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Research ArticleStudy of Model Predictive Control for Path-FollowingAutonomous Ground Vehicle Control under Crosswind Effect
Fitri Yakub1 Aminudin Abu1 Shamsul Sarip2 and Yasuchika Mori3
1Malaysia Japan International Institute of Technology Universiti Teknologi Malaysia Jalan Semarak 54100 Kuala Lumpur Malaysia2UTM Razak School of Engineering and Advanced Technology Universiti Teknologi Malaysia Jalan Semarak54100 Kuala Lumpur Malaysia3Graduate School of System Design Tokyo Metropolitan University 6-6 Asahigaoka Hino Tokyo 191-0065 Japan
Correspondence should be addressed to Fitri Yakub mfitriklutmmy
Received 4 November 2015 Revised 1 March 2016 Accepted 17 March 2016
Academic Editor Yongji Wang
Copyright copy 2016 Fitri Yakub et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We present a comparative study of model predictive control approaches of two-wheel steering four-wheel steering and a combi-nation of two-wheel steering with direct yaw moment control manoeuvres for path-following control in autonomous car vehicledynamics systems Single-track mode based on a linearized vehicle and tire model is used Based on a given trajectory we drovethe vehicle at low and high forward speeds and on low and high road friction surfaces for a double-lane change scenario in orderto follow the desired trajectory as close as possible while rejecting the effects of wind gusts We compared the controller based onboth simple and complex bicycle models without and with the roll vehicle dynamics for different types of model predictive controlmanoeuvres The simulation result showed that the model predictive control gave a better performance in terms of robustness forboth forward speeds and road surface variation in autonomous path-following control It also demonstrated that model predictivecontrol is useful to maintain vehicle stability along the desired path and has an ability to eliminate the crosswind effect
1 Introduction
Today model predictive control (MPC) is one of the morepopular optimal control techniqueswhich iswidely employedfor the control of constrained linear or nonlinear systemsMPC uses amathematical dynamics processmodel to predictthe future behaviour of the system and optimize the processcontrol performance over a prediction horizon [1] The MPCmodel can be used easily at different levels of the processcontrol structure such as multiple input and multiple outputdynamics systems that offer attractive solutions for regulationand tracking problems while guaranteeing stability [2] Sincethe end of the 1980s robust MPCs which explicitly takeaccount of model uncertainties constraints and faults in thecontrol actuator plant-model mismatch and disturbancesor noise have been studied for more practical applications[3] Numerous research studies have investigated the stabilityproperties of MPC for systems without uncertainty and foruncertain linear systems Several robust predictive formula-tions utilize the minndashmax approach where the manipulated
input trajectory is computed by solving an optimizationproblem that requires minimizing the objective function andsatisfying the input and state constraints over all possible real-izations of the uncertainty These have been applied mainlyto impulse response models and state-space approaches bysolving a finite horizon open-loop control optimization prob-lem [4] Due to its advantagesMPChas been implemented inseveral applications for automotive and other transportationactive safety systems such as active steering active tractionactive braking and active differentials or suspension systemsin order to coordinate and improve vehicle handling stabilityand ride comfort [5ndash8]
A vehicle capable of handling many things at oncewithout any human intervention can be termed as an auto-nomous vehicle Basic functions of autonomous brakingand steering however are insufficient the vehicle has tohave the ability to sense its surrounding plus being able todetermine desired location which can be achieved using avariety of instruments and pieces of equipment such as radarglobal positioning system on-board camera for vision and
Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2016 Article ID 6752671 18 pageshttpdxdoiorg10115520166752671
2 Journal of Control Science and Engineering
an independent operating unit All sensory data are thencomputed for obstacle identification avoidance is then exe-cuted using advanced control system In this paper we limitautonomous vehicles to ground vehicles which are increas-ingly being studied by several researchers from academiaindustry and military Several control methods are beingused including fuzzy logic [9] hybrid control [10] H-infinitycontrol [11] and linear quadratic regulators [12] The bestcomparative studies on model predictive control strategiesfor autonomous guidance vehicles can be found in Park et al[13] Yoon et al [14] and Falcone et al [15] where a nonlineardynamics model of a vehicle is used for the controller designof an active front steeringmanoeuvre in a double-lane changescenario Keviczky et al [16] studied the effect of side wind viaan active front steeringmanoeuvre for an autonomous vehicleusing nonlinear MPC Since nonlinear MPC poses consid-erable challenges due to its real-time implementation algo-rithmmany researchers opted for linearMPC for their study
The assumption of this study are as follows vehicle trajec-tory is known as per Borrelli et alrsquos [17] study and the cross-wind effect will be assigned as a step response since thecrosswind effect disturbance on the vehicle can be approx-imated as a constant In this paper we extend the conceptof MPC for application in vehicle manoeuvring problemswhere a trajectory optimization is solved at each time stepThe trajectorywas designed to be a double-lane changing sce-nario The autonomous vehicle manoeuvring was simulatedin a variety of conditions low (10msminus1) and high (30msminus1)forward speed and low (icy) and high (concrete wet) roadfriction surface with the intention of following predeter-mined trajectory as close as possible while maintainingvehicle stability The control inputs were front steering anglerear steering angle and direct yawmoment control while theoutput controls were the yaw angle and lateral vehicle posi-tion Two different controllers were compared to evaluatethe performance of the six-degree-of-freedom (6DoF) vehi-cle model a simple (2DoF) controller without vehicle rolldynamic and a complex (3DoF) bicycle model controllerwith roll dynamics Other performance indexes evaluatedwere efficacy and robustness of the MPC for the autonomousvehicle in terms of control and stability
Themain objective of this paper is to evaluate the robust-ness of model predictive control approaches for autonomouspath-following car dynamics control with simple and com-plex models of the vehicle while rejecting the effects of windgusts to the systemThere are many studies to be consulted instabilization of the vehicle examples are two-wheel steering(2WS) [15 18] four-wheel steering (4WS) [19] and 2WS withdirect yaw moment control (DYC) [5 20 21] with differentcontrol strategies The first contribution of this paper is theeffect of vehicle roll dynamics motion consideration to thesystem whereas most previous papers only focused on a2DoF vehicle model (lateral and yaw motion) Moreoverbased on the authorsrsquo knowledge there is no comparativestudy for autonomous path-following vehicle control usingMPC techniques for three control signalmanoeuvres (includ-ing the three control signals here) the robustness evaluationdiscussed here becomes the main novelty of this paperFurthermore we would like to investigate the effectiveness
of model predictive control manoeuvres in the case of thecrosswind effect to the system in a variety of forward speedsand road adhesions The rest of the paper is organized asfollows Section 2 describes the linear vehicle model lineartire model and wind model Next a linear model predictivecontrol algorithm concept is explained in Section 3 Section 4examines and describes the effectiveness of the linear MPCfor a car dynamics system on a two-lane change scenarioLastly conclusions and future works are given in Section 5
2 Vehicle Model
21 Bicycle Model Figure 1 shows a well-known vehiclemodel which is a single-track model based on the simplifica-tion that the right and leftwheels are lumped in a single wheelat the front and rear axles The simplified vehicle model usedin this paper illustrates the motion movement and dynamicsconcerning the car vehicle subject to the longitudinal lateralyaw roll and rotational dynamics of the front and rear wheelmotion represented as 6DoF The longitudinal lateral andyaw dynamics effects are shown in Figure 1(b) as a top viewof the car vehicle and in Figure 1(a) the roll dynamics effectis explained with the nomenclature for a front view of thevehicle In this paper the nonlinear vehicle was linearizedbased on the assumption that sin 120579 = 0 and cos 120579 = 1 for bothsteering angles the vehicle side slip angle and the roll angleWe also assumed that thewhole vehiclemass is sprung whichis ignoring the suspension and wheel weights for unsprungmass This linearized model still behaves and represents theactual nonlinear vehicle model at certain operating points ofthe regionThe details of themathematical calculation for thevehiclemodel are presented inChen andPeng [22] for furtherknowledge
In this paper we use the following nomenclature 119865119909
and 119865119910
represent the longitudinal and lateral tire forcesrespectively 119865
119911
is the normal tire load 119865119908
and 119872119908
rep-resent the force and moment exerted by the side windrespectively 119909 119910 and 119911 correspond to the coordinates of thebody frame of a car position 119897 is the vehicle wheelbase 119879
119887
isthe wheel torque V
119909
and V119910
are the longitudinal and lateralwheels velocities respectively 120575
119891
and 120575119903
express the steeringangle of the front and rear wheels respectively 119872
119891
and119872119903
represent the reaction yaw moment appearing at the frontand rear wheels respectively119872
119911
is the total reaction of yawmoment of the wheels produced by the DYC 120583 acts as thetrack friction coefficient 120595 is the yawheading angle and isthe yaw rate120573 is the vehicle side slip angle and120601 and are theroll and roll rate angles respectively The variable at the frontand rear wheels is denoted by lower subscripts (sdot)
119891
and (sdot)119903
The motion of longitudinal lateral yaw roll and rota-
tional dynamics of the front and rear wheels using a 6DoFsystem based on a linear vehicle model is described throughthe following differential equations [23]
119898 = 119898 + 2119865119909119891
+ 2119865119909119903
minus 2ℎ119898 (1a)
119898 = minus119898 + 2119865119910119891
+ 2119865119910119903
+ ℎ119898 + 119865119908
(1b)
119868119911119911
= 2119897119891
119865119910119891
minus 2119897119903
119865119910119891
+ 119868119909119911
+ 119872119891
+119872119903
+119872119908
(1c)
Journal of Control Science and Engineering 3
z
h
y
120601
mg
Mw
b120601
Fzl Fzr
k120601
Fw
(a) Front view
x
Fyf
120575fxf
120572f f
Mf
yflf
x120573
FwMw
lr
120575r
Fyr
Mr
Fxr
120595y
(b) Top view
Figure 1 Simplified bicycle model
(119868119909119909
+ 119898ℎ2
) = 119898119892ℎ120601 minus 2119896120601
120601 minus 2119887120601
+ 119898ℎ ( + ) + 119868119909119911
(1d)
119869119887
119908119894
= minus119903119908
119865119909119894
minus 119879119887119894
minus 119887119908
120596119894
119894 = (119891 119903) (1e)
The motion equations for the vehicle in an inertial frame oron 119910-119909 axis under the assumption of a small yaw angle maybe given as
= cos120595 minus sin120595 = V119909
minus 120595
= sin120595 + cos120595 = V119909
120595 +
(2)
22 Tire Model Tire dynamics must be considered for thevehicle model since the tires are the only contact that thevehicle has with the road surface Besides the forces ofgravity and aerodynamics all the forces are induced by thetires which may affect the vehicle chassis handling andstabilityTheir complexity and nonlinear behaviourmust alsoreflect the operating condition of the controller over theirwhole region throughout varied manoeuvring range in long-itudinal lateral and roll directionsThemost frequently usedexisting nonlinear tire models of application and structureare determined through key parameters and analytical con-siderations based on tire measurement data They are calledsemiempirical tire models or Pacejka tire model [24]
On the other hand the nonlinear tire model can alsobe described linearly for some parts of the operating con-ditions therefore in this paper we will consider the linear
Fx
x
Fy
120572
Fz
Tb
Figure 2 Tire model
tire model Figure 2 illustrates the terminology used fordescribing the longitudinal lateral and vertical tire forcesand their orientation Thus the linear tire model is validunder constant normal load forces on the tires constantlongitudinal slip of tires and neglected aerodynamic dragThe relationship between the longitudinal force verticalforce and the longitudinal tire slip ratio is given by thefollowing equations
119865119909
= 120583119909
(119904) 119865119911
(3a)
119904 =119903119908
120596119908
V119909
minus 1 (3b)
4 Journal of Control Science and Engineering
119865119911119891
=119897119903
119898119892
2119897
119865119911119903
=119897119891
119898119892
2119897
(3c)
where 119903119908
is the tirersquos geometric radius120596119908
is the angular veloc-ity of the tires V
119909
in (3b) is the tirersquos forward velocity 119865119909
isproportional to the normal force 119865
119911
and 120583119909
(119904) represents thelongitudinal wheel slip friction coefficient of road adhesionsand is a function of slip ratio 119904
The lateral forces on the front and rear tires are char-acterized and modelled by a linear function with the frontand rear tire slip angles 120572
119891
and 120572119903
denoted by 119865119910119891
and119865119910119903
respectively The linear tire model yields the followingexpression for the front and rear tire forces
119865119910119891
= 119862119891
120572119891
119865119910119903
= 119862119903
120572119903
(4)
where 119862119891
and 119862119903
are the tire cornering stiffness parametersfor the front and rear tires respectively The slip angles forthe front and rear wheels with a small angle assumption aregiven such that
120572119891
=V119910
+ 119897119891
V119909
minus 120575119891
120572119903
=V119910
minus 119897119903
V119909
minus 120575119903
(5)
The details of mathematical equations for a linear tire modelcan be read through in [25] Assumptions and approxi-mations presented in this paper are representative of thenonlinear tire model at certain regional points this providesa good balance between capturing the important features andregions of laterally unstable behaviour [26]
23 Wind Model The effect of wind on the stability of thevehicle is an important and primary safety consideration ofthis paper A strong gust of wind from the inward or outwardside will generate force and torque that could be large enoughfor a vehicle to roll over or go outside of the laneThe resultingwind pressure forces and torques acting on the rigid body ingeneral can be represented by three axes longitudinal lateraland vertical A general expression of force and torque is givenby the following equations
119865119908
=119862119865
120588119860V2119903
2
119872119908
=119862119872
120588119860119871V2119903
2
(6)
In the above equations 119860 is the vehicle area 119871 is the vehiclelength 119871 = 119897
119891
+ 119897119903
120588 is the density of air V119903
is the relativewind speed and 119862
119865
and 119862119872
are the force and momentnondimensional coefficients respectively The wind speedor crosswind is represented by V
119908
as shown in Figure 3 In
w
r
x
120595
Figure 3 Vehicle wind speed in crosswind situation
general crosswind can be at various angles but for simplicityin this paper we will assume the crosswind to be at a 90-degree angle andwill focus on the windrsquos impact on the lateralforces and yaw torques [27]
The vehicle motion in ((1a) (1b) (1c) (1d) (1e))ndash(6) canbe described by the following compact differential equation
= 119860119909 + 1198611
119906 + 1198612
119908 + 1198613
119903
119910 = 119862119909 + 119863119906
(7)
with 119909 isin 119877119909 119906 isin 119877119906119908 isin 119877119908 119903 isin 119877119903 and 119910 isin 119877119910 representing
the state vectors control input vectors crosswind effects as adisturbance vector desired trajectory vectors and measuredoutput vectors respectively We define
119909 = [V119910
119884 120595 120601 120596119891
120596119903
]119879
119906 = [120575119891
120575119903
119872119911
]119879
119908 = [119865119908
119872119908
]119879
119903 = [119884des 120595des]119879
ℎ (120585) = [119884 120595]119879
(8)
The state vectors represent the states for lateral velocity lateralposition yaw rate yaw angle roll rate roll angle and frontand rear wheels angular speed respectively and the outputvectors represent lateral position and yaw angle
3 Linear Model Predictive Control
MPC has been used as a highly effective and very successfulcontrol scheme with the simple design framework provingto be a practical controller able to combine multivariablesystems and utilize online optimization process control Thestability guarantees of existing predictive control approachesfor nonlinear systems with uncertainty however remaincontingent upon the assumption of the initial feasibility ofthe optimization and the set of initial conditions startingfrom where feasibility of the optimization problem andtherefore stability of the closed-loop system is guaranteed isnot explicitly characterized [28] For example the occurrenceof faults in control actuators adds another layer of complexityto the problem of controller design for nonlinear uncertainsystems Note that the stability guarantees provided by acontroller may no longer hold in the presence of faults inthe control actuators that prevent the implementation ofthe control action prescribed by the control law and these
Journal of Control Science and Engineering 5
Modelpredictivecontrol
Linear tiremodel
Referencetracking
Dist
urba
nce Linear vehicle
model
x
Yref120575f
120575r
Fx Fy
Y
Mz120595ref
120595
Figure 4 Predictive control structure
faults can have substantial negative ramifications owing to theinterconnected nature of processes [29]
In order to implement MPC with a receding horizoncontrol strategy the following strategy method is adopted
(1) A dynamics process model is used to predict thebehaviour of the plant and future plant outputs ateach instant 119896 based on both previous and the latestobservations of the systemrsquos inputs and outputs
(2) The control signal inputs are calculated by minimiz-ing the error of tracking between the predicted outputand desired trajectory signal to keep the process asclose as possible to following the trajectory whileconsidering the objective functions and constraints
(3) Only the first control signal is implemented on theplant whilst others are rejected in anticipation of thenext sampling instant where the future output will beknown
(4) Step (1) is repeated with updated values and all ordersare updated
In this paper we use the linear model of predictive controlThe hierarchical control structure is adopted in MPC asshown in Figure 4 Figure 4 illustrates the control structurefor 2WS that uses front steering only 120575
119891
for 4WS that usesfront and rear steering 120575
119891
and 120575119903
and for 2WS with DYCthat produces external yaw moment (119872
119911
) at both front andrear wheels as a control input to the system These systemsare used to control the vehicle in order to follow a givenreference trajectory They include the vehicle speed desiredreference trajectory modelled predictive control and linearvehicle model with linear tire model
Using the equations from the vehicle and tire model asexplained and defined in (7) the basic equations of linearvehicle motion can be given as follows
119898V119910
=1
119868119909119909
V119909
[minus (120583119862119903
+ 119862119891
) 119869119909119902
V119910
+ (120583 (119862119903
119897119903
minus 119862119891
119897119891
) 119869119909119902
minus 119868119909119909
119898V2119909
) ]
minus1
119868119909119909
[(ℎ119887120601
) minus ℎ (119898119892ℎ minus 119896120601
) 120601 minus 120583119862119891
119869119909119902
120575119891
+ 120583119862119903
119869119909119902
120575119903
]
(9a)
119868119911119911
=1
V119909
[120583 (119862119903
119897119903
minus 119862119891
119897119891
) V119910
minus 120583 (119862119891
119897119891
2
+ 119862119903
119897119903
2
) ] + 120583119862119891
119897119891
120575119891
minus 120583119862119903
119897119903
120575119903
+119872119911
(9b)
119868119909119909
=ℎ
V119909
[120583 (119862119903
119897119903
minus 119862119891
119897119891
) minus 120583 (119862119891
+ 119862119903
) V119910
]
minus 119887120601
+ (119898119892ℎ minus 119896120601
) 120601 + 120583119862119891
ℎ120575119891
+ 120583119862119903
ℎ120575119903
(9c)
where 119869119909119902
= 119868119909119909
+ 119898ℎ2 A DYC that produces the reaction
moment occurring at the front and rear wheels due to thesteering angle effect (as an external yaw moments (119872
119911
)) canbe approximated with the following equations
119872119891
asymp 2119897119891
119862119891
119872119911
119872119903
asymp 2119897119903
119862119903
119872119911
(10)
We define front steering angle rear steering angle and directyaw moment control as the inputs to the system Thus thevehicle motion can be represented in a given discrete state-space structure as follows
119909119897
(119896 + 1 | 119896) = 119860119897
119909119897
(119896 | 119896) + 119861119897
119906119897
(119896 | 119896)
+ 119861119903
119903119897
(119896 | 119896)
119910119897
(119896 | 119896) = 119862119897
119909119897
(119896 | 119896) + 119863119897
119906119897
(119896 | 119896)
(11)
where 119909119897
(119896 | 119896) is the state vector at time step 119896 and 119909119897
(119896 +
1 | 119896) is the state vector at time step 119896 + 1 with 119909119897
(119896 | 119896) isin
119877119909119897(119896|119896) 119906
119897
(119896 | 119896) isin 119877119906119897(119896|119896) 119903
119897
(119896 | 119896) isin 119877119903119897(119896|119896) and 119910
119897
(119896 | 119896) isin
119877119910119897(119896|119896) representing the state vectors control input vectors
reference vectors and measured output vectors respectivelyWe define
119909119897
(119896) = [V119910
119884 120595 120601]119879
119903119897
(119896) = [119884des 120595des]119879
119910119897
(119896) = [119884 120595]119879
(12)
For 2WS 4WS and 2WS with DYC the control signal to thesystems with the same tuning control parameters is given asfollows
119906119897
(119896) = [120575119891
]
119906119897
(119896) = [120575119891
120575119903
]119879
119906119897
(119896) = [120575119891
119872119911
]119879
(13)
We formulate the optimization of the predictive controlsystem which takes the constraints imposed on the input andinput rate respectively as given in the following form
minus119906119897
le 119906119897
(119896) le +119906119897
minusΔ119906119897
le Δ119906119897
(119896) le +Δ119906119897
(14)
6 Journal of Control Science and Engineering
One approach to maintain closed-loop stability would be touse all available control actuators (13) so that even if one ofthe control actuators fails the rest can maintain closed-loopstability which the reliable control approaches The use ofredundant control actuators however incurs possibly pre-ventable operation and maintenance costs These economicconsiderations dictate the use of only as many control loopsas is required at a time To achieve tolerance with respect tofaults control-loop reconfiguration can be carried out in theevent of failure of the primary control configuration based onthe assumption of a linear system
Based on the linear vehicle model in (11) and by definingthe outputs in (12) we consider the following cost functionsto the system
119869 (119909119897
(119896) 119880119897119896
) =
119867119901
sum
119894=1
1003817100381710038171003817119897 (119896 + 119894 | 119896) minus 119903 (119896 + 119894 | 119896)10038171003817100381710038172
119876119894
+
119867119888minus1
sum
119894=0
1003817100381710038171003817Δ119897 (119896 + 119894 | 119896)10038171003817100381710038172
119877119894
(15)
In (15) the first summation of the given cost functionreflects the reduction of trajectory tracking errors among thepredicted outputs
119897
(119896 + 119894 | 119896) (119894 = 0 119867119901
minus 1) and theoutput reference signals 119903(119896+119894 | 119896) (119894 = 0 119867
119901
minus1) and thesecond summation reflects on the penalization of the controlsignal effort Δ
119897
(119896 + 119894 | 119896) (119894 = 0 119867119888
minus 1) of the steeringcontrol manoeuvre The aim of the predictive control systemis to determine the optimal control input vector Δ
119897
(119896+ 119894 | 119896)
so that the error function between the reference signal andthe predicted output is reduced
The difference in the steering angle Δ119897
(119896 + 119894 | 119896) is givenin the case that the cost function is at a minimum valueThe weight matrices 119876(119894) and 119877(119894) are semipositive definiteand positive definite respectively and can be tuned for anydesired closed-loop performance 119876(119894) is defined as the statetracking weight since the error
119897
(119896 + 119894 | 119896) minus 119903(119896 + 119894 | 119896) canbecome as small as possible by increasing 119876(119894) In a similarfashion 119877(119894) represents the input tracking weight and thevariation of the input is decreased to slow down the responseof the system by increasing 119877(119894) The predictive and controlhorizon are typically considered to be 119867
119901
ge 119867119888
while thecontrol signal is considered constant for 119867
119888
le 119894 le 119867119901
Matrices 119876(119894) and 119877(119894) represent the matrix weight of theirappropriate dimensions for outputs and inputs respectivelygiving the state feedback control law as follows
119906119897
(119896 119910119897
(119896)) = 119906119897
(119896 minus 1) + Δ119906119897
(119910119897
(119896)) (16)
An optimal input was computed for the following time steprather than being calculated for the immediate time step byaddressing convex optimization problems during each timestep After that the first input is applied on the plant in (11)and the others are rejected which yield the optimal controlsequence as (16) The optimization problem in (15) is iteratedwith the new value at a time 119896 + 1 via new state 119909(119896 + 1) andall orders are then updated
The required direct yaw moment control119872119911
is obtainedfrom the variation between the two sides of the vehicle torque
as in (1e) In this research the braking torque is only usedbased on the yaw rate in otherwords it is only activated in thecase that the vehiclemoves toward instability or in emergencymanoeuvres since it impacts the longitudinal motion whilethe rear steering angle is assumed for the total manoeuvre tobe in control or in standard drivingmanoeuvresWe take intoaccount three control inputs that is front and rear steeringangles as well as differential braking at rear right and left tirebut only one control input is used at a time [30]
4 Simulation Results
41 Scenario Description The linear model predictive con-troller explained andpresentedwas implemented for a vehiclepath following a double-lane change scenario with a cross-wind effect present throughout the simulation A double-lanechange manoeuvre approximates as an emergency manoeu-vre case and generally demonstrates the agility and capabili-ties of the vehicle in lateral and roll dynamics During such amanoeuvre an understeering or oversteering or even a roll-over situation may occur In this scenario the vehicle is con-sidered travelling horizontally or straightforward followingthe path with a constant velocity of 10msminus1 and 30msminus1without braking or accelerating The drag force and torquegiven by (6) in the initial driving conditions are assumed toact in the direction of the path at time 119905 = 1 sec with a windvelocity of V
119908
= 10msminus1 The forces and torques arising fromthis sidewise acting wind gust are assumed to be persistentand are applied as step functions throughout the simulationtime as shown in Figure 5 Table 1 shows the sport utilitypassenger car model parameters and their definitions used inthis paper based on Solmaz et al [31]
For stability analysis we simulated the system with onlyone inputmdashin particular the front steering angle 120575
119891
or 2WSWe set the vehicle velocity to be constant at V
119909
= 15msminus1the road adhesion coefficient 120583 = 1 and wind velocity V
119908
=
10msminus1 Figure 6(a) shows a stability analysis based on theroot locus of the open-loop system in (7) showing matrix 119860without the crosswind effect (119861
2
= 0) and references signal(1198613
= 0) Those settings contribute to a stable region for bothoutputs (vehicle side slip and yaw rate responses) Matrix 119860is given in (8) with input signal being front steering angle(1198611
= 120575119891
) Since the root locus command inMATLAB is onlyworking on the single-input single-output therefore we eval-uate separately the output response with single input signal
Figure 6(b) illustrates the effect of these factors a cross-wind effect (119861
2
= 0) references signal (1198613
= 0) and con-trol signal (119861
1
= 0) with the same matrix 119860 as in (8) thatyields unstable responses for both vehicle side slip and yawrate We can also notice that the poles are different for bothcases due to 119861
2
matrices that are based on the crosswindeffect Furthermore Figure 6(b) shows that within a windvelocity V
119908
= 10msminus1 the system is stable with the appropri-ate matrices of 119860 119861
1
and 1198612
Figure 6(a) also illustrates thatpaired 119860 and 119861
1
are controllable based on the model para-meters listed in Table 1
In this paper the predictive controller was implementedby minimizing vehicle deviation from the target path to
Journal of Control Science and Engineering 7
0 5 10 150
5
10
Time (s)
Cros
swin
d sp
eed
w(m
s)
Time (s)0 5 10 15
0
100
200
300
400
Win
d fo
rce
Fw
(N)
Time (s)0 5 10 15
0
minus500
minus1000
Win
d to
rque
Mw
(Nm
)
Figure 5 Force and torque for crosswind speed of 10msminus1
Table 1 Car vehicle parameters
Parameter ValueCar mass [kg] 1300Inertia around the roll 119909-axis [kgm2] 370Inertia around 119911-axis [kgm2] 216756Distance of front and rear wheels from centre of gravity (CoG) [m] 145 107Distance between the vehicle CoG and the assumed roll axis [m] 045Equivalent suspension roll damping coefficient [Nms] 4200Equivalent suspension roll stiffness coefficient [Nm] 40000Linear approximation of tire stiffness for front and rear tire [Nrad] 133000 121000Gravitational constant [ms2] 98
achieve the main aim which is to follow the desired or refer-ence trajectory as close as possible The controllers are com-pared against each other for 2DoF and 3DoF bicycle modelswhich excludes vehicle roll dynamics at various speedsspecifically low (10msminus1) and high (30msminus1) and also underwet concrete (120583 = 07) and icy (120583 = 01) road surfaces
Since our study focused on an autonomous vehicle it isreasonable to assume that the controllers will adapt to theroad or environment such as in the case of road adhesionsadverse weather visual cues traffic and overall relevantdriving conditions Therefore both controllers (2DoF and3DoF) were designed and implemented with the parametersand conditions given in Tables 2 3 and 4 for different vehicleforward speeds and road adhesion coefficients The path-following tracking error is a measurement of how closelythe output responses follow the reference trajectory whichis a measure of the deviation from the benchmark In this
paper we use the standard deviation of the root-mean squareformula to calculate the tracking error given by the followingequation
119890119905
= radicVar (119910119900
minus 119910119903
) = radic1
(119899 minus 1)sum (119910119900
minus 119910119903
)2
(17)
where 119890119905
is the tracking error 119899 represents the number oftime periods and 119910
119900
and 119910119903
express the measured output andreference
42 Results and Discussions Prior to proposed simulationswe carried out standard vehicle manoeuvres tests to validateour model in the open-loop simulation scenario Standardtests available are the J-turn fishhook and double-lanechange [32] which are representative of on-roadmanoeuvreswithout tripping (caused by external force from road curb or
8 Journal of Control Science and Engineering
0 50
0
10
20
30
40
50Root locus
Imag
inar
y ax
is (sminus1)
minus10
minus20
minus30
minus40
minus50minus50minus100minus150minus200minus250minus300
Real axis (sminus1)
120595998400
0 20
0
2
4
6
8
10Root locus
Imag
inar
y ax
is (sminus1)
minus2
minus4
minus6
minus8
minus10minus20minus40minus60minus80minus100minus120minus140
Real axis (sminus1)
120573
(a) Without crosswind effect
0 20 40 60
0
2
4
6
8
10Root locus
Imag
inar
y ax
is (sminus1)
Real axis (sminus1)
minus2
minus4
minus6
minus8
minus10
120595998400
minus20minus40minus60minus800 20
0
5
10
15
20
25Root locus
Imag
inar
y ax
is (sminus1)
minus5
minus10
minus15
minus20
minus25
Real axis (sminus1)
120573
minus20minus40minus60minus80minus100minus120minus140
(b) With crosswind effect
Figure 6 Open-loop system in (7)
collisionwith other vehicles)We performed only the double-lane change and roll rate feedback fishhook test for vehiclevalidation purposes in the open-loop simulation as shown inFigures 7 and 8 The vehicle speed was set at 30msminus1 whichis suitable for both tests with a road surface coefficient of 01As illustrated in Figures 7 and 8 the vehicle response in terms
of yaw rate roll rate and lateral acceleration has been provento be validated thus being suitable for other simulations
