displacements and heading angleof vehicle gravity center in the ground coordinate system Itshould be noted that the preview point lies in 119889meter aheadof driver seat not vehicle gravity center According to 119883
119889
and the required route function the required lateral position119877119884119889is easily obtained Substituting 119877119884
119889and 119884
119889into (15) and
introducing time delay the front wheel angle can be obtainedIt should be noted that themodified drivermodel depends onthe vehicle longitudinal speed as strongly as Legouisrsquo drivermodel
The displacements and velocities in vehicle coordinatesystem (119909 119910 119911) can be gained from vehicle model andtransferred to the ground coordinate system (119883 119884 119885) by thefollowing relation
= 119881119909cos120595 minus 119881
119910sin120595
= 119881119909sin120595 + 119881
119910cos120595
(17)
Finally the vehicle model tire model and driver modelare coupled into the driver-vehicle closed-loop system Thelongitudinal slip ratios of six wheels are calculated in real-time with vehicle responses as inputThe front wheel steeringangle is obtained by the modified driver model and fed back
Mathematical Problems in Engineering 7
Coupled vehicle model
Wheel rotate equations
Braking toque
Three-directionaltire model
Slip ratio equation
Drivermodel
Output responses
Road surface properties
Requiredroute
Vehicle initial conditions
120596ij Mb
ztij
Ftxij FtyijFtzijMtzij
120575rij 120583ij
Vx
Ssij
RY119889(t minus Tr)
Yd(t minus Tr)
Xd(t minus Tr)
Figure 4 The driver-heavy-vehicle closed-loop model
to the tire model Then the vertical longitudinal and lateraltire forces are calculated by the tire model and input intothe vehicle model to gain vehicle responses and positionsin next time step The simulation process of this driver-vehicle closed-loop system is shown in Figure 4 Due to thetime variability nonlinearity and high-dimensional propertyof this system the closed-loop system equations are solvednumerically by the quick integration method [22] and theRunge-Kutta method of order four
3 Model Evaluation
In order to verify the presented TCLP vehicle model andthe new driver model simulation results of this driver-vehicle closed-loop system are obtained using different vehi-cle models or driver models During simulation the vehicleparameters are chosen for a DFL1250A9 truck manufacturedby Dongfeng Motor Group Company Limited [23ndash25] andthe B-class road roughness is selected referring to [26]
The parameters of double-lane change route for theheavy-duty vehicle are chosen referring to [27 28] and shownin Figure 5
31 Comparison with the Handling Stability Vehicle Model Atraditional two-degree of freedom (2DOF) handling stabilityvehicle model for a three-axle heavy vehicle is set up whichconsiders only the lateral and yaw motion [29] The ordinarydifferential equations of motion of this 2DOF model may beexpressed by
where 119898 119868119911 119881119909 119881119910 and 120596
119903are vehicle mass vehicle inertia
around 119911 axial and longitudinal lateral and yaw rate of thevehicle respectively 120575
119894119865120572119894 and119872
119911119894are steering angle lateral
tire force and self-aligning torque of wheels respectively 119897119909119894
and 119897119910119894are the distance from wheel center to vehicle gravity
center in longitudinal and lateral direction respectivelyThe double-lane change responses of this TCLP model
and the traditional 2DOF model at an entrance speedof 60 kmh are simulated respectively and compared inFigure 6 It can be seen from Figure 6 that the results of thesetwo models are very consistent in magnitude and trendsHence the two vehicle models verify each other The TCLPmodel has a worse path-following ability than the 2DOFmodel and the vehicle running speed of TCLP model fluctu-ates randomlyThe yaw rate and lateral acceleration obtainedfrom TCLP model is bigger than that from 2DOF modelThe reason for these differences between two models is thatthe TCLP vehicle model considers B-class road roughnessand the coupled effect of roll vertical longitudinal and pitchmotion on yaw and lateral motion while the 2DOF modelneglects them
32 Comparison with the Linear Vehicle Model When thenonlinearity of front suspension damper and tire force isneglected as shown in Figure 7 the vehicle responses becomesmaller and the tracking performance is better Thus thelinear vehicle model may predict more conservative results
8 Mathematical Problems in Engineering
20m
20m40m40m
35m35m
15m
30m
35
m
35
m
35
m
375
m
Figure 5 The lane change route for the heavy-duty vehicle
0 5 10
162
164
166
168
17
0 5 10 15
0
5
Coupled model2DOF model
Required route
minus5
50 100 150 200
0
2
4
Y(m
)
X (m)
u(m
s)
t (s)
0 5 10 15
0
02
04
Coupled model2DOF model
Required route
minus04
minus02
t (s) t (s)
wr (
rad
s)
ay(m
s2)
Figure 6 Double-lane change responses of TCLP model and 2DOF model at 60 kmh
In the meantime it is found that the difference between thelinear and nonlinear vehicle models in the maneuver of lanechange is much greater than that in the maneuver of drivingstraight The lateral load transfer yaw motion and lateralmotion while lane change destabilizes the vehicle and conse-quently aggravates other vehicle motions due to couple effectIn addition by simulating responses at different entrancespeeds we found that the difference between the linearand nonlinear vehicle models becomes greater with the riseof vehicle speed Therefore the nonlinear vehicle modelshould be used so as to simulate vehicle dynamics accurately
especially in lane change maneuver or high speed drivingcondition
33 Comparison with Test Data During double-lane changetest an empty loaded DFL1250A9 truck was selected to runat 30 kmh speed A three-axis piezoelectric accelerometer(frequency range from 1Hz to 500Hz) and a gyro (measur-ing range plusmn50 degs) were placed at the center of gravity ofwhole vehicle and a cable displacement sensor was placedat front wheel to measure front wheel steering angle asshown in Figure 8 The measured signals were amplified by
Mathematical Problems in Engineering 9
50 100 150 200 250 300
0
1
2
3
4
minus1
Y(m
)
X (m)0 5 10 15 20
162
164
166
168
u(m
s)
t (s)
5 10 15 20
0
2
4
minus4
minus2
t (s)
ay
(ms
2 )
5 10 15 20
0
01
02
03
minus02
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
001
minus002
minus001
t (s)
120573(r
ad)
7 8 9
0
10
20
minus20
minus10
t (s)
azb
(ms
2 )
5 10 15 20
0
01
02
minus02
minus01
NonlinearLinear
Requied route
t (s)
Φ(r
ad)
5 10 15 20
0
005
minus01
minus005
NonlinearLinear
Requied route
t (s)
120579(r
ad)
Figure 7 The double-lane change responses of the linear and nonlinear TCLP model at 60 kmh
10 Mathematical Problems in Engineering
Figure 8 Vehicle and sensors in field test
Table 1 Vehicle responses obtained from three driver models
Vehicle speed Driver model RMS value of vehicle responses119906 119908119903 119886119910 119886119911119887 119891119894 119909119894119905119886
a multichannel charge amplifier (YE5853A) and convertedto digital signals by an intelligent acquisition and processinganalyzer (INV360DF) The sampling frequency was 200Hz
Figure 9 gives the experimental time history of frontwheel steering angle Using this measured front wheel steer-ing angle as input the yaw rate and lateral acceleration are cal-culated from TCLP model 2DOF model and linear modelrespectively These simulation results are compared with
the test results as shown in Figure 10 It is obvious thatthe simulation results of TCLP model are most consistentwith the results of field test Due to neglecting of the three-directional coupling effects of vehiclemotions or nonlinearityof suspension and tire the time histories of yaw rate obtainedfrom 2DOF model and linear model are too smooth and thelateral accelerations of 2DOF model and linear model aresmaller than the test results Thus the verification of thisproposed TCLP model is verified
34 Comparison with Other Driver Models Using the pro-posed modified driver model the optimal curvature modelby Guo and nonlinear time delay driver model by Legouisthe trajectories of vehicle gravity center during double-lanechange maneuver at four entrance speeds of 20 40 60 and80 kmh are simulated respectively as shown in Figure 11Table 1 lists the root mean square (RMS) value of vehicleresponses in six directions at different speeds applying thesethree driver models As shown in Figure 11 it is obvious thatthe tracking performance of the modified driver model issuperior to the other two driver models at different entrancespeeds It is also found from Figure 11 that though an over-shoot of the newmodel at a vehicle speed of 80 kmh exists in119883 = [150 180]m the results of the other two models divergewhen the vehicle arrives at 119883 = 200m while the resultof the new model is stable From Table 1 we can see thatthe real vehicle speed of the modified driver model is the
Mathematical Problems in Engineering 11
Simulation results of TCLP model
Test results
Simulation results of 2DOF model
Simulation results of linear model
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
02
04
minus04
minus02
t (s)
ay(m
s2 )
5 10 15 20
0
05
1
minus05
t (s)
ay(m
s2 )
Figure 10 Comparison of simulation and test results
12 Mathematical Problems in Engineering
50 100 150 200 250 300
0
1
2
3
4
5
minus1
Y(m
)
X (m)
(a) 20 kmh
50 100 150 200 250
0
1
2
3
4
Y(m
)
X (m)
(b) 40 kmh
50 100 150 200 250 300
0
1
2
3
4
5
NewLiu
GuoRoute
minus1
Y(m
)
X (m)
(c) 60 kmh
50 100 150 200
0
2
4
NewLiu
GuoRoute
minus2
Y(m
)
X (m)
(d) 80 kmh
Figure 11 Comparison of modified driver model with other driver models at different speeds
closest to the assumed speed and the modified driver modelhas smaller responses except for vertical vibration and betterstability than the other models Though the modified drivermodel obtains greater vertical vehicle acceleration and worseride comfort than Legouisrsquos model at speeds lower than60 kmh the new driver model can give the vehicle betterpath-following ability and handling stability
4 Effects of Driver Model Parameters onFull Vehicle Dynamics and Stability
The important parameters in driver model include vehiclerunning speed time delay preview distance and permitposition error which decide the steering control effect andvehicle dynamics During the same driving maneuver theskilled drivers may use higher vehicle speed longer previewdistance shorter time delay and larger permit position error
than the new drivers The match of these four driverparameters also influences the vehicle dynamics RecentlyPauwelussen [30] researched the dependencies of driversteering control parameters on vehicle lateral properties andpath keeping and gives some useful conclusions This workfocuses on the effects of driver model parameters on bothpath keeping and full vehicle dynamics such as vehiclehandling stability ride comfort and roll stability
Since the vehicle lateral displacement is randomly causedby road roughness excitations and the lateral position devi-ation is always changing during lane change maneuvers astatistic index is introduced to evaluate the vehiclersquos path-following ability globally which is formulated by
Route error = radicsum (119877119884
119889(119905) minus 119884
119889(119905))2
119873
(20)
Mathematical Problems in Engineering 13
20 40 60 80 100 120 140 160 180 200
0
1
2
3
4
5
20 kmh30 kmh40 kmh
50 kmh60 kmhRoute
Y(m
)
X (m)
minus1
0 50 100 150 200
0
1
2
3
4
0 s002 s005 s
01 s02 sRoute
Y(m
)
X (m)
minus1
Figure 12 Routes of vehicle under different speed or time delay
where 119873 is the total number of data points 119877119884119889and 119884
119889are
the required lateral position and real lateral displacement ofthe preview point in the ground coordinate system
When a vehicle is undergoing a lane change its steeringstability and roll stability are vital problems The path-following error lateral acceleration and yaw rate can be usedto evaluate the steering stability The roll angle can be used toevaluate the roll stability Being proportional to the rolloverindex LTR (lateral-load transfer rate) the lateral accelerationis also able to reflect the vehicle roll stability [31] In additionthe vertical acceleration and pitch angle are simulated so asto research the whole-body dynamics in the vehicle Usingthese evaluation indexes the effects of parameters on steeringstability roll stability and riding comfort are discussed indetail
Figure 12 shows the vehicle trajectories under differentinitial vehicle speeds or driver time delays The effects ofvehicle speed and driver time delay on full vehicle dynamicsand stability are shown in Figure 13 As we can see fromFigures 12 and 13 a higher speed or a longer time delay willlead to a greater path-following error However low speed(20 kmh) and longtime delay (035 s) may enlarge the path-following error With the rise of vehicle speed the verticalacceleration lateral acceleration yaw rate and roll angleincrease while the pitch angle decreases Rise of time delaycauses the increase of lateral acceleration yaw rate and rollangle but hardly changes the vertical acceleration and pitchangle As for the effects of vehicle speed and driver time delayon full vehicle dynamics the following may be concluded
(1) A high vehicle speed during double-lane changeyields bad path keeping performance ride comfortsteering stability and roll stability When the vehiclespeed is higher than 60 kmh the ride comfort of thisvehicle gets worse greatly
(2) A long time delay is harmful to path keeping per-formance steering stability and roll stability and hasalmost no effect on the ride comfort
Thus a low vehicle speed ranging from 20 kmh to 60 kmh and a small time delay less than 035 s are recommended forimproving this truckrsquos stability when going through the lanechange
In case of varying the preview distance or permit positionerror the variation in vehicle trajectories at the speed of40 kmh is shown in Figure 14The effects of preview distanceor permit position error on the path-following ability andfull vehicle dynamics are shown in Figure 15 As shown inFigures 14 and 15 a large preview distance deteriorates thepath-following ability but has small effect on other dynamicresponses On the other hand a too small preview distanceleads to a hop of lateral acceleration yaw rate and roll anglewhich conflicts with closed-loop stability and is harmfulto steering control This conclusion is consistent with thatof Gillespie and MacAdam [15] and Pauwelussen [30] Theinfluence of permit position error on path-following ability issmall at short preview distance but great at long preview dis-tance As shown in Figure 14 a too small or too large permitposition error (001m or 08m)will worsen the vehiclersquos path-following ability In addition large permit position error mayincrease vertical acceleration lateral acceleration yaw rateand roll angle and reduce the handling stability and rollstability and ride comfort The effect of preview distance andpermit position error on pitch angle is small and shows fluc-tuation Thus a preview distance ranging from 10m to 30mand a small permit position error but more than 001m arerecommended for improving this truckrsquos stability when mov-ing through the lane change A smaller permit position errorshould be matched with a shorter preview distance
14 Mathematical Problems in Engineering
002
04
2040
60800
1
2
Rout
e err
or (m
)
u0 (kmh) 0
02
04
2040
60807
8
9
u0 (kmh)
002
04
204060800
02
04
wr (
rad
s)
u0 (kmh)
0
02
04
204060800
005
01
120579(r
ad)
u0 (kmh)002
04
2040
60800
01
02
Φ(r
ad)
u0 (kmh)
002
04
2040
60800
2
4
u0(kmh)
ay(m
s2 )
Tr(s) Tr
(s)
Tr(s)
Tr(s)
Tr(s) Tr
(s)
azb
(ms2 )
Figure 13 The effects of vehicle speed and driver time delay
0 50 100 150 200
0
1
2
3
4
5
001 m02 m04 m
06 m08 mRoute
Y(m
)
X (m)
minus10 50 100 150 200
0
2
4
6
8
10
12
2 m10 m20 m
30 m40 mRoute
Y(m
)
X (m)
minus2
minus4
Figure 14 Routes of vehicle under different preview distance and permit position error
Mathematical Problems in Engineering 15
020
4060
005
1150
1
2
Rout
e err
or (m
)
er (m) Ld (m)
020
4060
01
25
10
15
er (m) Ld (m)
020
4060
01
20
5
er (m) Ld (m) 020
4060
01
20
02
04
er (m) Ld (m)wr (
rad
s)
2040
601
20
02
04
00er (m) Ld (m)
Φ(r
ad)
020
40 60
01
20
005
01
er (m) Ld (m)
120579(r
ad)
ay(m
s2 )
azb
(ms2 )
Figure 15 The effects of preview distance and permit position error
5 Conclusions
In this paper a nonlinear three-directional coupled lumpedparameters (TCLP) model of heavy-duty vehicle consideringthe nonlinearity of suspension damping and tire stiffness isbuilt and connected with a proposedmodified preview drivermodel with nonlinear time delayThe proposed driver modelis simple and has a feedback gain 2119871119889
2 that is decided bywheelbase and previewdistanceThe validity of this presenteddriver-vehicle closed-loop system is verified by comparisonwith the test data the traditional steering stability vehiclemodel the linear vehicle model and other driver controlmodelsThe results show that the new drivermodel has betterlane keeping performances than the other two driver modelsThe coupling effects of vehicle motions and the nonlinear-ity of suspension damping and tire stiffness could not beneglected in turning or high speed driving situation
The effects of driver parameters on path-following abilityand full vehicle dynamics are also discussed It is foundthat the high vehicle speed large time delay long previewdistance or big permit position error will deteriorate path-following performances and handling stability Of course atoo short preview distance or a too small permit position
error is also harmful to path-following performances andhandling stability The ride comfort mainly depends on vehi-cle speed preview distance and permit position error Thekey factors influencing roll characteristics are vehicle speedand time delay
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The National Natural Science Foundation of China(11472180) theNatural Science Foundation ofHebei Province(E2012210025) and the New Century Talent Foundation ofMinistry of Education (NCET-13-0913) support this work
References
[1] M Plochl and J Edelmann ldquoDriver models in automobiledynamics applicationrdquoVehicle SystemDynamics vol 45 no 7-8pp 699ndash741 2007
16 Mathematical Problems in Engineering
[2] C C MacAdam ldquoUnderstanding and modeling the humandriverrdquo Vehicle System Dynamics vol 40 no 1ndash3 pp 101ndash1342003
[3] K Guo and H Guan ldquoModeling of drivervehicle directionalcontrol systemrdquo Vehicle System Dynamics vol 22 no 3-4 pp141ndash184 1993
[4] K GuoVehicle Handling DynamicsTheory Jiangsu Science andTechnology Publishing House Nanjing China 2011
[5] R S Sharp ldquoDriver steering control and a new perspective oncar handling qualitiesrdquo Proceedings of the Institution ofMechani-cal Engineers Part C Journal of Mechanical Engineering Sciencevol 219 no 10 pp 1041ndash1051 2005
[6] T Legouis A Laneville P Bourassa and G Payre ldquoCharac-terization of dynamic vehicle stability using two models of thehuman pilot behaviorrdquo Vehicle System Dynamics vol 15 no 1pp 1ndash18 1986
[7] Z Liu G Payre and P Bourassa ldquoNonlinear oscillations andchaotic motions in a road vehicle system with driver steeringcontrolrdquo Nonlinear Dynamics vol 9 no 3 pp 281ndash304 1996
[8] Z Liu G Payre and P Bourassa ldquoStability and oscillations ina time-delayed vehicle system with driver controlrdquo NonlinearDynamics vol 35 no 2 pp 159ndash173 2004
[9] X D Liu W Feng and N Kang ldquoResearch of simulationmethod of tractor-semitrailer carrying liquid considering liquidsloshingrdquo in Proceedings of the 10th National Conference onVibrationTheory and Application pp 581ndash590 Nanjing ChinaOctober 2011