For easier comparison between controllersrsquo perfor-mances in achieving vehicle stabilization when subjectedto various conditions controllers tuning parameters fromTables 3 and 4 were selected as suitable The weighting gains
Journal of Control Science and Engineering 9
0 5 10 15
0
05
1
Time (s)
minus05
minus10 5 10 15
0
01
02
Time (s)
minus01
minus02
minus030 5 10 15
0
05
1
Time (s)
minus1
minus05
Tire
slip
angl
e120572f
(rad
)
0 5 10 15
0
05
1
Time (s)
minus05Yaw
angl
e res
pons
e120595
(rad
)
0 5 10 15
0
5
10
Time (s)
minus5
minus10
Fron
t ste
erin
g an
gle120575f
(deg
)
5 10 15
0
01
02
Time (s)
minus01
minus02
minus03Roll
angl
e res
pons
e120601
(rad
)
Time (s)0 5 10 15
0
10
20
30
40
50
Late
ral p
ositi
onY
(m)
0 5 10 15
0
2000
4000
Time (s)
minus2000
minus4000
Late
ral f
orce
Fyf
(N)
0 5 10 15Time (s)
10
5
0
minus5
minus10
Late
ral a
ccel
erat
ion
a y(m
s2)
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 7 Vehicle manoeuvre test of single-lane change at 30msminus1 with 120583 = 01
Table 2 Model predictive control parameters
Parameter ValuePredictive horizon 20Control horizon 9Sampling time [s] 005Constraints on max and min steering angles [deg] plusmn20Constraints on max and min changes in steering angles [degs] plusmn10Constraints on max and min changes in moment torque [Nmrad] plusmn2000Constraints on max and min moment torque [Nmrads] plusmn1500Simulation time [s] 15
for controllers input and output were selected using trial-and-error method where selection criteria were based on thebest response output for both input and output for weightinggains
The first simulation revolves around various road surfacefriction coefficients from wet concrete (120583 = 07) to icy sur-face condition (120583 = 01) with a constant forward speed of10msminus1 The MPC weighting tuning parameters are listedin Table 3 We evaluated the controllerrsquos robustness for theoutput responses by comparing the performance of 2WS4WS and 2WS with DYC manoeuvres at a forward speed of
10msminus1 as shown in Figures 9ndash11 The controller trackingerrors based on (17) for lateral position and yaw angleresponses are summarized in Table 5 From Figure 9 thereare not many differences between both controllers underthe conditions of lowest speed and high road adhesioncoefficient both controllers yielded perfect response whenfollowing a given trajectory while maintaining vehicle stabil-ity and rejecting external crosswind effect It can be seen thatfor 4WS and 2WS with DYC manoeuvres the rear steeringangle and the direct yaw moment are almost unused sincethe front steering can provide sufficient control ability This
10 Journal of Control Science and Engineering
0 5 10 15
0
50
100
Time (s)
minus50
minus100
Late
ral a
ccel
erat
ion
a y(m
s2)
Time (s)0 5 10 15
0
1000
2000
3000
minus1000Late
ral p
ositi
onY
(m)
0 5 10 15
0
2
4
Time (s)
minus2
minus4
times104
Late
ral f
orce
Fyf
(N)
0 5 10 15
0
200
400
Time (s)
minus200
minus400
Fron
t ste
erin
g an
gle120575f
(deg
)
0 5 10 15
0
50
100
150
200
Time (s)
minus50
Yaw
angl
e res
pons
e120595
(rad
)
5 10 15
0
1
2
3
Time (s)
minus1
minus2Roll
angl
e res
pons
e120601
(rad
)
0 5 10 15
0
1
2
Time (s)
minus1
minus2Vehi
cle sl
ip an
gle120573
(rad
)
0 5 10 15
0
20
40
60
Time (s)
minus200 5 10 15
0
5
10
Time (s)
minus5
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 8 Vehicle manoeuvre test of roll rate feedback at 30msminus1 with 120583 = 01
Table 3 Model predictive control weighting matrices parameters for V119909
= 10msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 10msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 195 11987622
= 05 11987611
= 525 11987622
= 05
V119909
= 10msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 152 11987622
= 05 11987611
= 395 11987622
= 05
4WS
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 195 11987622
= 05 11987611
= 45 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 153 11987622
= 05 11987611
= 35 11987622
= 05
2WS + DYC
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 193 11987622
= 05 11987611
= 37 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 152 11987622
= 05 11987611
= 35 11987622
= 05
Journal of Control Science and Engineering 11
Time (s)0 5 10 15
0
05
1
4WS
minus05
minus1
minus15
times10minus5Re
ar st
eerin
g an
gle120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
1
2
3
Time (s)
2WS + DYC
minus1
minus2
minus3
times10minus3
Dire
ct y
aw m
omen
tM
z(N
m)
0 5 10 15
0
02
0 5 10 15
0
02
04
Time (s)Time (s)
Ref2WS
4WS2WS + DYC
Ref2WS
4WS2WS + DYC
minus02
minus04
minus02
minus04
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
Yaw
angl
e120595
(rad
)
Yaw
rate
120595998400
(rad
sminus1)
Figure 9 2DoF controller at 10msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
may be partly due to the fact that the rear steering and DYCwere not used in lower speed manoeuvres
Moreover when the road surface friction was set to icythe output responses from both controllers and all controlmanoeuvre techniques did not perfectly follow the desiredtrajectory as shown in Figures 10 and 11 2WS with DYCyielded the best performance output compared to the othersespecially in yaw rate response 4WSwas the next best follow-ing 2WS where the lateral position and yaw angle lookalmost identical It seems that the rear steering and direct yawmoment controls were used in conjunction with front steer-ing for vehicle stabilization along the trajectory Howeverfrom Table 5 the controller with 3DoF gives slightly bettertracking error performance manoeuvres compared to con-troller 2DoF due to the fact that roll dynamics will not havemuch influence during low speed manoeuvres Another factis that at low vehicle speed all control inputs (front steeringangle rear steering angle and direct yawmoment control) areunder constraints
Next we tested the vehicle at various road friction coef-ficients again however this time the vehicle was tested undera high forward speed of 30msminus1 The same MPC design wasused of which the parameter controls are listed in Table 4Wecompared the simulation results for 2WS 4WS and 2WSwithDYC control manoeuvres for both controllers and presentthe comparison in Figures 12ndash15The simulation results show
that for 2DoF controller at 30msminus1 and on wet concrete(120583 = 07) all control manoeuvres give slightly similar out-puts performances Better responses were achieved in lateralpositioning and although the tracking performance of yawangle and yaw rate deteriorated the system still allowedthe vehicle to track and follow the trajectory successfullyrejecting the effects of wind gust that impact the vehicle
As tabulated in Table 6 4WS and 2WS with DYC showslightly better tracking performance compared to 2WS onlyHowever in the case of the 3DoF controller it can be clearlyseen that 2WS with DYC and 4WS control manoeuvresoffered amuch higher performing response especially in yawangle and yaw rate response than 2WS This is shown inFigure 13 In this scenario the rear steering and direct yawmoment control was fully utilized to control and enhancevehicle stability It is therefore very important to consider rolldynamics in order to enhance vehicle stability and to followthe path trajectory From Figures 12 and 13 there are somestrong frequency oscillations in front of some responsesthat is front lateral force vehicle side slip angle and lateralacceleration trace based on authorsrsquo knowledge come fromthe numerical calculation or associated glitches and initialcondition of the system
Furthermore when we simulated vehicle manoeuvre forhigh speed (30msminus1) and icy road condition (120583 = 01) for2DoF controller it can be seen that all manoeuvres control
12 Journal of Control Science and Engineering
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
Time (s)0 5 10 15
0
01
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
Time (s)
2WS + DYC
minus01
minus02Dire
ct y
aw m
omen
tM
z(N
m)
Figure 10 2DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 4 Model predictive control weighting matrices parameters for V119909
= 30msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 30msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 565 11987622
= 05 11987611
= 355 11987622
= 05
V119909
= 30msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 0083 11987622
= 0042 11987611
= 003 11987622
= 045
4WS
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 155 11987622
= 25 11987611
= 135 11987622
= 35
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0005 11987622
= 0001 11987611
= 004 11987622
= 05
2WS + DYC
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 505 11987622
= 05 11987611
= 35 11987622
= 05
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0092 11987622
= 004 11987611
= 05 11987622
= 25
simulation the tracking responses become unstable andimpossible to control especially under the crosswind effectas shown in Figure 14 The most probable cause is the rolldynamic motion neglect in the 2DoF controller design whenthe vehicle itself was modelled by including roll dynamicfactor It can be said that the inclusion of roll dynamic factor
is important for vehicle manoeuvre in terms of stability andcontrollability at high speed and icy road condition Sincehigh speed coupled with icy road condition will render theequation to become highly nonlinear it is impossible fora linear tire model to react positively since the handlingpropertiesmay be significantly different from those generated
Journal of Control Science and Engineering 13
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
01
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
002
004
Time (s)
2WS + DYC
minus002
minus004
minus006Dire
ct y
aw m
omen
tM
z(N
m)
Figure 11 3DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 5 Path following tracking errors with road friction surface 120583 = 07
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00637 00085 00623 000744WS 00548 00054 00526 00051
2WS + DYC 00542 00051 00527 00051
302WS 00639 05348 00616 052154WS 00558 05239 00528 00023
2WS + DYC 00549 05226 00524 00254
by the linear tire model These results show that linear tiremodel is only suitable for analyzing a stable vehicle behaviourunder the assumption of small steering and acceleration
Next for the 3DoF controller simulation 2WS with DYCmanoeuvres provides a much better tracking response incomparison to 4WS particularly in the case of lateral outputOn the contrary 4WS manoeuvre demonstrates a much bet-ter tracking response in the yaw angle and yaw rate responsesas tabulated in Table 6 With appropriate weight tuning gainthe rear steering angle and direct yaw moment control arefully optimized in order to become stable along the given tra-jectory However for the 2WS manoeuvre the vehicleresponses become unstable despite several instances wherethe weighting tuning gains are adjusted for output and inputgains With the inclusion of another input control to the con-troller design we can enhance the vehicle responses for both
2WS with DYC and 4WS control manoeuvres show Thismeans that for 2WS controller design it may have to usemore than just front steering to stabilize the vehicle under theeffects of wind gusts Figure 15 shows that rear steering angleand direct yaw moment are fully utilized in order to stabilizethe vehicle for 2WS with DYC and 4WSmanoeuvres control
In contrast to the 2DoF controller as shown in Figure 14in the 3DoF controllers the 2WS control manoeuvre canstill be used to track a given trajectory despite not perfectlyfollowing the trajectory in a satisfactory manner especiallyunder crosswind effects In Figure 15 we can see some oscil-lations in few traces responses at the end of the responsesBased on the authorsrsquo knowledge these oscillationsmay comefrom the vehicle model and some initial conditions of thesystem with imperfect controllers tuning weighting gains Inthese conditions neither 2WS nor 4WS control manoeuvres
14 Journal of Control Science and Engineering
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
Time (s)
2WS4WS2WS + DYC
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
001
002
Time (s)
2WS4WS2WS + DYC
minus001Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
1
2
Time (s)
2WS4WS2WS + DYC
minus1
minus2
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
001
002
003
Time (s)
2WS4WS2WS + DYC
minus001
minus002
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
00005
0001
4WS
Time (s)
minus00005
minus0001
minus00015
Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
001
002
Time (s)
2WS + DYC
minus001
minus002
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 12 2DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
for lateral response and neither 2WS nor 2WS with DYCcontrol manoeuvres for yaw angle or yaw rate responseperformed well causing an increase in vehicle instabilityincreased vibration and tracking responses deteriorationWewill next focus on how to enhance the controller in order tostabilize vehicle manoeuvrability and handling stability
We can therefore conclude that for 4WS the rear wheelswere helping the car to steer by improving the vehiclehandling at high speed while decreasing the turning radiusat low speed as shown by the control signal in Figures 12 and13 Meanwhile for 2WS with DYC active front steering wasused in low speed manoeuvres for lateral acceleration whileinclusion of DYC was adopted for high lateral accelerationwhen the tires were saturated and could not produce enoughlateral force for vehicle control and stability as intendedIn 2WS vehicles the rear set of wheels do not play an
active role in controlling the steering From Table 5 it canbe seen that among the three controllers 4WS gave the bestperformance by reducing the tracking error for lateral andyaw angle responses when compared with 2WS with DYCand 2WS only Tables 5 and 6 tabulated the MPC robustnessand the tracking errors in (17) for all types of manoeuvre at10msminus1 and 30msminus1 vehicle speeds and under various roadsurfaces Furthermore all controlmanoeuvre signals for bothcontrollers were within the input constraints We would liketo highlight that in MPC approaches although there was anadvantage in multivariable systems in this study there was atrade-off between theweighting tuning parameter to focus oneither lateral position or yaw angle trajectory In this paper wehave chosen lateral position as the weighting tuning priorityso we may get a better response on yaw angle and yaw rateresponse by sacrificing lateral position precision
Journal of Control Science and Engineering 15
0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
Time (s)
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
2WS4WS2WS + DYC
Time (s)
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02
Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
5
10
2WS4WS2WS + DYC
Time (s)
minus5
minus10Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
03
4WS
Time (s)
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
2WS + DYC
Time (s)
minus005
minus01
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 13 3DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
Table 6 Path following tracking errors with road friction surface 120583 = 01
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00874 00115 00841 001044WS 00838 00109 00832 00094
2WS + DYC 00838 00110 00835 00083
302WS Uncontrolled Uncontrolled 21352 052644WS Uncontrolled Uncontrolled 08824 02982
2WS + DYC Uncontrolled Uncontrolled 09964 04580
16 Journal of Control Science and Engineering
0 10 20 30
02468
10
Time (s)
Ref2WS
4WS2WS + DYC
minus2
minus4Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0020406
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
minus06
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15
Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
0020406
Time (s)
2WS4WS2WS + DYC
minus02
minus04
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
02
04
06
Time (s)
4WS
minus02
Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
005
01
Time (s)
2WS + DYC
minus005
Dire
ct y
aw m
omen
tM
z(N
m)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15Fron
t lat
eral
forc
eFyf
(Nm
) times104
Figure 14 2DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
5 Conclusion
This paper presents the robustness of model predictive con-trol for an autonomous ground vehicle on a path-followingcontrol with consideration of wind gust effects subjectedto the variation of forward speeds and road surfaces condi-tion in a double-lane change scenario The predictive con-trollers were designed based on two- (lateral-yaw model)and three- (lateral-roll model) degree-of-freedom vehiclemodels The only difference between the models is the rolldynamics factor which is taken into consideration andboth controllers were implemented into a six-degree-of-free-dom vehicle model Based on linear vehicle and tire modelapproximations of a known trajectory we evaluated the effectand impact of roll dynamics at low and high forward vehicle
speeds and at various road frictions (wet concrete to icyroads) in order to follow a desired trajectory as close as pos-sible while rejecting the crosswind effects and maintainingvehicle stability We also evaluated and compared the effi-ciency of the different manoeuvres via two-wheel steeringfour-wheel steering and two-wheel steering with direct yawmoment control
The simulation results showed that with the inclusion ofroll dynamics in the linear vehicle model the vehicle stabilityand adherence to trajectory was considerably improved forthe case of high speed manoeuvres on a higher road surfacefriction coefficient (120583 = 07) and improved even more for thecase of a lower road surface coefficient (120583 = 01) Further-more the obtained results showed and proved that the rearwheels and direct yaw moment are beneficial to help steering
Journal of Control Science and Engineering 17
0 10 20 30
0
5
Time (s)
Ref2WS
4WS2WS + DYC
minus10
minus5
minus15Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0
025
05
Time (s)
Ref2WS
4WS2WS + DYC
minus025
minus05
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
0
500
1000
Time (s)
2WS4WS2WS + DYC
minus500
minus1000
minus1500Fron
t lat
eral
forc
eFyf
(Nm
)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
01
02
03
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS + DYC
minus2
minus4
times10minus4
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 15 3DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
the car by improving the vehicle handling at high speeddecreasing the turning radius at low speeds and reducing thetracking error for lateral position and yaw angle response
The simulation also demonstrated that model predictivecontrol has an ability to reject the side wind effect as adisturbance to the system which is particularly effective at10msminus1 for all cases except for the 2DoF controller duringsimulation of higher speeds on lower road surface adhesionHowever both controllers were not able to provide a betterresponse when the vehicle performed at a high forwardspeedwith a low road adhesion coefficient particularly when2DoF controller was simulated all results showed unstablevehicle response Thus for this scenario we are currentlyinvestigating new approaches with a real-time implementable
control scheme Further improvement for future works oncontrol methodology will also be considered to minimizetracking error and to enhance vehicle stability
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors would like to thank Professor Zidong Wangand Professor Manfred Ploechl for their comments andadvice Special thanks are also extended to Dr A Zahran MKhudzari for his help preparing this paper
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
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Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
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DistributedSensor Networks
International Journal of
2 Journal of Control Science and Engineering
an independent operating unit All sensory data are thencomputed for obstacle identification avoidance is then exe-cuted using advanced control system In this paper we limitautonomous vehicles to ground vehicles which are increas-ingly being studied by several researchers from academiaindustry and military Several control methods are beingused including fuzzy logic [9] hybrid control [10] H-infinitycontrol [11] and linear quadratic regulators [12] The bestcomparative studies on model predictive control strategiesfor autonomous guidance vehicles can be found in Park et al[13] Yoon et al [14] and Falcone et al [15] where a nonlineardynamics model of a vehicle is used for the controller designof an active front steeringmanoeuvre in a double-lane changescenario Keviczky et al [16] studied the effect of side wind viaan active front steeringmanoeuvre for an autonomous vehicleusing nonlinear MPC Since nonlinear MPC poses consid-erable challenges due to its real-time implementation algo-rithmmany researchers opted for linearMPC for their study
The assumption of this study are as follows vehicle trajec-tory is known as per Borrelli et alrsquos [17] study and the cross-wind effect will be assigned as a step response since thecrosswind effect disturbance on the vehicle can be approx-imated as a constant In this paper we extend the conceptof MPC for application in vehicle manoeuvring problemswhere a trajectory optimization is solved at each time stepThe trajectorywas designed to be a double-lane changing sce-nario The autonomous vehicle manoeuvring was simulatedin a variety of conditions low (10msminus1) and high (30msminus1)forward speed and low (icy) and high (concrete wet) roadfriction surface with the intention of following predeter-mined trajectory as close as possible while maintainingvehicle stability The control inputs were front steering anglerear steering angle and direct yawmoment control while theoutput controls were the yaw angle and lateral vehicle posi-tion Two different controllers were compared to evaluatethe performance of the six-degree-of-freedom (6DoF) vehi-cle model a simple (2DoF) controller without vehicle rolldynamic and a complex (3DoF) bicycle model controllerwith roll dynamics Other performance indexes evaluatedwere efficacy and robustness of the MPC for the autonomousvehicle in terms of control and stability
Themain objective of this paper is to evaluate the robust-ness of model predictive control approaches for autonomouspath-following car dynamics control with simple and com-plex models of the vehicle while rejecting the effects of windgusts to the systemThere are many studies to be consulted instabilization of the vehicle examples are two-wheel steering(2WS) [15 18] four-wheel steering (4WS) [19] and 2WS withdirect yaw moment control (DYC) [5 20 21] with differentcontrol strategies The first contribution of this paper is theeffect of vehicle roll dynamics motion consideration to thesystem whereas most previous papers only focused on a2DoF vehicle model (lateral and yaw motion) Moreoverbased on the authorsrsquo knowledge there is no comparativestudy for autonomous path-following vehicle control usingMPC techniques for three control signalmanoeuvres (includ-ing the three control signals here) the robustness evaluationdiscussed here becomes the main novelty of this paperFurthermore we would like to investigate the effectiveness
of model predictive control manoeuvres in the case of thecrosswind effect to the system in a variety of forward speedsand road adhesions The rest of the paper is organized asfollows Section 2 describes the linear vehicle model lineartire model and wind model Next a linear model predictivecontrol algorithm concept is explained in Section 3 Section 4examines and describes the effectiveness of the linear MPCfor a car dynamics system on a two-lane change scenarioLastly conclusions and future works are given in Section 5
2 Vehicle Model
21 Bicycle Model Figure 1 shows a well-known vehiclemodel which is a single-track model based on the simplifica-tion that the right and leftwheels are lumped in a single wheelat the front and rear axles The simplified vehicle model usedin this paper illustrates the motion movement and dynamicsconcerning the car vehicle subject to the longitudinal lateralyaw roll and rotational dynamics of the front and rear wheelmotion represented as 6DoF The longitudinal lateral andyaw dynamics effects are shown in Figure 1(b) as a top viewof the car vehicle and in Figure 1(a) the roll dynamics effectis explained with the nomenclature for a front view of thevehicle In this paper the nonlinear vehicle was linearizedbased on the assumption that sin 120579 = 0 and cos 120579 = 1 for bothsteering angles the vehicle side slip angle and the roll angleWe also assumed that thewhole vehiclemass is sprung whichis ignoring the suspension and wheel weights for unsprungmass This linearized model still behaves and represents theactual nonlinear vehicle model at certain operating points ofthe regionThe details of themathematical calculation for thevehiclemodel are presented inChen andPeng [22] for furtherknowledge
In this paper we use the following nomenclature 119865119909
and 119865119910
represent the longitudinal and lateral tire forcesrespectively 119865
119911
is the normal tire load 119865119908
and 119872119908
rep-resent the force and moment exerted by the side windrespectively 119909 119910 and 119911 correspond to the coordinates of thebody frame of a car position 119897 is the vehicle wheelbase 119879
119887
isthe wheel torque V
119909
and V119910
are the longitudinal and lateralwheels velocities respectively 120575
119891
and 120575119903
express the steeringangle of the front and rear wheels respectively 119872
119891
and119872119903
represent the reaction yaw moment appearing at the frontand rear wheels respectively119872
119911
is the total reaction of yawmoment of the wheels produced by the DYC 120583 acts as thetrack friction coefficient 120595 is the yawheading angle and isthe yaw rate120573 is the vehicle side slip angle and120601 and are theroll and roll rate angles respectively The variable at the frontand rear wheels is denoted by lower subscripts (sdot)
119891
and (sdot)119903
The motion of longitudinal lateral yaw roll and rota-
tional dynamics of the front and rear wheels using a 6DoFsystem based on a linear vehicle model is described throughthe following differential equations [23]
119898 = 119898 + 2119865119909119891
+ 2119865119909119903
minus 2ℎ119898 (1a)
119898 = minus119898 + 2119865119910119891
+ 2119865119910119903
+ ℎ119898 + 119865119908
(1b)
119868119911119911
= 2119897119891
119865119910119891
minus 2119897119903
119865119910119891
+ 119868119909119911
+ 119872119891
+119872119903
+119872119908
(1c)
Journal of Control Science and Engineering 3
z
h
y
120601
mg
Mw
b120601
Fzl Fzr
k120601
Fw
(a) Front view
x
Fyf
120575fxf
120572f f
Mf
yflf
x120573
FwMw
lr
120575r
Fyr
Mr
Fxr
120595y
(b) Top view
Figure 1 Simplified bicycle model
(119868119909119909
+ 119898ℎ2
) = 119898119892ℎ120601 minus 2119896120601
120601 minus 2119887120601
+ 119898ℎ ( + ) + 119868119909119911
(1d)
119869119887
119908119894
= minus119903119908
119865119909119894
minus 119879119887119894
minus 119887119908
120596119894
119894 = (119891 119903) (1e)
The motion equations for the vehicle in an inertial frame oron 119910-119909 axis under the assumption of a small yaw angle maybe given as
= cos120595 minus sin120595 = V119909
minus 120595
= sin120595 + cos120595 = V119909
120595 +
(2)
22 Tire Model Tire dynamics must be considered for thevehicle model since the tires are the only contact that thevehicle has with the road surface Besides the forces ofgravity and aerodynamics all the forces are induced by thetires which may affect the vehicle chassis handling andstabilityTheir complexity and nonlinear behaviourmust alsoreflect the operating condition of the controller over theirwhole region throughout varied manoeuvring range in long-itudinal lateral and roll directionsThemost frequently usedexisting nonlinear tire models of application and structureare determined through key parameters and analytical con-siderations based on tire measurement data They are calledsemiempirical tire models or Pacejka tire model [24]
On the other hand the nonlinear tire model can alsobe described linearly for some parts of the operating con-ditions therefore in this paper we will consider the linear
Fx
x
Fy
120572
Fz
Tb
Figure 2 Tire model
tire model Figure 2 illustrates the terminology used fordescribing the longitudinal lateral and vertical tire forcesand their orientation Thus the linear tire model is validunder constant normal load forces on the tires constantlongitudinal slip of tires and neglected aerodynamic dragThe relationship between the longitudinal force verticalforce and the longitudinal tire slip ratio is given by thefollowing equations
119865119909
= 120583119909
(119904) 119865119911
(3a)
119904 =119903119908
120596119908
V119909
minus 1 (3b)
4 Journal of Control Science and Engineering
119865119911119891
=119897119903
119898119892
2119897
119865119911119903
=119897119891
119898119892
2119897
(3c)
where 119903119908
is the tirersquos geometric radius120596119908
is the angular veloc-ity of the tires V
119909
in (3b) is the tirersquos forward velocity 119865119909
isproportional to the normal force 119865
119911
and 120583119909
(119904) represents thelongitudinal wheel slip friction coefficient of road adhesionsand is a function of slip ratio 119904
The lateral forces on the front and rear tires are char-acterized and modelled by a linear function with the frontand rear tire slip angles 120572
119891
and 120572119903
denoted by 119865119910119891
and119865119910119903
respectively The linear tire model yields the followingexpression for the front and rear tire forces
119865119910119891
= 119862119891
120572119891
119865119910119903
= 119862119903
120572119903
(4)
where 119862119891
and 119862119903
are the tire cornering stiffness parametersfor the front and rear tires respectively The slip angles forthe front and rear wheels with a small angle assumption aregiven such that
120572119891
=V119910
+ 119897119891
V119909
minus 120575119891
120572119903
=V119910
minus 119897119903
V119909
minus 120575119903
(5)
The details of mathematical equations for a linear tire modelcan be read through in [25] Assumptions and approxi-mations presented in this paper are representative of thenonlinear tire model at certain regional points this providesa good balance between capturing the important features andregions of laterally unstable behaviour [26]
23 Wind Model The effect of wind on the stability of thevehicle is an important and primary safety consideration ofthis paper A strong gust of wind from the inward or outwardside will generate force and torque that could be large enoughfor a vehicle to roll over or go outside of the laneThe resultingwind pressure forces and torques acting on the rigid body ingeneral can be represented by three axes longitudinal lateraland vertical A general expression of force and torque is givenby the following equations
119865119908
=119862119865
120588119860V2119903
2
119872119908
=119862119872
120588119860119871V2119903
2
(6)
In the above equations 119860 is the vehicle area 119871 is the vehiclelength 119871 = 119897
119891
+ 119897119903
120588 is the density of air V119903
is the relativewind speed and 119862
119865
and 119862119872
are the force and momentnondimensional coefficients respectively The wind speedor crosswind is represented by V
119908
as shown in Figure 3 In
w
r
x
120595
Figure 3 Vehicle wind speed in crosswind situation
general crosswind can be at various angles but for simplicityin this paper we will assume the crosswind to be at a 90-degree angle andwill focus on the windrsquos impact on the lateralforces and yaw torques [27]
The vehicle motion in ((1a) (1b) (1c) (1d) (1e))ndash(6) canbe described by the following compact differential equation
= 119860119909 + 1198611
119906 + 1198612
119908 + 1198613
119903
119910 = 119862119909 + 119863119906
(7)
with 119909 isin 119877119909 119906 isin 119877119906119908 isin 119877119908 119903 isin 119877119903 and 119910 isin 119877119910 representing
the state vectors control input vectors crosswind effects as adisturbance vector desired trajectory vectors and measuredoutput vectors respectively We define
119909 = [V119910
119884 120595 120601 120596119891
120596119903
]119879
119906 = [120575119891
120575119903
119872119911
]119879
119908 = [119865119908
119872119908
]119879
119903 = [119884des 120595des]119879
ℎ (120585) = [119884 120595]119879
(8)
The state vectors represent the states for lateral velocity lateralposition yaw rate yaw angle roll rate roll angle and frontand rear wheels angular speed respectively and the outputvectors represent lateral position and yaw angle
3 Linear Model Predictive Control
MPC has been used as a highly effective and very successfulcontrol scheme with the simple design framework provingto be a practical controller able to combine multivariablesystems and utilize online optimization process control Thestability guarantees of existing predictive control approachesfor nonlinear systems with uncertainty however remaincontingent upon the assumption of the initial feasibility ofthe optimization and the set of initial conditions startingfrom where feasibility of the optimization problem andtherefore stability of the closed-loop system is guaranteed isnot explicitly characterized [28] For example the occurrenceof faults in control actuators adds another layer of complexityto the problem of controller design for nonlinear uncertainsystems Note that the stability guarantees provided by acontroller may no longer hold in the presence of faults inthe control actuators that prevent the implementation ofthe control action prescribed by the control law and these
Journal of Control Science and Engineering 5
Modelpredictivecontrol
Linear tiremodel
Referencetracking
Dist
urba
nce Linear vehicle
model
x
Yref120575f
120575r
Fx Fy
Y
Mz120595ref
120595
Figure 4 Predictive control structure
faults can have substantial negative ramifications owing to theinterconnected nature of processes [29]
In order to implement MPC with a receding horizoncontrol strategy the following strategy method is adopted
(1) A dynamics process model is used to predict thebehaviour of the plant and future plant outputs ateach instant 119896 based on both previous and the latestobservations of the systemrsquos inputs and outputs
(2) The control signal inputs are calculated by minimiz-ing the error of tracking between the predicted outputand desired trajectory signal to keep the process asclose as possible to following the trajectory whileconsidering the objective functions and constraints
(3) Only the first control signal is implemented on theplant whilst others are rejected in anticipation of thenext sampling instant where the future output will beknown
(4) Step (1) is repeated with updated values and all ordersare updated
In this paper we use the linear model of predictive controlThe hierarchical control structure is adopted in MPC asshown in Figure 4 Figure 4 illustrates the control structurefor 2WS that uses front steering only 120575
119891
for 4WS that usesfront and rear steering 120575
119891
and 120575119903
and for 2WS with DYCthat produces external yaw moment (119872
119911
) at both front andrear wheels as a control input to the system These systemsare used to control the vehicle in order to follow a givenreference trajectory They include the vehicle speed desiredreference trajectory modelled predictive control and linearvehicle model with linear tire model
Using the equations from the vehicle and tire model asexplained and defined in (7) the basic equations of linearvehicle motion can be given as follows
119898V119910
=1
119868119909119909
V119909
[minus (120583119862119903
+ 119862119891
) 119869119909119902
V119910
+ (120583 (119862119903
119897119903
minus 119862119891
119897119891
) 119869119909119902
minus 119868119909119909
119898V2119909
) ]
minus1
119868119909119909
[(ℎ119887120601
) minus ℎ (119898119892ℎ minus 119896120601
) 120601 minus 120583119862119891
119869119909119902
120575119891
+ 120583119862119903
119869119909119902
120575119903
]
(9a)
119868119911119911
=1
V119909
[120583 (119862119903
119897119903
minus 119862119891
119897119891
) V119910
minus 120583 (119862119891
119897119891
2
+ 119862119903
119897119903
2
) ] + 120583119862119891
119897119891
120575119891
minus 120583119862119903
119897119903
120575119903
+119872119911
(9b)
119868119909119909
=ℎ
V119909
[120583 (119862119903
119897119903
minus 119862119891
119897119891
) minus 120583 (119862119891
+ 119862119903
) V119910
]
minus 119887120601
+ (119898119892ℎ minus 119896120601
) 120601 + 120583119862119891
ℎ120575119891
+ 120583119862119903
ℎ120575119903
(9c)
where 119869119909119902
= 119868119909119909
+ 119898ℎ2 A DYC that produces the reaction
moment occurring at the front and rear wheels due to thesteering angle effect (as an external yaw moments (119872
119911
)) canbe approximated with the following equations
119872119891
asymp 2119897119891
119862119891
119872119911
119872119903
asymp 2119897119903
119862119903
119872119911
(10)
We define front steering angle rear steering angle and directyaw moment control as the inputs to the system Thus thevehicle motion can be represented in a given discrete state-space structure as follows
119909119897
(119896 + 1 | 119896) = 119860119897
119909119897
(119896 | 119896) + 119861119897
119906119897
(119896 | 119896)
+ 119861119903
119903119897
(119896 | 119896)
119910119897
(119896 | 119896) = 119862119897
119909119897
(119896 | 119896) + 119863119897
119906119897
(119896 | 119896)
(11)
where 119909119897
(119896 | 119896) is the state vector at time step 119896 and 119909119897
(119896 +
1 | 119896) is the state vector at time step 119896 + 1 with 119909119897
(119896 | 119896) isin
119877119909119897(119896|119896) 119906
119897
(119896 | 119896) isin 119877119906119897(119896|119896) 119903
119897
(119896 | 119896) isin 119877119903119897(119896|119896) and 119910
119897
(119896 | 119896) isin
119877119910119897(119896|119896) representing the state vectors control input vectors
reference vectors and measured output vectors respectivelyWe define
119909119897
(119896) = [V119910
119884 120595 120601]119879
119903119897
(119896) = [119884des 120595des]119879
119910119897
(119896) = [119884 120595]119879
(12)
For 2WS 4WS and 2WS with DYC the control signal to thesystems with the same tuning control parameters is given asfollows
119906119897
(119896) = [120575119891
]
119906119897
(119896) = [120575119891
120575119903
]119879
119906119897
(119896) = [120575119891
119872119911
]119879
(13)
We formulate the optimization of the predictive controlsystem which takes the constraints imposed on the input andinput rate respectively as given in the following form
minus119906119897
le 119906119897
(119896) le +119906119897
minusΔ119906119897
le Δ119906119897
(119896) le +Δ119906119897
(14)
6 Journal of Control Science and Engineering
One approach to maintain closed-loop stability would be touse all available control actuators (13) so that even if one ofthe control actuators fails the rest can maintain closed-loopstability which the reliable control approaches The use ofredundant control actuators however incurs possibly pre-ventable operation and maintenance costs These economicconsiderations dictate the use of only as many control loopsas is required at a time To achieve tolerance with respect tofaults control-loop reconfiguration can be carried out in theevent of failure of the primary control configuration based onthe assumption of a linear system
Based on the linear vehicle model in (11) and by definingthe outputs in (12) we consider the following cost functionsto the system
119869 (119909119897
(119896) 119880119897119896
) =
119867119901
sum
119894=1
1003817100381710038171003817119897 (119896 + 119894 | 119896) minus 119903 (119896 + 119894 | 119896)10038171003817100381710038172
119876119894
+
119867119888minus1
sum
119894=0
1003817100381710038171003817Δ119897 (119896 + 119894 | 119896)10038171003817100381710038172
119877119894
(15)
In (15) the first summation of the given cost functionreflects the reduction of trajectory tracking errors among thepredicted outputs
119897
(119896 + 119894 | 119896) (119894 = 0 119867119901
minus 1) and theoutput reference signals 119903(119896+119894 | 119896) (119894 = 0 119867
119901
minus1) and thesecond summation reflects on the penalization of the controlsignal effort Δ
119897
(119896 + 119894 | 119896) (119894 = 0 119867119888
minus 1) of the steeringcontrol manoeuvre The aim of the predictive control systemis to determine the optimal control input vector Δ
119897
(119896+ 119894 | 119896)
so that the error function between the reference signal andthe predicted output is reduced
The difference in the steering angle Δ119897
(119896 + 119894 | 119896) is givenin the case that the cost function is at a minimum valueThe weight matrices 119876(119894) and 119877(119894) are semipositive definiteand positive definite respectively and can be tuned for anydesired closed-loop performance 119876(119894) is defined as the statetracking weight since the error
119897
(119896 + 119894 | 119896) minus 119903(119896 + 119894 | 119896) canbecome as small as possible by increasing 119876(119894) In a similarfashion 119877(119894) represents the input tracking weight and thevariation of the input is decreased to slow down the responseof the system by increasing 119877(119894) The predictive and controlhorizon are typically considered to be 119867
119901
ge 119867119888
while thecontrol signal is considered constant for 119867
119888
le 119894 le 119867119901
Matrices 119876(119894) and 119877(119894) represent the matrix weight of theirappropriate dimensions for outputs and inputs respectivelygiving the state feedback control law as follows
119906119897
(119896 119910119897
(119896)) = 119906119897
(119896 minus 1) + Δ119906119897
(119910119897
(119896)) (16)
An optimal input was computed for the following time steprather than being calculated for the immediate time step byaddressing convex optimization problems during each timestep After that the first input is applied on the plant in (11)and the others are rejected which yield the optimal controlsequence as (16) The optimization problem in (15) is iteratedwith the new value at a time 119896 + 1 via new state 119909(119896 + 1) andall orders are then updated
The required direct yaw moment control119872119911
is obtainedfrom the variation between the two sides of the vehicle torque
as in (1e) In this research the braking torque is only usedbased on the yaw rate in otherwords it is only activated in thecase that the vehiclemoves toward instability or in emergencymanoeuvres since it impacts the longitudinal motion whilethe rear steering angle is assumed for the total manoeuvre tobe in control or in standard drivingmanoeuvresWe take intoaccount three control inputs that is front and rear steeringangles as well as differential braking at rear right and left tirebut only one control input is used at a time [30]
4 Simulation Results
41 Scenario Description The linear model predictive con-troller explained andpresentedwas implemented for a vehiclepath following a double-lane change scenario with a cross-wind effect present throughout the simulation A double-lanechange manoeuvre approximates as an emergency manoeu-vre case and generally demonstrates the agility and capabili-ties of the vehicle in lateral and roll dynamics During such amanoeuvre an understeering or oversteering or even a roll-over situation may occur In this scenario the vehicle is con-sidered travelling horizontally or straightforward followingthe path with a constant velocity of 10msminus1 and 30msminus1without braking or accelerating The drag force and torquegiven by (6) in the initial driving conditions are assumed toact in the direction of the path at time 119905 = 1 sec with a windvelocity of V
119908
= 10msminus1 The forces and torques arising fromthis sidewise acting wind gust are assumed to be persistentand are applied as step functions throughout the simulationtime as shown in Figure 5 Table 1 shows the sport utilitypassenger car model parameters and their definitions used inthis paper based on Solmaz et al [31]
For stability analysis we simulated the system with onlyone inputmdashin particular the front steering angle 120575
119891
or 2WSWe set the vehicle velocity to be constant at V
119909
= 15msminus1the road adhesion coefficient 120583 = 1 and wind velocity V
119908
=
10msminus1 Figure 6(a) shows a stability analysis based on theroot locus of the open-loop system in (7) showing matrix 119860without the crosswind effect (119861
2
= 0) and references signal(1198613
= 0) Those settings contribute to a stable region for bothoutputs (vehicle side slip and yaw rate responses) Matrix 119860is given in (8) with input signal being front steering angle(1198611
= 120575119891
) Since the root locus command inMATLAB is onlyworking on the single-input single-output therefore we eval-uate separately the output response with single input signal
Figure 6(b) illustrates the effect of these factors a cross-wind effect (119861
2
= 0) references signal (1198613
= 0) and con-trol signal (119861
1
= 0) with the same matrix 119860 as in (8) thatyields unstable responses for both vehicle side slip and yawrate We can also notice that the poles are different for bothcases due to 119861
2
matrices that are based on the crosswindeffect Furthermore Figure 6(b) shows that within a windvelocity V
119908
= 10msminus1 the system is stable with the appropri-ate matrices of 119860 119861
1
and 1198612
Figure 6(a) also illustrates thatpaired 119860 and 119861
1
are controllable based on the model para-meters listed in Table 1
In this paper the predictive controller was implementedby minimizing vehicle deviation from the target path to
Journal of Control Science and Engineering 7
0 5 10 150
5
10
Time (s)
Cros
swin
d sp
eed
w(m
s)
Time (s)0 5 10 15
0
100
200
300
400
Win
d fo
rce
Fw
(N)
Time (s)0 5 10 15
0
minus500
minus1000
Win
d to
rque
Mw
(Nm
)
Figure 5 Force and torque for crosswind speed of 10msminus1
Table 1 Car vehicle parameters
Parameter ValueCar mass [kg] 1300Inertia around the roll 119909-axis [kgm2] 370Inertia around 119911-axis [kgm2] 216756Distance of front and rear wheels from centre of gravity (CoG) [m] 145 107Distance between the vehicle CoG and the assumed roll axis [m] 045Equivalent suspension roll damping coefficient [Nms] 4200Equivalent suspension roll stiffness coefficient [Nm] 40000Linear approximation of tire stiffness for front and rear tire [Nrad] 133000 121000Gravitational constant [ms2] 98
achieve the main aim which is to follow the desired or refer-ence trajectory as close as possible The controllers are com-pared against each other for 2DoF and 3DoF bicycle modelswhich excludes vehicle roll dynamics at various speedsspecifically low (10msminus1) and high (30msminus1) and also underwet concrete (120583 = 07) and icy (120583 = 01) road surfaces
Since our study focused on an autonomous vehicle it isreasonable to assume that the controllers will adapt to theroad or environment such as in the case of road adhesionsadverse weather visual cues traffic and overall relevantdriving conditions Therefore both controllers (2DoF and3DoF) were designed and implemented with the parametersand conditions given in Tables 2 3 and 4 for different vehicleforward speeds and road adhesion coefficients The path-following tracking error is a measurement of how closelythe output responses follow the reference trajectory whichis a measure of the deviation from the benchmark In this
paper we use the standard deviation of the root-mean squareformula to calculate the tracking error given by the followingequation
119890119905
= radicVar (119910119900
minus 119910119903
) = radic1
(119899 minus 1)sum (119910119900
minus 119910119903
)2
(17)
where 119890119905
is the tracking error 119899 represents the number oftime periods and 119910
119900
and 119910119903
express the measured output andreference
42 Results and Discussions Prior to proposed simulationswe carried out standard vehicle manoeuvres tests to validateour model in the open-loop simulation scenario Standardtests available are the J-turn fishhook and double-lanechange [32] which are representative of on-roadmanoeuvreswithout tripping (caused by external force from road curb or
8 Journal of Control Science and Engineering
0 50
0
10
20
30
40
50Root locus
Imag
inar
y ax
is (sminus1)
minus10
minus20
minus30
minus40
minus50minus50minus100minus150minus200minus250minus300
Real axis (sminus1)
120595998400
0 20
0
2
4
6
8
10Root locus
Imag
inar
y ax
is (sminus1)
minus2
minus4
minus6
minus8
minus10minus20minus40minus60minus80minus100minus120minus140
Real axis (sminus1)
120573
(a) Without crosswind effect
0 20 40 60
0
2
4
6
8
10Root locus
Imag
inar
y ax
is (sminus1)
Real axis (sminus1)
minus2
minus4
minus6
minus8
minus10
120595998400
minus20minus40minus60minus800 20
0
5
10
15
20
25Root locus
Imag
inar
y ax
is (sminus1)
minus5
minus10
minus15
minus20
minus25
Real axis (sminus1)
120573
minus20minus40minus60minus80minus100minus120minus140
(b) With crosswind effect
Figure 6 Open-loop system in (7)
collisionwith other vehicles)We performed only the double-lane change and roll rate feedback fishhook test for vehiclevalidation purposes in the open-loop simulation as shown inFigures 7 and 8 The vehicle speed was set at 30msminus1 whichis suitable for both tests with a road surface coefficient of 01As illustrated in Figures 7 and 8 the vehicle response in terms
of yaw rate roll rate and lateral acceleration has been provento be validated thus being suitable for other simulations
For easier comparison between controllersrsquo perfor-mances in achieving vehicle stabilization when subjectedto various conditions controllers tuning parameters fromTables 3 and 4 were selected as suitable The weighting gains
Journal of Control Science and Engineering 9
0 5 10 15
0
05
1
Time (s)
minus05
minus10 5 10 15
0
01
02
Time (s)
minus01
minus02
minus030 5 10 15
0
05
1
Time (s)
minus1
minus05
Tire
slip
angl
e120572f
(rad
)
0 5 10 15
0
05
1
Time (s)
minus05Yaw
angl
e res
pons
e120595
(rad
)
0 5 10 15
0
5
10
Time (s)
minus5
minus10
Fron
t ste
erin
g an
gle120575f
(deg
)
5 10 15
0
01
02
Time (s)
minus01
minus02
minus03Roll
angl
e res
pons
e120601
(rad
)
Time (s)0 5 10 15
0
10
20
30
40
50
Late
ral p
ositi
onY
(m)
0 5 10 15
0
2000
4000
Time (s)
minus2000
minus4000
Late
ral f
orce
Fyf
(N)
0 5 10 15Time (s)
10
5
0
minus5
minus10
Late
ral a
ccel
erat
ion
a y(m
s2)
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 7 Vehicle manoeuvre test of single-lane change at 30msminus1 with 120583 = 01
Table 2 Model predictive control parameters
Parameter ValuePredictive horizon 20Control horizon 9Sampling time [s] 005Constraints on max and min steering angles [deg] plusmn20Constraints on max and min changes in steering angles [degs] plusmn10Constraints on max and min changes in moment torque [Nmrad] plusmn2000Constraints on max and min moment torque [Nmrads] plusmn1500Simulation time [s] 15
for controllers input and output were selected using trial-and-error method where selection criteria were based on thebest response output for both input and output for weightinggains
The first simulation revolves around various road surfacefriction coefficients from wet concrete (120583 = 07) to icy sur-face condition (120583 = 01) with a constant forward speed of10msminus1 The MPC weighting tuning parameters are listedin Table 3 We evaluated the controllerrsquos robustness for theoutput responses by comparing the performance of 2WS4WS and 2WS with DYC manoeuvres at a forward speed of
10msminus1 as shown in Figures 9ndash11 The controller trackingerrors based on (17) for lateral position and yaw angleresponses are summarized in Table 5 From Figure 9 thereare not many differences between both controllers underthe conditions of lowest speed and high road adhesioncoefficient both controllers yielded perfect response whenfollowing a given trajectory while maintaining vehicle stabil-ity and rejecting external crosswind effect It can be seen thatfor 4WS and 2WS with DYC manoeuvres the rear steeringangle and the direct yaw moment are almost unused sincethe front steering can provide sufficient control ability This
10 Journal of Control Science and Engineering
0 5 10 15
0
50
100
Time (s)
minus50
minus100
Late
ral a
ccel
erat
ion
a y(m
s2)
Time (s)0 5 10 15
0
1000
2000
3000
minus1000Late
ral p
ositi
onY
(m)
0 5 10 15
0
2
4
Time (s)
minus2
minus4
times104
Late
ral f
orce
Fyf
(N)
0 5 10 15
0
200
400
Time (s)
minus200
minus400
Fron
t ste
erin
g an
gle120575f
(deg
)
0 5 10 15
0
50
100
150
200
Time (s)
minus50
Yaw
angl
e res
pons
e120595
(rad
)
5 10 15
0
1
2
3
Time (s)
minus1
minus2Roll
angl
e res
pons
e120601
(rad
)
0 5 10 15
0
1
2
Time (s)
minus1
minus2Vehi
cle sl
ip an
gle120573
(rad
)
0 5 10 15
0
20
40
60
Time (s)
minus200 5 10 15
0
5
10
Time (s)
minus5
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 8 Vehicle manoeuvre test of roll rate feedback at 30msminus1 with 120583 = 01
Table 3 Model predictive control weighting matrices parameters for V119909
= 10msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 10msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 195 11987622
= 05 11987611
= 525 11987622
= 05
V119909
= 10msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 152 11987622
= 05 11987611
= 395 11987622
= 05
4WS
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 195 11987622
= 05 11987611
= 45 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 153 11987622
= 05 11987611
= 35 11987622
= 05
2WS + DYC
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 193 11987622
= 05 11987611
= 37 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 152 11987622
= 05 11987611
= 35 11987622
= 05
Journal of Control Science and Engineering 11
Time (s)0 5 10 15
0
05
1
4WS
minus05
minus1
minus15
times10minus5Re
ar st
eerin
g an
gle120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
1
2
3
Time (s)
2WS + DYC
minus1
minus2
minus3
times10minus3
Dire
ct y
aw m
omen
tM
z(N
m)
0 5 10 15
0
02
0 5 10 15
0
02
04
Time (s)Time (s)
Ref2WS
4WS2WS + DYC
Ref2WS
4WS2WS + DYC
minus02
minus04
minus02
minus04
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
Yaw
angl
e120595
(rad
)
Yaw
rate
120595998400
(rad
sminus1)
Figure 9 2DoF controller at 10msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
may be partly due to the fact that the rear steering and DYCwere not used in lower speed manoeuvres
Moreover when the road surface friction was set to icythe output responses from both controllers and all controlmanoeuvre techniques did not perfectly follow the desiredtrajectory as shown in Figures 10 and 11 2WS with DYCyielded the best performance output compared to the othersespecially in yaw rate response 4WSwas the next best follow-ing 2WS where the lateral position and yaw angle lookalmost identical It seems that the rear steering and direct yawmoment controls were used in conjunction with front steer-ing for vehicle stabilization along the trajectory Howeverfrom Table 5 the controller with 3DoF gives slightly bettertracking error performance manoeuvres compared to con-troller 2DoF due to the fact that roll dynamics will not havemuch influence during low speed manoeuvres Another factis that at low vehicle speed all control inputs (front steeringangle rear steering angle and direct yawmoment control) areunder constraints
Next we tested the vehicle at various road friction coef-ficients again however this time the vehicle was tested undera high forward speed of 30msminus1 The same MPC design wasused of which the parameter controls are listed in Table 4Wecompared the simulation results for 2WS 4WS and 2WSwithDYC control manoeuvres for both controllers and presentthe comparison in Figures 12ndash15The simulation results show
that for 2DoF controller at 30msminus1 and on wet concrete(120583 = 07) all control manoeuvres give slightly similar out-puts performances Better responses were achieved in lateralpositioning and although the tracking performance of yawangle and yaw rate deteriorated the system still allowedthe vehicle to track and follow the trajectory successfullyrejecting the effects of wind gust that impact the vehicle
As tabulated in Table 6 4WS and 2WS with DYC showslightly better tracking performance compared to 2WS onlyHowever in the case of the 3DoF controller it can be clearlyseen that 2WS with DYC and 4WS control manoeuvresoffered amuch higher performing response especially in yawangle and yaw rate response than 2WS This is shown inFigure 13 In this scenario the rear steering and direct yawmoment control was fully utilized to control and enhancevehicle stability It is therefore very important to consider rolldynamics in order to enhance vehicle stability and to followthe path trajectory From Figures 12 and 13 there are somestrong frequency oscillations in front of some responsesthat is front lateral force vehicle side slip angle and lateralacceleration trace based on authorsrsquo knowledge come fromthe numerical calculation or associated glitches and initialcondition of the system
Furthermore when we simulated vehicle manoeuvre forhigh speed (30msminus1) and icy road condition (120583 = 01) for2DoF controller it can be seen that all manoeuvres control
12 Journal of Control Science and Engineering
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
Time (s)0 5 10 15
0
01
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
Time (s)
2WS + DYC
minus01
minus02Dire
ct y
aw m
omen
tM
z(N
m)
Figure 10 2DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 4 Model predictive control weighting matrices parameters for V119909
= 30msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 30msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 565 11987622
= 05 11987611
= 355 11987622
= 05
V119909
= 30msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 0083 11987622
= 0042 11987611
= 003 11987622
= 045
4WS
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 155 11987622
= 25 11987611
= 135 11987622
= 35
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0005 11987622
= 0001 11987611
= 004 11987622
= 05
2WS + DYC
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 505 11987622
= 05 11987611
= 35 11987622
= 05
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0092 11987622
= 004 11987611
= 05 11987622
= 25
simulation the tracking responses become unstable andimpossible to control especially under the crosswind effectas shown in Figure 14 The most probable cause is the rolldynamic motion neglect in the 2DoF controller design whenthe vehicle itself was modelled by including roll dynamicfactor It can be said that the inclusion of roll dynamic factor
is important for vehicle manoeuvre in terms of stability andcontrollability at high speed and icy road condition Sincehigh speed coupled with icy road condition will render theequation to become highly nonlinear it is impossible fora linear tire model to react positively since the handlingpropertiesmay be significantly different from those generated
Journal of Control Science and Engineering 13
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
01
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
002
004
Time (s)
2WS + DYC
minus002
minus004
minus006Dire
ct y
aw m
omen
tM
z(N
m)
Figure 11 3DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 5 Path following tracking errors with road friction surface 120583 = 07
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00637 00085 00623 000744WS 00548 00054 00526 00051
2WS + DYC 00542 00051 00527 00051
302WS 00639 05348 00616 052154WS 00558 05239 00528 00023
2WS + DYC 00549 05226 00524 00254
by the linear tire model These results show that linear tiremodel is only suitable for analyzing a stable vehicle behaviourunder the assumption of small steering and acceleration
Next for the 3DoF controller simulation 2WS with DYCmanoeuvres provides a much better tracking response incomparison to 4WS particularly in the case of lateral outputOn the contrary 4WS manoeuvre demonstrates a much bet-ter tracking response in the yaw angle and yaw rate responsesas tabulated in Table 6 With appropriate weight tuning gainthe rear steering angle and direct yaw moment control arefully optimized in order to become stable along the given tra-jectory However for the 2WS manoeuvre the vehicleresponses become unstable despite several instances wherethe weighting tuning gains are adjusted for output and inputgains With the inclusion of another input control to the con-troller design we can enhance the vehicle responses for both
2WS with DYC and 4WS control manoeuvres show Thismeans that for 2WS controller design it may have to usemore than just front steering to stabilize the vehicle under theeffects of wind gusts Figure 15 shows that rear steering angleand direct yaw moment are fully utilized in order to stabilizethe vehicle for 2WS with DYC and 4WSmanoeuvres control
In contrast to the 2DoF controller as shown in Figure 14in the 3DoF controllers the 2WS control manoeuvre canstill be used to track a given trajectory despite not perfectlyfollowing the trajectory in a satisfactory manner especiallyunder crosswind effects In Figure 15 we can see some oscil-lations in few traces responses at the end of the responsesBased on the authorsrsquo knowledge these oscillationsmay comefrom the vehicle model and some initial conditions of thesystem with imperfect controllers tuning weighting gains Inthese conditions neither 2WS nor 4WS control manoeuvres
14 Journal of Control Science and Engineering
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
Time (s)
2WS4WS2WS + DYC
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
001
002
Time (s)
2WS4WS2WS + DYC
minus001Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
1
2
Time (s)
2WS4WS2WS + DYC
minus1
minus2
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
001
002
003
Time (s)
2WS4WS2WS + DYC
minus001
minus002
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
00005
0001
4WS
Time (s)
minus00005
minus0001
minus00015
Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
001
002
Time (s)
2WS + DYC
minus001
minus002
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 12 2DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
for lateral response and neither 2WS nor 2WS with DYCcontrol manoeuvres for yaw angle or yaw rate responseperformed well causing an increase in vehicle instabilityincreased vibration and tracking responses deteriorationWewill next focus on how to enhance the controller in order tostabilize vehicle manoeuvrability and handling stability
We can therefore conclude that for 4WS the rear wheelswere helping the car to steer by improving the vehiclehandling at high speed while decreasing the turning radiusat low speed as shown by the control signal in Figures 12 and13 Meanwhile for 2WS with DYC active front steering wasused in low speed manoeuvres for lateral acceleration whileinclusion of DYC was adopted for high lateral accelerationwhen the tires were saturated and could not produce enoughlateral force for vehicle control and stability as intendedIn 2WS vehicles the rear set of wheels do not play an
active role in controlling the steering From Table 5 it canbe seen that among the three controllers 4WS gave the bestperformance by reducing the tracking error for lateral andyaw angle responses when compared with 2WS with DYCand 2WS only Tables 5 and 6 tabulated the MPC robustnessand the tracking errors in (17) for all types of manoeuvre at10msminus1 and 30msminus1 vehicle speeds and under various roadsurfaces Furthermore all controlmanoeuvre signals for bothcontrollers were within the input constraints We would liketo highlight that in MPC approaches although there was anadvantage in multivariable systems in this study there was atrade-off between theweighting tuning parameter to focus oneither lateral position or yaw angle trajectory In this paper wehave chosen lateral position as the weighting tuning priorityso we may get a better response on yaw angle and yaw rateresponse by sacrificing lateral position precision
Journal of Control Science and Engineering 15
0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
Time (s)
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
2WS4WS2WS + DYC
Time (s)
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02
Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
5
10
2WS4WS2WS + DYC
Time (s)
minus5
minus10Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
03
4WS
Time (s)
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
2WS + DYC
Time (s)
minus005
minus01
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 13 3DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
Table 6 Path following tracking errors with road friction surface 120583 = 01
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00874 00115 00841 001044WS 00838 00109 00832 00094
2WS + DYC 00838 00110 00835 00083
302WS Uncontrolled Uncontrolled 21352 052644WS Uncontrolled Uncontrolled 08824 02982
2WS + DYC Uncontrolled Uncontrolled 09964 04580
16 Journal of Control Science and Engineering
0 10 20 30
02468
10
Time (s)
Ref2WS
4WS2WS + DYC
minus2
minus4Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0020406
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
minus06
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15
Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
0020406
Time (s)
2WS4WS2WS + DYC
minus02
minus04
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
02
04
06
Time (s)
4WS
minus02
Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
005
01
Time (s)
2WS + DYC
minus005
Dire
ct y
aw m
omen
tM
z(N
m)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15Fron
t lat
eral
forc
eFyf
(Nm
) times104
Figure 14 2DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
5 Conclusion
This paper presents the robustness of model predictive con-trol for an autonomous ground vehicle on a path-followingcontrol with consideration of wind gust effects subjectedto the variation of forward speeds and road surfaces condi-tion in a double-lane change scenario The predictive con-trollers were designed based on two- (lateral-yaw model)and three- (lateral-roll model) degree-of-freedom vehiclemodels The only difference between the models is the rolldynamics factor which is taken into consideration andboth controllers were implemented into a six-degree-of-free-dom vehicle model Based on linear vehicle and tire modelapproximations of a known trajectory we evaluated the effectand impact of roll dynamics at low and high forward vehicle
speeds and at various road frictions (wet concrete to icyroads) in order to follow a desired trajectory as close as pos-sible while rejecting the crosswind effects and maintainingvehicle stability We also evaluated and compared the effi-ciency of the different manoeuvres via two-wheel steeringfour-wheel steering and two-wheel steering with direct yawmoment control
The simulation results showed that with the inclusion ofroll dynamics in the linear vehicle model the vehicle stabilityand adherence to trajectory was considerably improved forthe case of high speed manoeuvres on a higher road surfacefriction coefficient (120583 = 07) and improved even more for thecase of a lower road surface coefficient (120583 = 01) Further-more the obtained results showed and proved that the rearwheels and direct yaw moment are beneficial to help steering
Journal of Control Science and Engineering 17
0 10 20 30
0
5
Time (s)
Ref2WS
4WS2WS + DYC
minus10
minus5
minus15Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0
025
05
Time (s)
Ref2WS
4WS2WS + DYC
minus025
minus05
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
0
500
1000
Time (s)
2WS4WS2WS + DYC
minus500
minus1000
minus1500Fron
t lat
eral
forc
eFyf
(Nm
)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
01
02
03
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS + DYC
minus2
minus4
times10minus4
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 15 3DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
the car by improving the vehicle handling at high speeddecreasing the turning radius at low speeds and reducing thetracking error for lateral position and yaw angle response