[10] C C MacAdam ldquoApplication of an optimal preview control forsimulation of closed-loop automobile drivingrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 11 no 6 pp 393ndash399 1981
[11] A Y Ungoren and H Peng ldquoAn adaptive lateral preview drivermodelrdquo Vehicle System Dynamics vol 43 no 4 pp 245ndash2592005
[12] C I Chatzikomis and K N Spentzas ldquoA path-following drivermodel with longitudinal and lateral control of vehiclersquos motionrdquoForschung im Ingenieurwesen vol 73 no 4 pp 257ndash266 2009
[13] R S Sharp D Casanova and P Symonds ldquoA mathematicalmodel for driver steering control with design tuning andperformance resultsrdquo Vehicle System Dynamics vol 33 no 5pp 289ndash326 2000
[14] A J Pick and D J Cole ldquoNeuromuscular dynamics in thedriver-vehicle systemrdquo Vehicle System Dynamics vol 44 sup-plement 1 pp 624ndash631 2006
[15] T D Gillespie and C C MacAdam Constant Velocity YawRollProgram Userrsquos Manual The University of Michigan Trans-portation Research Institute Ann Arbor Mich USA 1982
[16] R I S Raj Influence of road roughness and directional maneuveron the dynamic performance of heavy vehicles [MS thesis]Department of Mechanical Engineering Concordia UniversityMontreal Canada 1998
[17] KWordenDHickeyMHaroon andD E Adams ldquoNonlinearsystem identification of automotive dampers a time and fre-quency-domain analysisrdquo Mechanical Systems and Signal Pro-cessing vol 23 no 1 pp 104ndash126 2009
[18] S H Li Y J Lu and L Y Li ldquoDynamical test and modelingfor hydraulic shock absorber on heavy vehicle under harmonicand random loadingsrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 13 pp 1903ndash1910 2012
[19] X W Ji and Y M Gao ldquoThe dynamic stiffness and dampingcharacteristics of the tirerdquo Automotive Engineering vol 5 pp315ndash321 1994
[20] G Gim and P E Nikravesh ldquoA three dimensional tire modelfor steady-state simulations of vehiclesrdquo SAE vol 102 no 2 pp150ndash159 1993
[21] J D Zhuang Principles of Automobile Tire Beijing Institute ofTechnology Press Beijing China 1996
[22] W-M Zhai ldquoTwo simple fast integration methods for large-scale dynamic problems in engineeringrdquo International Journalfor Numerical Methods in Engineering vol 39 no 24 pp 4199ndash4214 1996
[23] Dongfeng Motor Group Company Dongfeng DFL1250A8DFL1250A9 Truck Chassis Refit Manual Dongfeng MotorGroup Company 2008
[24] S Li S Yang L Chen and Y Lu ldquoEffects of parameters ondynamic responses for a heavy vehicle-pavement- foundationcoupled systemrdquo International Journal of Heavy Vehicle Systemsvol 19 no 2 pp 207ndash224 2012
[25] S Yang S Li andY Lu ldquoInvestigation on dynamical interactionbetween a heavy vehicle and road pavementrdquo Vehicle SystemDynamics vol 48 no 8 pp 923ndash944 2010
[26] GBT 7031-2005ISO 86081995 Mechanical Vibration-RoadSurface Profiles-Reporting of Measured Data 2005
[27] M Li X Pu and Z Changfu ldquoModeling and simulationof heavy-duty semi-trailer in extreme condition based onADAMSrdquo Automobile Technology vol 2 pp 22ndash25 2009
[28] ldquoPassenger carsmdashtest track for a severe lane-change manoeu-vremdashpart 1 double lane-changerdquo ISO 3888-11999
[29] S H Li S P Yang and N Chen ldquoDirectional control of adriver-heavy-vehicle closed-loop systemrdquo Advanced Engineer-ing Forum vol 2-3 pp 33ndash38 2012
[30] J Pauwelussen ldquoDependencies of driver steering controlparametersrdquo Vehicle System Dynamics vol 50 no 6 pp 939ndash959 2012
[31] H Imine and V Dolcemascolo ldquoRollover risk prediction ofheavy vehicle in interaction with infrastructurerdquo InternationalJournal ofHeavyVehicle Systems vol 14 no 3 pp 294ndash307 2007
Figure 4 The driver-heavy-vehicle closed-loop model
to the tire model Then the vertical longitudinal and lateraltire forces are calculated by the tire model and input intothe vehicle model to gain vehicle responses and positionsin next time step The simulation process of this driver-vehicle closed-loop system is shown in Figure 4 Due to thetime variability nonlinearity and high-dimensional propertyof this system the closed-loop system equations are solvednumerically by the quick integration method [22] and theRunge-Kutta method of order four
3 Model Evaluation
In order to verify the presented TCLP vehicle model andthe new driver model simulation results of this driver-vehicle closed-loop system are obtained using different vehi-cle models or driver models During simulation the vehicleparameters are chosen for a DFL1250A9 truck manufacturedby Dongfeng Motor Group Company Limited [23ndash25] andthe B-class road roughness is selected referring to [26]
The parameters of double-lane change route for theheavy-duty vehicle are chosen referring to [27 28] and shownin Figure 5
31 Comparison with the Handling Stability Vehicle Model Atraditional two-degree of freedom (2DOF) handling stabilityvehicle model for a three-axle heavy vehicle is set up whichconsiders only the lateral and yaw motion [29] The ordinarydifferential equations of motion of this 2DOF model may beexpressed by
where 119898 119868119911 119881119909 119881119910 and 120596
119903are vehicle mass vehicle inertia
around 119911 axial and longitudinal lateral and yaw rate of thevehicle respectively 120575
119894119865120572119894 and119872
119911119894are steering angle lateral
tire force and self-aligning torque of wheels respectively 119897119909119894
and 119897119910119894are the distance from wheel center to vehicle gravity
center in longitudinal and lateral direction respectivelyThe double-lane change responses of this TCLP model
and the traditional 2DOF model at an entrance speedof 60 kmh are simulated respectively and compared inFigure 6 It can be seen from Figure 6 that the results of thesetwo models are very consistent in magnitude and trendsHence the two vehicle models verify each other The TCLPmodel has a worse path-following ability than the 2DOFmodel and the vehicle running speed of TCLP model fluctu-ates randomlyThe yaw rate and lateral acceleration obtainedfrom TCLP model is bigger than that from 2DOF modelThe reason for these differences between two models is thatthe TCLP vehicle model considers B-class road roughnessand the coupled effect of roll vertical longitudinal and pitchmotion on yaw and lateral motion while the 2DOF modelneglects them
32 Comparison with the Linear Vehicle Model When thenonlinearity of front suspension damper and tire force isneglected as shown in Figure 7 the vehicle responses becomesmaller and the tracking performance is better Thus thelinear vehicle model may predict more conservative results
8 Mathematical Problems in Engineering
20m
20m40m40m
35m35m
15m
30m
35
m
35
m
35
m
375
m
Figure 5 The lane change route for the heavy-duty vehicle
0 5 10
162
164
166
168
17
0 5 10 15
0
5
Coupled model2DOF model
Required route
minus5
50 100 150 200
0
2
4
Y(m
)
X (m)
u(m
s)
t (s)
0 5 10 15
0
02
04
Coupled model2DOF model
Required route
minus04
minus02
t (s) t (s)
wr (
rad
s)
ay(m
s2)
Figure 6 Double-lane change responses of TCLP model and 2DOF model at 60 kmh
In the meantime it is found that the difference between thelinear and nonlinear vehicle models in the maneuver of lanechange is much greater than that in the maneuver of drivingstraight The lateral load transfer yaw motion and lateralmotion while lane change destabilizes the vehicle and conse-quently aggravates other vehicle motions due to couple effectIn addition by simulating responses at different entrancespeeds we found that the difference between the linearand nonlinear vehicle models becomes greater with the riseof vehicle speed Therefore the nonlinear vehicle modelshould be used so as to simulate vehicle dynamics accurately
especially in lane change maneuver or high speed drivingcondition
33 Comparison with Test Data During double-lane changetest an empty loaded DFL1250A9 truck was selected to runat 30 kmh speed A three-axis piezoelectric accelerometer(frequency range from 1Hz to 500Hz) and a gyro (measur-ing range plusmn50 degs) were placed at the center of gravity ofwhole vehicle and a cable displacement sensor was placedat front wheel to measure front wheel steering angle asshown in Figure 8 The measured signals were amplified by
Mathematical Problems in Engineering 9
50 100 150 200 250 300
0
1
2
3
4
minus1
Y(m
)
X (m)0 5 10 15 20
162
164
166
168
u(m
s)
t (s)
5 10 15 20
0
2
4
minus4
minus2
t (s)
ay
(ms
2 )
5 10 15 20
0
01
02
03
minus02
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
001
minus002
minus001
t (s)
120573(r
ad)
7 8 9
0
10
20
minus20
minus10
t (s)
azb
(ms
2 )
5 10 15 20
0
01
02
minus02
minus01
NonlinearLinear
Requied route
t (s)
Φ(r
ad)
5 10 15 20
0
005
minus01
minus005
NonlinearLinear
Requied route
t (s)
120579(r
ad)
Figure 7 The double-lane change responses of the linear and nonlinear TCLP model at 60 kmh
10 Mathematical Problems in Engineering
Figure 8 Vehicle and sensors in field test
Table 1 Vehicle responses obtained from three driver models
Vehicle speed Driver model RMS value of vehicle responses119906 119908119903 119886119910 119886119911119887 119891119894 119909119894119905119886
a multichannel charge amplifier (YE5853A) and convertedto digital signals by an intelligent acquisition and processinganalyzer (INV360DF) The sampling frequency was 200Hz
Figure 9 gives the experimental time history of frontwheel steering angle Using this measured front wheel steer-ing angle as input the yaw rate and lateral acceleration are cal-culated from TCLP model 2DOF model and linear modelrespectively These simulation results are compared with
the test results as shown in Figure 10 It is obvious thatthe simulation results of TCLP model are most consistentwith the results of field test Due to neglecting of the three-directional coupling effects of vehiclemotions or nonlinearityof suspension and tire the time histories of yaw rate obtainedfrom 2DOF model and linear model are too smooth and thelateral accelerations of 2DOF model and linear model aresmaller than the test results Thus the verification of thisproposed TCLP model is verified
34 Comparison with Other Driver Models Using the pro-posed modified driver model the optimal curvature modelby Guo and nonlinear time delay driver model by Legouisthe trajectories of vehicle gravity center during double-lanechange maneuver at four entrance speeds of 20 40 60 and80 kmh are simulated respectively as shown in Figure 11Table 1 lists the root mean square (RMS) value of vehicleresponses in six directions at different speeds applying thesethree driver models As shown in Figure 11 it is obvious thatthe tracking performance of the modified driver model issuperior to the other two driver models at different entrancespeeds It is also found from Figure 11 that though an over-shoot of the newmodel at a vehicle speed of 80 kmh exists in119883 = [150 180]m the results of the other two models divergewhen the vehicle arrives at 119883 = 200m while the resultof the new model is stable From Table 1 we can see thatthe real vehicle speed of the modified driver model is the
Mathematical Problems in Engineering 11
Simulation results of TCLP model
Test results
Simulation results of 2DOF model
Simulation results of linear model
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
02
04
minus04
minus02
t (s)
ay(m
s2 )
5 10 15 20
0
05
1
minus05
t (s)
ay(m
s2 )
Figure 10 Comparison of simulation and test results
12 Mathematical Problems in Engineering
50 100 150 200 250 300
0
1
2
3
4
5
minus1
Y(m
)
X (m)
(a) 20 kmh
50 100 150 200 250
0
1
2
3
4
Y(m
)
X (m)
(b) 40 kmh
50 100 150 200 250 300
0
1
2
3
4
5
NewLiu
GuoRoute
minus1
Y(m
)
X (m)
(c) 60 kmh
50 100 150 200
0
2
4
NewLiu
GuoRoute
minus2
Y(m
)
X (m)
(d) 80 kmh
Figure 11 Comparison of modified driver model with other driver models at different speeds
closest to the assumed speed and the modified driver modelhas smaller responses except for vertical vibration and betterstability than the other models Though the modified drivermodel obtains greater vertical vehicle acceleration and worseride comfort than Legouisrsquos model at speeds lower than60 kmh the new driver model can give the vehicle betterpath-following ability and handling stability
4 Effects of Driver Model Parameters onFull Vehicle Dynamics and Stability
The important parameters in driver model include vehiclerunning speed time delay preview distance and permitposition error which decide the steering control effect andvehicle dynamics During the same driving maneuver theskilled drivers may use higher vehicle speed longer previewdistance shorter time delay and larger permit position error
than the new drivers The match of these four driverparameters also influences the vehicle dynamics RecentlyPauwelussen [30] researched the dependencies of driversteering control parameters on vehicle lateral properties andpath keeping and gives some useful conclusions This workfocuses on the effects of driver model parameters on bothpath keeping and full vehicle dynamics such as vehiclehandling stability ride comfort and roll stability
Since the vehicle lateral displacement is randomly causedby road roughness excitations and the lateral position devi-ation is always changing during lane change maneuvers astatistic index is introduced to evaluate the vehiclersquos path-following ability globally which is formulated by
Route error = radicsum (119877119884
119889(119905) minus 119884
119889(119905))2
119873
(20)
Mathematical Problems in Engineering 13
20 40 60 80 100 120 140 160 180 200
0
1
2
3
4
5
20 kmh30 kmh40 kmh
50 kmh60 kmhRoute
Y(m
)
X (m)
minus1
0 50 100 150 200
0
1
2
3
4
0 s002 s005 s
01 s02 sRoute
Y(m
)
X (m)
minus1
Figure 12 Routes of vehicle under different speed or time delay
where 119873 is the total number of data points 119877119884119889and 119884
119889are
the required lateral position and real lateral displacement ofthe preview point in the ground coordinate system
When a vehicle is undergoing a lane change its steeringstability and roll stability are vital problems The path-following error lateral acceleration and yaw rate can be usedto evaluate the steering stability The roll angle can be used toevaluate the roll stability Being proportional to the rolloverindex LTR (lateral-load transfer rate) the lateral accelerationis also able to reflect the vehicle roll stability [31] In additionthe vertical acceleration and pitch angle are simulated so asto research the whole-body dynamics in the vehicle Usingthese evaluation indexes the effects of parameters on steeringstability roll stability and riding comfort are discussed indetail
Figure 12 shows the vehicle trajectories under differentinitial vehicle speeds or driver time delays The effects ofvehicle speed and driver time delay on full vehicle dynamicsand stability are shown in Figure 13 As we can see fromFigures 12 and 13 a higher speed or a longer time delay willlead to a greater path-following error However low speed(20 kmh) and longtime delay (035 s) may enlarge the path-following error With the rise of vehicle speed the verticalacceleration lateral acceleration yaw rate and roll angleincrease while the pitch angle decreases Rise of time delaycauses the increase of lateral acceleration yaw rate and rollangle but hardly changes the vertical acceleration and pitchangle As for the effects of vehicle speed and driver time delayon full vehicle dynamics the following may be concluded
(1) A high vehicle speed during double-lane changeyields bad path keeping performance ride comfortsteering stability and roll stability When the vehiclespeed is higher than 60 kmh the ride comfort of thisvehicle gets worse greatly
(2) A long time delay is harmful to path keeping per-formance steering stability and roll stability and hasalmost no effect on the ride comfort
Thus a low vehicle speed ranging from 20 kmh to 60 kmh and a small time delay less than 035 s are recommended forimproving this truckrsquos stability when going through the lanechange
In case of varying the preview distance or permit positionerror the variation in vehicle trajectories at the speed of40 kmh is shown in Figure 14The effects of preview distanceor permit position error on the path-following ability andfull vehicle dynamics are shown in Figure 15 As shown inFigures 14 and 15 a large preview distance deteriorates thepath-following ability but has small effect on other dynamicresponses On the other hand a too small preview distanceleads to a hop of lateral acceleration yaw rate and roll anglewhich conflicts with closed-loop stability and is harmfulto steering control This conclusion is consistent with thatof Gillespie and MacAdam [15] and Pauwelussen [30] Theinfluence of permit position error on path-following ability issmall at short preview distance but great at long preview dis-tance As shown in Figure 14 a too small or too large permitposition error (001m or 08m)will worsen the vehiclersquos path-following ability In addition large permit position error mayincrease vertical acceleration lateral acceleration yaw rateand roll angle and reduce the handling stability and rollstability and ride comfort The effect of preview distance andpermit position error on pitch angle is small and shows fluc-tuation Thus a preview distance ranging from 10m to 30mand a small permit position error but more than 001m arerecommended for improving this truckrsquos stability when mov-ing through the lane change A smaller permit position errorshould be matched with a shorter preview distance
14 Mathematical Problems in Engineering
002
04
2040
60800
1
2
Rout
e err
or (m
)
u0 (kmh) 0
02
04
2040
60807
8
9
u0 (kmh)
002
04
204060800
02
04
wr (
rad
s)
u0 (kmh)
0
02
04
204060800
005
01
120579(r
ad)
u0 (kmh)002
04
2040
60800
01
02
Φ(r
ad)
u0 (kmh)
002
04
2040
60800
2
4
u0(kmh)
ay(m
s2 )
Tr(s) Tr
(s)
Tr(s)
Tr(s)
Tr(s) Tr
(s)
azb
(ms2 )
Figure 13 The effects of vehicle speed and driver time delay
0 50 100 150 200
0
1
2
3
4
5
001 m02 m04 m
06 m08 mRoute
Y(m
)
X (m)
minus10 50 100 150 200
0
2
4
6
8
10
12
2 m10 m20 m
30 m40 mRoute
Y(m
)
X (m)
minus2
minus4
Figure 14 Routes of vehicle under different preview distance and permit position error
Mathematical Problems in Engineering 15
020
4060
005
1150
1
2
Rout
e err
or (m
)
er (m) Ld (m)
020
4060
01
25
10
15
er (m) Ld (m)
020
4060
01
20
5
er (m) Ld (m) 020
4060
01
20
02
04
er (m) Ld (m)wr (
rad
s)
2040
601
20
02
04
00er (m) Ld (m)
Φ(r
ad)
020
40 60
01
20
005
01
er (m) Ld (m)
120579(r
ad)
ay(m
s2 )
azb
(ms2 )
Figure 15 The effects of preview distance and permit position error
5 Conclusions
In this paper a nonlinear three-directional coupled lumpedparameters (TCLP) model of heavy-duty vehicle consideringthe nonlinearity of suspension damping and tire stiffness isbuilt and connected with a proposedmodified preview drivermodel with nonlinear time delayThe proposed driver modelis simple and has a feedback gain 2119871119889
2 that is decided bywheelbase and previewdistanceThe validity of this presenteddriver-vehicle closed-loop system is verified by comparisonwith the test data the traditional steering stability vehiclemodel the linear vehicle model and other driver controlmodelsThe results show that the new drivermodel has betterlane keeping performances than the other two driver modelsThe coupling effects of vehicle motions and the nonlinear-ity of suspension damping and tire stiffness could not beneglected in turning or high speed driving situation