The simulation also demonstrated that model predictivecontrol has an ability to reject the side wind effect as adisturbance to the system which is particularly effective at10msminus1 for all cases except for the 2DoF controller duringsimulation of higher speeds on lower road surface adhesionHowever both controllers were not able to provide a betterresponse when the vehicle performed at a high forwardspeedwith a low road adhesion coefficient particularly when2DoF controller was simulated all results showed unstablevehicle response Thus for this scenario we are currentlyinvestigating new approaches with a real-time implementable
control scheme Further improvement for future works oncontrol methodology will also be considered to minimizetracking error and to enhance vehicle stability
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors would like to thank Professor Zidong Wangand Professor Manfred Ploechl for their comments andadvice Special thanks are also extended to Dr A Zahran MKhudzari for his help preparing this paper
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
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DistributedSensor Networks
International Journal of
Journal of Control Science and Engineering 3
z
h
y
120601
mg
Mw
b120601
Fzl Fzr
k120601
Fw
(a) Front view
x
Fyf
120575fxf
120572f f
Mf
yflf
x120573
FwMw
lr
120575r
Fyr
Mr
Fxr
120595y
(b) Top view
Figure 1 Simplified bicycle model
(119868119909119909
+ 119898ℎ2
) = 119898119892ℎ120601 minus 2119896120601
120601 minus 2119887120601
+ 119898ℎ ( + ) + 119868119909119911
(1d)
119869119887
119908119894
= minus119903119908
119865119909119894
minus 119879119887119894
minus 119887119908
120596119894
119894 = (119891 119903) (1e)
The motion equations for the vehicle in an inertial frame oron 119910-119909 axis under the assumption of a small yaw angle maybe given as
= cos120595 minus sin120595 = V119909
minus 120595
= sin120595 + cos120595 = V119909
120595 +
(2)
22 Tire Model Tire dynamics must be considered for thevehicle model since the tires are the only contact that thevehicle has with the road surface Besides the forces ofgravity and aerodynamics all the forces are induced by thetires which may affect the vehicle chassis handling andstabilityTheir complexity and nonlinear behaviourmust alsoreflect the operating condition of the controller over theirwhole region throughout varied manoeuvring range in long-itudinal lateral and roll directionsThemost frequently usedexisting nonlinear tire models of application and structureare determined through key parameters and analytical con-siderations based on tire measurement data They are calledsemiempirical tire models or Pacejka tire model [24]
On the other hand the nonlinear tire model can alsobe described linearly for some parts of the operating con-ditions therefore in this paper we will consider the linear
Fx
x
Fy
120572
Fz
Tb
Figure 2 Tire model
tire model Figure 2 illustrates the terminology used fordescribing the longitudinal lateral and vertical tire forcesand their orientation Thus the linear tire model is validunder constant normal load forces on the tires constantlongitudinal slip of tires and neglected aerodynamic dragThe relationship between the longitudinal force verticalforce and the longitudinal tire slip ratio is given by thefollowing equations
119865119909
= 120583119909
(119904) 119865119911
(3a)
119904 =119903119908
120596119908
V119909
minus 1 (3b)
4 Journal of Control Science and Engineering
119865119911119891
=119897119903
119898119892
2119897
119865119911119903
=119897119891
119898119892
2119897
(3c)
where 119903119908
is the tirersquos geometric radius120596119908
is the angular veloc-ity of the tires V
119909
in (3b) is the tirersquos forward velocity 119865119909
isproportional to the normal force 119865
119911
and 120583119909
(119904) represents thelongitudinal wheel slip friction coefficient of road adhesionsand is a function of slip ratio 119904
The lateral forces on the front and rear tires are char-acterized and modelled by a linear function with the frontand rear tire slip angles 120572
119891
and 120572119903
denoted by 119865119910119891
and119865119910119903
respectively The linear tire model yields the followingexpression for the front and rear tire forces
119865119910119891
= 119862119891
120572119891
119865119910119903
= 119862119903
120572119903
(4)
where 119862119891
and 119862119903
are the tire cornering stiffness parametersfor the front and rear tires respectively The slip angles forthe front and rear wheels with a small angle assumption aregiven such that
120572119891
=V119910
+ 119897119891
V119909
minus 120575119891
120572119903
=V119910
minus 119897119903
V119909
minus 120575119903
(5)
The details of mathematical equations for a linear tire modelcan be read through in [25] Assumptions and approxi-mations presented in this paper are representative of thenonlinear tire model at certain regional points this providesa good balance between capturing the important features andregions of laterally unstable behaviour [26]
23 Wind Model The effect of wind on the stability of thevehicle is an important and primary safety consideration ofthis paper A strong gust of wind from the inward or outwardside will generate force and torque that could be large enoughfor a vehicle to roll over or go outside of the laneThe resultingwind pressure forces and torques acting on the rigid body ingeneral can be represented by three axes longitudinal lateraland vertical A general expression of force and torque is givenby the following equations
119865119908
=119862119865
120588119860V2119903
2
119872119908
=119862119872
120588119860119871V2119903
2
(6)
In the above equations 119860 is the vehicle area 119871 is the vehiclelength 119871 = 119897
119891
+ 119897119903
120588 is the density of air V119903
is the relativewind speed and 119862
119865
and 119862119872
are the force and momentnondimensional coefficients respectively The wind speedor crosswind is represented by V
119908
as shown in Figure 3 In
w
r
x
120595
Figure 3 Vehicle wind speed in crosswind situation
general crosswind can be at various angles but for simplicityin this paper we will assume the crosswind to be at a 90-degree angle andwill focus on the windrsquos impact on the lateralforces and yaw torques [27]
The vehicle motion in ((1a) (1b) (1c) (1d) (1e))ndash(6) canbe described by the following compact differential equation
= 119860119909 + 1198611
119906 + 1198612
119908 + 1198613
119903
119910 = 119862119909 + 119863119906
(7)
with 119909 isin 119877119909 119906 isin 119877119906119908 isin 119877119908 119903 isin 119877119903 and 119910 isin 119877119910 representing
the state vectors control input vectors crosswind effects as adisturbance vector desired trajectory vectors and measuredoutput vectors respectively We define
119909 = [V119910
119884 120595 120601 120596119891
120596119903
]119879
119906 = [120575119891
120575119903
119872119911
]119879
119908 = [119865119908
119872119908
]119879
119903 = [119884des 120595des]119879
ℎ (120585) = [119884 120595]119879
(8)
The state vectors represent the states for lateral velocity lateralposition yaw rate yaw angle roll rate roll angle and frontand rear wheels angular speed respectively and the outputvectors represent lateral position and yaw angle
3 Linear Model Predictive Control
MPC has been used as a highly effective and very successfulcontrol scheme with the simple design framework provingto be a practical controller able to combine multivariablesystems and utilize online optimization process control Thestability guarantees of existing predictive control approachesfor nonlinear systems with uncertainty however remaincontingent upon the assumption of the initial feasibility ofthe optimization and the set of initial conditions startingfrom where feasibility of the optimization problem andtherefore stability of the closed-loop system is guaranteed isnot explicitly characterized [28] For example the occurrenceof faults in control actuators adds another layer of complexityto the problem of controller design for nonlinear uncertainsystems Note that the stability guarantees provided by acontroller may no longer hold in the presence of faults inthe control actuators that prevent the implementation ofthe control action prescribed by the control law and these
Journal of Control Science and Engineering 5
Modelpredictivecontrol
Linear tiremodel
Referencetracking
Dist
urba
nce Linear vehicle
model
x
Yref120575f
120575r
Fx Fy
Y
Mz120595ref
120595
Figure 4 Predictive control structure
faults can have substantial negative ramifications owing to theinterconnected nature of processes [29]
In order to implement MPC with a receding horizoncontrol strategy the following strategy method is adopted
(1) A dynamics process model is used to predict thebehaviour of the plant and future plant outputs ateach instant 119896 based on both previous and the latestobservations of the systemrsquos inputs and outputs
(2) The control signal inputs are calculated by minimiz-ing the error of tracking between the predicted outputand desired trajectory signal to keep the process asclose as possible to following the trajectory whileconsidering the objective functions and constraints
(3) Only the first control signal is implemented on theplant whilst others are rejected in anticipation of thenext sampling instant where the future output will beknown
(4) Step (1) is repeated with updated values and all ordersare updated
In this paper we use the linear model of predictive controlThe hierarchical control structure is adopted in MPC asshown in Figure 4 Figure 4 illustrates the control structurefor 2WS that uses front steering only 120575
119891
for 4WS that usesfront and rear steering 120575
119891
and 120575119903
and for 2WS with DYCthat produces external yaw moment (119872
119911
) at both front andrear wheels as a control input to the system These systemsare used to control the vehicle in order to follow a givenreference trajectory They include the vehicle speed desiredreference trajectory modelled predictive control and linearvehicle model with linear tire model
Using the equations from the vehicle and tire model asexplained and defined in (7) the basic equations of linearvehicle motion can be given as follows
119898V119910
=1
119868119909119909
V119909
[minus (120583119862119903
+ 119862119891
) 119869119909119902
V119910
+ (120583 (119862119903
119897119903
minus 119862119891
119897119891
) 119869119909119902
minus 119868119909119909
119898V2119909
) ]
minus1
119868119909119909
[(ℎ119887120601
) minus ℎ (119898119892ℎ minus 119896120601
) 120601 minus 120583119862119891
119869119909119902
120575119891
+ 120583119862119903
119869119909119902
120575119903
]
(9a)
119868119911119911
=1
V119909
[120583 (119862119903
119897119903
minus 119862119891
119897119891
) V119910
minus 120583 (119862119891
119897119891
2
+ 119862119903
119897119903
2
) ] + 120583119862119891
119897119891
120575119891
minus 120583119862119903
119897119903
120575119903
+119872119911
(9b)
119868119909119909
=ℎ
V119909
[120583 (119862119903
119897119903
minus 119862119891
119897119891
) minus 120583 (119862119891
+ 119862119903
) V119910
]
minus 119887120601
+ (119898119892ℎ minus 119896120601
) 120601 + 120583119862119891
ℎ120575119891
+ 120583119862119903
ℎ120575119903
(9c)
where 119869119909119902
= 119868119909119909
+ 119898ℎ2 A DYC that produces the reaction
moment occurring at the front and rear wheels due to thesteering angle effect (as an external yaw moments (119872
119911
)) canbe approximated with the following equations
119872119891
asymp 2119897119891
119862119891
119872119911
119872119903
asymp 2119897119903
119862119903
119872119911
(10)
We define front steering angle rear steering angle and directyaw moment control as the inputs to the system Thus thevehicle motion can be represented in a given discrete state-space structure as follows
119909119897
(119896 + 1 | 119896) = 119860119897
119909119897
(119896 | 119896) + 119861119897
119906119897
(119896 | 119896)
+ 119861119903
119903119897
(119896 | 119896)
119910119897
(119896 | 119896) = 119862119897
119909119897
(119896 | 119896) + 119863119897
119906119897
(119896 | 119896)
(11)
where 119909119897
(119896 | 119896) is the state vector at time step 119896 and 119909119897
(119896 +
1 | 119896) is the state vector at time step 119896 + 1 with 119909119897
(119896 | 119896) isin
119877119909119897(119896|119896) 119906
119897
(119896 | 119896) isin 119877119906119897(119896|119896) 119903
119897
(119896 | 119896) isin 119877119903119897(119896|119896) and 119910
119897
(119896 | 119896) isin
119877119910119897(119896|119896) representing the state vectors control input vectors
reference vectors and measured output vectors respectivelyWe define
119909119897
(119896) = [V119910
119884 120595 120601]119879
119903119897
(119896) = [119884des 120595des]119879
119910119897
(119896) = [119884 120595]119879
(12)
For 2WS 4WS and 2WS with DYC the control signal to thesystems with the same tuning control parameters is given asfollows
119906119897
(119896) = [120575119891
]
119906119897
(119896) = [120575119891
120575119903
]119879
119906119897
(119896) = [120575119891
119872119911
]119879
(13)
We formulate the optimization of the predictive controlsystem which takes the constraints imposed on the input andinput rate respectively as given in the following form
minus119906119897
le 119906119897
(119896) le +119906119897
minusΔ119906119897
le Δ119906119897
(119896) le +Δ119906119897
(14)
6 Journal of Control Science and Engineering
One approach to maintain closed-loop stability would be touse all available control actuators (13) so that even if one ofthe control actuators fails the rest can maintain closed-loopstability which the reliable control approaches The use ofredundant control actuators however incurs possibly pre-ventable operation and maintenance costs These economicconsiderations dictate the use of only as many control loopsas is required at a time To achieve tolerance with respect tofaults control-loop reconfiguration can be carried out in theevent of failure of the primary control configuration based onthe assumption of a linear system
Based on the linear vehicle model in (11) and by definingthe outputs in (12) we consider the following cost functionsto the system
119869 (119909119897
(119896) 119880119897119896
) =
119867119901
sum
119894=1
1003817100381710038171003817119897 (119896 + 119894 | 119896) minus 119903 (119896 + 119894 | 119896)10038171003817100381710038172
119876119894
+
119867119888minus1
sum
119894=0
1003817100381710038171003817Δ119897 (119896 + 119894 | 119896)10038171003817100381710038172
119877119894
(15)
In (15) the first summation of the given cost functionreflects the reduction of trajectory tracking errors among thepredicted outputs
119897
(119896 + 119894 | 119896) (119894 = 0 119867119901
minus 1) and theoutput reference signals 119903(119896+119894 | 119896) (119894 = 0 119867
119901
minus1) and thesecond summation reflects on the penalization of the controlsignal effort Δ
119897
(119896 + 119894 | 119896) (119894 = 0 119867119888
minus 1) of the steeringcontrol manoeuvre The aim of the predictive control systemis to determine the optimal control input vector Δ
119897
(119896+ 119894 | 119896)
so that the error function between the reference signal andthe predicted output is reduced
The difference in the steering angle Δ119897
(119896 + 119894 | 119896) is givenin the case that the cost function is at a minimum valueThe weight matrices 119876(119894) and 119877(119894) are semipositive definiteand positive definite respectively and can be tuned for anydesired closed-loop performance 119876(119894) is defined as the statetracking weight since the error
119897
(119896 + 119894 | 119896) minus 119903(119896 + 119894 | 119896) canbecome as small as possible by increasing 119876(119894) In a similarfashion 119877(119894) represents the input tracking weight and thevariation of the input is decreased to slow down the responseof the system by increasing 119877(119894) The predictive and controlhorizon are typically considered to be 119867
119901
ge 119867119888
while thecontrol signal is considered constant for 119867
119888
le 119894 le 119867119901
Matrices 119876(119894) and 119877(119894) represent the matrix weight of theirappropriate dimensions for outputs and inputs respectivelygiving the state feedback control law as follows
119906119897
(119896 119910119897
(119896)) = 119906119897
(119896 minus 1) + Δ119906119897
(119910119897
(119896)) (16)
An optimal input was computed for the following time steprather than being calculated for the immediate time step byaddressing convex optimization problems during each timestep After that the first input is applied on the plant in (11)and the others are rejected which yield the optimal controlsequence as (16) The optimization problem in (15) is iteratedwith the new value at a time 119896 + 1 via new state 119909(119896 + 1) andall orders are then updated
The required direct yaw moment control119872119911
is obtainedfrom the variation between the two sides of the vehicle torque
as in (1e) In this research the braking torque is only usedbased on the yaw rate in otherwords it is only activated in thecase that the vehiclemoves toward instability or in emergencymanoeuvres since it impacts the longitudinal motion whilethe rear steering angle is assumed for the total manoeuvre tobe in control or in standard drivingmanoeuvresWe take intoaccount three control inputs that is front and rear steeringangles as well as differential braking at rear right and left tirebut only one control input is used at a time [30]
4 Simulation Results
41 Scenario Description The linear model predictive con-troller explained andpresentedwas implemented for a vehiclepath following a double-lane change scenario with a cross-wind effect present throughout the simulation A double-lanechange manoeuvre approximates as an emergency manoeu-vre case and generally demonstrates the agility and capabili-ties of the vehicle in lateral and roll dynamics During such amanoeuvre an understeering or oversteering or even a roll-over situation may occur In this scenario the vehicle is con-sidered travelling horizontally or straightforward followingthe path with a constant velocity of 10msminus1 and 30msminus1without braking or accelerating The drag force and torquegiven by (6) in the initial driving conditions are assumed toact in the direction of the path at time 119905 = 1 sec with a windvelocity of V
119908
= 10msminus1 The forces and torques arising fromthis sidewise acting wind gust are assumed to be persistentand are applied as step functions throughout the simulationtime as shown in Figure 5 Table 1 shows the sport utilitypassenger car model parameters and their definitions used inthis paper based on Solmaz et al [31]
For stability analysis we simulated the system with onlyone inputmdashin particular the front steering angle 120575
119891
or 2WSWe set the vehicle velocity to be constant at V
119909
= 15msminus1the road adhesion coefficient 120583 = 1 and wind velocity V
119908
=
10msminus1 Figure 6(a) shows a stability analysis based on theroot locus of the open-loop system in (7) showing matrix 119860without the crosswind effect (119861
2
= 0) and references signal(1198613
= 0) Those settings contribute to a stable region for bothoutputs (vehicle side slip and yaw rate responses) Matrix 119860is given in (8) with input signal being front steering angle(1198611
= 120575119891
) Since the root locus command inMATLAB is onlyworking on the single-input single-output therefore we eval-uate separately the output response with single input signal
Figure 6(b) illustrates the effect of these factors a cross-wind effect (119861
2
= 0) references signal (1198613
= 0) and con-trol signal (119861
1
= 0) with the same matrix 119860 as in (8) thatyields unstable responses for both vehicle side slip and yawrate We can also notice that the poles are different for bothcases due to 119861
2
matrices that are based on the crosswindeffect Furthermore Figure 6(b) shows that within a windvelocity V
119908
= 10msminus1 the system is stable with the appropri-ate matrices of 119860 119861
1
and 1198612
Figure 6(a) also illustrates thatpaired 119860 and 119861
1
are controllable based on the model para-meters listed in Table 1
In this paper the predictive controller was implementedby minimizing vehicle deviation from the target path to
Journal of Control Science and Engineering 7
0 5 10 150
5
10
Time (s)
Cros
swin
d sp
eed
w(m
s)
Time (s)0 5 10 15
0
100
200
300
400
Win
d fo
rce
Fw
(N)
Time (s)0 5 10 15
0
minus500
minus1000
Win
d to
rque
Mw
(Nm
)
Figure 5 Force and torque for crosswind speed of 10msminus1
Table 1 Car vehicle parameters
Parameter ValueCar mass [kg] 1300Inertia around the roll 119909-axis [kgm2] 370Inertia around 119911-axis [kgm2] 216756Distance of front and rear wheels from centre of gravity (CoG) [m] 145 107Distance between the vehicle CoG and the assumed roll axis [m] 045Equivalent suspension roll damping coefficient [Nms] 4200Equivalent suspension roll stiffness coefficient [Nm] 40000Linear approximation of tire stiffness for front and rear tire [Nrad] 133000 121000Gravitational constant [ms2] 98
achieve the main aim which is to follow the desired or refer-ence trajectory as close as possible The controllers are com-pared against each other for 2DoF and 3DoF bicycle modelswhich excludes vehicle roll dynamics at various speedsspecifically low (10msminus1) and high (30msminus1) and also underwet concrete (120583 = 07) and icy (120583 = 01) road surfaces
Since our study focused on an autonomous vehicle it isreasonable to assume that the controllers will adapt to theroad or environment such as in the case of road adhesionsadverse weather visual cues traffic and overall relevantdriving conditions Therefore both controllers (2DoF and3DoF) were designed and implemented with the parametersand conditions given in Tables 2 3 and 4 for different vehicleforward speeds and road adhesion coefficients The path-following tracking error is a measurement of how closelythe output responses follow the reference trajectory whichis a measure of the deviation from the benchmark In this
paper we use the standard deviation of the root-mean squareformula to calculate the tracking error given by the followingequation
119890119905
= radicVar (119910119900
minus 119910119903
) = radic1
(119899 minus 1)sum (119910119900
minus 119910119903
)2
(17)
where 119890119905
is the tracking error 119899 represents the number oftime periods and 119910
119900
and 119910119903
express the measured output andreference
42 Results and Discussions Prior to proposed simulationswe carried out standard vehicle manoeuvres tests to validateour model in the open-loop simulation scenario Standardtests available are the J-turn fishhook and double-lanechange [32] which are representative of on-roadmanoeuvreswithout tripping (caused by external force from road curb or
8 Journal of Control Science and Engineering
0 50
0
10
20
30
40
50Root locus
Imag
inar
y ax
is (sminus1)
minus10
minus20
minus30
minus40
minus50minus50minus100minus150minus200minus250minus300
Real axis (sminus1)
120595998400
0 20
0
2
4
6
8
10Root locus
Imag
inar
y ax
is (sminus1)
minus2
minus4
minus6
minus8
minus10minus20minus40minus60minus80minus100minus120minus140
Real axis (sminus1)
120573
(a) Without crosswind effect
0 20 40 60
0
2
4
6
8
10Root locus
Imag
inar
y ax
is (sminus1)
Real axis (sminus1)
minus2
minus4
minus6
minus8
minus10
120595998400
minus20minus40minus60minus800 20
0
5
10
15
20
25Root locus
Imag
inar
y ax
is (sminus1)
minus5
minus10
minus15
minus20
minus25
Real axis (sminus1)
120573
minus20minus40minus60minus80minus100minus120minus140
(b) With crosswind effect
Figure 6 Open-loop system in (7)
collisionwith other vehicles)We performed only the double-lane change and roll rate feedback fishhook test for vehiclevalidation purposes in the open-loop simulation as shown inFigures 7 and 8 The vehicle speed was set at 30msminus1 whichis suitable for both tests with a road surface coefficient of 01As illustrated in Figures 7 and 8 the vehicle response in terms
of yaw rate roll rate and lateral acceleration has been provento be validated thus being suitable for other simulations
For easier comparison between controllersrsquo perfor-mances in achieving vehicle stabilization when subjectedto various conditions controllers tuning parameters fromTables 3 and 4 were selected as suitable The weighting gains
Journal of Control Science and Engineering 9
0 5 10 15
0
05
1
Time (s)
minus05
minus10 5 10 15
0
01
02
Time (s)
minus01
minus02
minus030 5 10 15
0
05
1
Time (s)
minus1
minus05
Tire
slip
angl
e120572f
(rad
)
0 5 10 15
0
05
1
Time (s)
minus05Yaw
angl
e res
pons
e120595
(rad
)
0 5 10 15
0
5
10
Time (s)
minus5
minus10
Fron
t ste
erin
g an
gle120575f
(deg
)
5 10 15
0
01
02
Time (s)
minus01
minus02
minus03Roll
angl
e res
pons
e120601
(rad
)
Time (s)0 5 10 15
0
10
20
30
40
50
Late
ral p
ositi
onY
(m)
0 5 10 15
0
2000
4000
Time (s)
minus2000
minus4000
Late
ral f
orce
Fyf
(N)
0 5 10 15Time (s)
10
5
0
minus5
minus10
Late
ral a
ccel
erat
ion
a y(m
s2)
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 7 Vehicle manoeuvre test of single-lane change at 30msminus1 with 120583 = 01
Table 2 Model predictive control parameters
Parameter ValuePredictive horizon 20Control horizon 9Sampling time [s] 005Constraints on max and min steering angles [deg] plusmn20Constraints on max and min changes in steering angles [degs] plusmn10Constraints on max and min changes in moment torque [Nmrad] plusmn2000Constraints on max and min moment torque [Nmrads] plusmn1500Simulation time [s] 15
for controllers input and output were selected using trial-and-error method where selection criteria were based on thebest response output for both input and output for weightinggains
The first simulation revolves around various road surfacefriction coefficients from wet concrete (120583 = 07) to icy sur-face condition (120583 = 01) with a constant forward speed of10msminus1 The MPC weighting tuning parameters are listedin Table 3 We evaluated the controllerrsquos robustness for theoutput responses by comparing the performance of 2WS4WS and 2WS with DYC manoeuvres at a forward speed of
10msminus1 as shown in Figures 9ndash11 The controller trackingerrors based on (17) for lateral position and yaw angleresponses are summarized in Table 5 From Figure 9 thereare not many differences between both controllers underthe conditions of lowest speed and high road adhesioncoefficient both controllers yielded perfect response whenfollowing a given trajectory while maintaining vehicle stabil-ity and rejecting external crosswind effect It can be seen thatfor 4WS and 2WS with DYC manoeuvres the rear steeringangle and the direct yaw moment are almost unused sincethe front steering can provide sufficient control ability This
10 Journal of Control Science and Engineering
0 5 10 15
0
50
100
Time (s)
minus50
minus100
Late
ral a
ccel
erat
ion
a y(m
s2)
Time (s)0 5 10 15
0
1000
2000
3000
minus1000Late
ral p
ositi
onY
(m)
0 5 10 15
0
2
4
Time (s)
minus2
minus4
times104
Late
ral f
orce
Fyf
(N)
0 5 10 15
0
200
400
Time (s)
minus200
minus400
Fron
t ste
erin
g an
gle120575f
(deg
)
0 5 10 15
0
50
100
150
200
Time (s)
minus50
Yaw
angl
e res
pons
e120595
(rad
)
5 10 15
0
1
2
3
Time (s)
minus1
minus2Roll
angl
e res
pons
e120601
(rad
)
0 5 10 15
0
1
2
Time (s)
minus1
minus2Vehi
cle sl
ip an
gle120573
(rad
)
0 5 10 15
0
20
40
60
Time (s)
minus200 5 10 15
0
5
10
Time (s)
minus5
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 8 Vehicle manoeuvre test of roll rate feedback at 30msminus1 with 120583 = 01
Table 3 Model predictive control weighting matrices parameters for V119909
= 10msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 10msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 195 11987622
= 05 11987611
= 525 11987622
= 05
V119909
= 10msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 152 11987622
= 05 11987611
= 395 11987622
= 05
4WS
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 195 11987622
= 05 11987611
= 45 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 153 11987622
= 05 11987611
= 35 11987622
= 05
2WS + DYC
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 193 11987622
= 05 11987611
= 37 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 152 11987622
= 05 11987611
= 35 11987622
= 05
Journal of Control Science and Engineering 11
Time (s)0 5 10 15
0
05
1
4WS
minus05
minus1
minus15
times10minus5Re
ar st
eerin
g an
gle120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
1
2
3
Time (s)
2WS + DYC
minus1
minus2
minus3
times10minus3
Dire
ct y
aw m
omen
tM
z(N
m)
0 5 10 15
0
02
0 5 10 15
0
02
04
Time (s)Time (s)
Ref2WS
4WS2WS + DYC
Ref2WS
4WS2WS + DYC
minus02
minus04
minus02
minus04
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
Yaw
angl
e120595
(rad
)
Yaw
rate
120595998400
(rad
sminus1)
Figure 9 2DoF controller at 10msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
may be partly due to the fact that the rear steering and DYCwere not used in lower speed manoeuvres
Moreover when the road surface friction was set to icythe output responses from both controllers and all controlmanoeuvre techniques did not perfectly follow the desiredtrajectory as shown in Figures 10 and 11 2WS with DYCyielded the best performance output compared to the othersespecially in yaw rate response 4WSwas the next best follow-ing 2WS where the lateral position and yaw angle lookalmost identical It seems that the rear steering and direct yawmoment controls were used in conjunction with front steer-ing for vehicle stabilization along the trajectory Howeverfrom Table 5 the controller with 3DoF gives slightly bettertracking error performance manoeuvres compared to con-troller 2DoF due to the fact that roll dynamics will not havemuch influence during low speed manoeuvres Another factis that at low vehicle speed all control inputs (front steeringangle rear steering angle and direct yawmoment control) areunder constraints
Next we tested the vehicle at various road friction coef-ficients again however this time the vehicle was tested undera high forward speed of 30msminus1 The same MPC design wasused of which the parameter controls are listed in Table 4Wecompared the simulation results for 2WS 4WS and 2WSwithDYC control manoeuvres for both controllers and presentthe comparison in Figures 12ndash15The simulation results show
that for 2DoF controller at 30msminus1 and on wet concrete(120583 = 07) all control manoeuvres give slightly similar out-puts performances Better responses were achieved in lateralpositioning and although the tracking performance of yawangle and yaw rate deteriorated the system still allowedthe vehicle to track and follow the trajectory successfullyrejecting the effects of wind gust that impact the vehicle
As tabulated in Table 6 4WS and 2WS with DYC showslightly better tracking performance compared to 2WS onlyHowever in the case of the 3DoF controller it can be clearlyseen that 2WS with DYC and 4WS control manoeuvresoffered amuch higher performing response especially in yawangle and yaw rate response than 2WS This is shown inFigure 13 In this scenario the rear steering and direct yawmoment control was fully utilized to control and enhancevehicle stability It is therefore very important to consider rolldynamics in order to enhance vehicle stability and to followthe path trajectory From Figures 12 and 13 there are somestrong frequency oscillations in front of some responsesthat is front lateral force vehicle side slip angle and lateralacceleration trace based on authorsrsquo knowledge come fromthe numerical calculation or associated glitches and initialcondition of the system
Furthermore when we simulated vehicle manoeuvre forhigh speed (30msminus1) and icy road condition (120583 = 01) for2DoF controller it can be seen that all manoeuvres control
12 Journal of Control Science and Engineering
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
Time (s)0 5 10 15
0
01
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
Time (s)
2WS + DYC
minus01
minus02Dire
ct y
aw m
omen
tM
z(N
m)
Figure 10 2DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 4 Model predictive control weighting matrices parameters for V119909
= 30msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 30msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 565 11987622
= 05 11987611
= 355 11987622
= 05
V119909
= 30msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 0083 11987622
= 0042 11987611
= 003 11987622
= 045
4WS
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 155 11987622
= 25 11987611
= 135 11987622
= 35
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0005 11987622
= 0001 11987611
= 004 11987622
= 05
2WS + DYC
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 505 11987622
= 05 11987611
= 35 11987622
= 05
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0092 11987622
= 004 11987611
= 05 11987622
= 25
simulation the tracking responses become unstable andimpossible to control especially under the crosswind effectas shown in Figure 14 The most probable cause is the rolldynamic motion neglect in the 2DoF controller design whenthe vehicle itself was modelled by including roll dynamicfactor It can be said that the inclusion of roll dynamic factor
is important for vehicle manoeuvre in terms of stability andcontrollability at high speed and icy road condition Sincehigh speed coupled with icy road condition will render theequation to become highly nonlinear it is impossible fora linear tire model to react positively since the handlingpropertiesmay be significantly different from those generated
Journal of Control Science and Engineering 13
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
01
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
002
004
Time (s)
2WS + DYC
minus002
minus004
minus006Dire
ct y
aw m
omen
tM
z(N
m)
Figure 11 3DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 5 Path following tracking errors with road friction surface 120583 = 07
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00637 00085 00623 000744WS 00548 00054 00526 00051
2WS + DYC 00542 00051 00527 00051
302WS 00639 05348 00616 052154WS 00558 05239 00528 00023
2WS + DYC 00549 05226 00524 00254
by the linear tire model These results show that linear tiremodel is only suitable for analyzing a stable vehicle behaviourunder the assumption of small steering and acceleration
Next for the 3DoF controller simulation 2WS with DYCmanoeuvres provides a much better tracking response incomparison to 4WS particularly in the case of lateral outputOn the contrary 4WS manoeuvre demonstrates a much bet-ter tracking response in the yaw angle and yaw rate responsesas tabulated in Table 6 With appropriate weight tuning gainthe rear steering angle and direct yaw moment control arefully optimized in order to become stable along the given tra-jectory However for the 2WS manoeuvre the vehicleresponses become unstable despite several instances wherethe weighting tuning gains are adjusted for output and inputgains With the inclusion of another input control to the con-troller design we can enhance the vehicle responses for both
2WS with DYC and 4WS control manoeuvres show Thismeans that for 2WS controller design it may have to usemore than just front steering to stabilize the vehicle under theeffects of wind gusts Figure 15 shows that rear steering angleand direct yaw moment are fully utilized in order to stabilizethe vehicle for 2WS with DYC and 4WSmanoeuvres control
In contrast to the 2DoF controller as shown in Figure 14in the 3DoF controllers the 2WS control manoeuvre canstill be used to track a given trajectory despite not perfectlyfollowing the trajectory in a satisfactory manner especiallyunder crosswind effects In Figure 15 we can see some oscil-lations in few traces responses at the end of the responsesBased on the authorsrsquo knowledge these oscillationsmay comefrom the vehicle model and some initial conditions of thesystem with imperfect controllers tuning weighting gains Inthese conditions neither 2WS nor 4WS control manoeuvres
14 Journal of Control Science and Engineering
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
Time (s)
2WS4WS2WS + DYC
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
001
002
Time (s)
2WS4WS2WS + DYC
minus001Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
1
2
Time (s)
2WS4WS2WS + DYC
minus1
minus2
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
001
002
003
Time (s)
2WS4WS2WS + DYC
minus001
minus002
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
00005
0001
4WS
Time (s)
minus00005
minus0001
minus00015
Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
001
002
Time (s)
2WS + DYC
minus001
minus002
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 12 2DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
for lateral response and neither 2WS nor 2WS with DYCcontrol manoeuvres for yaw angle or yaw rate responseperformed well causing an increase in vehicle instabilityincreased vibration and tracking responses deteriorationWewill next focus on how to enhance the controller in order tostabilize vehicle manoeuvrability and handling stability
We can therefore conclude that for 4WS the rear wheelswere helping the car to steer by improving the vehiclehandling at high speed while decreasing the turning radiusat low speed as shown by the control signal in Figures 12 and13 Meanwhile for 2WS with DYC active front steering wasused in low speed manoeuvres for lateral acceleration whileinclusion of DYC was adopted for high lateral accelerationwhen the tires were saturated and could not produce enoughlateral force for vehicle control and stability as intendedIn 2WS vehicles the rear set of wheels do not play an
active role in controlling the steering From Table 5 it canbe seen that among the three controllers 4WS gave the bestperformance by reducing the tracking error for lateral andyaw angle responses when compared with 2WS with DYCand 2WS only Tables 5 and 6 tabulated the MPC robustnessand the tracking errors in (17) for all types of manoeuvre at10msminus1 and 30msminus1 vehicle speeds and under various roadsurfaces Furthermore all controlmanoeuvre signals for bothcontrollers were within the input constraints We would liketo highlight that in MPC approaches although there was anadvantage in multivariable systems in this study there was atrade-off between theweighting tuning parameter to focus oneither lateral position or yaw angle trajectory In this paper wehave chosen lateral position as the weighting tuning priorityso we may get a better response on yaw angle and yaw rateresponse by sacrificing lateral position precision
Journal of Control Science and Engineering 15
0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
Time (s)
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
2WS4WS2WS + DYC
Time (s)
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02
Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
5
10
2WS4WS2WS + DYC
Time (s)
minus5
minus10Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
03
4WS
Time (s)
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
2WS + DYC
Time (s)
minus005
minus01
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 13 3DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
Table 6 Path following tracking errors with road friction surface 120583 = 01
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00874 00115 00841 001044WS 00838 00109 00832 00094
2WS + DYC 00838 00110 00835 00083
302WS Uncontrolled Uncontrolled 21352 052644WS Uncontrolled Uncontrolled 08824 02982
2WS + DYC Uncontrolled Uncontrolled 09964 04580
16 Journal of Control Science and Engineering
0 10 20 30
02468
10
Time (s)
Ref2WS
4WS2WS + DYC
minus2
minus4Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0020406
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
minus06
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15
Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
0020406
Time (s)
2WS4WS2WS + DYC
minus02
minus04
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
02
04
06
Time (s)
4WS
minus02
Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
005
01
Time (s)
2WS + DYC
minus005
Dire
ct y
aw m
omen
tM
z(N
m)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15Fron
t lat
eral
forc
eFyf
(Nm
) times104
Figure 14 2DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
5 Conclusion
This paper presents the robustness of model predictive con-trol for an autonomous ground vehicle on a path-followingcontrol with consideration of wind gust effects subjectedto the variation of forward speeds and road surfaces condi-tion in a double-lane change scenario The predictive con-trollers were designed based on two- (lateral-yaw model)and three- (lateral-roll model) degree-of-freedom vehiclemodels The only difference between the models is the rolldynamics factor which is taken into consideration andboth controllers were implemented into a six-degree-of-free-dom vehicle model Based on linear vehicle and tire modelapproximations of a known trajectory we evaluated the effectand impact of roll dynamics at low and high forward vehicle
speeds and at various road frictions (wet concrete to icyroads) in order to follow a desired trajectory as close as pos-sible while rejecting the crosswind effects and maintainingvehicle stability We also evaluated and compared the effi-ciency of the different manoeuvres via two-wheel steeringfour-wheel steering and two-wheel steering with direct yawmoment control
The simulation results showed that with the inclusion ofroll dynamics in the linear vehicle model the vehicle stabilityand adherence to trajectory was considerably improved forthe case of high speed manoeuvres on a higher road surfacefriction coefficient (120583 = 07) and improved even more for thecase of a lower road surface coefficient (120583 = 01) Further-more the obtained results showed and proved that the rearwheels and direct yaw moment are beneficial to help steering
Journal of Control Science and Engineering 17
0 10 20 30
0
5
Time (s)
Ref2WS
4WS2WS + DYC
minus10
minus5
minus15Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0
025
05
Time (s)
Ref2WS
4WS2WS + DYC
minus025
minus05
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
0
500
1000
Time (s)
2WS4WS2WS + DYC
minus500
minus1000
minus1500Fron
t lat
eral
forc
eFyf
(Nm
)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
01
02
03
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS + DYC
minus2
minus4
times10minus4
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 15 3DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
the car by improving the vehicle handling at high speeddecreasing the turning radius at low speeds and reducing thetracking error for lateral position and yaw angle response
The simulation also demonstrated that model predictivecontrol has an ability to reject the side wind effect as adisturbance to the system which is particularly effective at10msminus1 for all cases except for the 2DoF controller duringsimulation of higher speeds on lower road surface adhesionHowever both controllers were not able to provide a betterresponse when the vehicle performed at a high forwardspeedwith a low road adhesion coefficient particularly when2DoF controller was simulated all results showed unstablevehicle response Thus for this scenario we are currentlyinvestigating new approaches with a real-time implementable
control scheme Further improvement for future works oncontrol methodology will also be considered to minimizetracking error and to enhance vehicle stability
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors would like to thank Professor Zidong Wangand Professor Manfred Ploechl for their comments andadvice Special thanks are also extended to Dr A Zahran MKhudzari for his help preparing this paper
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
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4 Journal of Control Science and Engineering
119865119911119891
=119897119903
119898119892
2119897
119865119911119903
=119897119891
119898119892
2119897
(3c)
where 119903119908
is the tirersquos geometric radius120596119908
is the angular veloc-ity of the tires V
119909
in (3b) is the tirersquos forward velocity 119865119909
isproportional to the normal force 119865
119911
and 120583119909
(119904) represents thelongitudinal wheel slip friction coefficient of road adhesionsand is a function of slip ratio 119904
The lateral forces on the front and rear tires are char-acterized and modelled by a linear function with the frontand rear tire slip angles 120572
119891
and 120572119903
denoted by 119865119910119891
and119865119910119903
respectively The linear tire model yields the followingexpression for the front and rear tire forces
119865119910119891
= 119862119891
120572119891
119865119910119903
= 119862119903
120572119903
(4)
where 119862119891
and 119862119903
are the tire cornering stiffness parametersfor the front and rear tires respectively The slip angles forthe front and rear wheels with a small angle assumption aregiven such that
120572119891
=V119910
+ 119897119891
V119909
minus 120575119891
120572119903
=V119910
minus 119897119903
V119909
minus 120575119903
(5)
The details of mathematical equations for a linear tire modelcan be read through in [25] Assumptions and approxi-mations presented in this paper are representative of thenonlinear tire model at certain regional points this providesa good balance between capturing the important features andregions of laterally unstable behaviour [26]
23 Wind Model The effect of wind on the stability of thevehicle is an important and primary safety consideration ofthis paper A strong gust of wind from the inward or outwardside will generate force and torque that could be large enoughfor a vehicle to roll over or go outside of the laneThe resultingwind pressure forces and torques acting on the rigid body ingeneral can be represented by three axes longitudinal lateraland vertical A general expression of force and torque is givenby the following equations
119865119908
=119862119865
120588119860V2119903
2
119872119908
=119862119872
120588119860119871V2119903
2
(6)
In the above equations 119860 is the vehicle area 119871 is the vehiclelength 119871 = 119897
119891
+ 119897119903
120588 is the density of air V119903
is the relativewind speed and 119862
119865
and 119862119872
are the force and momentnondimensional coefficients respectively The wind speedor crosswind is represented by V
119908
as shown in Figure 3 In
w
r
x
120595
Figure 3 Vehicle wind speed in crosswind situation
general crosswind can be at various angles but for simplicityin this paper we will assume the crosswind to be at a 90-degree angle andwill focus on the windrsquos impact on the lateralforces and yaw torques [27]
The vehicle motion in ((1a) (1b) (1c) (1d) (1e))ndash(6) canbe described by the following compact differential equation
= 119860119909 + 1198611
119906 + 1198612
119908 + 1198613
119903
119910 = 119862119909 + 119863119906
(7)
with 119909 isin 119877119909 119906 isin 119877119906119908 isin 119877119908 119903 isin 119877119903 and 119910 isin 119877119910 representing
the state vectors control input vectors crosswind effects as adisturbance vector desired trajectory vectors and measuredoutput vectors respectively We define
119909 = [V119910
119884 120595 120601 120596119891
120596119903
]119879
119906 = [120575119891
120575119903
119872119911
]119879
119908 = [119865119908
119872119908
]119879
119903 = [119884des 120595des]119879
ℎ (120585) = [119884 120595]119879
(8)
The state vectors represent the states for lateral velocity lateralposition yaw rate yaw angle roll rate roll angle and frontand rear wheels angular speed respectively and the outputvectors represent lateral position and yaw angle
3 Linear Model Predictive Control
MPC has been used as a highly effective and very successfulcontrol scheme with the simple design framework provingto be a practical controller able to combine multivariablesystems and utilize online optimization process control Thestability guarantees of existing predictive control approachesfor nonlinear systems with uncertainty however remaincontingent upon the assumption of the initial feasibility ofthe optimization and the set of initial conditions startingfrom where feasibility of the optimization problem andtherefore stability of the closed-loop system is guaranteed isnot explicitly characterized [28] For example the occurrenceof faults in control actuators adds another layer of complexityto the problem of controller design for nonlinear uncertainsystems Note that the stability guarantees provided by acontroller may no longer hold in the presence of faults inthe control actuators that prevent the implementation ofthe control action prescribed by the control law and these
Journal of Control Science and Engineering 5
Modelpredictivecontrol
Linear tiremodel
Referencetracking
Dist
urba
nce Linear vehicle
model
x
Yref120575f
120575r
Fx Fy
Y
Mz120595ref
120595
Figure 4 Predictive control structure
faults can have substantial negative ramifications owing to theinterconnected nature of processes [29]
In order to implement MPC with a receding horizoncontrol strategy the following strategy method is adopted
(1) A dynamics process model is used to predict thebehaviour of the plant and future plant outputs ateach instant 119896 based on both previous and the latestobservations of the systemrsquos inputs and outputs
(2) The control signal inputs are calculated by minimiz-ing the error of tracking between the predicted outputand desired trajectory signal to keep the process asclose as possible to following the trajectory whileconsidering the objective functions and constraints
(3) Only the first control signal is implemented on theplant whilst others are rejected in anticipation of thenext sampling instant where the future output will beknown
(4) Step (1) is repeated with updated values and all ordersare updated
In this paper we use the linear model of predictive controlThe hierarchical control structure is adopted in MPC asshown in Figure 4 Figure 4 illustrates the control structurefor 2WS that uses front steering only 120575
119891
for 4WS that usesfront and rear steering 120575
119891
and 120575119903
and for 2WS with DYCthat produces external yaw moment (119872
119911
) at both front andrear wheels as a control input to the system These systemsare used to control the vehicle in order to follow a givenreference trajectory They include the vehicle speed desiredreference trajectory modelled predictive control and linearvehicle model with linear tire model
Using the equations from the vehicle and tire model asexplained and defined in (7) the basic equations of linearvehicle motion can be given as follows
119898V119910
=1
119868119909119909
V119909
[minus (120583119862119903
+ 119862119891
) 119869119909119902
V119910
+ (120583 (119862119903
119897119903
minus 119862119891
119897119891
) 119869119909119902
minus 119868119909119909
119898V2119909
) ]
minus1
119868119909119909
[(ℎ119887120601
) minus ℎ (119898119892ℎ minus 119896120601
) 120601 minus 120583119862119891
119869119909119902
120575119891
+ 120583119862119903
119869119909119902
120575119903
]
(9a)
119868119911119911
=1
V119909
[120583 (119862119903
119897119903
minus 119862119891
119897119891
) V119910
minus 120583 (119862119891
119897119891
2
+ 119862119903
119897119903
2
) ] + 120583119862119891
119897119891
120575119891
minus 120583119862119903
119897119903
120575119903
+119872119911
(9b)
119868119909119909
=ℎ
V119909
[120583 (119862119903
119897119903
minus 119862119891
119897119891
) minus 120583 (119862119891
+ 119862119903
) V119910
]
minus 119887120601
+ (119898119892ℎ minus 119896120601
) 120601 + 120583119862119891
ℎ120575119891
+ 120583119862119903
ℎ120575119903
(9c)
where 119869119909119902
= 119868119909119909
+ 119898ℎ2 A DYC that produces the reaction
moment occurring at the front and rear wheels due to thesteering angle effect (as an external yaw moments (119872
119911
)) canbe approximated with the following equations
119872119891
asymp 2119897119891
119862119891
119872119911
119872119903
asymp 2119897119903
119862119903
119872119911
(10)
We define front steering angle rear steering angle and directyaw moment control as the inputs to the system Thus thevehicle motion can be represented in a given discrete state-space structure as follows
119909119897
(119896 + 1 | 119896) = 119860119897
119909119897
(119896 | 119896) + 119861119897
119906119897
(119896 | 119896)
+ 119861119903
119903119897
(119896 | 119896)
119910119897
(119896 | 119896) = 119862119897
119909119897
(119896 | 119896) + 119863119897
119906119897
(119896 | 119896)
(11)
where 119909119897
(119896 | 119896) is the state vector at time step 119896 and 119909119897
(119896 +
1 | 119896) is the state vector at time step 119896 + 1 with 119909119897
(119896 | 119896) isin
119877119909119897(119896|119896) 119906
119897
(119896 | 119896) isin 119877119906119897(119896|119896) 119903
119897
(119896 | 119896) isin 119877119903119897(119896|119896) and 119910
119897
(119896 | 119896) isin
119877119910119897(119896|119896) representing the state vectors control input vectors
reference vectors and measured output vectors respectivelyWe define
119909119897
(119896) = [V119910
119884 120595 120601]119879
119903119897
(119896) = [119884des 120595des]119879
119910119897
(119896) = [119884 120595]119879
(12)
For 2WS 4WS and 2WS with DYC the control signal to thesystems with the same tuning control parameters is given asfollows
119906119897
(119896) = [120575119891
]
119906119897
(119896) = [120575119891
120575119903
]119879
119906119897
(119896) = [120575119891
119872119911
]119879
(13)
We formulate the optimization of the predictive controlsystem which takes the constraints imposed on the input andinput rate respectively as given in the following form
minus119906119897
le 119906119897
(119896) le +119906119897
minusΔ119906119897
le Δ119906119897
(119896) le +Δ119906119897
(14)
6 Journal of Control Science and Engineering
One approach to maintain closed-loop stability would be touse all available control actuators (13) so that even if one ofthe control actuators fails the rest can maintain closed-loopstability which the reliable control approaches The use ofredundant control actuators however incurs possibly pre-ventable operation and maintenance costs These economicconsiderations dictate the use of only as many control loopsas is required at a time To achieve tolerance with respect tofaults control-loop reconfiguration can be carried out in theevent of failure of the primary control configuration based onthe assumption of a linear system
Based on the linear vehicle model in (11) and by definingthe outputs in (12) we consider the following cost functionsto the system
119869 (119909119897
(119896) 119880119897119896
) =
119867119901
sum
119894=1
1003817100381710038171003817119897 (119896 + 119894 | 119896) minus 119903 (119896 + 119894 | 119896)10038171003817100381710038172
119876119894
+
119867119888minus1
sum
119894=0
1003817100381710038171003817Δ119897 (119896 + 119894 | 119896)10038171003817100381710038172
119877119894
(15)
In (15) the first summation of the given cost functionreflects the reduction of trajectory tracking errors among thepredicted outputs
119897
(119896 + 119894 | 119896) (119894 = 0 119867119901
minus 1) and theoutput reference signals 119903(119896+119894 | 119896) (119894 = 0 119867
119901
minus1) and thesecond summation reflects on the penalization of the controlsignal effort Δ
119897
(119896 + 119894 | 119896) (119894 = 0 119867119888
minus 1) of the steeringcontrol manoeuvre The aim of the predictive control systemis to determine the optimal control input vector Δ
119897
(119896+ 119894 | 119896)
so that the error function between the reference signal andthe predicted output is reduced
The difference in the steering angle Δ119897
(119896 + 119894 | 119896) is givenin the case that the cost function is at a minimum valueThe weight matrices 119876(119894) and 119877(119894) are semipositive definiteand positive definite respectively and can be tuned for anydesired closed-loop performance 119876(119894) is defined as the statetracking weight since the error
119897
(119896 + 119894 | 119896) minus 119903(119896 + 119894 | 119896) canbecome as small as possible by increasing 119876(119894) In a similarfashion 119877(119894) represents the input tracking weight and thevariation of the input is decreased to slow down the responseof the system by increasing 119877(119894) The predictive and controlhorizon are typically considered to be 119867
119901
ge 119867119888
while thecontrol signal is considered constant for 119867
119888
le 119894 le 119867119901
Matrices 119876(119894) and 119877(119894) represent the matrix weight of theirappropriate dimensions for outputs and inputs respectivelygiving the state feedback control law as follows
119906119897
(119896 119910119897
(119896)) = 119906119897
(119896 minus 1) + Δ119906119897
(119910119897
(119896)) (16)
An optimal input was computed for the following time steprather than being calculated for the immediate time step byaddressing convex optimization problems during each timestep After that the first input is applied on the plant in (11)and the others are rejected which yield the optimal controlsequence as (16) The optimization problem in (15) is iteratedwith the new value at a time 119896 + 1 via new state 119909(119896 + 1) andall orders are then updated
The required direct yaw moment control119872119911
is obtainedfrom the variation between the two sides of the vehicle torque
as in (1e) In this research the braking torque is only usedbased on the yaw rate in otherwords it is only activated in thecase that the vehiclemoves toward instability or in emergencymanoeuvres since it impacts the longitudinal motion whilethe rear steering angle is assumed for the total manoeuvre tobe in control or in standard drivingmanoeuvresWe take intoaccount three control inputs that is front and rear steeringangles as well as differential braking at rear right and left tirebut only one control input is used at a time [30]
4 Simulation Results
41 Scenario Description The linear model predictive con-troller explained andpresentedwas implemented for a vehiclepath following a double-lane change scenario with a cross-wind effect present throughout the simulation A double-lanechange manoeuvre approximates as an emergency manoeu-vre case and generally demonstrates the agility and capabili-ties of the vehicle in lateral and roll dynamics During such amanoeuvre an understeering or oversteering or even a roll-over situation may occur In this scenario the vehicle is con-sidered travelling horizontally or straightforward followingthe path with a constant velocity of 10msminus1 and 30msminus1without braking or accelerating The drag force and torquegiven by (6) in the initial driving conditions are assumed toact in the direction of the path at time 119905 = 1 sec with a windvelocity of V
119908
= 10msminus1 The forces and torques arising fromthis sidewise acting wind gust are assumed to be persistentand are applied as step functions throughout the simulationtime as shown in Figure 5 Table 1 shows the sport utilitypassenger car model parameters and their definitions used inthis paper based on Solmaz et al [31]
For stability analysis we simulated the system with onlyone inputmdashin particular the front steering angle 120575
119891
or 2WSWe set the vehicle velocity to be constant at V
119909
= 15msminus1the road adhesion coefficient 120583 = 1 and wind velocity V
119908
=
10msminus1 Figure 6(a) shows a stability analysis based on theroot locus of the open-loop system in (7) showing matrix 119860without the crosswind effect (119861
2
= 0) and references signal(1198613
= 0) Those settings contribute to a stable region for bothoutputs (vehicle side slip and yaw rate responses) Matrix 119860is given in (8) with input signal being front steering angle(1198611
= 120575119891
) Since the root locus command inMATLAB is onlyworking on the single-input single-output therefore we eval-uate separately the output response with single input signal
Figure 6(b) illustrates the effect of these factors a cross-wind effect (119861
2
= 0) references signal (1198613
= 0) and con-trol signal (119861
1
= 0) with the same matrix 119860 as in (8) thatyields unstable responses for both vehicle side slip and yawrate We can also notice that the poles are different for bothcases due to 119861
2
matrices that are based on the crosswindeffect Furthermore Figure 6(b) shows that within a windvelocity V
119908
= 10msminus1 the system is stable with the appropri-ate matrices of 119860 119861
1
and 1198612
Figure 6(a) also illustrates thatpaired 119860 and 119861
1
are controllable based on the model para-meters listed in Table 1
In this paper the predictive controller was implementedby minimizing vehicle deviation from the target path to
Journal of Control Science and Engineering 7
0 5 10 150
5
10
Time (s)
Cros
swin
d sp
eed
w(m
s)
Time (s)0 5 10 15
0
100
200
300
400
Win
d fo
rce
Fw
(N)
Time (s)0 5 10 15
0
minus500
minus1000
Win
d to
rque
Mw
(Nm
)
Figure 5 Force and torque for crosswind speed of 10msminus1
Table 1 Car vehicle parameters
Parameter ValueCar mass [kg] 1300Inertia around the roll 119909-axis [kgm2] 370Inertia around 119911-axis [kgm2] 216756Distance of front and rear wheels from centre of gravity (CoG) [m] 145 107Distance between the vehicle CoG and the assumed roll axis [m] 045Equivalent suspension roll damping coefficient [Nms] 4200Equivalent suspension roll stiffness coefficient [Nm] 40000Linear approximation of tire stiffness for front and rear tire [Nrad] 133000 121000Gravitational constant [ms2] 98
achieve the main aim which is to follow the desired or refer-ence trajectory as close as possible The controllers are com-pared against each other for 2DoF and 3DoF bicycle modelswhich excludes vehicle roll dynamics at various speedsspecifically low (10msminus1) and high (30msminus1) and also underwet concrete (120583 = 07) and icy (120583 = 01) road surfaces
Since our study focused on an autonomous vehicle it isreasonable to assume that the controllers will adapt to theroad or environment such as in the case of road adhesionsadverse weather visual cues traffic and overall relevantdriving conditions Therefore both controllers (2DoF and3DoF) were designed and implemented with the parametersand conditions given in Tables 2 3 and 4 for different vehicleforward speeds and road adhesion coefficients The path-following tracking error is a measurement of how closelythe output responses follow the reference trajectory whichis a measure of the deviation from the benchmark In this
paper we use the standard deviation of the root-mean squareformula to calculate the tracking error given by the followingequation
119890119905
= radicVar (119910119900
minus 119910119903
) = radic1
(119899 minus 1)sum (119910119900
minus 119910119903
)2
(17)
where 119890119905
is the tracking error 119899 represents the number oftime periods and 119910
119900
and 119910119903
express the measured output andreference
42 Results and Discussions Prior to proposed simulationswe carried out standard vehicle manoeuvres tests to validateour model in the open-loop simulation scenario Standardtests available are the J-turn fishhook and double-lanechange [32] which are representative of on-roadmanoeuvreswithout tripping (caused by external force from road curb or
8 Journal of Control Science and Engineering
0 50
0
10
20
30
40
50Root locus
Imag
inar
y ax
is (sminus1)
minus10
minus20
minus30
minus40
minus50minus50minus100minus150minus200minus250minus300
Real axis (sminus1)
120595998400
0 20
0
2
4
6
8
10Root locus
Imag
inar
y ax
is (sminus1)
minus2
minus4
minus6
minus8
minus10minus20minus40minus60minus80minus100minus120minus140
Real axis (sminus1)
120573
(a) Without crosswind effect
0 20 40 60
0
2
4
6
8
10Root locus
Imag
inar
y ax
is (sminus1)
Real axis (sminus1)
minus2
minus4
minus6
minus8
minus10
120595998400
minus20minus40minus60minus800 20
0
5
10
15
20
25Root locus
Imag
inar
y ax
is (sminus1)
minus5
minus10
minus15
minus20
minus25
Real axis (sminus1)
120573
minus20minus40minus60minus80minus100minus120minus140
(b) With crosswind effect
Figure 6 Open-loop system in (7)
collisionwith other vehicles)We performed only the double-lane change and roll rate feedback fishhook test for vehiclevalidation purposes in the open-loop simulation as shown inFigures 7 and 8 The vehicle speed was set at 30msminus1 whichis suitable for both tests with a road surface coefficient of 01As illustrated in Figures 7 and 8 the vehicle response in terms
of yaw rate roll rate and lateral acceleration has been provento be validated thus being suitable for other simulations
For easier comparison between controllersrsquo perfor-mances in achieving vehicle stabilization when subjectedto various conditions controllers tuning parameters fromTables 3 and 4 were selected as suitable The weighting gains
Journal of Control Science and Engineering 9
0 5 10 15
0
05
1
Time (s)
minus05
minus10 5 10 15
0
01
02
Time (s)
minus01
minus02
minus030 5 10 15
0
05
1
Time (s)
minus1
minus05
Tire
slip
angl
e120572f
(rad
)
0 5 10 15
0
05
1
Time (s)
minus05Yaw
angl
e res
pons
e120595
(rad
)
0 5 10 15
0
5
10
Time (s)
minus5
minus10
Fron
t ste
erin
g an
gle120575f
(deg
)
5 10 15
0
01
02
Time (s)
minus01
minus02
minus03Roll
angl
e res
pons
e120601
(rad
)
Time (s)0 5 10 15
0
10
20
30
40
50
Late
ral p
ositi
onY
(m)
0 5 10 15
0
2000
4000
Time (s)
minus2000
minus4000
Late
ral f
orce
Fyf
(N)
0 5 10 15Time (s)
10
5
0
minus5
minus10
Late
ral a
ccel
erat
ion
a y(m
s2)
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 7 Vehicle manoeuvre test of single-lane change at 30msminus1 with 120583 = 01
Table 2 Model predictive control parameters
Parameter ValuePredictive horizon 20Control horizon 9Sampling time [s] 005Constraints on max and min steering angles [deg] plusmn20Constraints on max and min changes in steering angles [degs] plusmn10Constraints on max and min changes in moment torque [Nmrad] plusmn2000Constraints on max and min moment torque [Nmrads] plusmn1500Simulation time [s] 15
for controllers input and output were selected using trial-and-error method where selection criteria were based on thebest response output for both input and output for weightinggains
The first simulation revolves around various road surfacefriction coefficients from wet concrete (120583 = 07) to icy sur-face condition (120583 = 01) with a constant forward speed of10msminus1 The MPC weighting tuning parameters are listedin Table 3 We evaluated the controllerrsquos robustness for theoutput responses by comparing the performance of 2WS4WS and 2WS with DYC manoeuvres at a forward speed of
10msminus1 as shown in Figures 9ndash11 The controller trackingerrors based on (17) for lateral position and yaw angleresponses are summarized in Table 5 From Figure 9 thereare not many differences between both controllers underthe conditions of lowest speed and high road adhesioncoefficient both controllers yielded perfect response whenfollowing a given trajectory while maintaining vehicle stabil-ity and rejecting external crosswind effect It can be seen thatfor 4WS and 2WS with DYC manoeuvres the rear steeringangle and the direct yaw moment are almost unused sincethe front steering can provide sufficient control ability This
10 Journal of Control Science and Engineering
0 5 10 15
0
50
100
Time (s)
minus50
minus100
Late
ral a
ccel
erat
ion
a y(m
s2)
Time (s)0 5 10 15
0
1000
2000
3000
minus1000Late
ral p
ositi
onY
(m)
0 5 10 15
0
2
4
Time (s)
minus2
minus4
times104
Late
ral f
orce
Fyf
(N)
0 5 10 15
0
200
400
Time (s)
minus200
minus400
Fron
t ste
erin
g an
gle120575f
(deg
)
0 5 10 15
0
50
100
150
200
Time (s)
minus50
Yaw
angl
e res
pons
e120595
(rad
)
5 10 15
0
1
2
3
Time (s)
minus1
minus2Roll
angl
e res
pons
e120601
(rad
)
0 5 10 15
0
1
2
Time (s)
minus1
minus2Vehi
cle sl
ip an
gle120573
(rad
)
0 5 10 15
0
20
40
60
Time (s)
minus200 5 10 15
0
5
10
Time (s)
minus5
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 8 Vehicle manoeuvre test of roll rate feedback at 30msminus1 with 120583 = 01
Table 3 Model predictive control weighting matrices parameters for V119909