The effects of driver parameters on path-following abilityand full vehicle dynamics are also discussed It is foundthat the high vehicle speed large time delay long previewdistance or big permit position error will deteriorate path-following performances and handling stability Of course atoo short preview distance or a too small permit position
error is also harmful to path-following performances andhandling stability The ride comfort mainly depends on vehi-cle speed preview distance and permit position error Thekey factors influencing roll characteristics are vehicle speedand time delay
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The National Natural Science Foundation of China(11472180) theNatural Science Foundation ofHebei Province(E2012210025) and the New Century Talent Foundation ofMinistry of Education (NCET-13-0913) support this work
References
[1] M Plochl and J Edelmann ldquoDriver models in automobiledynamics applicationrdquoVehicle SystemDynamics vol 45 no 7-8pp 699ndash741 2007
16 Mathematical Problems in Engineering
[2] C C MacAdam ldquoUnderstanding and modeling the humandriverrdquo Vehicle System Dynamics vol 40 no 1ndash3 pp 101ndash1342003
[3] K Guo and H Guan ldquoModeling of drivervehicle directionalcontrol systemrdquo Vehicle System Dynamics vol 22 no 3-4 pp141ndash184 1993
[4] K GuoVehicle Handling DynamicsTheory Jiangsu Science andTechnology Publishing House Nanjing China 2011
[5] R S Sharp ldquoDriver steering control and a new perspective oncar handling qualitiesrdquo Proceedings of the Institution ofMechani-cal Engineers Part C Journal of Mechanical Engineering Sciencevol 219 no 10 pp 1041ndash1051 2005
[6] T Legouis A Laneville P Bourassa and G Payre ldquoCharac-terization of dynamic vehicle stability using two models of thehuman pilot behaviorrdquo Vehicle System Dynamics vol 15 no 1pp 1ndash18 1986
[7] Z Liu G Payre and P Bourassa ldquoNonlinear oscillations andchaotic motions in a road vehicle system with driver steeringcontrolrdquo Nonlinear Dynamics vol 9 no 3 pp 281ndash304 1996
[8] Z Liu G Payre and P Bourassa ldquoStability and oscillations ina time-delayed vehicle system with driver controlrdquo NonlinearDynamics vol 35 no 2 pp 159ndash173 2004
[9] X D Liu W Feng and N Kang ldquoResearch of simulationmethod of tractor-semitrailer carrying liquid considering liquidsloshingrdquo in Proceedings of the 10th National Conference onVibrationTheory and Application pp 581ndash590 Nanjing ChinaOctober 2011
[10] C C MacAdam ldquoApplication of an optimal preview control forsimulation of closed-loop automobile drivingrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 11 no 6 pp 393ndash399 1981
[11] A Y Ungoren and H Peng ldquoAn adaptive lateral preview drivermodelrdquo Vehicle System Dynamics vol 43 no 4 pp 245ndash2592005
[12] C I Chatzikomis and K N Spentzas ldquoA path-following drivermodel with longitudinal and lateral control of vehiclersquos motionrdquoForschung im Ingenieurwesen vol 73 no 4 pp 257ndash266 2009
[13] R S Sharp D Casanova and P Symonds ldquoA mathematicalmodel for driver steering control with design tuning andperformance resultsrdquo Vehicle System Dynamics vol 33 no 5pp 289ndash326 2000
[14] A J Pick and D J Cole ldquoNeuromuscular dynamics in thedriver-vehicle systemrdquo Vehicle System Dynamics vol 44 sup-plement 1 pp 624ndash631 2006
[15] T D Gillespie and C C MacAdam Constant Velocity YawRollProgram Userrsquos Manual The University of Michigan Trans-portation Research Institute Ann Arbor Mich USA 1982
[16] R I S Raj Influence of road roughness and directional maneuveron the dynamic performance of heavy vehicles [MS thesis]Department of Mechanical Engineering Concordia UniversityMontreal Canada 1998
[17] KWordenDHickeyMHaroon andD E Adams ldquoNonlinearsystem identification of automotive dampers a time and fre-quency-domain analysisrdquo Mechanical Systems and Signal Pro-cessing vol 23 no 1 pp 104ndash126 2009
[18] S H Li Y J Lu and L Y Li ldquoDynamical test and modelingfor hydraulic shock absorber on heavy vehicle under harmonicand random loadingsrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 13 pp 1903ndash1910 2012
[19] X W Ji and Y M Gao ldquoThe dynamic stiffness and dampingcharacteristics of the tirerdquo Automotive Engineering vol 5 pp315ndash321 1994
[20] G Gim and P E Nikravesh ldquoA three dimensional tire modelfor steady-state simulations of vehiclesrdquo SAE vol 102 no 2 pp150ndash159 1993
[21] J D Zhuang Principles of Automobile Tire Beijing Institute ofTechnology Press Beijing China 1996
[22] W-M Zhai ldquoTwo simple fast integration methods for large-scale dynamic problems in engineeringrdquo International Journalfor Numerical Methods in Engineering vol 39 no 24 pp 4199ndash4214 1996
[23] Dongfeng Motor Group Company Dongfeng DFL1250A8DFL1250A9 Truck Chassis Refit Manual Dongfeng MotorGroup Company 2008
[24] S Li S Yang L Chen and Y Lu ldquoEffects of parameters ondynamic responses for a heavy vehicle-pavement- foundationcoupled systemrdquo International Journal of Heavy Vehicle Systemsvol 19 no 2 pp 207ndash224 2012
[25] S Yang S Li andY Lu ldquoInvestigation on dynamical interactionbetween a heavy vehicle and road pavementrdquo Vehicle SystemDynamics vol 48 no 8 pp 923ndash944 2010
[26] GBT 7031-2005ISO 86081995 Mechanical Vibration-RoadSurface Profiles-Reporting of Measured Data 2005
[27] M Li X Pu and Z Changfu ldquoModeling and simulationof heavy-duty semi-trailer in extreme condition based onADAMSrdquo Automobile Technology vol 2 pp 22ndash25 2009
[28] ldquoPassenger carsmdashtest track for a severe lane-change manoeu-vremdashpart 1 double lane-changerdquo ISO 3888-11999
[29] S H Li S P Yang and N Chen ldquoDirectional control of adriver-heavy-vehicle closed-loop systemrdquo Advanced Engineer-ing Forum vol 2-3 pp 33ndash38 2012
[30] J Pauwelussen ldquoDependencies of driver steering controlparametersrdquo Vehicle System Dynamics vol 50 no 6 pp 939ndash959 2012
[31] H Imine and V Dolcemascolo ldquoRollover risk prediction ofheavy vehicle in interaction with infrastructurerdquo InternationalJournal ofHeavyVehicle Systems vol 14 no 3 pp 294ndash307 2007
Figure 5 The lane change route for the heavy-duty vehicle
0 5 10
162
164
166
168
17
0 5 10 15
0
5
Coupled model2DOF model
Required route
minus5
50 100 150 200
0
2
4
Y(m
)
X (m)
u(m
s)
t (s)
0 5 10 15
0
02
04
Coupled model2DOF model
Required route
minus04
minus02
t (s) t (s)
wr (
rad
s)
ay(m
s2)
Figure 6 Double-lane change responses of TCLP model and 2DOF model at 60 kmh
In the meantime it is found that the difference between thelinear and nonlinear vehicle models in the maneuver of lanechange is much greater than that in the maneuver of drivingstraight The lateral load transfer yaw motion and lateralmotion while lane change destabilizes the vehicle and conse-quently aggravates other vehicle motions due to couple effectIn addition by simulating responses at different entrancespeeds we found that the difference between the linearand nonlinear vehicle models becomes greater with the riseof vehicle speed Therefore the nonlinear vehicle modelshould be used so as to simulate vehicle dynamics accurately
especially in lane change maneuver or high speed drivingcondition
33 Comparison with Test Data During double-lane changetest an empty loaded DFL1250A9 truck was selected to runat 30 kmh speed A three-axis piezoelectric accelerometer(frequency range from 1Hz to 500Hz) and a gyro (measur-ing range plusmn50 degs) were placed at the center of gravity ofwhole vehicle and a cable displacement sensor was placedat front wheel to measure front wheel steering angle asshown in Figure 8 The measured signals were amplified by
Mathematical Problems in Engineering 9
50 100 150 200 250 300
0
1
2
3
4
minus1
Y(m
)
X (m)0 5 10 15 20
162
164
166
168
u(m
s)
t (s)
5 10 15 20
0
2
4
minus4
minus2
t (s)
ay
(ms
2 )
5 10 15 20
0
01
02
03
minus02
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
001
minus002
minus001
t (s)
120573(r
ad)
7 8 9
0
10
20
minus20
minus10
t (s)
azb
(ms
2 )
5 10 15 20
0
01
02
minus02
minus01
NonlinearLinear
Requied route
t (s)
Φ(r
ad)
5 10 15 20
0
005
minus01
minus005
NonlinearLinear
Requied route
t (s)
120579(r
ad)
Figure 7 The double-lane change responses of the linear and nonlinear TCLP model at 60 kmh
10 Mathematical Problems in Engineering
Figure 8 Vehicle and sensors in field test
Table 1 Vehicle responses obtained from three driver models
Vehicle speed Driver model RMS value of vehicle responses119906 119908119903 119886119910 119886119911119887 119891119894 119909119894119905119886
a multichannel charge amplifier (YE5853A) and convertedto digital signals by an intelligent acquisition and processinganalyzer (INV360DF) The sampling frequency was 200Hz
Figure 9 gives the experimental time history of frontwheel steering angle Using this measured front wheel steer-ing angle as input the yaw rate and lateral acceleration are cal-culated from TCLP model 2DOF model and linear modelrespectively These simulation results are compared with
the test results as shown in Figure 10 It is obvious thatthe simulation results of TCLP model are most consistentwith the results of field test Due to neglecting of the three-directional coupling effects of vehiclemotions or nonlinearityof suspension and tire the time histories of yaw rate obtainedfrom 2DOF model and linear model are too smooth and thelateral accelerations of 2DOF model and linear model aresmaller than the test results Thus the verification of thisproposed TCLP model is verified
34 Comparison with Other Driver Models Using the pro-posed modified driver model the optimal curvature modelby Guo and nonlinear time delay driver model by Legouisthe trajectories of vehicle gravity center during double-lanechange maneuver at four entrance speeds of 20 40 60 and80 kmh are simulated respectively as shown in Figure 11Table 1 lists the root mean square (RMS) value of vehicleresponses in six directions at different speeds applying thesethree driver models As shown in Figure 11 it is obvious thatthe tracking performance of the modified driver model issuperior to the other two driver models at different entrancespeeds It is also found from Figure 11 that though an over-shoot of the newmodel at a vehicle speed of 80 kmh exists in119883 = [150 180]m the results of the other two models divergewhen the vehicle arrives at 119883 = 200m while the resultof the new model is stable From Table 1 we can see thatthe real vehicle speed of the modified driver model is the
Mathematical Problems in Engineering 11
Simulation results of TCLP model
Test results
Simulation results of 2DOF model
Simulation results of linear model
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
02
04
minus04
minus02
t (s)
ay(m
s2 )
5 10 15 20
0
05
1
minus05
t (s)
ay(m
s2 )
Figure 10 Comparison of simulation and test results
12 Mathematical Problems in Engineering
50 100 150 200 250 300
0
1
2
3
4
5
minus1
Y(m
)
X (m)
(a) 20 kmh
50 100 150 200 250
0
1
2
3
4
Y(m
)
X (m)
(b) 40 kmh
50 100 150 200 250 300
0
1
2
3
4
5
NewLiu
GuoRoute
minus1
Y(m
)
X (m)
(c) 60 kmh
50 100 150 200
0
2
4
NewLiu
GuoRoute
minus2
Y(m
)
X (m)
(d) 80 kmh
Figure 11 Comparison of modified driver model with other driver models at different speeds
closest to the assumed speed and the modified driver modelhas smaller responses except for vertical vibration and betterstability than the other models Though the modified drivermodel obtains greater vertical vehicle acceleration and worseride comfort than Legouisrsquos model at speeds lower than60 kmh the new driver model can give the vehicle betterpath-following ability and handling stability
4 Effects of Driver Model Parameters onFull Vehicle Dynamics and Stability
The important parameters in driver model include vehiclerunning speed time delay preview distance and permitposition error which decide the steering control effect andvehicle dynamics During the same driving maneuver theskilled drivers may use higher vehicle speed longer previewdistance shorter time delay and larger permit position error
than the new drivers The match of these four driverparameters also influences the vehicle dynamics RecentlyPauwelussen [30] researched the dependencies of driversteering control parameters on vehicle lateral properties andpath keeping and gives some useful conclusions This workfocuses on the effects of driver model parameters on bothpath keeping and full vehicle dynamics such as vehiclehandling stability ride comfort and roll stability
Since the vehicle lateral displacement is randomly causedby road roughness excitations and the lateral position devi-ation is always changing during lane change maneuvers astatistic index is introduced to evaluate the vehiclersquos path-following ability globally which is formulated by
Route error = radicsum (119877119884
119889(119905) minus 119884
119889(119905))2
119873
(20)
Mathematical Problems in Engineering 13
20 40 60 80 100 120 140 160 180 200
0
1
2
3
4
5
20 kmh30 kmh40 kmh
50 kmh60 kmhRoute
Y(m
)
X (m)
minus1
0 50 100 150 200
0
1
2
3
4
0 s002 s005 s
01 s02 sRoute
Y(m
)
X (m)
minus1
Figure 12 Routes of vehicle under different speed or time delay
where 119873 is the total number of data points 119877119884119889and 119884
119889are
the required lateral position and real lateral displacement ofthe preview point in the ground coordinate system
When a vehicle is undergoing a lane change its steeringstability and roll stability are vital problems The path-following error lateral acceleration and yaw rate can be usedto evaluate the steering stability The roll angle can be used toevaluate the roll stability Being proportional to the rolloverindex LTR (lateral-load transfer rate) the lateral accelerationis also able to reflect the vehicle roll stability [31] In additionthe vertical acceleration and pitch angle are simulated so asto research the whole-body dynamics in the vehicle Usingthese evaluation indexes the effects of parameters on steeringstability roll stability and riding comfort are discussed indetail
Figure 12 shows the vehicle trajectories under differentinitial vehicle speeds or driver time delays The effects ofvehicle speed and driver time delay on full vehicle dynamicsand stability are shown in Figure 13 As we can see fromFigures 12 and 13 a higher speed or a longer time delay willlead to a greater path-following error However low speed(20 kmh) and longtime delay (035 s) may enlarge the path-following error With the rise of vehicle speed the verticalacceleration lateral acceleration yaw rate and roll angleincrease while the pitch angle decreases Rise of time delaycauses the increase of lateral acceleration yaw rate and rollangle but hardly changes the vertical acceleration and pitchangle As for the effects of vehicle speed and driver time delayon full vehicle dynamics the following may be concluded
(1) A high vehicle speed during double-lane changeyields bad path keeping performance ride comfortsteering stability and roll stability When the vehiclespeed is higher than 60 kmh the ride comfort of thisvehicle gets worse greatly
(2) A long time delay is harmful to path keeping per-formance steering stability and roll stability and hasalmost no effect on the ride comfort
Thus a low vehicle speed ranging from 20 kmh to 60 kmh and a small time delay less than 035 s are recommended forimproving this truckrsquos stability when going through the lanechange
In case of varying the preview distance or permit positionerror the variation in vehicle trajectories at the speed of40 kmh is shown in Figure 14The effects of preview distanceor permit position error on the path-following ability andfull vehicle dynamics are shown in Figure 15 As shown inFigures 14 and 15 a large preview distance deteriorates thepath-following ability but has small effect on other dynamicresponses On the other hand a too small preview distanceleads to a hop of lateral acceleration yaw rate and roll anglewhich conflicts with closed-loop stability and is harmfulto steering control This conclusion is consistent with thatof Gillespie and MacAdam [15] and Pauwelussen [30] Theinfluence of permit position error on path-following ability issmall at short preview distance but great at long preview dis-tance As shown in Figure 14 a too small or too large permitposition error (001m or 08m)will worsen the vehiclersquos path-following ability In addition large permit position error mayincrease vertical acceleration lateral acceleration yaw rateand roll angle and reduce the handling stability and rollstability and ride comfort The effect of preview distance andpermit position error on pitch angle is small and shows fluc-tuation Thus a preview distance ranging from 10m to 30mand a small permit position error but more than 001m arerecommended for improving this truckrsquos stability when mov-ing through the lane change A smaller permit position errorshould be matched with a shorter preview distance
14 Mathematical Problems in Engineering
002
04
2040
60800
1
2
Rout
e err
or (m
)
u0 (kmh) 0
02
04
2040
60807
8
9
u0 (kmh)
002
04
204060800
02
04
wr (
rad
s)
u0 (kmh)
0
02
04
204060800
005
01
120579(r
ad)
u0 (kmh)002
04
2040
60800
01
02
Φ(r
ad)
u0 (kmh)
002
04
2040
60800
2
4
u0(kmh)
ay(m
s2 )
Tr(s) Tr
(s)
Tr(s)
Tr(s)
Tr(s) Tr
(s)
azb
(ms2 )
Figure 13 The effects of vehicle speed and driver time delay
0 50 100 150 200
0
1
2
3
4
5
001 m02 m04 m
06 m08 mRoute
Y(m
)
X (m)
minus10 50 100 150 200
0
2
4
6
8
10
12
2 m10 m20 m
30 m40 mRoute
Y(m
)
X (m)
minus2
minus4
Figure 14 Routes of vehicle under different preview distance and permit position error
Mathematical Problems in Engineering 15
020
4060
005
1150
1
2
Rout
e err
or (m
)
er (m) Ld (m)
020
4060
01
25
10
15
er (m) Ld (m)
020
4060
01
20
5
er (m) Ld (m) 020
4060
01
20
02
04
er (m) Ld (m)wr (
rad
s)
2040
601
20
02
04
00er (m) Ld (m)
Φ(r
ad)
020
40 60
01
20
005
01
er (m) Ld (m)
120579(r
ad)
ay(m
s2 )
azb
(ms2 )
Figure 15 The effects of preview distance and permit position error
5 Conclusions
In this paper a nonlinear three-directional coupled lumpedparameters (TCLP) model of heavy-duty vehicle consideringthe nonlinearity of suspension damping and tire stiffness isbuilt and connected with a proposedmodified preview drivermodel with nonlinear time delayThe proposed driver modelis simple and has a feedback gain 2119871119889
2 that is decided bywheelbase and previewdistanceThe validity of this presenteddriver-vehicle closed-loop system is verified by comparisonwith the test data the traditional steering stability vehiclemodel the linear vehicle model and other driver controlmodelsThe results show that the new drivermodel has betterlane keeping performances than the other two driver modelsThe coupling effects of vehicle motions and the nonlinear-ity of suspension damping and tire stiffness could not beneglected in turning or high speed driving situation
The effects of driver parameters on path-following abilityand full vehicle dynamics are also discussed It is foundthat the high vehicle speed large time delay long previewdistance or big permit position error will deteriorate path-following performances and handling stability Of course atoo short preview distance or a too small permit position
error is also harmful to path-following performances andhandling stability The ride comfort mainly depends on vehi-cle speed preview distance and permit position error Thekey factors influencing roll characteristics are vehicle speedand time delay
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The National Natural Science Foundation of China(11472180) theNatural Science Foundation ofHebei Province(E2012210025) and the New Century Talent Foundation ofMinistry of Education (NCET-13-0913) support this work
References
[1] M Plochl and J Edelmann ldquoDriver models in automobiledynamics applicationrdquoVehicle SystemDynamics vol 45 no 7-8pp 699ndash741 2007
16 Mathematical Problems in Engineering
[2] C C MacAdam ldquoUnderstanding and modeling the humandriverrdquo Vehicle System Dynamics vol 40 no 1ndash3 pp 101ndash1342003
[3] K Guo and H Guan ldquoModeling of drivervehicle directionalcontrol systemrdquo Vehicle System Dynamics vol 22 no 3-4 pp141ndash184 1993
[4] K GuoVehicle Handling DynamicsTheory Jiangsu Science andTechnology Publishing House Nanjing China 2011
[5] R S Sharp ldquoDriver steering control and a new perspective oncar handling qualitiesrdquo Proceedings of the Institution ofMechani-cal Engineers Part C Journal of Mechanical Engineering Sciencevol 219 no 10 pp 1041ndash1051 2005
[6] T Legouis A Laneville P Bourassa and G Payre ldquoCharac-terization of dynamic vehicle stability using two models of thehuman pilot behaviorrdquo Vehicle System Dynamics vol 15 no 1pp 1ndash18 1986
[7] Z Liu G Payre and P Bourassa ldquoNonlinear oscillations andchaotic motions in a road vehicle system with driver steeringcontrolrdquo Nonlinear Dynamics vol 9 no 3 pp 281ndash304 1996
[8] Z Liu G Payre and P Bourassa ldquoStability and oscillations ina time-delayed vehicle system with driver controlrdquo NonlinearDynamics vol 35 no 2 pp 159ndash173 2004
[9] X D Liu W Feng and N Kang ldquoResearch of simulationmethod of tractor-semitrailer carrying liquid considering liquidsloshingrdquo in Proceedings of the 10th National Conference onVibrationTheory and Application pp 581ndash590 Nanjing ChinaOctober 2011
[10] C C MacAdam ldquoApplication of an optimal preview control forsimulation of closed-loop automobile drivingrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 11 no 6 pp 393ndash399 1981
[11] A Y Ungoren and H Peng ldquoAn adaptive lateral preview drivermodelrdquo Vehicle System Dynamics vol 43 no 4 pp 245ndash2592005
[12] C I Chatzikomis and K N Spentzas ldquoA path-following drivermodel with longitudinal and lateral control of vehiclersquos motionrdquoForschung im Ingenieurwesen vol 73 no 4 pp 257ndash266 2009
[13] R S Sharp D Casanova and P Symonds ldquoA mathematicalmodel for driver steering control with design tuning andperformance resultsrdquo Vehicle System Dynamics vol 33 no 5pp 289ndash326 2000
[14] A J Pick and D J Cole ldquoNeuromuscular dynamics in thedriver-vehicle systemrdquo Vehicle System Dynamics vol 44 sup-plement 1 pp 624ndash631 2006
[15] T D Gillespie and C C MacAdam Constant Velocity YawRollProgram Userrsquos Manual The University of Michigan Trans-portation Research Institute Ann Arbor Mich USA 1982
[16] R I S Raj Influence of road roughness and directional maneuveron the dynamic performance of heavy vehicles [MS thesis]Department of Mechanical Engineering Concordia UniversityMontreal Canada 1998
[17] KWordenDHickeyMHaroon andD E Adams ldquoNonlinearsystem identification of automotive dampers a time and fre-quency-domain analysisrdquo Mechanical Systems and Signal Pro-cessing vol 23 no 1 pp 104ndash126 2009
[18] S H Li Y J Lu and L Y Li ldquoDynamical test and modelingfor hydraulic shock absorber on heavy vehicle under harmonicand random loadingsrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 13 pp 1903ndash1910 2012
[19] X W Ji and Y M Gao ldquoThe dynamic stiffness and dampingcharacteristics of the tirerdquo Automotive Engineering vol 5 pp315ndash321 1994
[20] G Gim and P E Nikravesh ldquoA three dimensional tire modelfor steady-state simulations of vehiclesrdquo SAE vol 102 no 2 pp150ndash159 1993
[21] J D Zhuang Principles of Automobile Tire Beijing Institute ofTechnology Press Beijing China 1996
[22] W-M Zhai ldquoTwo simple fast integration methods for large-scale dynamic problems in engineeringrdquo International Journalfor Numerical Methods in Engineering vol 39 no 24 pp 4199ndash4214 1996
[23] Dongfeng Motor Group Company Dongfeng DFL1250A8DFL1250A9 Truck Chassis Refit Manual Dongfeng MotorGroup Company 2008
[24] S Li S Yang L Chen and Y Lu ldquoEffects of parameters ondynamic responses for a heavy vehicle-pavement- foundationcoupled systemrdquo International Journal of Heavy Vehicle Systemsvol 19 no 2 pp 207ndash224 2012
[25] S Yang S Li andY Lu ldquoInvestigation on dynamical interactionbetween a heavy vehicle and road pavementrdquo Vehicle SystemDynamics vol 48 no 8 pp 923ndash944 2010
[26] GBT 7031-2005ISO 86081995 Mechanical Vibration-RoadSurface Profiles-Reporting of Measured Data 2005
[27] M Li X Pu and Z Changfu ldquoModeling and simulationof heavy-duty semi-trailer in extreme condition based onADAMSrdquo Automobile Technology vol 2 pp 22ndash25 2009
[28] ldquoPassenger carsmdashtest track for a severe lane-change manoeu-vremdashpart 1 double lane-changerdquo ISO 3888-11999
[29] S H Li S P Yang and N Chen ldquoDirectional control of adriver-heavy-vehicle closed-loop systemrdquo Advanced Engineer-ing Forum vol 2-3 pp 33ndash38 2012
[30] J Pauwelussen ldquoDependencies of driver steering controlparametersrdquo Vehicle System Dynamics vol 50 no 6 pp 939ndash959 2012
[31] H Imine and V Dolcemascolo ldquoRollover risk prediction ofheavy vehicle in interaction with infrastructurerdquo InternationalJournal ofHeavyVehicle Systems vol 14 no 3 pp 294ndash307 2007
a multichannel charge amplifier (YE5853A) and convertedto digital signals by an intelligent acquisition and processinganalyzer (INV360DF) The sampling frequency was 200Hz
Figure 9 gives the experimental time history of frontwheel steering angle Using this measured front wheel steer-ing angle as input the yaw rate and lateral acceleration are cal-culated from TCLP model 2DOF model and linear modelrespectively These simulation results are compared with
the test results as shown in Figure 10 It is obvious thatthe simulation results of TCLP model are most consistentwith the results of field test Due to neglecting of the three-directional coupling effects of vehiclemotions or nonlinearityof suspension and tire the time histories of yaw rate obtainedfrom 2DOF model and linear model are too smooth and thelateral accelerations of 2DOF model and linear model aresmaller than the test results Thus the verification of thisproposed TCLP model is verified
34 Comparison with Other Driver Models Using the pro-posed modified driver model the optimal curvature modelby Guo and nonlinear time delay driver model by Legouisthe trajectories of vehicle gravity center during double-lanechange maneuver at four entrance speeds of 20 40 60 and80 kmh are simulated respectively as shown in Figure 11Table 1 lists the root mean square (RMS) value of vehicleresponses in six directions at different speeds applying thesethree driver models As shown in Figure 11 it is obvious thatthe tracking performance of the modified driver model issuperior to the other two driver models at different entrancespeeds It is also found from Figure 11 that though an over-shoot of the newmodel at a vehicle speed of 80 kmh exists in119883 = [150 180]m the results of the other two models divergewhen the vehicle arrives at 119883 = 200m while the resultof the new model is stable From Table 1 we can see thatthe real vehicle speed of the modified driver model is the
Mathematical Problems in Engineering 11
Simulation results of TCLP model
Test results
Simulation results of 2DOF model
Simulation results of linear model
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
02
04
minus04
minus02
t (s)
ay(m
s2 )
5 10 15 20
0
05
1
minus05
t (s)
ay(m
s2 )
Figure 10 Comparison of simulation and test results
12 Mathematical Problems in Engineering
50 100 150 200 250 300
0
1
2
3
4
5
minus1
Y(m
)
X (m)
(a) 20 kmh
50 100 150 200 250
0
1
2
3
4
Y(m
)
X (m)
(b) 40 kmh
50 100 150 200 250 300
0
1
2
3
4
5
NewLiu
GuoRoute
minus1
Y(m
)
X (m)
(c) 60 kmh
50 100 150 200
0
2
4
NewLiu
GuoRoute
minus2
Y(m
)
X (m)
(d) 80 kmh
Figure 11 Comparison of modified driver model with other driver models at different speeds
closest to the assumed speed and the modified driver modelhas smaller responses except for vertical vibration and betterstability than the other models Though the modified drivermodel obtains greater vertical vehicle acceleration and worseride comfort than Legouisrsquos model at speeds lower than60 kmh the new driver model can give the vehicle betterpath-following ability and handling stability
4 Effects of Driver Model Parameters onFull Vehicle Dynamics and Stability
The important parameters in driver model include vehiclerunning speed time delay preview distance and permitposition error which decide the steering control effect andvehicle dynamics During the same driving maneuver theskilled drivers may use higher vehicle speed longer previewdistance shorter time delay and larger permit position error
than the new drivers The match of these four driverparameters also influences the vehicle dynamics RecentlyPauwelussen [30] researched the dependencies of driversteering control parameters on vehicle lateral properties andpath keeping and gives some useful conclusions This workfocuses on the effects of driver model parameters on bothpath keeping and full vehicle dynamics such as vehiclehandling stability ride comfort and roll stability
Since the vehicle lateral displacement is randomly causedby road roughness excitations and the lateral position devi-ation is always changing during lane change maneuvers astatistic index is introduced to evaluate the vehiclersquos path-following ability globally which is formulated by
Route error = radicsum (119877119884
119889(119905) minus 119884
119889(119905))2
119873
(20)
Mathematical Problems in Engineering 13
20 40 60 80 100 120 140 160 180 200
0
1
2
3
4
5
20 kmh30 kmh40 kmh
50 kmh60 kmhRoute
Y(m
)
X (m)
minus1
0 50 100 150 200
0
1
2
3
4
0 s002 s005 s
01 s02 sRoute
Y(m
)
X (m)
minus1
Figure 12 Routes of vehicle under different speed or time delay
where 119873 is the total number of data points 119877119884119889and 119884
119889are
the required lateral position and real lateral displacement ofthe preview point in the ground coordinate system
When a vehicle is undergoing a lane change its steeringstability and roll stability are vital problems The path-following error lateral acceleration and yaw rate can be usedto evaluate the steering stability The roll angle can be used toevaluate the roll stability Being proportional to the rolloverindex LTR (lateral-load transfer rate) the lateral accelerationis also able to reflect the vehicle roll stability [31] In additionthe vertical acceleration and pitch angle are simulated so asto research the whole-body dynamics in the vehicle Usingthese evaluation indexes the effects of parameters on steeringstability roll stability and riding comfort are discussed indetail
Figure 12 shows the vehicle trajectories under differentinitial vehicle speeds or driver time delays The effects ofvehicle speed and driver time delay on full vehicle dynamicsand stability are shown in Figure 13 As we can see fromFigures 12 and 13 a higher speed or a longer time delay willlead to a greater path-following error However low speed(20 kmh) and longtime delay (035 s) may enlarge the path-following error With the rise of vehicle speed the verticalacceleration lateral acceleration yaw rate and roll angleincrease while the pitch angle decreases Rise of time delaycauses the increase of lateral acceleration yaw rate and rollangle but hardly changes the vertical acceleration and pitchangle As for the effects of vehicle speed and driver time delayon full vehicle dynamics the following may be concluded
(1) A high vehicle speed during double-lane changeyields bad path keeping performance ride comfortsteering stability and roll stability When the vehiclespeed is higher than 60 kmh the ride comfort of thisvehicle gets worse greatly
(2) A long time delay is harmful to path keeping per-formance steering stability and roll stability and hasalmost no effect on the ride comfort
Thus a low vehicle speed ranging from 20 kmh to 60 kmh and a small time delay less than 035 s are recommended forimproving this truckrsquos stability when going through the lanechange
In case of varying the preview distance or permit positionerror the variation in vehicle trajectories at the speed of40 kmh is shown in Figure 14The effects of preview distanceor permit position error on the path-following ability andfull vehicle dynamics are shown in Figure 15 As shown inFigures 14 and 15 a large preview distance deteriorates thepath-following ability but has small effect on other dynamicresponses On the other hand a too small preview distanceleads to a hop of lateral acceleration yaw rate and roll anglewhich conflicts with closed-loop stability and is harmfulto steering control This conclusion is consistent with thatof Gillespie and MacAdam [15] and Pauwelussen [30] Theinfluence of permit position error on path-following ability issmall at short preview distance but great at long preview dis-tance As shown in Figure 14 a too small or too large permitposition error (001m or 08m)will worsen the vehiclersquos path-following ability In addition large permit position error mayincrease vertical acceleration lateral acceleration yaw rateand roll angle and reduce the handling stability and rollstability and ride comfort The effect of preview distance andpermit position error on pitch angle is small and shows fluc-tuation Thus a preview distance ranging from 10m to 30mand a small permit position error but more than 001m arerecommended for improving this truckrsquos stability when mov-ing through the lane change A smaller permit position errorshould be matched with a shorter preview distance
14 Mathematical Problems in Engineering
002
04
2040
60800
1
2
Rout
e err
or (m
)
u0 (kmh) 0
02
04
2040
60807
8
9
u0 (kmh)
002
04
204060800
02
04
wr (
rad
s)
u0 (kmh)
0
02
04
204060800
005
01
120579(r
ad)
u0 (kmh)002
04
2040
60800
01
02
Φ(r
ad)
u0 (kmh)
002
04
2040
60800
2
4
u0(kmh)
ay(m
s2 )
Tr(s) Tr
(s)
Tr(s)
Tr(s)
Tr(s) Tr
(s)
azb
(ms2 )
Figure 13 The effects of vehicle speed and driver time delay
0 50 100 150 200
0
1
2
3
4
5
001 m02 m04 m
06 m08 mRoute
Y(m
)
X (m)
minus10 50 100 150 200
0
2
4
6
8
10
12
2 m10 m20 m
30 m40 mRoute
Y(m
)
X (m)
minus2
minus4
Figure 14 Routes of vehicle under different preview distance and permit position error
Mathematical Problems in Engineering 15
020
4060
005
1150
1
2
Rout
e err
or (m
)
er (m) Ld (m)
020
4060
01
25
10
15
er (m) Ld (m)
020
4060
01
20
5
er (m) Ld (m) 020
4060
01
20
02
04
er (m) Ld (m)wr (
rad
s)
2040
601
20
02
04
00er (m) Ld (m)
Φ(r
ad)
020
40 60
01
20
005
01
er (m) Ld (m)
120579(r
ad)
ay(m
s2 )
azb
(ms2 )
Figure 15 The effects of preview distance and permit position error
5 Conclusions
In this paper a nonlinear three-directional coupled lumpedparameters (TCLP) model of heavy-duty vehicle consideringthe nonlinearity of suspension damping and tire stiffness isbuilt and connected with a proposedmodified preview drivermodel with nonlinear time delayThe proposed driver modelis simple and has a feedback gain 2119871119889
2 that is decided bywheelbase and previewdistanceThe validity of this presenteddriver-vehicle closed-loop system is verified by comparisonwith the test data the traditional steering stability vehiclemodel the linear vehicle model and other driver controlmodelsThe results show that the new drivermodel has betterlane keeping performances than the other two driver modelsThe coupling effects of vehicle motions and the nonlinear-ity of suspension damping and tire stiffness could not beneglected in turning or high speed driving situation
The effects of driver parameters on path-following abilityand full vehicle dynamics are also discussed It is foundthat the high vehicle speed large time delay long previewdistance or big permit position error will deteriorate path-following performances and handling stability Of course atoo short preview distance or a too small permit position
error is also harmful to path-following performances andhandling stability The ride comfort mainly depends on vehi-cle speed preview distance and permit position error Thekey factors influencing roll characteristics are vehicle speedand time delay
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The National Natural Science Foundation of China(11472180) theNatural Science Foundation ofHebei Province(E2012210025) and the New Century Talent Foundation ofMinistry of Education (NCET-13-0913) support this work
References
[1] M Plochl and J Edelmann ldquoDriver models in automobiledynamics applicationrdquoVehicle SystemDynamics vol 45 no 7-8pp 699ndash741 2007
16 Mathematical Problems in Engineering
[2] C C MacAdam ldquoUnderstanding and modeling the humandriverrdquo Vehicle System Dynamics vol 40 no 1ndash3 pp 101ndash1342003
[3] K Guo and H Guan ldquoModeling of drivervehicle directionalcontrol systemrdquo Vehicle System Dynamics vol 22 no 3-4 pp141ndash184 1993
[4] K GuoVehicle Handling DynamicsTheory Jiangsu Science andTechnology Publishing House Nanjing China 2011
[5] R S Sharp ldquoDriver steering control and a new perspective oncar handling qualitiesrdquo Proceedings of the Institution ofMechani-cal Engineers Part C Journal of Mechanical Engineering Sciencevol 219 no 10 pp 1041ndash1051 2005
[6] T Legouis A Laneville P Bourassa and G Payre ldquoCharac-terization of dynamic vehicle stability using two models of thehuman pilot behaviorrdquo Vehicle System Dynamics vol 15 no 1pp 1ndash18 1986
[7] Z Liu G Payre and P Bourassa ldquoNonlinear oscillations andchaotic motions in a road vehicle system with driver steeringcontrolrdquo Nonlinear Dynamics vol 9 no 3 pp 281ndash304 1996
[8] Z Liu G Payre and P Bourassa ldquoStability and oscillations ina time-delayed vehicle system with driver controlrdquo NonlinearDynamics vol 35 no 2 pp 159ndash173 2004
[9] X D Liu W Feng and N Kang ldquoResearch of simulationmethod of tractor-semitrailer carrying liquid considering liquidsloshingrdquo in Proceedings of the 10th National Conference onVibrationTheory and Application pp 581ndash590 Nanjing ChinaOctober 2011
[10] C C MacAdam ldquoApplication of an optimal preview control forsimulation of closed-loop automobile drivingrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 11 no 6 pp 393ndash399 1981
[11] A Y Ungoren and H Peng ldquoAn adaptive lateral preview drivermodelrdquo Vehicle System Dynamics vol 43 no 4 pp 245ndash2592005
[12] C I Chatzikomis and K N Spentzas ldquoA path-following drivermodel with longitudinal and lateral control of vehiclersquos motionrdquoForschung im Ingenieurwesen vol 73 no 4 pp 257ndash266 2009
[13] R S Sharp D Casanova and P Symonds ldquoA mathematicalmodel for driver steering control with design tuning andperformance resultsrdquo Vehicle System Dynamics vol 33 no 5pp 289ndash326 2000
[14] A J Pick and D J Cole ldquoNeuromuscular dynamics in thedriver-vehicle systemrdquo Vehicle System Dynamics vol 44 sup-plement 1 pp 624ndash631 2006
[15] T D Gillespie and C C MacAdam Constant Velocity YawRollProgram Userrsquos Manual The University of Michigan Trans-portation Research Institute Ann Arbor Mich USA 1982
[16] R I S Raj Influence of road roughness and directional maneuveron the dynamic performance of heavy vehicles [MS thesis]Department of Mechanical Engineering Concordia UniversityMontreal Canada 1998
[17] KWordenDHickeyMHaroon andD E Adams ldquoNonlinearsystem identification of automotive dampers a time and fre-quency-domain analysisrdquo Mechanical Systems and Signal Pro-cessing vol 23 no 1 pp 104ndash126 2009
[18] S H Li Y J Lu and L Y Li ldquoDynamical test and modelingfor hydraulic shock absorber on heavy vehicle under harmonicand random loadingsrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 13 pp 1903ndash1910 2012