= 10msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 10msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 195 11987622
= 05 11987611
= 525 11987622
= 05
V119909
= 10msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 152 11987622
= 05 11987611
= 395 11987622
= 05
4WS
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 195 11987622
= 05 11987611
= 45 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 153 11987622
= 05 11987611
= 35 11987622
= 05
2WS + DYC
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 193 11987622
= 05 11987611
= 37 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 152 11987622
= 05 11987611
= 35 11987622
= 05
Journal of Control Science and Engineering 11
Time (s)0 5 10 15
0
05
1
4WS
minus05
minus1
minus15
times10minus5Re
ar st
eerin
g an
gle120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
1
2
3
Time (s)
2WS + DYC
minus1
minus2
minus3
times10minus3
Dire
ct y
aw m
omen
tM
z(N
m)
0 5 10 15
0
02
0 5 10 15
0
02
04
Time (s)Time (s)
Ref2WS
4WS2WS + DYC
Ref2WS
4WS2WS + DYC
minus02
minus04
minus02
minus04
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
Yaw
angl
e120595
(rad
)
Yaw
rate
120595998400
(rad
sminus1)
Figure 9 2DoF controller at 10msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
may be partly due to the fact that the rear steering and DYCwere not used in lower speed manoeuvres
Moreover when the road surface friction was set to icythe output responses from both controllers and all controlmanoeuvre techniques did not perfectly follow the desiredtrajectory as shown in Figures 10 and 11 2WS with DYCyielded the best performance output compared to the othersespecially in yaw rate response 4WSwas the next best follow-ing 2WS where the lateral position and yaw angle lookalmost identical It seems that the rear steering and direct yawmoment controls were used in conjunction with front steer-ing for vehicle stabilization along the trajectory Howeverfrom Table 5 the controller with 3DoF gives slightly bettertracking error performance manoeuvres compared to con-troller 2DoF due to the fact that roll dynamics will not havemuch influence during low speed manoeuvres Another factis that at low vehicle speed all control inputs (front steeringangle rear steering angle and direct yawmoment control) areunder constraints
Next we tested the vehicle at various road friction coef-ficients again however this time the vehicle was tested undera high forward speed of 30msminus1 The same MPC design wasused of which the parameter controls are listed in Table 4Wecompared the simulation results for 2WS 4WS and 2WSwithDYC control manoeuvres for both controllers and presentthe comparison in Figures 12ndash15The simulation results show
that for 2DoF controller at 30msminus1 and on wet concrete(120583 = 07) all control manoeuvres give slightly similar out-puts performances Better responses were achieved in lateralpositioning and although the tracking performance of yawangle and yaw rate deteriorated the system still allowedthe vehicle to track and follow the trajectory successfullyrejecting the effects of wind gust that impact the vehicle
As tabulated in Table 6 4WS and 2WS with DYC showslightly better tracking performance compared to 2WS onlyHowever in the case of the 3DoF controller it can be clearlyseen that 2WS with DYC and 4WS control manoeuvresoffered amuch higher performing response especially in yawangle and yaw rate response than 2WS This is shown inFigure 13 In this scenario the rear steering and direct yawmoment control was fully utilized to control and enhancevehicle stability It is therefore very important to consider rolldynamics in order to enhance vehicle stability and to followthe path trajectory From Figures 12 and 13 there are somestrong frequency oscillations in front of some responsesthat is front lateral force vehicle side slip angle and lateralacceleration trace based on authorsrsquo knowledge come fromthe numerical calculation or associated glitches and initialcondition of the system
Furthermore when we simulated vehicle manoeuvre forhigh speed (30msminus1) and icy road condition (120583 = 01) for2DoF controller it can be seen that all manoeuvres control
12 Journal of Control Science and Engineering
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
Time (s)0 5 10 15
0
01
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
Time (s)
2WS + DYC
minus01
minus02Dire
ct y
aw m
omen
tM
z(N
m)
Figure 10 2DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 4 Model predictive control weighting matrices parameters for V119909
= 30msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 30msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 565 11987622
= 05 11987611
= 355 11987622
= 05
V119909
= 30msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 0083 11987622
= 0042 11987611
= 003 11987622
= 045
4WS
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 155 11987622
= 25 11987611
= 135 11987622
= 35
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0005 11987622
= 0001 11987611
= 004 11987622
= 05
2WS + DYC
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 505 11987622
= 05 11987611
= 35 11987622
= 05
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0092 11987622
= 004 11987611
= 05 11987622
= 25
simulation the tracking responses become unstable andimpossible to control especially under the crosswind effectas shown in Figure 14 The most probable cause is the rolldynamic motion neglect in the 2DoF controller design whenthe vehicle itself was modelled by including roll dynamicfactor It can be said that the inclusion of roll dynamic factor
is important for vehicle manoeuvre in terms of stability andcontrollability at high speed and icy road condition Sincehigh speed coupled with icy road condition will render theequation to become highly nonlinear it is impossible fora linear tire model to react positively since the handlingpropertiesmay be significantly different from those generated
Journal of Control Science and Engineering 13
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
01
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
002
004
Time (s)
2WS + DYC
minus002
minus004
minus006Dire
ct y
aw m
omen
tM
z(N
m)
Figure 11 3DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 5 Path following tracking errors with road friction surface 120583 = 07
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00637 00085 00623 000744WS 00548 00054 00526 00051
2WS + DYC 00542 00051 00527 00051
302WS 00639 05348 00616 052154WS 00558 05239 00528 00023
2WS + DYC 00549 05226 00524 00254
by the linear tire model These results show that linear tiremodel is only suitable for analyzing a stable vehicle behaviourunder the assumption of small steering and acceleration
Next for the 3DoF controller simulation 2WS with DYCmanoeuvres provides a much better tracking response incomparison to 4WS particularly in the case of lateral outputOn the contrary 4WS manoeuvre demonstrates a much bet-ter tracking response in the yaw angle and yaw rate responsesas tabulated in Table 6 With appropriate weight tuning gainthe rear steering angle and direct yaw moment control arefully optimized in order to become stable along the given tra-jectory However for the 2WS manoeuvre the vehicleresponses become unstable despite several instances wherethe weighting tuning gains are adjusted for output and inputgains With the inclusion of another input control to the con-troller design we can enhance the vehicle responses for both
2WS with DYC and 4WS control manoeuvres show Thismeans that for 2WS controller design it may have to usemore than just front steering to stabilize the vehicle under theeffects of wind gusts Figure 15 shows that rear steering angleand direct yaw moment are fully utilized in order to stabilizethe vehicle for 2WS with DYC and 4WSmanoeuvres control
In contrast to the 2DoF controller as shown in Figure 14in the 3DoF controllers the 2WS control manoeuvre canstill be used to track a given trajectory despite not perfectlyfollowing the trajectory in a satisfactory manner especiallyunder crosswind effects In Figure 15 we can see some oscil-lations in few traces responses at the end of the responsesBased on the authorsrsquo knowledge these oscillationsmay comefrom the vehicle model and some initial conditions of thesystem with imperfect controllers tuning weighting gains Inthese conditions neither 2WS nor 4WS control manoeuvres
14 Journal of Control Science and Engineering
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
Time (s)
2WS4WS2WS + DYC
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
001
002
Time (s)
2WS4WS2WS + DYC
minus001Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
1
2
Time (s)
2WS4WS2WS + DYC
minus1
minus2
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
001
002
003
Time (s)
2WS4WS2WS + DYC
minus001
minus002
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
00005
0001
4WS
Time (s)
minus00005
minus0001
minus00015
Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
001
002
Time (s)
2WS + DYC
minus001
minus002
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 12 2DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
for lateral response and neither 2WS nor 2WS with DYCcontrol manoeuvres for yaw angle or yaw rate responseperformed well causing an increase in vehicle instabilityincreased vibration and tracking responses deteriorationWewill next focus on how to enhance the controller in order tostabilize vehicle manoeuvrability and handling stability
We can therefore conclude that for 4WS the rear wheelswere helping the car to steer by improving the vehiclehandling at high speed while decreasing the turning radiusat low speed as shown by the control signal in Figures 12 and13 Meanwhile for 2WS with DYC active front steering wasused in low speed manoeuvres for lateral acceleration whileinclusion of DYC was adopted for high lateral accelerationwhen the tires were saturated and could not produce enoughlateral force for vehicle control and stability as intendedIn 2WS vehicles the rear set of wheels do not play an
active role in controlling the steering From Table 5 it canbe seen that among the three controllers 4WS gave the bestperformance by reducing the tracking error for lateral andyaw angle responses when compared with 2WS with DYCand 2WS only Tables 5 and 6 tabulated the MPC robustnessand the tracking errors in (17) for all types of manoeuvre at10msminus1 and 30msminus1 vehicle speeds and under various roadsurfaces Furthermore all controlmanoeuvre signals for bothcontrollers were within the input constraints We would liketo highlight that in MPC approaches although there was anadvantage in multivariable systems in this study there was atrade-off between theweighting tuning parameter to focus oneither lateral position or yaw angle trajectory In this paper wehave chosen lateral position as the weighting tuning priorityso we may get a better response on yaw angle and yaw rateresponse by sacrificing lateral position precision
Journal of Control Science and Engineering 15
0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
Time (s)
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
2WS4WS2WS + DYC
Time (s)
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02
Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
5
10
2WS4WS2WS + DYC
Time (s)
minus5
minus10Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
03
4WS
Time (s)
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
2WS + DYC
Time (s)
minus005
minus01
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 13 3DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
Table 6 Path following tracking errors with road friction surface 120583 = 01
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00874 00115 00841 001044WS 00838 00109 00832 00094
2WS + DYC 00838 00110 00835 00083
302WS Uncontrolled Uncontrolled 21352 052644WS Uncontrolled Uncontrolled 08824 02982
2WS + DYC Uncontrolled Uncontrolled 09964 04580
16 Journal of Control Science and Engineering
0 10 20 30
02468
10
Time (s)
Ref2WS
4WS2WS + DYC
minus2
minus4Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0020406
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
minus06
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15
Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
0020406
Time (s)
2WS4WS2WS + DYC
minus02
minus04
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
02
04
06
Time (s)
4WS
minus02
Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
005
01
Time (s)
2WS + DYC
minus005
Dire
ct y
aw m
omen
tM
z(N
m)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15Fron
t lat
eral
forc
eFyf
(Nm
) times104
Figure 14 2DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
5 Conclusion
This paper presents the robustness of model predictive con-trol for an autonomous ground vehicle on a path-followingcontrol with consideration of wind gust effects subjectedto the variation of forward speeds and road surfaces condi-tion in a double-lane change scenario The predictive con-trollers were designed based on two- (lateral-yaw model)and three- (lateral-roll model) degree-of-freedom vehiclemodels The only difference between the models is the rolldynamics factor which is taken into consideration andboth controllers were implemented into a six-degree-of-free-dom vehicle model Based on linear vehicle and tire modelapproximations of a known trajectory we evaluated the effectand impact of roll dynamics at low and high forward vehicle
speeds and at various road frictions (wet concrete to icyroads) in order to follow a desired trajectory as close as pos-sible while rejecting the crosswind effects and maintainingvehicle stability We also evaluated and compared the effi-ciency of the different manoeuvres via two-wheel steeringfour-wheel steering and two-wheel steering with direct yawmoment control
The simulation results showed that with the inclusion ofroll dynamics in the linear vehicle model the vehicle stabilityand adherence to trajectory was considerably improved forthe case of high speed manoeuvres on a higher road surfacefriction coefficient (120583 = 07) and improved even more for thecase of a lower road surface coefficient (120583 = 01) Further-more the obtained results showed and proved that the rearwheels and direct yaw moment are beneficial to help steering
Journal of Control Science and Engineering 17
0 10 20 30
0
5
Time (s)
Ref2WS
4WS2WS + DYC
minus10
minus5
minus15Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0
025
05
Time (s)
Ref2WS
4WS2WS + DYC
minus025
minus05
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
0
500
1000
Time (s)
2WS4WS2WS + DYC
minus500
minus1000
minus1500Fron
t lat
eral
forc
eFyf
(Nm
)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
01
02
03
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS + DYC
minus2
minus4
times10minus4
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 15 3DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
the car by improving the vehicle handling at high speeddecreasing the turning radius at low speeds and reducing thetracking error for lateral position and yaw angle response
The simulation also demonstrated that model predictivecontrol has an ability to reject the side wind effect as adisturbance to the system which is particularly effective at10msminus1 for all cases except for the 2DoF controller duringsimulation of higher speeds on lower road surface adhesionHowever both controllers were not able to provide a betterresponse when the vehicle performed at a high forwardspeedwith a low road adhesion coefficient particularly when2DoF controller was simulated all results showed unstablevehicle response Thus for this scenario we are currentlyinvestigating new approaches with a real-time implementable
control scheme Further improvement for future works oncontrol methodology will also be considered to minimizetracking error and to enhance vehicle stability
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors would like to thank Professor Zidong Wangand Professor Manfred Ploechl for their comments andadvice Special thanks are also extended to Dr A Zahran MKhudzari for his help preparing this paper
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
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DistributedSensor Networks
International Journal of
Journal of Control Science and Engineering 5
Modelpredictivecontrol
Linear tiremodel
Referencetracking
Dist
urba
nce Linear vehicle
model
x
Yref120575f
120575r
Fx Fy
Y
Mz120595ref
120595
Figure 4 Predictive control structure
faults can have substantial negative ramifications owing to theinterconnected nature of processes [29]
In order to implement MPC with a receding horizoncontrol strategy the following strategy method is adopted
(1) A dynamics process model is used to predict thebehaviour of the plant and future plant outputs ateach instant 119896 based on both previous and the latestobservations of the systemrsquos inputs and outputs
(2) The control signal inputs are calculated by minimiz-ing the error of tracking between the predicted outputand desired trajectory signal to keep the process asclose as possible to following the trajectory whileconsidering the objective functions and constraints
(3) Only the first control signal is implemented on theplant whilst others are rejected in anticipation of thenext sampling instant where the future output will beknown
(4) Step (1) is repeated with updated values and all ordersare updated
In this paper we use the linear model of predictive controlThe hierarchical control structure is adopted in MPC asshown in Figure 4 Figure 4 illustrates the control structurefor 2WS that uses front steering only 120575
119891
for 4WS that usesfront and rear steering 120575
119891
and 120575119903
and for 2WS with DYCthat produces external yaw moment (119872
119911
) at both front andrear wheels as a control input to the system These systemsare used to control the vehicle in order to follow a givenreference trajectory They include the vehicle speed desiredreference trajectory modelled predictive control and linearvehicle model with linear tire model
Using the equations from the vehicle and tire model asexplained and defined in (7) the basic equations of linearvehicle motion can be given as follows
119898V119910
=1
119868119909119909
V119909
[minus (120583119862119903
+ 119862119891
) 119869119909119902
V119910
+ (120583 (119862119903
119897119903
minus 119862119891
119897119891
) 119869119909119902
minus 119868119909119909
119898V2119909
) ]
minus1
119868119909119909
[(ℎ119887120601
) minus ℎ (119898119892ℎ minus 119896120601
) 120601 minus 120583119862119891
119869119909119902
120575119891
+ 120583119862119903
119869119909119902
120575119903
]
(9a)
119868119911119911
=1
V119909
[120583 (119862119903
119897119903
minus 119862119891
119897119891
) V119910
minus 120583 (119862119891
119897119891
2
+ 119862119903
119897119903
2
) ] + 120583119862119891
119897119891
120575119891
minus 120583119862119903
119897119903
120575119903
+119872119911
(9b)
119868119909119909
=ℎ
V119909
[120583 (119862119903
119897119903
minus 119862119891
119897119891
) minus 120583 (119862119891
+ 119862119903
) V119910
]
minus 119887120601
+ (119898119892ℎ minus 119896120601
) 120601 + 120583119862119891
ℎ120575119891
+ 120583119862119903
ℎ120575119903
(9c)
where 119869119909119902
= 119868119909119909
+ 119898ℎ2 A DYC that produces the reaction
moment occurring at the front and rear wheels due to thesteering angle effect (as an external yaw moments (119872
119911
)) canbe approximated with the following equations
119872119891
asymp 2119897119891
119862119891
119872119911
119872119903
asymp 2119897119903
119862119903
119872119911
(10)
We define front steering angle rear steering angle and directyaw moment control as the inputs to the system Thus thevehicle motion can be represented in a given discrete state-space structure as follows
119909119897
(119896 + 1 | 119896) = 119860119897
119909119897
(119896 | 119896) + 119861119897
119906119897
(119896 | 119896)
+ 119861119903
119903119897
(119896 | 119896)
119910119897
(119896 | 119896) = 119862119897
119909119897
(119896 | 119896) + 119863119897
119906119897
(119896 | 119896)
(11)
where 119909119897
(119896 | 119896) is the state vector at time step 119896 and 119909119897
(119896 +
1 | 119896) is the state vector at time step 119896 + 1 with 119909119897
(119896 | 119896) isin
119877119909119897(119896|119896) 119906
119897
(119896 | 119896) isin 119877119906119897(119896|119896) 119903
119897
(119896 | 119896) isin 119877119903119897(119896|119896) and 119910
119897
(119896 | 119896) isin
119877119910119897(119896|119896) representing the state vectors control input vectors
reference vectors and measured output vectors respectivelyWe define
119909119897
(119896) = [V119910
119884 120595 120601]119879
119903119897
(119896) = [119884des 120595des]119879
119910119897
(119896) = [119884 120595]119879
(12)
For 2WS 4WS and 2WS with DYC the control signal to thesystems with the same tuning control parameters is given asfollows
119906119897
(119896) = [120575119891
]
119906119897
(119896) = [120575119891
120575119903
]119879
119906119897
(119896) = [120575119891
119872119911
]119879
(13)
We formulate the optimization of the predictive controlsystem which takes the constraints imposed on the input andinput rate respectively as given in the following form
minus119906119897
le 119906119897
(119896) le +119906119897
minusΔ119906119897
le Δ119906119897
(119896) le +Δ119906119897
(14)
6 Journal of Control Science and Engineering
One approach to maintain closed-loop stability would be touse all available control actuators (13) so that even if one ofthe control actuators fails the rest can maintain closed-loopstability which the reliable control approaches The use ofredundant control actuators however incurs possibly pre-ventable operation and maintenance costs These economicconsiderations dictate the use of only as many control loopsas is required at a time To achieve tolerance with respect tofaults control-loop reconfiguration can be carried out in theevent of failure of the primary control configuration based onthe assumption of a linear system
Based on the linear vehicle model in (11) and by definingthe outputs in (12) we consider the following cost functionsto the system
119869 (119909119897
(119896) 119880119897119896
) =
119867119901
sum
119894=1
1003817100381710038171003817119897 (119896 + 119894 | 119896) minus 119903 (119896 + 119894 | 119896)10038171003817100381710038172
119876119894
+
119867119888minus1
sum
119894=0
1003817100381710038171003817Δ119897 (119896 + 119894 | 119896)10038171003817100381710038172
119877119894
(15)
In (15) the first summation of the given cost functionreflects the reduction of trajectory tracking errors among thepredicted outputs
119897
(119896 + 119894 | 119896) (119894 = 0 119867119901
minus 1) and theoutput reference signals 119903(119896+119894 | 119896) (119894 = 0 119867
119901
minus1) and thesecond summation reflects on the penalization of the controlsignal effort Δ
119897
(119896 + 119894 | 119896) (119894 = 0 119867119888
minus 1) of the steeringcontrol manoeuvre The aim of the predictive control systemis to determine the optimal control input vector Δ
119897
(119896+ 119894 | 119896)
so that the error function between the reference signal andthe predicted output is reduced
The difference in the steering angle Δ119897
(119896 + 119894 | 119896) is givenin the case that the cost function is at a minimum valueThe weight matrices 119876(119894) and 119877(119894) are semipositive definiteand positive definite respectively and can be tuned for anydesired closed-loop performance 119876(119894) is defined as the statetracking weight since the error
119897
(119896 + 119894 | 119896) minus 119903(119896 + 119894 | 119896) canbecome as small as possible by increasing 119876(119894) In a similarfashion 119877(119894) represents the input tracking weight and thevariation of the input is decreased to slow down the responseof the system by increasing 119877(119894) The predictive and controlhorizon are typically considered to be 119867
119901
ge 119867119888
while thecontrol signal is considered constant for 119867
119888
le 119894 le 119867119901
Matrices 119876(119894) and 119877(119894) represent the matrix weight of theirappropriate dimensions for outputs and inputs respectivelygiving the state feedback control law as follows
119906119897
(119896 119910119897
(119896)) = 119906119897
(119896 minus 1) + Δ119906119897
(119910119897
(119896)) (16)
An optimal input was computed for the following time steprather than being calculated for the immediate time step byaddressing convex optimization problems during each timestep After that the first input is applied on the plant in (11)and the others are rejected which yield the optimal controlsequence as (16) The optimization problem in (15) is iteratedwith the new value at a time 119896 + 1 via new state 119909(119896 + 1) andall orders are then updated
The required direct yaw moment control119872119911
is obtainedfrom the variation between the two sides of the vehicle torque
as in (1e) In this research the braking torque is only usedbased on the yaw rate in otherwords it is only activated in thecase that the vehiclemoves toward instability or in emergencymanoeuvres since it impacts the longitudinal motion whilethe rear steering angle is assumed for the total manoeuvre tobe in control or in standard drivingmanoeuvresWe take intoaccount three control inputs that is front and rear steeringangles as well as differential braking at rear right and left tirebut only one control input is used at a time [30]
4 Simulation Results
41 Scenario Description The linear model predictive con-troller explained andpresentedwas implemented for a vehiclepath following a double-lane change scenario with a cross-wind effect present throughout the simulation A double-lanechange manoeuvre approximates as an emergency manoeu-vre case and generally demonstrates the agility and capabili-ties of the vehicle in lateral and roll dynamics During such amanoeuvre an understeering or oversteering or even a roll-over situation may occur In this scenario the vehicle is con-sidered travelling horizontally or straightforward followingthe path with a constant velocity of 10msminus1 and 30msminus1without braking or accelerating The drag force and torquegiven by (6) in the initial driving conditions are assumed toact in the direction of the path at time 119905 = 1 sec with a windvelocity of V
119908
= 10msminus1 The forces and torques arising fromthis sidewise acting wind gust are assumed to be persistentand are applied as step functions throughout the simulationtime as shown in Figure 5 Table 1 shows the sport utilitypassenger car model parameters and their definitions used inthis paper based on Solmaz et al [31]
For stability analysis we simulated the system with onlyone inputmdashin particular the front steering angle 120575
119891
or 2WSWe set the vehicle velocity to be constant at V
119909
= 15msminus1the road adhesion coefficient 120583 = 1 and wind velocity V
119908
=
10msminus1 Figure 6(a) shows a stability analysis based on theroot locus of the open-loop system in (7) showing matrix 119860without the crosswind effect (119861
2
= 0) and references signal(1198613
= 0) Those settings contribute to a stable region for bothoutputs (vehicle side slip and yaw rate responses) Matrix 119860is given in (8) with input signal being front steering angle(1198611
= 120575119891
) Since the root locus command inMATLAB is onlyworking on the single-input single-output therefore we eval-uate separately the output response with single input signal
Figure 6(b) illustrates the effect of these factors a cross-wind effect (119861
2
= 0) references signal (1198613
= 0) and con-trol signal (119861
1
= 0) with the same matrix 119860 as in (8) thatyields unstable responses for both vehicle side slip and yawrate We can also notice that the poles are different for bothcases due to 119861
2
matrices that are based on the crosswindeffect Furthermore Figure 6(b) shows that within a windvelocity V
119908
= 10msminus1 the system is stable with the appropri-ate matrices of 119860 119861
1
and 1198612
Figure 6(a) also illustrates thatpaired 119860 and 119861
1
are controllable based on the model para-meters listed in Table 1
In this paper the predictive controller was implementedby minimizing vehicle deviation from the target path to
Journal of Control Science and Engineering 7
0 5 10 150
5
10
Time (s)
Cros
swin
d sp
eed
w(m
s)
Time (s)0 5 10 15
0
100
200
300
400
Win
d fo
rce
Fw
(N)
Time (s)0 5 10 15
0
minus500
minus1000
Win
d to
rque
Mw
(Nm
)
Figure 5 Force and torque for crosswind speed of 10msminus1
Table 1 Car vehicle parameters
Parameter ValueCar mass [kg] 1300Inertia around the roll 119909-axis [kgm2] 370Inertia around 119911-axis [kgm2] 216756Distance of front and rear wheels from centre of gravity (CoG) [m] 145 107Distance between the vehicle CoG and the assumed roll axis [m] 045Equivalent suspension roll damping coefficient [Nms] 4200Equivalent suspension roll stiffness coefficient [Nm] 40000Linear approximation of tire stiffness for front and rear tire [Nrad] 133000 121000Gravitational constant [ms2] 98
achieve the main aim which is to follow the desired or refer-ence trajectory as close as possible The controllers are com-pared against each other for 2DoF and 3DoF bicycle modelswhich excludes vehicle roll dynamics at various speedsspecifically low (10msminus1) and high (30msminus1) and also underwet concrete (120583 = 07) and icy (120583 = 01) road surfaces
Since our study focused on an autonomous vehicle it isreasonable to assume that the controllers will adapt to theroad or environment such as in the case of road adhesionsadverse weather visual cues traffic and overall relevantdriving conditions Therefore both controllers (2DoF and3DoF) were designed and implemented with the parametersand conditions given in Tables 2 3 and 4 for different vehicleforward speeds and road adhesion coefficients The path-following tracking error is a measurement of how closelythe output responses follow the reference trajectory whichis a measure of the deviation from the benchmark In this
paper we use the standard deviation of the root-mean squareformula to calculate the tracking error given by the followingequation
119890119905
= radicVar (119910119900
minus 119910119903
) = radic1
(119899 minus 1)sum (119910119900
minus 119910119903
)2
(17)
where 119890119905
is the tracking error 119899 represents the number oftime periods and 119910
119900
and 119910119903
express the measured output andreference
42 Results and Discussions Prior to proposed simulationswe carried out standard vehicle manoeuvres tests to validateour model in the open-loop simulation scenario Standardtests available are the J-turn fishhook and double-lanechange [32] which are representative of on-roadmanoeuvreswithout tripping (caused by external force from road curb or
8 Journal of Control Science and Engineering
0 50
0
10
20
30
40
50Root locus
Imag
inar
y ax
is (sminus1)
minus10
minus20
minus30
minus40
minus50minus50minus100minus150minus200minus250minus300
Real axis (sminus1)
120595998400
0 20
0
2
4
6
8
10Root locus
Imag
inar
y ax
is (sminus1)
minus2
minus4
minus6
minus8
minus10minus20minus40minus60minus80minus100minus120minus140
Real axis (sminus1)
120573
(a) Without crosswind effect
0 20 40 60
0
2
4
6
8
10Root locus
Imag
inar
y ax
is (sminus1)
Real axis (sminus1)
minus2
minus4
minus6
minus8
minus10
120595998400
minus20minus40minus60minus800 20
0
5
10
15
20
25Root locus
Imag
inar
y ax
is (sminus1)
minus5
minus10
minus15
minus20
minus25
Real axis (sminus1)
120573
minus20minus40minus60minus80minus100minus120minus140
(b) With crosswind effect
Figure 6 Open-loop system in (7)
collisionwith other vehicles)We performed only the double-lane change and roll rate feedback fishhook test for vehiclevalidation purposes in the open-loop simulation as shown inFigures 7 and 8 The vehicle speed was set at 30msminus1 whichis suitable for both tests with a road surface coefficient of 01As illustrated in Figures 7 and 8 the vehicle response in terms
of yaw rate roll rate and lateral acceleration has been provento be validated thus being suitable for other simulations
For easier comparison between controllersrsquo perfor-mances in achieving vehicle stabilization when subjectedto various conditions controllers tuning parameters fromTables 3 and 4 were selected as suitable The weighting gains
Journal of Control Science and Engineering 9
0 5 10 15
0
05
1
Time (s)
minus05
minus10 5 10 15
0
01
02
Time (s)
minus01
minus02
minus030 5 10 15
0
05
1
Time (s)
minus1
minus05
Tire
slip
angl
e120572f
(rad
)
0 5 10 15
0
05
1
Time (s)
minus05Yaw
angl
e res
pons
e120595
(rad
)
0 5 10 15
0
5
10
Time (s)
minus5
minus10
Fron
t ste
erin
g an
gle120575f
(deg
)
5 10 15
0
01
02
Time (s)
minus01
minus02
minus03Roll
angl
e res
pons
e120601
(rad
)
Time (s)0 5 10 15
0
10
20
30
40
50
Late
ral p
ositi
onY
(m)
0 5 10 15
0
2000
4000
Time (s)
minus2000
minus4000
Late
ral f
orce
Fyf
(N)
0 5 10 15Time (s)
10
5
0
minus5
minus10
Late
ral a
ccel
erat
ion
a y(m
s2)
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 7 Vehicle manoeuvre test of single-lane change at 30msminus1 with 120583 = 01
Table 2 Model predictive control parameters
Parameter ValuePredictive horizon 20Control horizon 9Sampling time [s] 005Constraints on max and min steering angles [deg] plusmn20Constraints on max and min changes in steering angles [degs] plusmn10Constraints on max and min changes in moment torque [Nmrad] plusmn2000Constraints on max and min moment torque [Nmrads] plusmn1500Simulation time [s] 15
for controllers input and output were selected using trial-and-error method where selection criteria were based on thebest response output for both input and output for weightinggains
The first simulation revolves around various road surfacefriction coefficients from wet concrete (120583 = 07) to icy sur-face condition (120583 = 01) with a constant forward speed of10msminus1 The MPC weighting tuning parameters are listedin Table 3 We evaluated the controllerrsquos robustness for theoutput responses by comparing the performance of 2WS4WS and 2WS with DYC manoeuvres at a forward speed of