[19] X W Ji and Y M Gao ldquoThe dynamic stiffness and dampingcharacteristics of the tirerdquo Automotive Engineering vol 5 pp315ndash321 1994
[20] G Gim and P E Nikravesh ldquoA three dimensional tire modelfor steady-state simulations of vehiclesrdquo SAE vol 102 no 2 pp150ndash159 1993
[21] J D Zhuang Principles of Automobile Tire Beijing Institute ofTechnology Press Beijing China 1996
[22] W-M Zhai ldquoTwo simple fast integration methods for large-scale dynamic problems in engineeringrdquo International Journalfor Numerical Methods in Engineering vol 39 no 24 pp 4199ndash4214 1996
[23] Dongfeng Motor Group Company Dongfeng DFL1250A8DFL1250A9 Truck Chassis Refit Manual Dongfeng MotorGroup Company 2008
[24] S Li S Yang L Chen and Y Lu ldquoEffects of parameters ondynamic responses for a heavy vehicle-pavement- foundationcoupled systemrdquo International Journal of Heavy Vehicle Systemsvol 19 no 2 pp 207ndash224 2012
[25] S Yang S Li andY Lu ldquoInvestigation on dynamical interactionbetween a heavy vehicle and road pavementrdquo Vehicle SystemDynamics vol 48 no 8 pp 923ndash944 2010
[26] GBT 7031-2005ISO 86081995 Mechanical Vibration-RoadSurface Profiles-Reporting of Measured Data 2005
[27] M Li X Pu and Z Changfu ldquoModeling and simulationof heavy-duty semi-trailer in extreme condition based onADAMSrdquo Automobile Technology vol 2 pp 22ndash25 2009
[28] ldquoPassenger carsmdashtest track for a severe lane-change manoeu-vremdashpart 1 double lane-changerdquo ISO 3888-11999
[29] S H Li S P Yang and N Chen ldquoDirectional control of adriver-heavy-vehicle closed-loop systemrdquo Advanced Engineer-ing Forum vol 2-3 pp 33ndash38 2012
[30] J Pauwelussen ldquoDependencies of driver steering controlparametersrdquo Vehicle System Dynamics vol 50 no 6 pp 939ndash959 2012
[31] H Imine and V Dolcemascolo ldquoRollover risk prediction ofheavy vehicle in interaction with infrastructurerdquo InternationalJournal ofHeavyVehicle Systems vol 14 no 3 pp 294ndash307 2007
a multichannel charge amplifier (YE5853A) and convertedto digital signals by an intelligent acquisition and processinganalyzer (INV360DF) The sampling frequency was 200Hz
Figure 9 gives the experimental time history of frontwheel steering angle Using this measured front wheel steer-ing angle as input the yaw rate and lateral acceleration are cal-culated from TCLP model 2DOF model and linear modelrespectively These simulation results are compared with
the test results as shown in Figure 10 It is obvious thatthe simulation results of TCLP model are most consistentwith the results of field test Due to neglecting of the three-directional coupling effects of vehiclemotions or nonlinearityof suspension and tire the time histories of yaw rate obtainedfrom 2DOF model and linear model are too smooth and thelateral accelerations of 2DOF model and linear model aresmaller than the test results Thus the verification of thisproposed TCLP model is verified
34 Comparison with Other Driver Models Using the pro-posed modified driver model the optimal curvature modelby Guo and nonlinear time delay driver model by Legouisthe trajectories of vehicle gravity center during double-lanechange maneuver at four entrance speeds of 20 40 60 and80 kmh are simulated respectively as shown in Figure 11Table 1 lists the root mean square (RMS) value of vehicleresponses in six directions at different speeds applying thesethree driver models As shown in Figure 11 it is obvious thatthe tracking performance of the modified driver model issuperior to the other two driver models at different entrancespeeds It is also found from Figure 11 that though an over-shoot of the newmodel at a vehicle speed of 80 kmh exists in119883 = [150 180]m the results of the other two models divergewhen the vehicle arrives at 119883 = 200m while the resultof the new model is stable From Table 1 we can see thatthe real vehicle speed of the modified driver model is the
Mathematical Problems in Engineering 11
Simulation results of TCLP model
Test results
Simulation results of 2DOF model
Simulation results of linear model
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
01
minus01
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
005
01
015
minus005
t (s)
wr (
rad
s)
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
1
2
minus2
minus1
t (s)
ay(m
s2 )
5 10 15 20
0
02
04
minus04
minus02
t (s)
ay(m
s2 )
5 10 15 20
0
05
1
minus05
t (s)
ay(m
s2 )
Figure 10 Comparison of simulation and test results
12 Mathematical Problems in Engineering
50 100 150 200 250 300
0
1
2
3
4
5
minus1
Y(m
)
X (m)
(a) 20 kmh
50 100 150 200 250
0
1
2
3
4
Y(m
)
X (m)
(b) 40 kmh
50 100 150 200 250 300
0
1
2
3
4
5
NewLiu
GuoRoute
minus1
Y(m
)
X (m)
(c) 60 kmh
50 100 150 200
0
2
4
NewLiu
GuoRoute
minus2
Y(m
)
X (m)
(d) 80 kmh
Figure 11 Comparison of modified driver model with other driver models at different speeds
closest to the assumed speed and the modified driver modelhas smaller responses except for vertical vibration and betterstability than the other models Though the modified drivermodel obtains greater vertical vehicle acceleration and worseride comfort than Legouisrsquos model at speeds lower than60 kmh the new driver model can give the vehicle betterpath-following ability and handling stability
4 Effects of Driver Model Parameters onFull Vehicle Dynamics and Stability
The important parameters in driver model include vehiclerunning speed time delay preview distance and permitposition error which decide the steering control effect andvehicle dynamics During the same driving maneuver theskilled drivers may use higher vehicle speed longer previewdistance shorter time delay and larger permit position error
than the new drivers The match of these four driverparameters also influences the vehicle dynamics RecentlyPauwelussen [30] researched the dependencies of driversteering control parameters on vehicle lateral properties andpath keeping and gives some useful conclusions This workfocuses on the effects of driver model parameters on bothpath keeping and full vehicle dynamics such as vehiclehandling stability ride comfort and roll stability
Since the vehicle lateral displacement is randomly causedby road roughness excitations and the lateral position devi-ation is always changing during lane change maneuvers astatistic index is introduced to evaluate the vehiclersquos path-following ability globally which is formulated by
Route error = radicsum (119877119884
119889(119905) minus 119884
119889(119905))2
119873
(20)
Mathematical Problems in Engineering 13
20 40 60 80 100 120 140 160 180 200
0
1
2
3
4
5
20 kmh30 kmh40 kmh
50 kmh60 kmhRoute
Y(m
)
X (m)
minus1
0 50 100 150 200
0
1
2
3
4
0 s002 s005 s
01 s02 sRoute
Y(m
)
X (m)
minus1
Figure 12 Routes of vehicle under different speed or time delay
where 119873 is the total number of data points 119877119884119889and 119884
119889are
the required lateral position and real lateral displacement ofthe preview point in the ground coordinate system
When a vehicle is undergoing a lane change its steeringstability and roll stability are vital problems The path-following error lateral acceleration and yaw rate can be usedto evaluate the steering stability The roll angle can be used toevaluate the roll stability Being proportional to the rolloverindex LTR (lateral-load transfer rate) the lateral accelerationis also able to reflect the vehicle roll stability [31] In additionthe vertical acceleration and pitch angle are simulated so asto research the whole-body dynamics in the vehicle Usingthese evaluation indexes the effects of parameters on steeringstability roll stability and riding comfort are discussed indetail
Figure 12 shows the vehicle trajectories under differentinitial vehicle speeds or driver time delays The effects ofvehicle speed and driver time delay on full vehicle dynamicsand stability are shown in Figure 13 As we can see fromFigures 12 and 13 a higher speed or a longer time delay willlead to a greater path-following error However low speed(20 kmh) and longtime delay (035 s) may enlarge the path-following error With the rise of vehicle speed the verticalacceleration lateral acceleration yaw rate and roll angleincrease while the pitch angle decreases Rise of time delaycauses the increase of lateral acceleration yaw rate and rollangle but hardly changes the vertical acceleration and pitchangle As for the effects of vehicle speed and driver time delayon full vehicle dynamics the following may be concluded
(1) A high vehicle speed during double-lane changeyields bad path keeping performance ride comfortsteering stability and roll stability When the vehiclespeed is higher than 60 kmh the ride comfort of thisvehicle gets worse greatly
(2) A long time delay is harmful to path keeping per-formance steering stability and roll stability and hasalmost no effect on the ride comfort
Thus a low vehicle speed ranging from 20 kmh to 60 kmh and a small time delay less than 035 s are recommended forimproving this truckrsquos stability when going through the lanechange
In case of varying the preview distance or permit positionerror the variation in vehicle trajectories at the speed of40 kmh is shown in Figure 14The effects of preview distanceor permit position error on the path-following ability andfull vehicle dynamics are shown in Figure 15 As shown inFigures 14 and 15 a large preview distance deteriorates thepath-following ability but has small effect on other dynamicresponses On the other hand a too small preview distanceleads to a hop of lateral acceleration yaw rate and roll anglewhich conflicts with closed-loop stability and is harmfulto steering control This conclusion is consistent with thatof Gillespie and MacAdam [15] and Pauwelussen [30] Theinfluence of permit position error on path-following ability issmall at short preview distance but great at long preview dis-tance As shown in Figure 14 a too small or too large permitposition error (001m or 08m)will worsen the vehiclersquos path-following ability In addition large permit position error mayincrease vertical acceleration lateral acceleration yaw rateand roll angle and reduce the handling stability and rollstability and ride comfort The effect of preview distance andpermit position error on pitch angle is small and shows fluc-tuation Thus a preview distance ranging from 10m to 30mand a small permit position error but more than 001m arerecommended for improving this truckrsquos stability when mov-ing through the lane change A smaller permit position errorshould be matched with a shorter preview distance
14 Mathematical Problems in Engineering
002
04
2040
60800
1
2
Rout
e err
or (m
)
u0 (kmh) 0
02
04
2040
60807
8
9
u0 (kmh)
002
04
204060800
02
04
wr (
rad
s)
u0 (kmh)
0
02
04
204060800
005
01
120579(r
ad)
u0 (kmh)002
04
2040
60800
01
02
Φ(r
ad)
u0 (kmh)
002
04
2040
60800
2
4
u0(kmh)
ay(m
s2 )
Tr(s) Tr
(s)
Tr(s)
Tr(s)
Tr(s) Tr
(s)
azb
(ms2 )
Figure 13 The effects of vehicle speed and driver time delay
0 50 100 150 200
0
1
2
3
4
5
001 m02 m04 m
06 m08 mRoute
Y(m
)
X (m)
minus10 50 100 150 200
0
2
4
6
8
10
12
2 m10 m20 m
30 m40 mRoute
Y(m
)
X (m)
minus2
minus4
Figure 14 Routes of vehicle under different preview distance and permit position error
Mathematical Problems in Engineering 15
020
4060
005
1150
1
2
Rout
e err
or (m
)
er (m) Ld (m)
020
4060
01
25
10
15
er (m) Ld (m)
020
4060
01
20
5
er (m) Ld (m) 020
4060
01
20
02
04
er (m) Ld (m)wr (
rad
s)
2040
601
20
02
04
00er (m) Ld (m)
Φ(r
ad)
020
40 60
01
20
005
01
er (m) Ld (m)
120579(r
ad)
ay(m
s2 )
azb
(ms2 )
Figure 15 The effects of preview distance and permit position error
5 Conclusions
In this paper a nonlinear three-directional coupled lumpedparameters (TCLP) model of heavy-duty vehicle consideringthe nonlinearity of suspension damping and tire stiffness isbuilt and connected with a proposedmodified preview drivermodel with nonlinear time delayThe proposed driver modelis simple and has a feedback gain 2119871119889
2 that is decided bywheelbase and previewdistanceThe validity of this presenteddriver-vehicle closed-loop system is verified by comparisonwith the test data the traditional steering stability vehiclemodel the linear vehicle model and other driver controlmodelsThe results show that the new drivermodel has betterlane keeping performances than the other two driver modelsThe coupling effects of vehicle motions and the nonlinear-ity of suspension damping and tire stiffness could not beneglected in turning or high speed driving situation
The effects of driver parameters on path-following abilityand full vehicle dynamics are also discussed It is foundthat the high vehicle speed large time delay long previewdistance or big permit position error will deteriorate path-following performances and handling stability Of course atoo short preview distance or a too small permit position
error is also harmful to path-following performances andhandling stability The ride comfort mainly depends on vehi-cle speed preview distance and permit position error Thekey factors influencing roll characteristics are vehicle speedand time delay
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The National Natural Science Foundation of China(11472180) theNatural Science Foundation ofHebei Province(E2012210025) and the New Century Talent Foundation ofMinistry of Education (NCET-13-0913) support this work
References
[1] M Plochl and J Edelmann ldquoDriver models in automobiledynamics applicationrdquoVehicle SystemDynamics vol 45 no 7-8pp 699ndash741 2007
16 Mathematical Problems in Engineering
[2] C C MacAdam ldquoUnderstanding and modeling the humandriverrdquo Vehicle System Dynamics vol 40 no 1ndash3 pp 101ndash1342003
[3] K Guo and H Guan ldquoModeling of drivervehicle directionalcontrol systemrdquo Vehicle System Dynamics vol 22 no 3-4 pp141ndash184 1993
[4] K GuoVehicle Handling DynamicsTheory Jiangsu Science andTechnology Publishing House Nanjing China 2011
[5] R S Sharp ldquoDriver steering control and a new perspective oncar handling qualitiesrdquo Proceedings of the Institution ofMechani-cal Engineers Part C Journal of Mechanical Engineering Sciencevol 219 no 10 pp 1041ndash1051 2005
[6] T Legouis A Laneville P Bourassa and G Payre ldquoCharac-terization of dynamic vehicle stability using two models of thehuman pilot behaviorrdquo Vehicle System Dynamics vol 15 no 1pp 1ndash18 1986
[7] Z Liu G Payre and P Bourassa ldquoNonlinear oscillations andchaotic motions in a road vehicle system with driver steeringcontrolrdquo Nonlinear Dynamics vol 9 no 3 pp 281ndash304 1996
[8] Z Liu G Payre and P Bourassa ldquoStability and oscillations ina time-delayed vehicle system with driver controlrdquo NonlinearDynamics vol 35 no 2 pp 159ndash173 2004
[9] X D Liu W Feng and N Kang ldquoResearch of simulationmethod of tractor-semitrailer carrying liquid considering liquidsloshingrdquo in Proceedings of the 10th National Conference onVibrationTheory and Application pp 581ndash590 Nanjing ChinaOctober 2011
[10] C C MacAdam ldquoApplication of an optimal preview control forsimulation of closed-loop automobile drivingrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 11 no 6 pp 393ndash399 1981
[11] A Y Ungoren and H Peng ldquoAn adaptive lateral preview drivermodelrdquo Vehicle System Dynamics vol 43 no 4 pp 245ndash2592005
[12] C I Chatzikomis and K N Spentzas ldquoA path-following drivermodel with longitudinal and lateral control of vehiclersquos motionrdquoForschung im Ingenieurwesen vol 73 no 4 pp 257ndash266 2009
[13] R S Sharp D Casanova and P Symonds ldquoA mathematicalmodel for driver steering control with design tuning andperformance resultsrdquo Vehicle System Dynamics vol 33 no 5pp 289ndash326 2000
[14] A J Pick and D J Cole ldquoNeuromuscular dynamics in thedriver-vehicle systemrdquo Vehicle System Dynamics vol 44 sup-plement 1 pp 624ndash631 2006
[15] T D Gillespie and C C MacAdam Constant Velocity YawRollProgram Userrsquos Manual The University of Michigan Trans-portation Research Institute Ann Arbor Mich USA 1982
[16] R I S Raj Influence of road roughness and directional maneuveron the dynamic performance of heavy vehicles [MS thesis]Department of Mechanical Engineering Concordia UniversityMontreal Canada 1998
[17] KWordenDHickeyMHaroon andD E Adams ldquoNonlinearsystem identification of automotive dampers a time and fre-quency-domain analysisrdquo Mechanical Systems and Signal Pro-cessing vol 23 no 1 pp 104ndash126 2009
[18] S H Li Y J Lu and L Y Li ldquoDynamical test and modelingfor hydraulic shock absorber on heavy vehicle under harmonicand random loadingsrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 13 pp 1903ndash1910 2012
[19] X W Ji and Y M Gao ldquoThe dynamic stiffness and dampingcharacteristics of the tirerdquo Automotive Engineering vol 5 pp315ndash321 1994
[20] G Gim and P E Nikravesh ldquoA three dimensional tire modelfor steady-state simulations of vehiclesrdquo SAE vol 102 no 2 pp150ndash159 1993
[21] J D Zhuang Principles of Automobile Tire Beijing Institute ofTechnology Press Beijing China 1996
[22] W-M Zhai ldquoTwo simple fast integration methods for large-scale dynamic problems in engineeringrdquo International Journalfor Numerical Methods in Engineering vol 39 no 24 pp 4199ndash4214 1996
[23] Dongfeng Motor Group Company Dongfeng DFL1250A8DFL1250A9 Truck Chassis Refit Manual Dongfeng MotorGroup Company 2008
[24] S Li S Yang L Chen and Y Lu ldquoEffects of parameters ondynamic responses for a heavy vehicle-pavement- foundationcoupled systemrdquo International Journal of Heavy Vehicle Systemsvol 19 no 2 pp 207ndash224 2012
[25] S Yang S Li andY Lu ldquoInvestigation on dynamical interactionbetween a heavy vehicle and road pavementrdquo Vehicle SystemDynamics vol 48 no 8 pp 923ndash944 2010
[26] GBT 7031-2005ISO 86081995 Mechanical Vibration-RoadSurface Profiles-Reporting of Measured Data 2005
[27] M Li X Pu and Z Changfu ldquoModeling and simulationof heavy-duty semi-trailer in extreme condition based onADAMSrdquo Automobile Technology vol 2 pp 22ndash25 2009
[28] ldquoPassenger carsmdashtest track for a severe lane-change manoeu-vremdashpart 1 double lane-changerdquo ISO 3888-11999
[29] S H Li S P Yang and N Chen ldquoDirectional control of adriver-heavy-vehicle closed-loop systemrdquo Advanced Engineer-ing Forum vol 2-3 pp 33ndash38 2012
[30] J Pauwelussen ldquoDependencies of driver steering controlparametersrdquo Vehicle System Dynamics vol 50 no 6 pp 939ndash959 2012
[31] H Imine and V Dolcemascolo ldquoRollover risk prediction ofheavy vehicle in interaction with infrastructurerdquo InternationalJournal ofHeavyVehicle Systems vol 14 no 3 pp 294ndash307 2007
Figure 10 Comparison of simulation and test results
12 Mathematical Problems in Engineering
50 100 150 200 250 300
0
1
2
3
4
5
minus1
Y(m
)
X (m)
(a) 20 kmh
50 100 150 200 250
0
1
2
3
4
Y(m
)
X (m)
(b) 40 kmh
50 100 150 200 250 300
0
1
2
3
4
5
NewLiu
GuoRoute
minus1
Y(m
)
X (m)
(c) 60 kmh
50 100 150 200
0
2
4
NewLiu
GuoRoute
minus2
Y(m
)
X (m)
(d) 80 kmh
Figure 11 Comparison of modified driver model with other driver models at different speeds
closest to the assumed speed and the modified driver modelhas smaller responses except for vertical vibration and betterstability than the other models Though the modified drivermodel obtains greater vertical vehicle acceleration and worseride comfort than Legouisrsquos model at speeds lower than60 kmh the new driver model can give the vehicle betterpath-following ability and handling stability
4 Effects of Driver Model Parameters onFull Vehicle Dynamics and Stability