10msminus1 as shown in Figures 9ndash11 The controller trackingerrors based on (17) for lateral position and yaw angleresponses are summarized in Table 5 From Figure 9 thereare not many differences between both controllers underthe conditions of lowest speed and high road adhesioncoefficient both controllers yielded perfect response whenfollowing a given trajectory while maintaining vehicle stabil-ity and rejecting external crosswind effect It can be seen thatfor 4WS and 2WS with DYC manoeuvres the rear steeringangle and the direct yaw moment are almost unused sincethe front steering can provide sufficient control ability This
10 Journal of Control Science and Engineering
0 5 10 15
0
50
100
Time (s)
minus50
minus100
Late
ral a
ccel
erat
ion
a y(m
s2)
Time (s)0 5 10 15
0
1000
2000
3000
minus1000Late
ral p
ositi
onY
(m)
0 5 10 15
0
2
4
Time (s)
minus2
minus4
times104
Late
ral f
orce
Fyf
(N)
0 5 10 15
0
200
400
Time (s)
minus200
minus400
Fron
t ste
erin
g an
gle120575f
(deg
)
0 5 10 15
0
50
100
150
200
Time (s)
minus50
Yaw
angl
e res
pons
e120595
(rad
)
5 10 15
0
1
2
3
Time (s)
minus1
minus2Roll
angl
e res
pons
e120601
(rad
)
0 5 10 15
0
1
2
Time (s)
minus1
minus2Vehi
cle sl
ip an
gle120573
(rad
)
0 5 10 15
0
20
40
60
Time (s)
minus200 5 10 15
0
5
10
Time (s)
minus5
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 8 Vehicle manoeuvre test of roll rate feedback at 30msminus1 with 120583 = 01
Table 3 Model predictive control weighting matrices parameters for V119909
= 10msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 10msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 195 11987622
= 05 11987611
= 525 11987622
= 05
V119909
= 10msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 152 11987622
= 05 11987611
= 395 11987622
= 05
4WS
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 195 11987622
= 05 11987611
= 45 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 153 11987622
= 05 11987611
= 35 11987622
= 05
2WS + DYC
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 193 11987622
= 05 11987611
= 37 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 152 11987622
= 05 11987611
= 35 11987622
= 05
Journal of Control Science and Engineering 11
Time (s)0 5 10 15
0
05
1
4WS
minus05
minus1
minus15
times10minus5Re
ar st
eerin
g an
gle120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
1
2
3
Time (s)
2WS + DYC
minus1
minus2
minus3
times10minus3
Dire
ct y
aw m
omen
tM
z(N
m)
0 5 10 15
0
02
0 5 10 15
0
02
04
Time (s)Time (s)
Ref2WS
4WS2WS + DYC
Ref2WS
4WS2WS + DYC
minus02
minus04
minus02
minus04
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
Yaw
angl
e120595
(rad
)
Yaw
rate
120595998400
(rad
sminus1)
Figure 9 2DoF controller at 10msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
may be partly due to the fact that the rear steering and DYCwere not used in lower speed manoeuvres
Moreover when the road surface friction was set to icythe output responses from both controllers and all controlmanoeuvre techniques did not perfectly follow the desiredtrajectory as shown in Figures 10 and 11 2WS with DYCyielded the best performance output compared to the othersespecially in yaw rate response 4WSwas the next best follow-ing 2WS where the lateral position and yaw angle lookalmost identical It seems that the rear steering and direct yawmoment controls were used in conjunction with front steer-ing for vehicle stabilization along the trajectory Howeverfrom Table 5 the controller with 3DoF gives slightly bettertracking error performance manoeuvres compared to con-troller 2DoF due to the fact that roll dynamics will not havemuch influence during low speed manoeuvres Another factis that at low vehicle speed all control inputs (front steeringangle rear steering angle and direct yawmoment control) areunder constraints
Next we tested the vehicle at various road friction coef-ficients again however this time the vehicle was tested undera high forward speed of 30msminus1 The same MPC design wasused of which the parameter controls are listed in Table 4Wecompared the simulation results for 2WS 4WS and 2WSwithDYC control manoeuvres for both controllers and presentthe comparison in Figures 12ndash15The simulation results show
that for 2DoF controller at 30msminus1 and on wet concrete(120583 = 07) all control manoeuvres give slightly similar out-puts performances Better responses were achieved in lateralpositioning and although the tracking performance of yawangle and yaw rate deteriorated the system still allowedthe vehicle to track and follow the trajectory successfullyrejecting the effects of wind gust that impact the vehicle
As tabulated in Table 6 4WS and 2WS with DYC showslightly better tracking performance compared to 2WS onlyHowever in the case of the 3DoF controller it can be clearlyseen that 2WS with DYC and 4WS control manoeuvresoffered amuch higher performing response especially in yawangle and yaw rate response than 2WS This is shown inFigure 13 In this scenario the rear steering and direct yawmoment control was fully utilized to control and enhancevehicle stability It is therefore very important to consider rolldynamics in order to enhance vehicle stability and to followthe path trajectory From Figures 12 and 13 there are somestrong frequency oscillations in front of some responsesthat is front lateral force vehicle side slip angle and lateralacceleration trace based on authorsrsquo knowledge come fromthe numerical calculation or associated glitches and initialcondition of the system
Furthermore when we simulated vehicle manoeuvre forhigh speed (30msminus1) and icy road condition (120583 = 01) for2DoF controller it can be seen that all manoeuvres control
12 Journal of Control Science and Engineering
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
Time (s)0 5 10 15
0
01
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
Time (s)
2WS + DYC
minus01
minus02Dire
ct y
aw m
omen
tM
z(N
m)
Figure 10 2DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 4 Model predictive control weighting matrices parameters for V119909
= 30msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 30msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 565 11987622
= 05 11987611
= 355 11987622
= 05
V119909
= 30msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 0083 11987622
= 0042 11987611
= 003 11987622
= 045
4WS
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 155 11987622
= 25 11987611
= 135 11987622
= 35
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0005 11987622
= 0001 11987611
= 004 11987622
= 05
2WS + DYC
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 505 11987622
= 05 11987611
= 35 11987622
= 05
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0092 11987622
= 004 11987611
= 05 11987622
= 25
simulation the tracking responses become unstable andimpossible to control especially under the crosswind effectas shown in Figure 14 The most probable cause is the rolldynamic motion neglect in the 2DoF controller design whenthe vehicle itself was modelled by including roll dynamicfactor It can be said that the inclusion of roll dynamic factor
is important for vehicle manoeuvre in terms of stability andcontrollability at high speed and icy road condition Sincehigh speed coupled with icy road condition will render theequation to become highly nonlinear it is impossible fora linear tire model to react positively since the handlingpropertiesmay be significantly different from those generated
Journal of Control Science and Engineering 13
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
01
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
002
004
Time (s)
2WS + DYC
minus002
minus004
minus006Dire
ct y
aw m
omen
tM
z(N
m)
Figure 11 3DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 5 Path following tracking errors with road friction surface 120583 = 07
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00637 00085 00623 000744WS 00548 00054 00526 00051
2WS + DYC 00542 00051 00527 00051
302WS 00639 05348 00616 052154WS 00558 05239 00528 00023
2WS + DYC 00549 05226 00524 00254
by the linear tire model These results show that linear tiremodel is only suitable for analyzing a stable vehicle behaviourunder the assumption of small steering and acceleration
Next for the 3DoF controller simulation 2WS with DYCmanoeuvres provides a much better tracking response incomparison to 4WS particularly in the case of lateral outputOn the contrary 4WS manoeuvre demonstrates a much bet-ter tracking response in the yaw angle and yaw rate responsesas tabulated in Table 6 With appropriate weight tuning gainthe rear steering angle and direct yaw moment control arefully optimized in order to become stable along the given tra-jectory However for the 2WS manoeuvre the vehicleresponses become unstable despite several instances wherethe weighting tuning gains are adjusted for output and inputgains With the inclusion of another input control to the con-troller design we can enhance the vehicle responses for both
2WS with DYC and 4WS control manoeuvres show Thismeans that for 2WS controller design it may have to usemore than just front steering to stabilize the vehicle under theeffects of wind gusts Figure 15 shows that rear steering angleand direct yaw moment are fully utilized in order to stabilizethe vehicle for 2WS with DYC and 4WSmanoeuvres control
In contrast to the 2DoF controller as shown in Figure 14in the 3DoF controllers the 2WS control manoeuvre canstill be used to track a given trajectory despite not perfectlyfollowing the trajectory in a satisfactory manner especiallyunder crosswind effects In Figure 15 we can see some oscil-lations in few traces responses at the end of the responsesBased on the authorsrsquo knowledge these oscillationsmay comefrom the vehicle model and some initial conditions of thesystem with imperfect controllers tuning weighting gains Inthese conditions neither 2WS nor 4WS control manoeuvres
14 Journal of Control Science and Engineering
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
Time (s)
2WS4WS2WS + DYC
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
001
002
Time (s)
2WS4WS2WS + DYC
minus001Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
1
2
Time (s)
2WS4WS2WS + DYC
minus1
minus2
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
001
002
003
Time (s)
2WS4WS2WS + DYC
minus001
minus002
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
00005
0001
4WS
Time (s)
minus00005
minus0001
minus00015
Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
001
002
Time (s)
2WS + DYC
minus001
minus002
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 12 2DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
for lateral response and neither 2WS nor 2WS with DYCcontrol manoeuvres for yaw angle or yaw rate responseperformed well causing an increase in vehicle instabilityincreased vibration and tracking responses deteriorationWewill next focus on how to enhance the controller in order tostabilize vehicle manoeuvrability and handling stability
We can therefore conclude that for 4WS the rear wheelswere helping the car to steer by improving the vehiclehandling at high speed while decreasing the turning radiusat low speed as shown by the control signal in Figures 12 and13 Meanwhile for 2WS with DYC active front steering wasused in low speed manoeuvres for lateral acceleration whileinclusion of DYC was adopted for high lateral accelerationwhen the tires were saturated and could not produce enoughlateral force for vehicle control and stability as intendedIn 2WS vehicles the rear set of wheels do not play an
active role in controlling the steering From Table 5 it canbe seen that among the three controllers 4WS gave the bestperformance by reducing the tracking error for lateral andyaw angle responses when compared with 2WS with DYCand 2WS only Tables 5 and 6 tabulated the MPC robustnessand the tracking errors in (17) for all types of manoeuvre at10msminus1 and 30msminus1 vehicle speeds and under various roadsurfaces Furthermore all controlmanoeuvre signals for bothcontrollers were within the input constraints We would liketo highlight that in MPC approaches although there was anadvantage in multivariable systems in this study there was atrade-off between theweighting tuning parameter to focus oneither lateral position or yaw angle trajectory In this paper wehave chosen lateral position as the weighting tuning priorityso we may get a better response on yaw angle and yaw rateresponse by sacrificing lateral position precision
Journal of Control Science and Engineering 15
0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
Time (s)
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
2WS4WS2WS + DYC
Time (s)
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02
Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
5
10
2WS4WS2WS + DYC
Time (s)
minus5
minus10Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
03
4WS
Time (s)
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
2WS + DYC
Time (s)
minus005
minus01
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 13 3DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
Table 6 Path following tracking errors with road friction surface 120583 = 01
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00874 00115 00841 001044WS 00838 00109 00832 00094
2WS + DYC 00838 00110 00835 00083
302WS Uncontrolled Uncontrolled 21352 052644WS Uncontrolled Uncontrolled 08824 02982
2WS + DYC Uncontrolled Uncontrolled 09964 04580
16 Journal of Control Science and Engineering
0 10 20 30
02468
10
Time (s)
Ref2WS
4WS2WS + DYC
minus2
minus4Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0020406
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
minus06
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15
Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
0020406
Time (s)
2WS4WS2WS + DYC
minus02
minus04
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
02
04
06
Time (s)
4WS
minus02
Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
005
01
Time (s)
2WS + DYC
minus005
Dire
ct y
aw m
omen
tM
z(N
m)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15Fron
t lat
eral
forc
eFyf
(Nm
) times104
Figure 14 2DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
5 Conclusion
This paper presents the robustness of model predictive con-trol for an autonomous ground vehicle on a path-followingcontrol with consideration of wind gust effects subjectedto the variation of forward speeds and road surfaces condi-tion in a double-lane change scenario The predictive con-trollers were designed based on two- (lateral-yaw model)and three- (lateral-roll model) degree-of-freedom vehiclemodels The only difference between the models is the rolldynamics factor which is taken into consideration andboth controllers were implemented into a six-degree-of-free-dom vehicle model Based on linear vehicle and tire modelapproximations of a known trajectory we evaluated the effectand impact of roll dynamics at low and high forward vehicle
speeds and at various road frictions (wet concrete to icyroads) in order to follow a desired trajectory as close as pos-sible while rejecting the crosswind effects and maintainingvehicle stability We also evaluated and compared the effi-ciency of the different manoeuvres via two-wheel steeringfour-wheel steering and two-wheel steering with direct yawmoment control
The simulation results showed that with the inclusion ofroll dynamics in the linear vehicle model the vehicle stabilityand adherence to trajectory was considerably improved forthe case of high speed manoeuvres on a higher road surfacefriction coefficient (120583 = 07) and improved even more for thecase of a lower road surface coefficient (120583 = 01) Further-more the obtained results showed and proved that the rearwheels and direct yaw moment are beneficial to help steering
Journal of Control Science and Engineering 17
0 10 20 30
0
5
Time (s)
Ref2WS
4WS2WS + DYC
minus10
minus5
minus15Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0
025
05
Time (s)
Ref2WS
4WS2WS + DYC
minus025
minus05
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
0
500
1000
Time (s)
2WS4WS2WS + DYC
minus500
minus1000
minus1500Fron
t lat
eral
forc
eFyf
(Nm
)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
01
02
03
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS + DYC
minus2
minus4
times10minus4
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 15 3DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
the car by improving the vehicle handling at high speeddecreasing the turning radius at low speeds and reducing thetracking error for lateral position and yaw angle response
The simulation also demonstrated that model predictivecontrol has an ability to reject the side wind effect as adisturbance to the system which is particularly effective at10msminus1 for all cases except for the 2DoF controller duringsimulation of higher speeds on lower road surface adhesionHowever both controllers were not able to provide a betterresponse when the vehicle performed at a high forwardspeedwith a low road adhesion coefficient particularly when2DoF controller was simulated all results showed unstablevehicle response Thus for this scenario we are currentlyinvestigating new approaches with a real-time implementable
control scheme Further improvement for future works oncontrol methodology will also be considered to minimizetracking error and to enhance vehicle stability
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors would like to thank Professor Zidong Wangand Professor Manfred Ploechl for their comments andadvice Special thanks are also extended to Dr A Zahran MKhudzari for his help preparing this paper
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
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DistributedSensor Networks
International Journal of
6 Journal of Control Science and Engineering
One approach to maintain closed-loop stability would be touse all available control actuators (13) so that even if one ofthe control actuators fails the rest can maintain closed-loopstability which the reliable control approaches The use ofredundant control actuators however incurs possibly pre-ventable operation and maintenance costs These economicconsiderations dictate the use of only as many control loopsas is required at a time To achieve tolerance with respect tofaults control-loop reconfiguration can be carried out in theevent of failure of the primary control configuration based onthe assumption of a linear system
Based on the linear vehicle model in (11) and by definingthe outputs in (12) we consider the following cost functionsto the system
119869 (119909119897
(119896) 119880119897119896
) =
119867119901
sum
119894=1
1003817100381710038171003817119897 (119896 + 119894 | 119896) minus 119903 (119896 + 119894 | 119896)10038171003817100381710038172
119876119894
+
119867119888minus1
sum
119894=0
1003817100381710038171003817Δ119897 (119896 + 119894 | 119896)10038171003817100381710038172
119877119894
(15)
In (15) the first summation of the given cost functionreflects the reduction of trajectory tracking errors among thepredicted outputs
119897
(119896 + 119894 | 119896) (119894 = 0 119867119901
minus 1) and theoutput reference signals 119903(119896+119894 | 119896) (119894 = 0 119867
119901
minus1) and thesecond summation reflects on the penalization of the controlsignal effort Δ
119897
(119896 + 119894 | 119896) (119894 = 0 119867119888
minus 1) of the steeringcontrol manoeuvre The aim of the predictive control systemis to determine the optimal control input vector Δ
119897
(119896+ 119894 | 119896)
so that the error function between the reference signal andthe predicted output is reduced
The difference in the steering angle Δ119897
(119896 + 119894 | 119896) is givenin the case that the cost function is at a minimum valueThe weight matrices 119876(119894) and 119877(119894) are semipositive definiteand positive definite respectively and can be tuned for anydesired closed-loop performance 119876(119894) is defined as the statetracking weight since the error
119897
(119896 + 119894 | 119896) minus 119903(119896 + 119894 | 119896) canbecome as small as possible by increasing 119876(119894) In a similarfashion 119877(119894) represents the input tracking weight and thevariation of the input is decreased to slow down the responseof the system by increasing 119877(119894) The predictive and controlhorizon are typically considered to be 119867
119901
ge 119867119888
while thecontrol signal is considered constant for 119867
119888
le 119894 le 119867119901
Matrices 119876(119894) and 119877(119894) represent the matrix weight of theirappropriate dimensions for outputs and inputs respectivelygiving the state feedback control law as follows
119906119897
(119896 119910119897
(119896)) = 119906119897
(119896 minus 1) + Δ119906119897
(119910119897
(119896)) (16)
An optimal input was computed for the following time steprather than being calculated for the immediate time step byaddressing convex optimization problems during each timestep After that the first input is applied on the plant in (11)and the others are rejected which yield the optimal controlsequence as (16) The optimization problem in (15) is iteratedwith the new value at a time 119896 + 1 via new state 119909(119896 + 1) andall orders are then updated
The required direct yaw moment control119872119911
is obtainedfrom the variation between the two sides of the vehicle torque
as in (1e) In this research the braking torque is only usedbased on the yaw rate in otherwords it is only activated in thecase that the vehiclemoves toward instability or in emergencymanoeuvres since it impacts the longitudinal motion whilethe rear steering angle is assumed for the total manoeuvre tobe in control or in standard drivingmanoeuvresWe take intoaccount three control inputs that is front and rear steeringangles as well as differential braking at rear right and left tirebut only one control input is used at a time [30]
4 Simulation Results
41 Scenario Description The linear model predictive con-troller explained andpresentedwas implemented for a vehiclepath following a double-lane change scenario with a cross-wind effect present throughout the simulation A double-lanechange manoeuvre approximates as an emergency manoeu-vre case and generally demonstrates the agility and capabili-ties of the vehicle in lateral and roll dynamics During such amanoeuvre an understeering or oversteering or even a roll-over situation may occur In this scenario the vehicle is con-sidered travelling horizontally or straightforward followingthe path with a constant velocity of 10msminus1 and 30msminus1without braking or accelerating The drag force and torquegiven by (6) in the initial driving conditions are assumed toact in the direction of the path at time 119905 = 1 sec with a windvelocity of V
119908
= 10msminus1 The forces and torques arising fromthis sidewise acting wind gust are assumed to be persistentand are applied as step functions throughout the simulationtime as shown in Figure 5 Table 1 shows the sport utilitypassenger car model parameters and their definitions used inthis paper based on Solmaz et al [31]
For stability analysis we simulated the system with onlyone inputmdashin particular the front steering angle 120575
119891
or 2WSWe set the vehicle velocity to be constant at V
119909
= 15msminus1the road adhesion coefficient 120583 = 1 and wind velocity V
119908
=
10msminus1 Figure 6(a) shows a stability analysis based on theroot locus of the open-loop system in (7) showing matrix 119860without the crosswind effect (119861
2
= 0) and references signal(1198613
= 0) Those settings contribute to a stable region for bothoutputs (vehicle side slip and yaw rate responses) Matrix 119860is given in (8) with input signal being front steering angle(1198611
= 120575119891
) Since the root locus command inMATLAB is onlyworking on the single-input single-output therefore we eval-uate separately the output response with single input signal
Figure 6(b) illustrates the effect of these factors a cross-wind effect (119861
2
= 0) references signal (1198613
= 0) and con-trol signal (119861
1
= 0) with the same matrix 119860 as in (8) thatyields unstable responses for both vehicle side slip and yawrate We can also notice that the poles are different for bothcases due to 119861
2
matrices that are based on the crosswindeffect Furthermore Figure 6(b) shows that within a windvelocity V
119908
= 10msminus1 the system is stable with the appropri-ate matrices of 119860 119861
1
and 1198612
Figure 6(a) also illustrates thatpaired 119860 and 119861
1
are controllable based on the model para-meters listed in Table 1
In this paper the predictive controller was implementedby minimizing vehicle deviation from the target path to
Journal of Control Science and Engineering 7
0 5 10 150
5
10
Time (s)
Cros
swin
d sp
eed
w(m
s)
Time (s)0 5 10 15
0
100
200
300
400
Win
d fo
rce
Fw
(N)
Time (s)0 5 10 15
0
minus500
minus1000
Win
d to
rque
Mw
(Nm
)
Figure 5 Force and torque for crosswind speed of 10msminus1
Table 1 Car vehicle parameters
Parameter ValueCar mass [kg] 1300Inertia around the roll 119909-axis [kgm2] 370Inertia around 119911-axis [kgm2] 216756Distance of front and rear wheels from centre of gravity (CoG) [m] 145 107Distance between the vehicle CoG and the assumed roll axis [m] 045Equivalent suspension roll damping coefficient [Nms] 4200Equivalent suspension roll stiffness coefficient [Nm] 40000Linear approximation of tire stiffness for front and rear tire [Nrad] 133000 121000Gravitational constant [ms2] 98
achieve the main aim which is to follow the desired or refer-ence trajectory as close as possible The controllers are com-pared against each other for 2DoF and 3DoF bicycle modelswhich excludes vehicle roll dynamics at various speedsspecifically low (10msminus1) and high (30msminus1) and also underwet concrete (120583 = 07) and icy (120583 = 01) road surfaces
Since our study focused on an autonomous vehicle it isreasonable to assume that the controllers will adapt to theroad or environment such as in the case of road adhesionsadverse weather visual cues traffic and overall relevantdriving conditions Therefore both controllers (2DoF and3DoF) were designed and implemented with the parametersand conditions given in Tables 2 3 and 4 for different vehicleforward speeds and road adhesion coefficients The path-following tracking error is a measurement of how closelythe output responses follow the reference trajectory whichis a measure of the deviation from the benchmark In this
paper we use the standard deviation of the root-mean squareformula to calculate the tracking error given by the followingequation
119890119905
= radicVar (119910119900
minus 119910119903
) = radic1
(119899 minus 1)sum (119910119900
minus 119910119903
)2
(17)
where 119890119905
is the tracking error 119899 represents the number oftime periods and 119910
119900
and 119910119903
express the measured output andreference
42 Results and Discussions Prior to proposed simulationswe carried out standard vehicle manoeuvres tests to validateour model in the open-loop simulation scenario Standardtests available are the J-turn fishhook and double-lanechange [32] which are representative of on-roadmanoeuvreswithout tripping (caused by external force from road curb or
8 Journal of Control Science and Engineering
0 50
0
10
20
30
40
50Root locus
Imag
inar
y ax
is (sminus1)
minus10
minus20
minus30
minus40
minus50minus50minus100minus150minus200minus250minus300
Real axis (sminus1)
120595998400
0 20
0
2
4
6
8
10Root locus
Imag
inar
y ax
is (sminus1)
minus2
minus4
minus6
minus8
minus10minus20minus40minus60minus80minus100minus120minus140
Real axis (sminus1)
120573
(a) Without crosswind effect
0 20 40 60
0
2
4
6
8
10Root locus
Imag
inar
y ax
is (sminus1)
Real axis (sminus1)
minus2
minus4
minus6
minus8
minus10
120595998400
minus20minus40minus60minus800 20
0
5
10
15
20
25Root locus
Imag
inar
y ax
is (sminus1)
minus5
minus10
minus15
minus20
minus25
Real axis (sminus1)
120573
minus20minus40minus60minus80minus100minus120minus140
(b) With crosswind effect
Figure 6 Open-loop system in (7)
collisionwith other vehicles)We performed only the double-lane change and roll rate feedback fishhook test for vehiclevalidation purposes in the open-loop simulation as shown inFigures 7 and 8 The vehicle speed was set at 30msminus1 whichis suitable for both tests with a road surface coefficient of 01As illustrated in Figures 7 and 8 the vehicle response in terms
of yaw rate roll rate and lateral acceleration has been provento be validated thus being suitable for other simulations
For easier comparison between controllersrsquo perfor-mances in achieving vehicle stabilization when subjectedto various conditions controllers tuning parameters fromTables 3 and 4 were selected as suitable The weighting gains
Journal of Control Science and Engineering 9
0 5 10 15
0
05
1
Time (s)
minus05
minus10 5 10 15
0
01
02
Time (s)
minus01
minus02
minus030 5 10 15
0
05
1
Time (s)
minus1
minus05
Tire
slip
angl
e120572f
(rad
)
0 5 10 15
0
05
1
Time (s)
minus05Yaw
angl
e res
pons
e120595
(rad
)
0 5 10 15
0
5
10
Time (s)
minus5
minus10
Fron
t ste
erin
g an
gle120575f
(deg
)
5 10 15
0
01
02
Time (s)
minus01
minus02
minus03Roll
angl
e res
pons
e120601
(rad
)
Time (s)0 5 10 15
0
10
20
30
40
50
Late
ral p
ositi
onY
(m)
0 5 10 15
0
2000
4000
Time (s)
minus2000
minus4000
Late
ral f
orce
Fyf
(N)
0 5 10 15Time (s)
10
5
0
minus5
minus10
Late
ral a
ccel
erat
ion
a y(m
s2)
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 7 Vehicle manoeuvre test of single-lane change at 30msminus1 with 120583 = 01
Table 2 Model predictive control parameters
Parameter ValuePredictive horizon 20Control horizon 9Sampling time [s] 005Constraints on max and min steering angles [deg] plusmn20Constraints on max and min changes in steering angles [degs] plusmn10Constraints on max and min changes in moment torque [Nmrad] plusmn2000Constraints on max and min moment torque [Nmrads] plusmn1500Simulation time [s] 15
for controllers input and output were selected using trial-and-error method where selection criteria were based on thebest response output for both input and output for weightinggains
The first simulation revolves around various road surfacefriction coefficients from wet concrete (120583 = 07) to icy sur-face condition (120583 = 01) with a constant forward speed of10msminus1 The MPC weighting tuning parameters are listedin Table 3 We evaluated the controllerrsquos robustness for theoutput responses by comparing the performance of 2WS4WS and 2WS with DYC manoeuvres at a forward speed of
10msminus1 as shown in Figures 9ndash11 The controller trackingerrors based on (17) for lateral position and yaw angleresponses are summarized in Table 5 From Figure 9 thereare not many differences between both controllers underthe conditions of lowest speed and high road adhesioncoefficient both controllers yielded perfect response whenfollowing a given trajectory while maintaining vehicle stabil-ity and rejecting external crosswind effect It can be seen thatfor 4WS and 2WS with DYC manoeuvres the rear steeringangle and the direct yaw moment are almost unused sincethe front steering can provide sufficient control ability This
10 Journal of Control Science and Engineering
0 5 10 15
0
50
100
Time (s)
minus50
minus100
Late
ral a
ccel
erat
ion
a y(m
s2)
Time (s)0 5 10 15
0
1000
2000
3000
minus1000Late
ral p
ositi
onY
(m)
0 5 10 15
0
2
4
Time (s)
minus2
minus4
times104
Late
ral f
orce
Fyf
(N)
0 5 10 15
0
200
400
Time (s)
minus200
minus400
Fron
t ste
erin
g an
gle120575f
(deg
)
0 5 10 15
0
50
100
150
200
Time (s)
minus50
Yaw
angl
e res
pons
e120595
(rad
)
5 10 15
0
1
2
3
Time (s)
minus1
minus2Roll
angl
e res
pons
e120601
(rad
)
0 5 10 15
0
1
2
Time (s)
minus1
minus2Vehi
cle sl
ip an
gle120573
(rad
)
0 5 10 15
0
20
40
60
Time (s)
minus200 5 10 15
0
5
10
Time (s)
minus5
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 8 Vehicle manoeuvre test of roll rate feedback at 30msminus1 with 120583 = 01
Table 3 Model predictive control weighting matrices parameters for V119909
= 10msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 10msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 195 11987622
= 05 11987611
= 525 11987622
= 05
V119909
= 10msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 152 11987622
= 05 11987611
= 395 11987622
= 05
4WS
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 195 11987622
= 05 11987611
= 45 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 153 11987622
= 05 11987611
= 35 11987622
= 05
2WS + DYC
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 193 11987622
= 05 11987611
= 37 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 152 11987622
= 05 11987611
= 35 11987622
= 05
Journal of Control Science and Engineering 11
Time (s)0 5 10 15
0
05
1
4WS
minus05
minus1
minus15
times10minus5Re
ar st
eerin
g an
gle120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
1
2
3
Time (s)
2WS + DYC
minus1
minus2
minus3
times10minus3
Dire
ct y
aw m
omen
tM
z(N
m)
0 5 10 15
0
02
0 5 10 15
0
02
04
Time (s)Time (s)
Ref2WS
4WS2WS + DYC
Ref2WS
4WS2WS + DYC
minus02
minus04
minus02
minus04
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
Yaw
angl
e120595
(rad
)
Yaw
rate
120595998400
(rad
sminus1)
Figure 9 2DoF controller at 10msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
may be partly due to the fact that the rear steering and DYCwere not used in lower speed manoeuvres
Moreover when the road surface friction was set to icythe output responses from both controllers and all controlmanoeuvre techniques did not perfectly follow the desiredtrajectory as shown in Figures 10 and 11 2WS with DYCyielded the best performance output compared to the othersespecially in yaw rate response 4WSwas the next best follow-ing 2WS where the lateral position and yaw angle lookalmost identical It seems that the rear steering and direct yawmoment controls were used in conjunction with front steer-ing for vehicle stabilization along the trajectory Howeverfrom Table 5 the controller with 3DoF gives slightly bettertracking error performance manoeuvres compared to con-troller 2DoF due to the fact that roll dynamics will not havemuch influence during low speed manoeuvres Another factis that at low vehicle speed all control inputs (front steeringangle rear steering angle and direct yawmoment control) areunder constraints