The important parameters in driver model include vehiclerunning speed time delay preview distance and permitposition error which decide the steering control effect andvehicle dynamics During the same driving maneuver theskilled drivers may use higher vehicle speed longer previewdistance shorter time delay and larger permit position error
than the new drivers The match of these four driverparameters also influences the vehicle dynamics RecentlyPauwelussen [30] researched the dependencies of driversteering control parameters on vehicle lateral properties andpath keeping and gives some useful conclusions This workfocuses on the effects of driver model parameters on bothpath keeping and full vehicle dynamics such as vehiclehandling stability ride comfort and roll stability
Since the vehicle lateral displacement is randomly causedby road roughness excitations and the lateral position devi-ation is always changing during lane change maneuvers astatistic index is introduced to evaluate the vehiclersquos path-following ability globally which is formulated by
Route error = radicsum (119877119884
119889(119905) minus 119884
119889(119905))2
119873
(20)
Mathematical Problems in Engineering 13
20 40 60 80 100 120 140 160 180 200
0
1
2
3
4
5
20 kmh30 kmh40 kmh
50 kmh60 kmhRoute
Y(m
)
X (m)
minus1
0 50 100 150 200
0
1
2
3
4
0 s002 s005 s
01 s02 sRoute
Y(m
)
X (m)
minus1
Figure 12 Routes of vehicle under different speed or time delay
where 119873 is the total number of data points 119877119884119889and 119884
119889are
the required lateral position and real lateral displacement ofthe preview point in the ground coordinate system
When a vehicle is undergoing a lane change its steeringstability and roll stability are vital problems The path-following error lateral acceleration and yaw rate can be usedto evaluate the steering stability The roll angle can be used toevaluate the roll stability Being proportional to the rolloverindex LTR (lateral-load transfer rate) the lateral accelerationis also able to reflect the vehicle roll stability [31] In additionthe vertical acceleration and pitch angle are simulated so asto research the whole-body dynamics in the vehicle Usingthese evaluation indexes the effects of parameters on steeringstability roll stability and riding comfort are discussed indetail
Figure 12 shows the vehicle trajectories under differentinitial vehicle speeds or driver time delays The effects ofvehicle speed and driver time delay on full vehicle dynamicsand stability are shown in Figure 13 As we can see fromFigures 12 and 13 a higher speed or a longer time delay willlead to a greater path-following error However low speed(20 kmh) and longtime delay (035 s) may enlarge the path-following error With the rise of vehicle speed the verticalacceleration lateral acceleration yaw rate and roll angleincrease while the pitch angle decreases Rise of time delaycauses the increase of lateral acceleration yaw rate and rollangle but hardly changes the vertical acceleration and pitchangle As for the effects of vehicle speed and driver time delayon full vehicle dynamics the following may be concluded
(1) A high vehicle speed during double-lane changeyields bad path keeping performance ride comfortsteering stability and roll stability When the vehiclespeed is higher than 60 kmh the ride comfort of thisvehicle gets worse greatly
(2) A long time delay is harmful to path keeping per-formance steering stability and roll stability and hasalmost no effect on the ride comfort
Thus a low vehicle speed ranging from 20 kmh to 60 kmh and a small time delay less than 035 s are recommended forimproving this truckrsquos stability when going through the lanechange
In case of varying the preview distance or permit positionerror the variation in vehicle trajectories at the speed of40 kmh is shown in Figure 14The effects of preview distanceor permit position error on the path-following ability andfull vehicle dynamics are shown in Figure 15 As shown inFigures 14 and 15 a large preview distance deteriorates thepath-following ability but has small effect on other dynamicresponses On the other hand a too small preview distanceleads to a hop of lateral acceleration yaw rate and roll anglewhich conflicts with closed-loop stability and is harmfulto steering control This conclusion is consistent with thatof Gillespie and MacAdam [15] and Pauwelussen [30] Theinfluence of permit position error on path-following ability issmall at short preview distance but great at long preview dis-tance As shown in Figure 14 a too small or too large permitposition error (001m or 08m)will worsen the vehiclersquos path-following ability In addition large permit position error mayincrease vertical acceleration lateral acceleration yaw rateand roll angle and reduce the handling stability and rollstability and ride comfort The effect of preview distance andpermit position error on pitch angle is small and shows fluc-tuation Thus a preview distance ranging from 10m to 30mand a small permit position error but more than 001m arerecommended for improving this truckrsquos stability when mov-ing through the lane change A smaller permit position errorshould be matched with a shorter preview distance
14 Mathematical Problems in Engineering
002
04
2040
60800
1
2
Rout
e err
or (m
)
u0 (kmh) 0
02
04
2040
60807
8
9
u0 (kmh)
002
04
204060800
02
04
wr (
rad
s)
u0 (kmh)
0
02
04
204060800
005
01
120579(r
ad)
u0 (kmh)002
04
2040
60800
01
02
Φ(r
ad)
u0 (kmh)
002
04
2040
60800
2
4
u0(kmh)
ay(m
s2 )
Tr(s) Tr
(s)
Tr(s)
Tr(s)
Tr(s) Tr
(s)
azb
(ms2 )
Figure 13 The effects of vehicle speed and driver time delay
0 50 100 150 200
0
1
2
3
4
5
001 m02 m04 m
06 m08 mRoute
Y(m
)
X (m)
minus10 50 100 150 200
0
2
4
6
8
10
12
2 m10 m20 m
30 m40 mRoute
Y(m
)
X (m)
minus2
minus4
Figure 14 Routes of vehicle under different preview distance and permit position error
Mathematical Problems in Engineering 15
020
4060
005
1150
1
2
Rout
e err
or (m
)
er (m) Ld (m)
020
4060
01
25
10
15
er (m) Ld (m)
020
4060
01
20
5
er (m) Ld (m) 020
4060
01
20
02
04
er (m) Ld (m)wr (
rad
s)
2040
601
20
02
04
00er (m) Ld (m)
Φ(r
ad)
020
40 60
01
20
005
01
er (m) Ld (m)
120579(r
ad)
ay(m
s2 )
azb
(ms2 )
Figure 15 The effects of preview distance and permit position error
5 Conclusions
In this paper a nonlinear three-directional coupled lumpedparameters (TCLP) model of heavy-duty vehicle consideringthe nonlinearity of suspension damping and tire stiffness isbuilt and connected with a proposedmodified preview drivermodel with nonlinear time delayThe proposed driver modelis simple and has a feedback gain 2119871119889
2 that is decided bywheelbase and previewdistanceThe validity of this presenteddriver-vehicle closed-loop system is verified by comparisonwith the test data the traditional steering stability vehiclemodel the linear vehicle model and other driver controlmodelsThe results show that the new drivermodel has betterlane keeping performances than the other two driver modelsThe coupling effects of vehicle motions and the nonlinear-ity of suspension damping and tire stiffness could not beneglected in turning or high speed driving situation
The effects of driver parameters on path-following abilityand full vehicle dynamics are also discussed It is foundthat the high vehicle speed large time delay long previewdistance or big permit position error will deteriorate path-following performances and handling stability Of course atoo short preview distance or a too small permit position
error is also harmful to path-following performances andhandling stability The ride comfort mainly depends on vehi-cle speed preview distance and permit position error Thekey factors influencing roll characteristics are vehicle speedand time delay
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The National Natural Science Foundation of China(11472180) theNatural Science Foundation ofHebei Province(E2012210025) and the New Century Talent Foundation ofMinistry of Education (NCET-13-0913) support this work
References
[1] M Plochl and J Edelmann ldquoDriver models in automobiledynamics applicationrdquoVehicle SystemDynamics vol 45 no 7-8pp 699ndash741 2007
16 Mathematical Problems in Engineering
[2] C C MacAdam ldquoUnderstanding and modeling the humandriverrdquo Vehicle System Dynamics vol 40 no 1ndash3 pp 101ndash1342003
[3] K Guo and H Guan ldquoModeling of drivervehicle directionalcontrol systemrdquo Vehicle System Dynamics vol 22 no 3-4 pp141ndash184 1993
[4] K GuoVehicle Handling DynamicsTheory Jiangsu Science andTechnology Publishing House Nanjing China 2011
[5] R S Sharp ldquoDriver steering control and a new perspective oncar handling qualitiesrdquo Proceedings of the Institution ofMechani-cal Engineers Part C Journal of Mechanical Engineering Sciencevol 219 no 10 pp 1041ndash1051 2005
[6] T Legouis A Laneville P Bourassa and G Payre ldquoCharac-terization of dynamic vehicle stability using two models of thehuman pilot behaviorrdquo Vehicle System Dynamics vol 15 no 1pp 1ndash18 1986
[7] Z Liu G Payre and P Bourassa ldquoNonlinear oscillations andchaotic motions in a road vehicle system with driver steeringcontrolrdquo Nonlinear Dynamics vol 9 no 3 pp 281ndash304 1996
[8] Z Liu G Payre and P Bourassa ldquoStability and oscillations ina time-delayed vehicle system with driver controlrdquo NonlinearDynamics vol 35 no 2 pp 159ndash173 2004
[9] X D Liu W Feng and N Kang ldquoResearch of simulationmethod of tractor-semitrailer carrying liquid considering liquidsloshingrdquo in Proceedings of the 10th National Conference onVibrationTheory and Application pp 581ndash590 Nanjing ChinaOctober 2011
[10] C C MacAdam ldquoApplication of an optimal preview control forsimulation of closed-loop automobile drivingrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 11 no 6 pp 393ndash399 1981
[11] A Y Ungoren and H Peng ldquoAn adaptive lateral preview drivermodelrdquo Vehicle System Dynamics vol 43 no 4 pp 245ndash2592005
[12] C I Chatzikomis and K N Spentzas ldquoA path-following drivermodel with longitudinal and lateral control of vehiclersquos motionrdquoForschung im Ingenieurwesen vol 73 no 4 pp 257ndash266 2009
[13] R S Sharp D Casanova and P Symonds ldquoA mathematicalmodel for driver steering control with design tuning andperformance resultsrdquo Vehicle System Dynamics vol 33 no 5pp 289ndash326 2000
[14] A J Pick and D J Cole ldquoNeuromuscular dynamics in thedriver-vehicle systemrdquo Vehicle System Dynamics vol 44 sup-plement 1 pp 624ndash631 2006
[15] T D Gillespie and C C MacAdam Constant Velocity YawRollProgram Userrsquos Manual The University of Michigan Trans-portation Research Institute Ann Arbor Mich USA 1982
[16] R I S Raj Influence of road roughness and directional maneuveron the dynamic performance of heavy vehicles [MS thesis]Department of Mechanical Engineering Concordia UniversityMontreal Canada 1998
[17] KWordenDHickeyMHaroon andD E Adams ldquoNonlinearsystem identification of automotive dampers a time and fre-quency-domain analysisrdquo Mechanical Systems and Signal Pro-cessing vol 23 no 1 pp 104ndash126 2009
[18] S H Li Y J Lu and L Y Li ldquoDynamical test and modelingfor hydraulic shock absorber on heavy vehicle under harmonicand random loadingsrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 13 pp 1903ndash1910 2012
[19] X W Ji and Y M Gao ldquoThe dynamic stiffness and dampingcharacteristics of the tirerdquo Automotive Engineering vol 5 pp315ndash321 1994
[20] G Gim and P E Nikravesh ldquoA three dimensional tire modelfor steady-state simulations of vehiclesrdquo SAE vol 102 no 2 pp150ndash159 1993
[21] J D Zhuang Principles of Automobile Tire Beijing Institute ofTechnology Press Beijing China 1996
[22] W-M Zhai ldquoTwo simple fast integration methods for large-scale dynamic problems in engineeringrdquo International Journalfor Numerical Methods in Engineering vol 39 no 24 pp 4199ndash4214 1996
[23] Dongfeng Motor Group Company Dongfeng DFL1250A8DFL1250A9 Truck Chassis Refit Manual Dongfeng MotorGroup Company 2008
[24] S Li S Yang L Chen and Y Lu ldquoEffects of parameters ondynamic responses for a heavy vehicle-pavement- foundationcoupled systemrdquo International Journal of Heavy Vehicle Systemsvol 19 no 2 pp 207ndash224 2012
[25] S Yang S Li andY Lu ldquoInvestigation on dynamical interactionbetween a heavy vehicle and road pavementrdquo Vehicle SystemDynamics vol 48 no 8 pp 923ndash944 2010
[26] GBT 7031-2005ISO 86081995 Mechanical Vibration-RoadSurface Profiles-Reporting of Measured Data 2005
[27] M Li X Pu and Z Changfu ldquoModeling and simulationof heavy-duty semi-trailer in extreme condition based onADAMSrdquo Automobile Technology vol 2 pp 22ndash25 2009
[28] ldquoPassenger carsmdashtest track for a severe lane-change manoeu-vremdashpart 1 double lane-changerdquo ISO 3888-11999
[29] S H Li S P Yang and N Chen ldquoDirectional control of adriver-heavy-vehicle closed-loop systemrdquo Advanced Engineer-ing Forum vol 2-3 pp 33ndash38 2012
[30] J Pauwelussen ldquoDependencies of driver steering controlparametersrdquo Vehicle System Dynamics vol 50 no 6 pp 939ndash959 2012
[31] H Imine and V Dolcemascolo ldquoRollover risk prediction ofheavy vehicle in interaction with infrastructurerdquo InternationalJournal ofHeavyVehicle Systems vol 14 no 3 pp 294ndash307 2007
Figure 11 Comparison of modified driver model with other driver models at different speeds
closest to the assumed speed and the modified driver modelhas smaller responses except for vertical vibration and betterstability than the other models Though the modified drivermodel obtains greater vertical vehicle acceleration and worseride comfort than Legouisrsquos model at speeds lower than60 kmh the new driver model can give the vehicle betterpath-following ability and handling stability
4 Effects of Driver Model Parameters onFull Vehicle Dynamics and Stability
The important parameters in driver model include vehiclerunning speed time delay preview distance and permitposition error which decide the steering control effect andvehicle dynamics During the same driving maneuver theskilled drivers may use higher vehicle speed longer previewdistance shorter time delay and larger permit position error
than the new drivers The match of these four driverparameters also influences the vehicle dynamics RecentlyPauwelussen [30] researched the dependencies of driversteering control parameters on vehicle lateral properties andpath keeping and gives some useful conclusions This workfocuses on the effects of driver model parameters on bothpath keeping and full vehicle dynamics such as vehiclehandling stability ride comfort and roll stability
Since the vehicle lateral displacement is randomly causedby road roughness excitations and the lateral position devi-ation is always changing during lane change maneuvers astatistic index is introduced to evaluate the vehiclersquos path-following ability globally which is formulated by
Route error = radicsum (119877119884
119889(119905) minus 119884
119889(119905))2
119873
(20)
Mathematical Problems in Engineering 13
20 40 60 80 100 120 140 160 180 200
0
1
2
3
4
5
20 kmh30 kmh40 kmh
50 kmh60 kmhRoute
Y(m
)
X (m)
minus1
0 50 100 150 200
0
1
2
3
4
0 s002 s005 s
01 s02 sRoute
Y(m
)
X (m)
minus1
Figure 12 Routes of vehicle under different speed or time delay
where 119873 is the total number of data points 119877119884119889and 119884
119889are
the required lateral position and real lateral displacement ofthe preview point in the ground coordinate system
When a vehicle is undergoing a lane change its steeringstability and roll stability are vital problems The path-following error lateral acceleration and yaw rate can be usedto evaluate the steering stability The roll angle can be used toevaluate the roll stability Being proportional to the rolloverindex LTR (lateral-load transfer rate) the lateral accelerationis also able to reflect the vehicle roll stability [31] In additionthe vertical acceleration and pitch angle are simulated so asto research the whole-body dynamics in the vehicle Usingthese evaluation indexes the effects of parameters on steeringstability roll stability and riding comfort are discussed indetail
Figure 12 shows the vehicle trajectories under differentinitial vehicle speeds or driver time delays The effects ofvehicle speed and driver time delay on full vehicle dynamicsand stability are shown in Figure 13 As we can see fromFigures 12 and 13 a higher speed or a longer time delay willlead to a greater path-following error However low speed(20 kmh) and longtime delay (035 s) may enlarge the path-following error With the rise of vehicle speed the verticalacceleration lateral acceleration yaw rate and roll angleincrease while the pitch angle decreases Rise of time delaycauses the increase of lateral acceleration yaw rate and rollangle but hardly changes the vertical acceleration and pitchangle As for the effects of vehicle speed and driver time delayon full vehicle dynamics the following may be concluded
(1) A high vehicle speed during double-lane changeyields bad path keeping performance ride comfortsteering stability and roll stability When the vehiclespeed is higher than 60 kmh the ride comfort of thisvehicle gets worse greatly
(2) A long time delay is harmful to path keeping per-formance steering stability and roll stability and hasalmost no effect on the ride comfort
Thus a low vehicle speed ranging from 20 kmh to 60 kmh and a small time delay less than 035 s are recommended forimproving this truckrsquos stability when going through the lanechange
In case of varying the preview distance or permit positionerror the variation in vehicle trajectories at the speed of40 kmh is shown in Figure 14The effects of preview distanceor permit position error on the path-following ability andfull vehicle dynamics are shown in Figure 15 As shown inFigures 14 and 15 a large preview distance deteriorates thepath-following ability but has small effect on other dynamicresponses On the other hand a too small preview distanceleads to a hop of lateral acceleration yaw rate and roll anglewhich conflicts with closed-loop stability and is harmfulto steering control This conclusion is consistent with thatof Gillespie and MacAdam [15] and Pauwelussen [30] Theinfluence of permit position error on path-following ability issmall at short preview distance but great at long preview dis-tance As shown in Figure 14 a too small or too large permitposition error (001m or 08m)will worsen the vehiclersquos path-following ability In addition large permit position error mayincrease vertical acceleration lateral acceleration yaw rateand roll angle and reduce the handling stability and rollstability and ride comfort The effect of preview distance andpermit position error on pitch angle is small and shows fluc-tuation Thus a preview distance ranging from 10m to 30mand a small permit position error but more than 001m arerecommended for improving this truckrsquos stability when mov-ing through the lane change A smaller permit position errorshould be matched with a shorter preview distance
14 Mathematical Problems in Engineering
002
04
2040
60800
1
2
Rout
e err
or (m
)
u0 (kmh) 0
02
04
2040
60807
8
9
u0 (kmh)
002
04
204060800
02
04
wr (
rad
s)
u0 (kmh)
0
02
04
204060800
005
01
120579(r
ad)
u0 (kmh)002
04
2040
60800
01
02
Φ(r
ad)
u0 (kmh)
002
04
2040
60800
2
4
u0(kmh)
ay(m
s2 )
Tr(s) Tr
(s)
Tr(s)
Tr(s)
Tr(s) Tr
(s)
azb
(ms2 )
Figure 13 The effects of vehicle speed and driver time delay
0 50 100 150 200
0
1
2
3
4
5
001 m02 m04 m
06 m08 mRoute
Y(m
)
X (m)
minus10 50 100 150 200
0
2
4
6
8
10
12
2 m10 m20 m
30 m40 mRoute
Y(m
)
X (m)
minus2
minus4
Figure 14 Routes of vehicle under different preview distance and permit position error
Mathematical Problems in Engineering 15
020
4060
005
1150
1
2
Rout
e err
or (m
)
er (m) Ld (m)
020
4060
01
25
10
15
er (m) Ld (m)
020
4060
01
20
5
er (m) Ld (m) 020
4060
01
20
02
04
er (m) Ld (m)wr (
rad
s)
2040
601
20
02
04
00er (m) Ld (m)
Φ(r
ad)
020
40 60
01
20
005
01
er (m) Ld (m)
120579(r
ad)
ay(m
s2 )
azb
(ms2 )
Figure 15 The effects of preview distance and permit position error
5 Conclusions
In this paper a nonlinear three-directional coupled lumpedparameters (TCLP) model of heavy-duty vehicle consideringthe nonlinearity of suspension damping and tire stiffness isbuilt and connected with a proposedmodified preview drivermodel with nonlinear time delayThe proposed driver modelis simple and has a feedback gain 2119871119889
2 that is decided bywheelbase and previewdistanceThe validity of this presenteddriver-vehicle closed-loop system is verified by comparisonwith the test data the traditional steering stability vehiclemodel the linear vehicle model and other driver controlmodelsThe results show that the new drivermodel has betterlane keeping performances than the other two driver modelsThe coupling effects of vehicle motions and the nonlinear-ity of suspension damping and tire stiffness could not beneglected in turning or high speed driving situation
The effects of driver parameters on path-following abilityand full vehicle dynamics are also discussed It is foundthat the high vehicle speed large time delay long previewdistance or big permit position error will deteriorate path-following performances and handling stability Of course atoo short preview distance or a too small permit position
error is also harmful to path-following performances andhandling stability The ride comfort mainly depends on vehi-cle speed preview distance and permit position error Thekey factors influencing roll characteristics are vehicle speedand time delay
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The National Natural Science Foundation of China(11472180) theNatural Science Foundation ofHebei Province(E2012210025) and the New Century Talent Foundation ofMinistry of Education (NCET-13-0913) support this work
References
[1] M Plochl and J Edelmann ldquoDriver models in automobiledynamics applicationrdquoVehicle SystemDynamics vol 45 no 7-8pp 699ndash741 2007
16 Mathematical Problems in Engineering
[2] C C MacAdam ldquoUnderstanding and modeling the humandriverrdquo Vehicle System Dynamics vol 40 no 1ndash3 pp 101ndash1342003
[3] K Guo and H Guan ldquoModeling of drivervehicle directionalcontrol systemrdquo Vehicle System Dynamics vol 22 no 3-4 pp141ndash184 1993
[4] K GuoVehicle Handling DynamicsTheory Jiangsu Science andTechnology Publishing House Nanjing China 2011
[5] R S Sharp ldquoDriver steering control and a new perspective oncar handling qualitiesrdquo Proceedings of the Institution ofMechani-cal Engineers Part C Journal of Mechanical Engineering Sciencevol 219 no 10 pp 1041ndash1051 2005
[6] T Legouis A Laneville P Bourassa and G Payre ldquoCharac-terization of dynamic vehicle stability using two models of thehuman pilot behaviorrdquo Vehicle System Dynamics vol 15 no 1pp 1ndash18 1986
[7] Z Liu G Payre and P Bourassa ldquoNonlinear oscillations andchaotic motions in a road vehicle system with driver steeringcontrolrdquo Nonlinear Dynamics vol 9 no 3 pp 281ndash304 1996
[8] Z Liu G Payre and P Bourassa ldquoStability and oscillations ina time-delayed vehicle system with driver controlrdquo NonlinearDynamics vol 35 no 2 pp 159ndash173 2004
[9] X D Liu W Feng and N Kang ldquoResearch of simulationmethod of tractor-semitrailer carrying liquid considering liquidsloshingrdquo in Proceedings of the 10th National Conference onVibrationTheory and Application pp 581ndash590 Nanjing ChinaOctober 2011
[10] C C MacAdam ldquoApplication of an optimal preview control forsimulation of closed-loop automobile drivingrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 11 no 6 pp 393ndash399 1981
[11] A Y Ungoren and H Peng ldquoAn adaptive lateral preview drivermodelrdquo Vehicle System Dynamics vol 43 no 4 pp 245ndash2592005
[12] C I Chatzikomis and K N Spentzas ldquoA path-following drivermodel with longitudinal and lateral control of vehiclersquos motionrdquoForschung im Ingenieurwesen vol 73 no 4 pp 257ndash266 2009
[13] R S Sharp D Casanova and P Symonds ldquoA mathematicalmodel for driver steering control with design tuning andperformance resultsrdquo Vehicle System Dynamics vol 33 no 5pp 289ndash326 2000
[14] A J Pick and D J Cole ldquoNeuromuscular dynamics in thedriver-vehicle systemrdquo Vehicle System Dynamics vol 44 sup-plement 1 pp 624ndash631 2006
[15] T D Gillespie and C C MacAdam Constant Velocity YawRollProgram Userrsquos Manual The University of Michigan Trans-portation Research Institute Ann Arbor Mich USA 1982
[16] R I S Raj Influence of road roughness and directional maneuveron the dynamic performance of heavy vehicles [MS thesis]Department of Mechanical Engineering Concordia UniversityMontreal Canada 1998
[17] KWordenDHickeyMHaroon andD E Adams ldquoNonlinearsystem identification of automotive dampers a time and fre-quency-domain analysisrdquo Mechanical Systems and Signal Pro-cessing vol 23 no 1 pp 104ndash126 2009
[18] S H Li Y J Lu and L Y Li ldquoDynamical test and modelingfor hydraulic shock absorber on heavy vehicle under harmonicand random loadingsrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 13 pp 1903ndash1910 2012
[19] X W Ji and Y M Gao ldquoThe dynamic stiffness and dampingcharacteristics of the tirerdquo Automotive Engineering vol 5 pp315ndash321 1994
[20] G Gim and P E Nikravesh ldquoA three dimensional tire modelfor steady-state simulations of vehiclesrdquo SAE vol 102 no 2 pp150ndash159 1993
[21] J D Zhuang Principles of Automobile Tire Beijing Institute ofTechnology Press Beijing China 1996
[22] W-M Zhai ldquoTwo simple fast integration methods for large-scale dynamic problems in engineeringrdquo International Journalfor Numerical Methods in Engineering vol 39 no 24 pp 4199ndash4214 1996
[23] Dongfeng Motor Group Company Dongfeng DFL1250A8DFL1250A9 Truck Chassis Refit Manual Dongfeng MotorGroup Company 2008
[24] S Li S Yang L Chen and Y Lu ldquoEffects of parameters ondynamic responses for a heavy vehicle-pavement- foundationcoupled systemrdquo International Journal of Heavy Vehicle Systemsvol 19 no 2 pp 207ndash224 2012
[25] S Yang S Li andY Lu ldquoInvestigation on dynamical interactionbetween a heavy vehicle and road pavementrdquo Vehicle SystemDynamics vol 48 no 8 pp 923ndash944 2010
[26] GBT 7031-2005ISO 86081995 Mechanical Vibration-RoadSurface Profiles-Reporting of Measured Data 2005
[27] M Li X Pu and Z Changfu ldquoModeling and simulationof heavy-duty semi-trailer in extreme condition based onADAMSrdquo Automobile Technology vol 2 pp 22ndash25 2009
[28] ldquoPassenger carsmdashtest track for a severe lane-change manoeu-vremdashpart 1 double lane-changerdquo ISO 3888-11999
[29] S H Li S P Yang and N Chen ldquoDirectional control of adriver-heavy-vehicle closed-loop systemrdquo Advanced Engineer-ing Forum vol 2-3 pp 33ndash38 2012
[30] J Pauwelussen ldquoDependencies of driver steering controlparametersrdquo Vehicle System Dynamics vol 50 no 6 pp 939ndash959 2012
[31] H Imine and V Dolcemascolo ldquoRollover risk prediction ofheavy vehicle in interaction with infrastructurerdquo InternationalJournal ofHeavyVehicle Systems vol 14 no 3 pp 294ndash307 2007
Figure 12 Routes of vehicle under different speed or time delay
where 119873 is the total number of data points 119877119884119889and 119884
119889are
the required lateral position and real lateral displacement ofthe preview point in the ground coordinate system
When a vehicle is undergoing a lane change its steeringstability and roll stability are vital problems The path-following error lateral acceleration and yaw rate can be usedto evaluate the steering stability The roll angle can be used toevaluate the roll stability Being proportional to the rolloverindex LTR (lateral-load transfer rate) the lateral accelerationis also able to reflect the vehicle roll stability [31] In additionthe vertical acceleration and pitch angle are simulated so asto research the whole-body dynamics in the vehicle Usingthese evaluation indexes the effects of parameters on steeringstability roll stability and riding comfort are discussed indetail
Figure 12 shows the vehicle trajectories under differentinitial vehicle speeds or driver time delays The effects ofvehicle speed and driver time delay on full vehicle dynamicsand stability are shown in Figure 13 As we can see fromFigures 12 and 13 a higher speed or a longer time delay willlead to a greater path-following error However low speed(20 kmh) and longtime delay (035 s) may enlarge the path-following error With the rise of vehicle speed the verticalacceleration lateral acceleration yaw rate and roll angleincrease while the pitch angle decreases Rise of time delaycauses the increase of lateral acceleration yaw rate and rollangle but hardly changes the vertical acceleration and pitchangle As for the effects of vehicle speed and driver time delayon full vehicle dynamics the following may be concluded
(1) A high vehicle speed during double-lane changeyields bad path keeping performance ride comfortsteering stability and roll stability When the vehiclespeed is higher than 60 kmh the ride comfort of thisvehicle gets worse greatly
(2) A long time delay is harmful to path keeping per-formance steering stability and roll stability and hasalmost no effect on the ride comfort
Thus a low vehicle speed ranging from 20 kmh to 60 kmh and a small time delay less than 035 s are recommended forimproving this truckrsquos stability when going through the lanechange
In case of varying the preview distance or permit positionerror the variation in vehicle trajectories at the speed of40 kmh is shown in Figure 14The effects of preview distanceor permit position error on the path-following ability andfull vehicle dynamics are shown in Figure 15 As shown inFigures 14 and 15 a large preview distance deteriorates thepath-following ability but has small effect on other dynamicresponses On the other hand a too small preview distanceleads to a hop of lateral acceleration yaw rate and roll anglewhich conflicts with closed-loop stability and is harmfulto steering control This conclusion is consistent with thatof Gillespie and MacAdam [15] and Pauwelussen [30] Theinfluence of permit position error on path-following ability issmall at short preview distance but great at long preview dis-tance As shown in Figure 14 a too small or too large permitposition error (001m or 08m)will worsen the vehiclersquos path-following ability In addition large permit position error mayincrease vertical acceleration lateral acceleration yaw rateand roll angle and reduce the handling stability and rollstability and ride comfort The effect of preview distance andpermit position error on pitch angle is small and shows fluc-tuation Thus a preview distance ranging from 10m to 30mand a small permit position error but more than 001m arerecommended for improving this truckrsquos stability when mov-ing through the lane change A smaller permit position errorshould be matched with a shorter preview distance
14 Mathematical Problems in Engineering
002
04
2040
60800
1
2
Rout
e err
or (m
)
u0 (kmh) 0
02
04
2040
60807
8
9
u0 (kmh)
002
04
204060800
02
04
wr (
rad
s)
u0 (kmh)
0
02
04
204060800
005
01
120579(r
ad)
u0 (kmh)002
04
2040
60800
01
02
Φ(r
ad)
u0 (kmh)
002
04
2040
60800
2
4
u0(kmh)
ay(m
s2 )
Tr(s) Tr
(s)
Tr(s)
Tr(s)
Tr(s) Tr
(s)
azb
(ms2 )
Figure 13 The effects of vehicle speed and driver time delay
0 50 100 150 200
0
1
2
3
4
5
001 m02 m04 m
06 m08 mRoute
Y(m
)
X (m)
minus10 50 100 150 200
0
2
4
6
8
10
12
2 m10 m20 m
30 m40 mRoute
Y(m
)
X (m)
minus2
minus4
Figure 14 Routes of vehicle under different preview distance and permit position error
Mathematical Problems in Engineering 15
020
4060
005
1150
1
2
Rout
e err
or (m
)
er (m) Ld (m)
020
4060
01
25
10
15
er (m) Ld (m)
020
4060
01
20
5
er (m) Ld (m) 020
4060
01
20
02
04
er (m) Ld (m)wr (
rad
s)
2040
601
20
02
04
00er (m) Ld (m)
Φ(r
ad)
020
40 60
01
20
005
01
er (m) Ld (m)
120579(r
ad)
ay(m
s2 )
azb
(ms2 )
Figure 15 The effects of preview distance and permit position error
5 Conclusions
In this paper a nonlinear three-directional coupled lumpedparameters (TCLP) model of heavy-duty vehicle consideringthe nonlinearity of suspension damping and tire stiffness isbuilt and connected with a proposedmodified preview drivermodel with nonlinear time delayThe proposed driver modelis simple and has a feedback gain 2119871119889
2 that is decided bywheelbase and previewdistanceThe validity of this presenteddriver-vehicle closed-loop system is verified by comparisonwith the test data the traditional steering stability vehiclemodel the linear vehicle model and other driver controlmodelsThe results show that the new drivermodel has betterlane keeping performances than the other two driver modelsThe coupling effects of vehicle motions and the nonlinear-ity of suspension damping and tire stiffness could not beneglected in turning or high speed driving situation
The effects of driver parameters on path-following abilityand full vehicle dynamics are also discussed It is foundthat the high vehicle speed large time delay long previewdistance or big permit position error will deteriorate path-following performances and handling stability Of course atoo short preview distance or a too small permit position
error is also harmful to path-following performances andhandling stability The ride comfort mainly depends on vehi-cle speed preview distance and permit position error Thekey factors influencing roll characteristics are vehicle speedand time delay
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The National Natural Science Foundation of China(11472180) theNatural Science Foundation ofHebei Province(E2012210025) and the New Century Talent Foundation ofMinistry of Education (NCET-13-0913) support this work
References
[1] M Plochl and J Edelmann ldquoDriver models in automobiledynamics applicationrdquoVehicle SystemDynamics vol 45 no 7-8pp 699ndash741 2007
16 Mathematical Problems in Engineering
[2] C C MacAdam ldquoUnderstanding and modeling the humandriverrdquo Vehicle System Dynamics vol 40 no 1ndash3 pp 101ndash1342003
[3] K Guo and H Guan ldquoModeling of drivervehicle directionalcontrol systemrdquo Vehicle System Dynamics vol 22 no 3-4 pp141ndash184 1993
[4] K GuoVehicle Handling DynamicsTheory Jiangsu Science andTechnology Publishing House Nanjing China 2011
[5] R S Sharp ldquoDriver steering control and a new perspective oncar handling qualitiesrdquo Proceedings of the Institution ofMechani-cal Engineers Part C Journal of Mechanical Engineering Sciencevol 219 no 10 pp 1041ndash1051 2005
[6] T Legouis A Laneville P Bourassa and G Payre ldquoCharac-terization of dynamic vehicle stability using two models of thehuman pilot behaviorrdquo Vehicle System Dynamics vol 15 no 1pp 1ndash18 1986
[7] Z Liu G Payre and P Bourassa ldquoNonlinear oscillations andchaotic motions in a road vehicle system with driver steeringcontrolrdquo Nonlinear Dynamics vol 9 no 3 pp 281ndash304 1996
[8] Z Liu G Payre and P Bourassa ldquoStability and oscillations ina time-delayed vehicle system with driver controlrdquo NonlinearDynamics vol 35 no 2 pp 159ndash173 2004
[9] X D Liu W Feng and N Kang ldquoResearch of simulationmethod of tractor-semitrailer carrying liquid considering liquidsloshingrdquo in Proceedings of the 10th National Conference onVibrationTheory and Application pp 581ndash590 Nanjing ChinaOctober 2011
[10] C C MacAdam ldquoApplication of an optimal preview control forsimulation of closed-loop automobile drivingrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 11 no 6 pp 393ndash399 1981
[11] A Y Ungoren and H Peng ldquoAn adaptive lateral preview drivermodelrdquo Vehicle System Dynamics vol 43 no 4 pp 245ndash2592005
[12] C I Chatzikomis and K N Spentzas ldquoA path-following drivermodel with longitudinal and lateral control of vehiclersquos motionrdquoForschung im Ingenieurwesen vol 73 no 4 pp 257ndash266 2009
[13] R S Sharp D Casanova and P Symonds ldquoA mathematicalmodel for driver steering control with design tuning andperformance resultsrdquo Vehicle System Dynamics vol 33 no 5pp 289ndash326 2000
[14] A J Pick and D J Cole ldquoNeuromuscular dynamics in thedriver-vehicle systemrdquo Vehicle System Dynamics vol 44 sup-plement 1 pp 624ndash631 2006
[15] T D Gillespie and C C MacAdam Constant Velocity YawRollProgram Userrsquos Manual The University of Michigan Trans-portation Research Institute Ann Arbor Mich USA 1982
[16] R I S Raj Influence of road roughness and directional maneuveron the dynamic performance of heavy vehicles [MS thesis]Department of Mechanical Engineering Concordia UniversityMontreal Canada 1998
[17] KWordenDHickeyMHaroon andD E Adams ldquoNonlinearsystem identification of automotive dampers a time and fre-quency-domain analysisrdquo Mechanical Systems and Signal Pro-cessing vol 23 no 1 pp 104ndash126 2009
[18] S H Li Y J Lu and L Y Li ldquoDynamical test and modelingfor hydraulic shock absorber on heavy vehicle under harmonicand random loadingsrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 13 pp 1903ndash1910 2012
[19] X W Ji and Y M Gao ldquoThe dynamic stiffness and dampingcharacteristics of the tirerdquo Automotive Engineering vol 5 pp315ndash321 1994
[20] G Gim and P E Nikravesh ldquoA three dimensional tire modelfor steady-state simulations of vehiclesrdquo SAE vol 102 no 2 pp150ndash159 1993
[21] J D Zhuang Principles of Automobile Tire Beijing Institute ofTechnology Press Beijing China 1996
[22] W-M Zhai ldquoTwo simple fast integration methods for large-scale dynamic problems in engineeringrdquo International Journalfor Numerical Methods in Engineering vol 39 no 24 pp 4199ndash4214 1996
[23] Dongfeng Motor Group Company Dongfeng DFL1250A8DFL1250A9 Truck Chassis Refit Manual Dongfeng MotorGroup Company 2008
[24] S Li S Yang L Chen and Y Lu ldquoEffects of parameters ondynamic responses for a heavy vehicle-pavement- foundationcoupled systemrdquo International Journal of Heavy Vehicle Systemsvol 19 no 2 pp 207ndash224 2012
[25] S Yang S Li andY Lu ldquoInvestigation on dynamical interactionbetween a heavy vehicle and road pavementrdquo Vehicle SystemDynamics vol 48 no 8 pp 923ndash944 2010
[26] GBT 7031-2005ISO 86081995 Mechanical Vibration-RoadSurface Profiles-Reporting of Measured Data 2005
[27] M Li X Pu and Z Changfu ldquoModeling and simulationof heavy-duty semi-trailer in extreme condition based onADAMSrdquo Automobile Technology vol 2 pp 22ndash25 2009
[28] ldquoPassenger carsmdashtest track for a severe lane-change manoeu-vremdashpart 1 double lane-changerdquo ISO 3888-11999
[29] S H Li S P Yang and N Chen ldquoDirectional control of adriver-heavy-vehicle closed-loop systemrdquo Advanced Engineer-ing Forum vol 2-3 pp 33ndash38 2012
[30] J Pauwelussen ldquoDependencies of driver steering controlparametersrdquo Vehicle System Dynamics vol 50 no 6 pp 939ndash959 2012
[31] H Imine and V Dolcemascolo ldquoRollover risk prediction ofheavy vehicle in interaction with infrastructurerdquo InternationalJournal ofHeavyVehicle Systems vol 14 no 3 pp 294ndash307 2007
Figure 13 The effects of vehicle speed and driver time delay
0 50 100 150 200
0
1
2
3
4
5
001 m02 m04 m
06 m08 mRoute
Y(m
)
X (m)
minus10 50 100 150 200
0
2
4
6
8
10
12
2 m10 m20 m
30 m40 mRoute
Y(m
)
X (m)
minus2
minus4
Figure 14 Routes of vehicle under different preview distance and permit position error
Mathematical Problems in Engineering 15
020
4060
005
1150
1
2
Rout
e err
or (m
)
er (m) Ld (m)
020
4060
01
25
10
15
er (m) Ld (m)
020
4060
01
20
5
er (m) Ld (m) 020
4060
01
20
02
04
er (m) Ld (m)wr (
rad
s)
2040
601
20
02
04
00er (m) Ld (m)
Φ(r
ad)
020
40 60
01
20
005
01
er (m) Ld (m)
120579(r
ad)
ay(m
s2 )
azb
(ms2 )
Figure 15 The effects of preview distance and permit position error
5 Conclusions
In this paper a nonlinear three-directional coupled lumpedparameters (TCLP) model of heavy-duty vehicle consideringthe nonlinearity of suspension damping and tire stiffness isbuilt and connected with a proposedmodified preview drivermodel with nonlinear time delayThe proposed driver modelis simple and has a feedback gain 2119871119889
2 that is decided bywheelbase and previewdistanceThe validity of this presenteddriver-vehicle closed-loop system is verified by comparisonwith the test data the traditional steering stability vehiclemodel the linear vehicle model and other driver controlmodelsThe results show that the new drivermodel has betterlane keeping performances than the other two driver modelsThe coupling effects of vehicle motions and the nonlinear-ity of suspension damping and tire stiffness could not beneglected in turning or high speed driving situation
The effects of driver parameters on path-following abilityand full vehicle dynamics are also discussed It is foundthat the high vehicle speed large time delay long previewdistance or big permit position error will deteriorate path-following performances and handling stability Of course atoo short preview distance or a too small permit position
error is also harmful to path-following performances andhandling stability The ride comfort mainly depends on vehi-cle speed preview distance and permit position error Thekey factors influencing roll characteristics are vehicle speedand time delay
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The National Natural Science Foundation of China(11472180) theNatural Science Foundation ofHebei Province(E2012210025) and the New Century Talent Foundation ofMinistry of Education (NCET-13-0913) support this work
References
[1] M Plochl and J Edelmann ldquoDriver models in automobiledynamics applicationrdquoVehicle SystemDynamics vol 45 no 7-8pp 699ndash741 2007
16 Mathematical Problems in Engineering
[2] C C MacAdam ldquoUnderstanding and modeling the humandriverrdquo Vehicle System Dynamics vol 40 no 1ndash3 pp 101ndash1342003
[3] K Guo and H Guan ldquoModeling of drivervehicle directionalcontrol systemrdquo Vehicle System Dynamics vol 22 no 3-4 pp141ndash184 1993
[4] K GuoVehicle Handling DynamicsTheory Jiangsu Science andTechnology Publishing House Nanjing China 2011
[5] R S Sharp ldquoDriver steering control and a new perspective oncar handling qualitiesrdquo Proceedings of the Institution ofMechani-cal Engineers Part C Journal of Mechanical Engineering Sciencevol 219 no 10 pp 1041ndash1051 2005
[6] T Legouis A Laneville P Bourassa and G Payre ldquoCharac-terization of dynamic vehicle stability using two models of thehuman pilot behaviorrdquo Vehicle System Dynamics vol 15 no 1pp 1ndash18 1986
[7] Z Liu G Payre and P Bourassa ldquoNonlinear oscillations andchaotic motions in a road vehicle system with driver steeringcontrolrdquo Nonlinear Dynamics vol 9 no 3 pp 281ndash304 1996
[8] Z Liu G Payre and P Bourassa ldquoStability and oscillations ina time-delayed vehicle system with driver controlrdquo NonlinearDynamics vol 35 no 2 pp 159ndash173 2004
[9] X D Liu W Feng and N Kang ldquoResearch of simulationmethod of tractor-semitrailer carrying liquid considering liquidsloshingrdquo in Proceedings of the 10th National Conference onVibrationTheory and Application pp 581ndash590 Nanjing ChinaOctober 2011
[10] C C MacAdam ldquoApplication of an optimal preview control forsimulation of closed-loop automobile drivingrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 11 no 6 pp 393ndash399 1981
[11] A Y Ungoren and H Peng ldquoAn adaptive lateral preview drivermodelrdquo Vehicle System Dynamics vol 43 no 4 pp 245ndash2592005
[12] C I Chatzikomis and K N Spentzas ldquoA path-following drivermodel with longitudinal and lateral control of vehiclersquos motionrdquoForschung im Ingenieurwesen vol 73 no 4 pp 257ndash266 2009
[13] R S Sharp D Casanova and P Symonds ldquoA mathematicalmodel for driver steering control with design tuning andperformance resultsrdquo Vehicle System Dynamics vol 33 no 5pp 289ndash326 2000
[14] A J Pick and D J Cole ldquoNeuromuscular dynamics in thedriver-vehicle systemrdquo Vehicle System Dynamics vol 44 sup-plement 1 pp 624ndash631 2006
[15] T D Gillespie and C C MacAdam Constant Velocity YawRollProgram Userrsquos Manual The University of Michigan Trans-portation Research Institute Ann Arbor Mich USA 1982
[16] R I S Raj Influence of road roughness and directional maneuveron the dynamic performance of heavy vehicles [MS thesis]Department of Mechanical Engineering Concordia UniversityMontreal Canada 1998
[17] KWordenDHickeyMHaroon andD E Adams ldquoNonlinearsystem identification of automotive dampers a time and fre-quency-domain analysisrdquo Mechanical Systems and Signal Pro-cessing vol 23 no 1 pp 104ndash126 2009
[18] S H Li Y J Lu and L Y Li ldquoDynamical test and modelingfor hydraulic shock absorber on heavy vehicle under harmonicand random loadingsrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 13 pp 1903ndash1910 2012
[19] X W Ji and Y M Gao ldquoThe dynamic stiffness and dampingcharacteristics of the tirerdquo Automotive Engineering vol 5 pp315ndash321 1994
[20] G Gim and P E Nikravesh ldquoA three dimensional tire modelfor steady-state simulations of vehiclesrdquo SAE vol 102 no 2 pp150ndash159 1993
[21] J D Zhuang Principles of Automobile Tire Beijing Institute ofTechnology Press Beijing China 1996
[22] W-M Zhai ldquoTwo simple fast integration methods for large-scale dynamic problems in engineeringrdquo International Journalfor Numerical Methods in Engineering vol 39 no 24 pp 4199ndash4214 1996
[23] Dongfeng Motor Group Company Dongfeng DFL1250A8DFL1250A9 Truck Chassis Refit Manual Dongfeng MotorGroup Company 2008
[24] S Li S Yang L Chen and Y Lu ldquoEffects of parameters ondynamic responses for a heavy vehicle-pavement- foundationcoupled systemrdquo International Journal of Heavy Vehicle Systemsvol 19 no 2 pp 207ndash224 2012
[25] S Yang S Li andY Lu ldquoInvestigation on dynamical interactionbetween a heavy vehicle and road pavementrdquo Vehicle SystemDynamics vol 48 no 8 pp 923ndash944 2010
[26] GBT 7031-2005ISO 86081995 Mechanical Vibration-RoadSurface Profiles-Reporting of Measured Data 2005
[27] M Li X Pu and Z Changfu ldquoModeling and simulationof heavy-duty semi-trailer in extreme condition based onADAMSrdquo Automobile Technology vol 2 pp 22ndash25 2009
[28] ldquoPassenger carsmdashtest track for a severe lane-change manoeu-vremdashpart 1 double lane-changerdquo ISO 3888-11999
[29] S H Li S P Yang and N Chen ldquoDirectional control of adriver-heavy-vehicle closed-loop systemrdquo Advanced Engineer-ing Forum vol 2-3 pp 33ndash38 2012
[30] J Pauwelussen ldquoDependencies of driver steering controlparametersrdquo Vehicle System Dynamics vol 50 no 6 pp 939ndash959 2012
[31] H Imine and V Dolcemascolo ldquoRollover risk prediction ofheavy vehicle in interaction with infrastructurerdquo InternationalJournal ofHeavyVehicle Systems vol 14 no 3 pp 294ndash307 2007
Figure 15 The effects of preview distance and permit position error
5 Conclusions
In this paper a nonlinear three-directional coupled lumpedparameters (TCLP) model of heavy-duty vehicle consideringthe nonlinearity of suspension damping and tire stiffness isbuilt and connected with a proposedmodified preview drivermodel with nonlinear time delayThe proposed driver modelis simple and has a feedback gain 2119871119889
2 that is decided bywheelbase and previewdistanceThe validity of this presenteddriver-vehicle closed-loop system is verified by comparisonwith the test data the traditional steering stability vehiclemodel the linear vehicle model and other driver controlmodelsThe results show that the new drivermodel has betterlane keeping performances than the other two driver modelsThe coupling effects of vehicle motions and the nonlinear-ity of suspension damping and tire stiffness could not beneglected in turning or high speed driving situation
The effects of driver parameters on path-following abilityand full vehicle dynamics are also discussed It is foundthat the high vehicle speed large time delay long previewdistance or big permit position error will deteriorate path-following performances and handling stability Of course atoo short preview distance or a too small permit position
error is also harmful to path-following performances andhandling stability The ride comfort mainly depends on vehi-cle speed preview distance and permit position error Thekey factors influencing roll characteristics are vehicle speedand time delay
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The National Natural Science Foundation of China(11472180) theNatural Science Foundation ofHebei Province(E2012210025) and the New Century Talent Foundation ofMinistry of Education (NCET-13-0913) support this work
References
[1] M Plochl and J Edelmann ldquoDriver models in automobiledynamics applicationrdquoVehicle SystemDynamics vol 45 no 7-8pp 699ndash741 2007
16 Mathematical Problems in Engineering
[2] C C MacAdam ldquoUnderstanding and modeling the humandriverrdquo Vehicle System Dynamics vol 40 no 1ndash3 pp 101ndash1342003
[3] K Guo and H Guan ldquoModeling of drivervehicle directionalcontrol systemrdquo Vehicle System Dynamics vol 22 no 3-4 pp141ndash184 1993
[4] K GuoVehicle Handling DynamicsTheory Jiangsu Science andTechnology Publishing House Nanjing China 2011
[5] R S Sharp ldquoDriver steering control and a new perspective oncar handling qualitiesrdquo Proceedings of the Institution ofMechani-cal Engineers Part C Journal of Mechanical Engineering Sciencevol 219 no 10 pp 1041ndash1051 2005
[6] T Legouis A Laneville P Bourassa and G Payre ldquoCharac-terization of dynamic vehicle stability using two models of thehuman pilot behaviorrdquo Vehicle System Dynamics vol 15 no 1pp 1ndash18 1986
[7] Z Liu G Payre and P Bourassa ldquoNonlinear oscillations andchaotic motions in a road vehicle system with driver steeringcontrolrdquo Nonlinear Dynamics vol 9 no 3 pp 281ndash304 1996
[8] Z Liu G Payre and P Bourassa ldquoStability and oscillations ina time-delayed vehicle system with driver controlrdquo NonlinearDynamics vol 35 no 2 pp 159ndash173 2004
[9] X D Liu W Feng and N Kang ldquoResearch of simulationmethod of tractor-semitrailer carrying liquid considering liquidsloshingrdquo in Proceedings of the 10th National Conference onVibrationTheory and Application pp 581ndash590 Nanjing ChinaOctober 2011
[10] C C MacAdam ldquoApplication of an optimal preview control forsimulation of closed-loop automobile drivingrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 11 no 6 pp 393ndash399 1981
[11] A Y Ungoren and H Peng ldquoAn adaptive lateral preview drivermodelrdquo Vehicle System Dynamics vol 43 no 4 pp 245ndash2592005
[12] C I Chatzikomis and K N Spentzas ldquoA path-following drivermodel with longitudinal and lateral control of vehiclersquos motionrdquoForschung im Ingenieurwesen vol 73 no 4 pp 257ndash266 2009
[13] R S Sharp D Casanova and P Symonds ldquoA mathematicalmodel for driver steering control with design tuning andperformance resultsrdquo Vehicle System Dynamics vol 33 no 5pp 289ndash326 2000
[14] A J Pick and D J Cole ldquoNeuromuscular dynamics in thedriver-vehicle systemrdquo Vehicle System Dynamics vol 44 sup-plement 1 pp 624ndash631 2006
[15] T D Gillespie and C C MacAdam Constant Velocity YawRollProgram Userrsquos Manual The University of Michigan Trans-portation Research Institute Ann Arbor Mich USA 1982
[16] R I S Raj Influence of road roughness and directional maneuveron the dynamic performance of heavy vehicles [MS thesis]Department of Mechanical Engineering Concordia UniversityMontreal Canada 1998
[17] KWordenDHickeyMHaroon andD E Adams ldquoNonlinearsystem identification of automotive dampers a time and fre-quency-domain analysisrdquo Mechanical Systems and Signal Pro-cessing vol 23 no 1 pp 104ndash126 2009
[18] S H Li Y J Lu and L Y Li ldquoDynamical test and modelingfor hydraulic shock absorber on heavy vehicle under harmonicand random loadingsrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 13 pp 1903ndash1910 2012
[19] X W Ji and Y M Gao ldquoThe dynamic stiffness and dampingcharacteristics of the tirerdquo Automotive Engineering vol 5 pp315ndash321 1994
[20] G Gim and P E Nikravesh ldquoA three dimensional tire modelfor steady-state simulations of vehiclesrdquo SAE vol 102 no 2 pp150ndash159 1993
[21] J D Zhuang Principles of Automobile Tire Beijing Institute ofTechnology Press Beijing China 1996
[22] W-M Zhai ldquoTwo simple fast integration methods for large-scale dynamic problems in engineeringrdquo International Journalfor Numerical Methods in Engineering vol 39 no 24 pp 4199ndash4214 1996
[23] Dongfeng Motor Group Company Dongfeng DFL1250A8DFL1250A9 Truck Chassis Refit Manual Dongfeng MotorGroup Company 2008
[24] S Li S Yang L Chen and Y Lu ldquoEffects of parameters ondynamic responses for a heavy vehicle-pavement- foundationcoupled systemrdquo International Journal of Heavy Vehicle Systemsvol 19 no 2 pp 207ndash224 2012
[25] S Yang S Li andY Lu ldquoInvestigation on dynamical interactionbetween a heavy vehicle and road pavementrdquo Vehicle SystemDynamics vol 48 no 8 pp 923ndash944 2010
[26] GBT 7031-2005ISO 86081995 Mechanical Vibration-RoadSurface Profiles-Reporting of Measured Data 2005
[27] M Li X Pu and Z Changfu ldquoModeling and simulationof heavy-duty semi-trailer in extreme condition based onADAMSrdquo Automobile Technology vol 2 pp 22ndash25 2009
[28] ldquoPassenger carsmdashtest track for a severe lane-change manoeu-vremdashpart 1 double lane-changerdquo ISO 3888-11999
[29] S H Li S P Yang and N Chen ldquoDirectional control of adriver-heavy-vehicle closed-loop systemrdquo Advanced Engineer-ing Forum vol 2-3 pp 33ndash38 2012
[30] J Pauwelussen ldquoDependencies of driver steering controlparametersrdquo Vehicle System Dynamics vol 50 no 6 pp 939ndash959 2012
[31] H Imine and V Dolcemascolo ldquoRollover risk prediction ofheavy vehicle in interaction with infrastructurerdquo InternationalJournal ofHeavyVehicle Systems vol 14 no 3 pp 294ndash307 2007
[2] C C MacAdam ldquoUnderstanding and modeling the humandriverrdquo Vehicle System Dynamics vol 40 no 1ndash3 pp 101ndash1342003
[3] K Guo and H Guan ldquoModeling of drivervehicle directionalcontrol systemrdquo Vehicle System Dynamics vol 22 no 3-4 pp141ndash184 1993
[4] K GuoVehicle Handling DynamicsTheory Jiangsu Science andTechnology Publishing House Nanjing China 2011
[5] R S Sharp ldquoDriver steering control and a new perspective oncar handling qualitiesrdquo Proceedings of the Institution ofMechani-cal Engineers Part C Journal of Mechanical Engineering Sciencevol 219 no 10 pp 1041ndash1051 2005
[6] T Legouis A Laneville P Bourassa and G Payre ldquoCharac-terization of dynamic vehicle stability using two models of thehuman pilot behaviorrdquo Vehicle System Dynamics vol 15 no 1pp 1ndash18 1986
[7] Z Liu G Payre and P Bourassa ldquoNonlinear oscillations andchaotic motions in a road vehicle system with driver steeringcontrolrdquo Nonlinear Dynamics vol 9 no 3 pp 281ndash304 1996
[8] Z Liu G Payre and P Bourassa ldquoStability and oscillations ina time-delayed vehicle system with driver controlrdquo NonlinearDynamics vol 35 no 2 pp 159ndash173 2004
[9] X D Liu W Feng and N Kang ldquoResearch of simulationmethod of tractor-semitrailer carrying liquid considering liquidsloshingrdquo in Proceedings of the 10th National Conference onVibrationTheory and Application pp 581ndash590 Nanjing ChinaOctober 2011
[10] C C MacAdam ldquoApplication of an optimal preview control forsimulation of closed-loop automobile drivingrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 11 no 6 pp 393ndash399 1981
[11] A Y Ungoren and H Peng ldquoAn adaptive lateral preview drivermodelrdquo Vehicle System Dynamics vol 43 no 4 pp 245ndash2592005
[12] C I Chatzikomis and K N Spentzas ldquoA path-following drivermodel with longitudinal and lateral control of vehiclersquos motionrdquoForschung im Ingenieurwesen vol 73 no 4 pp 257ndash266 2009
[13] R S Sharp D Casanova and P Symonds ldquoA mathematicalmodel for driver steering control with design tuning andperformance resultsrdquo Vehicle System Dynamics vol 33 no 5pp 289ndash326 2000
[14] A J Pick and D J Cole ldquoNeuromuscular dynamics in thedriver-vehicle systemrdquo Vehicle System Dynamics vol 44 sup-plement 1 pp 624ndash631 2006
[15] T D Gillespie and C C MacAdam Constant Velocity YawRollProgram Userrsquos Manual The University of Michigan Trans-portation Research Institute Ann Arbor Mich USA 1982
[16] R I S Raj Influence of road roughness and directional maneuveron the dynamic performance of heavy vehicles [MS thesis]Department of Mechanical Engineering Concordia UniversityMontreal Canada 1998
[17] KWordenDHickeyMHaroon andD E Adams ldquoNonlinearsystem identification of automotive dampers a time and fre-quency-domain analysisrdquo Mechanical Systems and Signal Pro-cessing vol 23 no 1 pp 104ndash126 2009
[18] S H Li Y J Lu and L Y Li ldquoDynamical test and modelingfor hydraulic shock absorber on heavy vehicle under harmonicand random loadingsrdquo Research Journal of Applied SciencesEngineering and Technology vol 4 no 13 pp 1903ndash1910 2012
[19] X W Ji and Y M Gao ldquoThe dynamic stiffness and dampingcharacteristics of the tirerdquo Automotive Engineering vol 5 pp315ndash321 1994
[20] G Gim and P E Nikravesh ldquoA three dimensional tire modelfor steady-state simulations of vehiclesrdquo SAE vol 102 no 2 pp150ndash159 1993
[21] J D Zhuang Principles of Automobile Tire Beijing Institute ofTechnology Press Beijing China 1996
[22] W-M Zhai ldquoTwo simple fast integration methods for large-scale dynamic problems in engineeringrdquo International Journalfor Numerical Methods in Engineering vol 39 no 24 pp 4199ndash4214 1996
[23] Dongfeng Motor Group Company Dongfeng DFL1250A8DFL1250A9 Truck Chassis Refit Manual Dongfeng MotorGroup Company 2008
[24] S Li S Yang L Chen and Y Lu ldquoEffects of parameters ondynamic responses for a heavy vehicle-pavement- foundationcoupled systemrdquo International Journal of Heavy Vehicle Systemsvol 19 no 2 pp 207ndash224 2012
[25] S Yang S Li andY Lu ldquoInvestigation on dynamical interactionbetween a heavy vehicle and road pavementrdquo Vehicle SystemDynamics vol 48 no 8 pp 923ndash944 2010
[26] GBT 7031-2005ISO 86081995 Mechanical Vibration-RoadSurface Profiles-Reporting of Measured Data 2005
[27] M Li X Pu and Z Changfu ldquoModeling and simulationof heavy-duty semi-trailer in extreme condition based onADAMSrdquo Automobile Technology vol 2 pp 22ndash25 2009
[28] ldquoPassenger carsmdashtest track for a severe lane-change manoeu-vremdashpart 1 double lane-changerdquo ISO 3888-11999
[29] S H Li S P Yang and N Chen ldquoDirectional control of adriver-heavy-vehicle closed-loop systemrdquo Advanced Engineer-ing Forum vol 2-3 pp 33ndash38 2012
[30] J Pauwelussen ldquoDependencies of driver steering controlparametersrdquo Vehicle System Dynamics vol 50 no 6 pp 939ndash959 2012
[31] H Imine and V Dolcemascolo ldquoRollover risk prediction ofheavy vehicle in interaction with infrastructurerdquo InternationalJournal ofHeavyVehicle Systems vol 14 no 3 pp 294ndash307 2007