Next we tested the vehicle at various road friction coef-ficients again however this time the vehicle was tested undera high forward speed of 30msminus1 The same MPC design wasused of which the parameter controls are listed in Table 4Wecompared the simulation results for 2WS 4WS and 2WSwithDYC control manoeuvres for both controllers and presentthe comparison in Figures 12ndash15The simulation results show
that for 2DoF controller at 30msminus1 and on wet concrete(120583 = 07) all control manoeuvres give slightly similar out-puts performances Better responses were achieved in lateralpositioning and although the tracking performance of yawangle and yaw rate deteriorated the system still allowedthe vehicle to track and follow the trajectory successfullyrejecting the effects of wind gust that impact the vehicle
As tabulated in Table 6 4WS and 2WS with DYC showslightly better tracking performance compared to 2WS onlyHowever in the case of the 3DoF controller it can be clearlyseen that 2WS with DYC and 4WS control manoeuvresoffered amuch higher performing response especially in yawangle and yaw rate response than 2WS This is shown inFigure 13 In this scenario the rear steering and direct yawmoment control was fully utilized to control and enhancevehicle stability It is therefore very important to consider rolldynamics in order to enhance vehicle stability and to followthe path trajectory From Figures 12 and 13 there are somestrong frequency oscillations in front of some responsesthat is front lateral force vehicle side slip angle and lateralacceleration trace based on authorsrsquo knowledge come fromthe numerical calculation or associated glitches and initialcondition of the system
Furthermore when we simulated vehicle manoeuvre forhigh speed (30msminus1) and icy road condition (120583 = 01) for2DoF controller it can be seen that all manoeuvres control
12 Journal of Control Science and Engineering
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
Time (s)0 5 10 15
0
01
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
Time (s)
2WS + DYC
minus01
minus02Dire
ct y
aw m
omen
tM
z(N
m)
Figure 10 2DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 4 Model predictive control weighting matrices parameters for V119909
= 30msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 30msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 565 11987622
= 05 11987611
= 355 11987622
= 05
V119909
= 30msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 0083 11987622
= 0042 11987611
= 003 11987622
= 045
4WS
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 155 11987622
= 25 11987611
= 135 11987622
= 35
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0005 11987622
= 0001 11987611
= 004 11987622
= 05
2WS + DYC
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 505 11987622
= 05 11987611
= 35 11987622
= 05
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0092 11987622
= 004 11987611
= 05 11987622
= 25
simulation the tracking responses become unstable andimpossible to control especially under the crosswind effectas shown in Figure 14 The most probable cause is the rolldynamic motion neglect in the 2DoF controller design whenthe vehicle itself was modelled by including roll dynamicfactor It can be said that the inclusion of roll dynamic factor
is important for vehicle manoeuvre in terms of stability andcontrollability at high speed and icy road condition Sincehigh speed coupled with icy road condition will render theequation to become highly nonlinear it is impossible fora linear tire model to react positively since the handlingpropertiesmay be significantly different from those generated
Journal of Control Science and Engineering 13
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
01
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
002
004
Time (s)
2WS + DYC
minus002
minus004
minus006Dire
ct y
aw m
omen
tM
z(N
m)
Figure 11 3DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 5 Path following tracking errors with road friction surface 120583 = 07
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00637 00085 00623 000744WS 00548 00054 00526 00051
2WS + DYC 00542 00051 00527 00051
302WS 00639 05348 00616 052154WS 00558 05239 00528 00023
2WS + DYC 00549 05226 00524 00254
by the linear tire model These results show that linear tiremodel is only suitable for analyzing a stable vehicle behaviourunder the assumption of small steering and acceleration
Next for the 3DoF controller simulation 2WS with DYCmanoeuvres provides a much better tracking response incomparison to 4WS particularly in the case of lateral outputOn the contrary 4WS manoeuvre demonstrates a much bet-ter tracking response in the yaw angle and yaw rate responsesas tabulated in Table 6 With appropriate weight tuning gainthe rear steering angle and direct yaw moment control arefully optimized in order to become stable along the given tra-jectory However for the 2WS manoeuvre the vehicleresponses become unstable despite several instances wherethe weighting tuning gains are adjusted for output and inputgains With the inclusion of another input control to the con-troller design we can enhance the vehicle responses for both
2WS with DYC and 4WS control manoeuvres show Thismeans that for 2WS controller design it may have to usemore than just front steering to stabilize the vehicle under theeffects of wind gusts Figure 15 shows that rear steering angleand direct yaw moment are fully utilized in order to stabilizethe vehicle for 2WS with DYC and 4WSmanoeuvres control
In contrast to the 2DoF controller as shown in Figure 14in the 3DoF controllers the 2WS control manoeuvre canstill be used to track a given trajectory despite not perfectlyfollowing the trajectory in a satisfactory manner especiallyunder crosswind effects In Figure 15 we can see some oscil-lations in few traces responses at the end of the responsesBased on the authorsrsquo knowledge these oscillationsmay comefrom the vehicle model and some initial conditions of thesystem with imperfect controllers tuning weighting gains Inthese conditions neither 2WS nor 4WS control manoeuvres
14 Journal of Control Science and Engineering
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
Time (s)
2WS4WS2WS + DYC
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
001
002
Time (s)
2WS4WS2WS + DYC
minus001Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
1
2
Time (s)
2WS4WS2WS + DYC
minus1
minus2
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
001
002
003
Time (s)
2WS4WS2WS + DYC
minus001
minus002
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
00005
0001
4WS
Time (s)
minus00005
minus0001
minus00015
Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
001
002
Time (s)
2WS + DYC
minus001
minus002
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 12 2DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
for lateral response and neither 2WS nor 2WS with DYCcontrol manoeuvres for yaw angle or yaw rate responseperformed well causing an increase in vehicle instabilityincreased vibration and tracking responses deteriorationWewill next focus on how to enhance the controller in order tostabilize vehicle manoeuvrability and handling stability
We can therefore conclude that for 4WS the rear wheelswere helping the car to steer by improving the vehiclehandling at high speed while decreasing the turning radiusat low speed as shown by the control signal in Figures 12 and13 Meanwhile for 2WS with DYC active front steering wasused in low speed manoeuvres for lateral acceleration whileinclusion of DYC was adopted for high lateral accelerationwhen the tires were saturated and could not produce enoughlateral force for vehicle control and stability as intendedIn 2WS vehicles the rear set of wheels do not play an
active role in controlling the steering From Table 5 it canbe seen that among the three controllers 4WS gave the bestperformance by reducing the tracking error for lateral andyaw angle responses when compared with 2WS with DYCand 2WS only Tables 5 and 6 tabulated the MPC robustnessand the tracking errors in (17) for all types of manoeuvre at10msminus1 and 30msminus1 vehicle speeds and under various roadsurfaces Furthermore all controlmanoeuvre signals for bothcontrollers were within the input constraints We would liketo highlight that in MPC approaches although there was anadvantage in multivariable systems in this study there was atrade-off between theweighting tuning parameter to focus oneither lateral position or yaw angle trajectory In this paper wehave chosen lateral position as the weighting tuning priorityso we may get a better response on yaw angle and yaw rateresponse by sacrificing lateral position precision
Journal of Control Science and Engineering 15
0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
Time (s)
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
2WS4WS2WS + DYC
Time (s)
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02
Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
5
10
2WS4WS2WS + DYC
Time (s)
minus5
minus10Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
03
4WS
Time (s)
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
2WS + DYC
Time (s)
minus005
minus01
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 13 3DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
Table 6 Path following tracking errors with road friction surface 120583 = 01
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00874 00115 00841 001044WS 00838 00109 00832 00094
2WS + DYC 00838 00110 00835 00083
302WS Uncontrolled Uncontrolled 21352 052644WS Uncontrolled Uncontrolled 08824 02982
2WS + DYC Uncontrolled Uncontrolled 09964 04580
16 Journal of Control Science and Engineering
0 10 20 30
02468
10
Time (s)
Ref2WS
4WS2WS + DYC
minus2
minus4Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0020406
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
minus06
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15
Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
0020406
Time (s)
2WS4WS2WS + DYC
minus02
minus04
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
02
04
06
Time (s)
4WS
minus02
Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
005
01
Time (s)
2WS + DYC
minus005
Dire
ct y
aw m
omen
tM
z(N
m)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15Fron
t lat
eral
forc
eFyf
(Nm
) times104
Figure 14 2DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
5 Conclusion
This paper presents the robustness of model predictive con-trol for an autonomous ground vehicle on a path-followingcontrol with consideration of wind gust effects subjectedto the variation of forward speeds and road surfaces condi-tion in a double-lane change scenario The predictive con-trollers were designed based on two- (lateral-yaw model)and three- (lateral-roll model) degree-of-freedom vehiclemodels The only difference between the models is the rolldynamics factor which is taken into consideration andboth controllers were implemented into a six-degree-of-free-dom vehicle model Based on linear vehicle and tire modelapproximations of a known trajectory we evaluated the effectand impact of roll dynamics at low and high forward vehicle
speeds and at various road frictions (wet concrete to icyroads) in order to follow a desired trajectory as close as pos-sible while rejecting the crosswind effects and maintainingvehicle stability We also evaluated and compared the effi-ciency of the different manoeuvres via two-wheel steeringfour-wheel steering and two-wheel steering with direct yawmoment control
The simulation results showed that with the inclusion ofroll dynamics in the linear vehicle model the vehicle stabilityand adherence to trajectory was considerably improved forthe case of high speed manoeuvres on a higher road surfacefriction coefficient (120583 = 07) and improved even more for thecase of a lower road surface coefficient (120583 = 01) Further-more the obtained results showed and proved that the rearwheels and direct yaw moment are beneficial to help steering
Journal of Control Science and Engineering 17
0 10 20 30
0
5
Time (s)
Ref2WS
4WS2WS + DYC
minus10
minus5
minus15Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0
025
05
Time (s)
Ref2WS
4WS2WS + DYC
minus025
minus05
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
0
500
1000
Time (s)
2WS4WS2WS + DYC
minus500
minus1000
minus1500Fron
t lat
eral
forc
eFyf
(Nm
)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
01
02
03
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS + DYC
minus2
minus4
times10minus4
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 15 3DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
the car by improving the vehicle handling at high speeddecreasing the turning radius at low speeds and reducing thetracking error for lateral position and yaw angle response
The simulation also demonstrated that model predictivecontrol has an ability to reject the side wind effect as adisturbance to the system which is particularly effective at10msminus1 for all cases except for the 2DoF controller duringsimulation of higher speeds on lower road surface adhesionHowever both controllers were not able to provide a betterresponse when the vehicle performed at a high forwardspeedwith a low road adhesion coefficient particularly when2DoF controller was simulated all results showed unstablevehicle response Thus for this scenario we are currentlyinvestigating new approaches with a real-time implementable
control scheme Further improvement for future works oncontrol methodology will also be considered to minimizetracking error and to enhance vehicle stability
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors would like to thank Professor Zidong Wangand Professor Manfred Ploechl for their comments andadvice Special thanks are also extended to Dr A Zahran MKhudzari for his help preparing this paper
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
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Journal of Control Science and Engineering 7
0 5 10 150
5
10
Time (s)
Cros
swin
d sp
eed
w(m
s)
Time (s)0 5 10 15
0
100
200
300
400
Win
d fo
rce
Fw
(N)
Time (s)0 5 10 15
0
minus500
minus1000
Win
d to
rque
Mw
(Nm
)
Figure 5 Force and torque for crosswind speed of 10msminus1
Table 1 Car vehicle parameters
Parameter ValueCar mass [kg] 1300Inertia around the roll 119909-axis [kgm2] 370Inertia around 119911-axis [kgm2] 216756Distance of front and rear wheels from centre of gravity (CoG) [m] 145 107Distance between the vehicle CoG and the assumed roll axis [m] 045Equivalent suspension roll damping coefficient [Nms] 4200Equivalent suspension roll stiffness coefficient [Nm] 40000Linear approximation of tire stiffness for front and rear tire [Nrad] 133000 121000Gravitational constant [ms2] 98
achieve the main aim which is to follow the desired or refer-ence trajectory as close as possible The controllers are com-pared against each other for 2DoF and 3DoF bicycle modelswhich excludes vehicle roll dynamics at various speedsspecifically low (10msminus1) and high (30msminus1) and also underwet concrete (120583 = 07) and icy (120583 = 01) road surfaces
Since our study focused on an autonomous vehicle it isreasonable to assume that the controllers will adapt to theroad or environment such as in the case of road adhesionsadverse weather visual cues traffic and overall relevantdriving conditions Therefore both controllers (2DoF and3DoF) were designed and implemented with the parametersand conditions given in Tables 2 3 and 4 for different vehicleforward speeds and road adhesion coefficients The path-following tracking error is a measurement of how closelythe output responses follow the reference trajectory whichis a measure of the deviation from the benchmark In this
paper we use the standard deviation of the root-mean squareformula to calculate the tracking error given by the followingequation
119890119905
= radicVar (119910119900
minus 119910119903
) = radic1
(119899 minus 1)sum (119910119900
minus 119910119903
)2
(17)
where 119890119905
is the tracking error 119899 represents the number oftime periods and 119910
119900
and 119910119903
express the measured output andreference
42 Results and Discussions Prior to proposed simulationswe carried out standard vehicle manoeuvres tests to validateour model in the open-loop simulation scenario Standardtests available are the J-turn fishhook and double-lanechange [32] which are representative of on-roadmanoeuvreswithout tripping (caused by external force from road curb or
8 Journal of Control Science and Engineering
0 50
0
10
20
30
40
50Root locus
Imag
inar
y ax
is (sminus1)
minus10
minus20
minus30
minus40
minus50minus50minus100minus150minus200minus250minus300
Real axis (sminus1)
120595998400
0 20
0
2
4
6
8
10Root locus
Imag
inar
y ax
is (sminus1)
minus2
minus4
minus6
minus8
minus10minus20minus40minus60minus80minus100minus120minus140
Real axis (sminus1)
120573
(a) Without crosswind effect
0 20 40 60
0
2
4
6
8
10Root locus
Imag
inar
y ax
is (sminus1)
Real axis (sminus1)
minus2
minus4
minus6
minus8
minus10
120595998400
minus20minus40minus60minus800 20
0
5
10
15
20
25Root locus
Imag
inar
y ax
is (sminus1)
minus5
minus10
minus15
minus20
minus25
Real axis (sminus1)
120573
minus20minus40minus60minus80minus100minus120minus140
(b) With crosswind effect
Figure 6 Open-loop system in (7)
collisionwith other vehicles)We performed only the double-lane change and roll rate feedback fishhook test for vehiclevalidation purposes in the open-loop simulation as shown inFigures 7 and 8 The vehicle speed was set at 30msminus1 whichis suitable for both tests with a road surface coefficient of 01As illustrated in Figures 7 and 8 the vehicle response in terms
of yaw rate roll rate and lateral acceleration has been provento be validated thus being suitable for other simulations
For easier comparison between controllersrsquo perfor-mances in achieving vehicle stabilization when subjectedto various conditions controllers tuning parameters fromTables 3 and 4 were selected as suitable The weighting gains
Journal of Control Science and Engineering 9
0 5 10 15
0
05
1
Time (s)
minus05
minus10 5 10 15
0
01
02
Time (s)
minus01
minus02
minus030 5 10 15
0
05
1
Time (s)
minus1
minus05
Tire
slip
angl
e120572f
(rad
)
0 5 10 15
0
05
1
Time (s)
minus05Yaw
angl
e res
pons
e120595
(rad
)
0 5 10 15
0
5
10
Time (s)
minus5
minus10
Fron
t ste
erin
g an
gle120575f
(deg
)
5 10 15
0
01
02
Time (s)
minus01
minus02
minus03Roll
angl
e res
pons
e120601
(rad
)
Time (s)0 5 10 15
0
10
20
30
40
50
Late
ral p
ositi
onY
(m)
0 5 10 15
0
2000
4000
Time (s)
minus2000
minus4000
Late
ral f
orce
Fyf
(N)
0 5 10 15Time (s)
10
5
0
minus5
minus10
Late
ral a
ccel
erat
ion
a y(m
s2)
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 7 Vehicle manoeuvre test of single-lane change at 30msminus1 with 120583 = 01
Table 2 Model predictive control parameters
Parameter ValuePredictive horizon 20Control horizon 9Sampling time [s] 005Constraints on max and min steering angles [deg] plusmn20Constraints on max and min changes in steering angles [degs] plusmn10Constraints on max and min changes in moment torque [Nmrad] plusmn2000Constraints on max and min moment torque [Nmrads] plusmn1500Simulation time [s] 15
for controllers input and output were selected using trial-and-error method where selection criteria were based on thebest response output for both input and output for weightinggains
The first simulation revolves around various road surfacefriction coefficients from wet concrete (120583 = 07) to icy sur-face condition (120583 = 01) with a constant forward speed of10msminus1 The MPC weighting tuning parameters are listedin Table 3 We evaluated the controllerrsquos robustness for theoutput responses by comparing the performance of 2WS4WS and 2WS with DYC manoeuvres at a forward speed of
10msminus1 as shown in Figures 9ndash11 The controller trackingerrors based on (17) for lateral position and yaw angleresponses are summarized in Table 5 From Figure 9 thereare not many differences between both controllers underthe conditions of lowest speed and high road adhesioncoefficient both controllers yielded perfect response whenfollowing a given trajectory while maintaining vehicle stabil-ity and rejecting external crosswind effect It can be seen thatfor 4WS and 2WS with DYC manoeuvres the rear steeringangle and the direct yaw moment are almost unused sincethe front steering can provide sufficient control ability This
10 Journal of Control Science and Engineering
0 5 10 15
0
50
100
Time (s)
minus50
minus100
Late
ral a
ccel
erat
ion
a y(m
s2)
Time (s)0 5 10 15
0
1000
2000
3000
minus1000Late
ral p
ositi
onY
(m)
0 5 10 15
0
2
4
Time (s)
minus2
minus4
times104
Late
ral f
orce
Fyf
(N)
0 5 10 15
0
200
400
Time (s)
minus200
minus400
Fron
t ste
erin
g an
gle120575f
(deg
)
0 5 10 15
0
50
100
150
200
Time (s)
minus50
Yaw
angl
e res
pons
e120595
(rad
)
5 10 15
0
1
2
3
Time (s)
minus1
minus2Roll
angl
e res
pons
e120601
(rad
)
0 5 10 15
0
1
2
Time (s)
minus1
minus2Vehi
cle sl
ip an
gle120573
(rad
)
0 5 10 15
0
20
40
60
Time (s)
minus200 5 10 15
0
5
10
Time (s)
minus5
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 8 Vehicle manoeuvre test of roll rate feedback at 30msminus1 with 120583 = 01
Table 3 Model predictive control weighting matrices parameters for V119909
= 10msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 10msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 195 11987622
= 05 11987611
= 525 11987622
= 05
V119909
= 10msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 152 11987622
= 05 11987611
= 395 11987622
= 05
4WS
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 195 11987622
= 05 11987611
= 45 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 153 11987622
= 05 11987611
= 35 11987622
= 05
2WS + DYC
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 193 11987622
= 05 11987611
= 37 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 152 11987622
= 05 11987611
= 35 11987622
= 05
Journal of Control Science and Engineering 11
Time (s)0 5 10 15
0
05
1
4WS
minus05
minus1
minus15
times10minus5Re
ar st
eerin
g an
gle120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
1
2
3
Time (s)
2WS + DYC
minus1
minus2
minus3
times10minus3
Dire
ct y
aw m
omen
tM
z(N
m)
0 5 10 15
0
02
0 5 10 15
0
02
04
Time (s)Time (s)
Ref2WS
4WS2WS + DYC
Ref2WS
4WS2WS + DYC
minus02
minus04
minus02
minus04
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
Yaw
angl
e120595
(rad
)
Yaw
rate
120595998400
(rad
sminus1)
Figure 9 2DoF controller at 10msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
may be partly due to the fact that the rear steering and DYCwere not used in lower speed manoeuvres
Moreover when the road surface friction was set to icythe output responses from both controllers and all controlmanoeuvre techniques did not perfectly follow the desiredtrajectory as shown in Figures 10 and 11 2WS with DYCyielded the best performance output compared to the othersespecially in yaw rate response 4WSwas the next best follow-ing 2WS where the lateral position and yaw angle lookalmost identical It seems that the rear steering and direct yawmoment controls were used in conjunction with front steer-ing for vehicle stabilization along the trajectory Howeverfrom Table 5 the controller with 3DoF gives slightly bettertracking error performance manoeuvres compared to con-troller 2DoF due to the fact that roll dynamics will not havemuch influence during low speed manoeuvres Another factis that at low vehicle speed all control inputs (front steeringangle rear steering angle and direct yawmoment control) areunder constraints
Next we tested the vehicle at various road friction coef-ficients again however this time the vehicle was tested undera high forward speed of 30msminus1 The same MPC design wasused of which the parameter controls are listed in Table 4Wecompared the simulation results for 2WS 4WS and 2WSwithDYC control manoeuvres for both controllers and presentthe comparison in Figures 12ndash15The simulation results show
that for 2DoF controller at 30msminus1 and on wet concrete(120583 = 07) all control manoeuvres give slightly similar out-puts performances Better responses were achieved in lateralpositioning and although the tracking performance of yawangle and yaw rate deteriorated the system still allowedthe vehicle to track and follow the trajectory successfullyrejecting the effects of wind gust that impact the vehicle
As tabulated in Table 6 4WS and 2WS with DYC showslightly better tracking performance compared to 2WS onlyHowever in the case of the 3DoF controller it can be clearlyseen that 2WS with DYC and 4WS control manoeuvresoffered amuch higher performing response especially in yawangle and yaw rate response than 2WS This is shown inFigure 13 In this scenario the rear steering and direct yawmoment control was fully utilized to control and enhancevehicle stability It is therefore very important to consider rolldynamics in order to enhance vehicle stability and to followthe path trajectory From Figures 12 and 13 there are somestrong frequency oscillations in front of some responsesthat is front lateral force vehicle side slip angle and lateralacceleration trace based on authorsrsquo knowledge come fromthe numerical calculation or associated glitches and initialcondition of the system
Furthermore when we simulated vehicle manoeuvre forhigh speed (30msminus1) and icy road condition (120583 = 01) for2DoF controller it can be seen that all manoeuvres control
12 Journal of Control Science and Engineering
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
Time (s)0 5 10 15
0
01
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
Time (s)
2WS + DYC
minus01
minus02Dire
ct y
aw m
omen
tM
z(N
m)
Figure 10 2DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 4 Model predictive control weighting matrices parameters for V119909
= 30msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 30msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 565 11987622
= 05 11987611
= 355 11987622
= 05
V119909
= 30msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 0083 11987622
= 0042 11987611
= 003 11987622
= 045
4WS
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 155 11987622
= 25 11987611
= 135 11987622
= 35
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0005 11987622
= 0001 11987611
= 004 11987622
= 05
2WS + DYC
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 505 11987622
= 05 11987611
= 35 11987622
= 05
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0092 11987622
= 004 11987611
= 05 11987622
= 25
simulation the tracking responses become unstable andimpossible to control especially under the crosswind effectas shown in Figure 14 The most probable cause is the rolldynamic motion neglect in the 2DoF controller design whenthe vehicle itself was modelled by including roll dynamicfactor It can be said that the inclusion of roll dynamic factor
is important for vehicle manoeuvre in terms of stability andcontrollability at high speed and icy road condition Sincehigh speed coupled with icy road condition will render theequation to become highly nonlinear it is impossible fora linear tire model to react positively since the handlingpropertiesmay be significantly different from those generated
Journal of Control Science and Engineering 13
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
01
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
002
004
Time (s)
2WS + DYC
minus002
minus004
minus006Dire
ct y
aw m
omen
tM
z(N
m)
Figure 11 3DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 5 Path following tracking errors with road friction surface 120583 = 07
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00637 00085 00623 000744WS 00548 00054 00526 00051
2WS + DYC 00542 00051 00527 00051
302WS 00639 05348 00616 052154WS 00558 05239 00528 00023
2WS + DYC 00549 05226 00524 00254
by the linear tire model These results show that linear tiremodel is only suitable for analyzing a stable vehicle behaviourunder the assumption of small steering and acceleration
Next for the 3DoF controller simulation 2WS with DYCmanoeuvres provides a much better tracking response incomparison to 4WS particularly in the case of lateral outputOn the contrary 4WS manoeuvre demonstrates a much bet-ter tracking response in the yaw angle and yaw rate responsesas tabulated in Table 6 With appropriate weight tuning gainthe rear steering angle and direct yaw moment control arefully optimized in order to become stable along the given tra-jectory However for the 2WS manoeuvre the vehicleresponses become unstable despite several instances wherethe weighting tuning gains are adjusted for output and inputgains With the inclusion of another input control to the con-troller design we can enhance the vehicle responses for both
2WS with DYC and 4WS control manoeuvres show Thismeans that for 2WS controller design it may have to usemore than just front steering to stabilize the vehicle under theeffects of wind gusts Figure 15 shows that rear steering angleand direct yaw moment are fully utilized in order to stabilizethe vehicle for 2WS with DYC and 4WSmanoeuvres control
In contrast to the 2DoF controller as shown in Figure 14in the 3DoF controllers the 2WS control manoeuvre canstill be used to track a given trajectory despite not perfectlyfollowing the trajectory in a satisfactory manner especiallyunder crosswind effects In Figure 15 we can see some oscil-lations in few traces responses at the end of the responsesBased on the authorsrsquo knowledge these oscillationsmay comefrom the vehicle model and some initial conditions of thesystem with imperfect controllers tuning weighting gains Inthese conditions neither 2WS nor 4WS control manoeuvres
14 Journal of Control Science and Engineering
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
Time (s)
2WS4WS2WS + DYC
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
001
002
Time (s)
2WS4WS2WS + DYC
minus001Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
1
2
Time (s)
2WS4WS2WS + DYC
minus1
minus2
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
001
002
003
Time (s)
2WS4WS2WS + DYC
minus001
minus002
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
00005
0001
4WS
Time (s)
minus00005
minus0001
minus00015
Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
001
002
Time (s)
2WS + DYC
minus001
minus002
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 12 2DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
for lateral response and neither 2WS nor 2WS with DYCcontrol manoeuvres for yaw angle or yaw rate responseperformed well causing an increase in vehicle instabilityincreased vibration and tracking responses deteriorationWewill next focus on how to enhance the controller in order tostabilize vehicle manoeuvrability and handling stability
We can therefore conclude that for 4WS the rear wheelswere helping the car to steer by improving the vehiclehandling at high speed while decreasing the turning radiusat low speed as shown by the control signal in Figures 12 and13 Meanwhile for 2WS with DYC active front steering wasused in low speed manoeuvres for lateral acceleration whileinclusion of DYC was adopted for high lateral accelerationwhen the tires were saturated and could not produce enoughlateral force for vehicle control and stability as intendedIn 2WS vehicles the rear set of wheels do not play an
active role in controlling the steering From Table 5 it canbe seen that among the three controllers 4WS gave the bestperformance by reducing the tracking error for lateral andyaw angle responses when compared with 2WS with DYCand 2WS only Tables 5 and 6 tabulated the MPC robustnessand the tracking errors in (17) for all types of manoeuvre at10msminus1 and 30msminus1 vehicle speeds and under various roadsurfaces Furthermore all controlmanoeuvre signals for bothcontrollers were within the input constraints We would liketo highlight that in MPC approaches although there was anadvantage in multivariable systems in this study there was atrade-off between theweighting tuning parameter to focus oneither lateral position or yaw angle trajectory In this paper wehave chosen lateral position as the weighting tuning priorityso we may get a better response on yaw angle and yaw rateresponse by sacrificing lateral position precision
Journal of Control Science and Engineering 15
0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
Time (s)
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
2WS4WS2WS + DYC
Time (s)
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02
Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
5
10
2WS4WS2WS + DYC
Time (s)
minus5
minus10Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
03
4WS
Time (s)
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
2WS + DYC
Time (s)
minus005
minus01
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 13 3DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
Table 6 Path following tracking errors with road friction surface 120583 = 01
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00874 00115 00841 001044WS 00838 00109 00832 00094
2WS + DYC 00838 00110 00835 00083
302WS Uncontrolled Uncontrolled 21352 052644WS Uncontrolled Uncontrolled 08824 02982
2WS + DYC Uncontrolled Uncontrolled 09964 04580
16 Journal of Control Science and Engineering
0 10 20 30
02468
10
Time (s)
Ref2WS
4WS2WS + DYC
minus2
minus4Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0020406
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
minus06
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15
Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
0020406
Time (s)
2WS4WS2WS + DYC
minus02
minus04
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
02
04
06
Time (s)
4WS
minus02
Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
005
01
Time (s)
2WS + DYC
minus005
Dire
ct y
aw m
omen
tM
z(N
m)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15Fron
t lat
eral
forc
eFyf
(Nm
) times104
Figure 14 2DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
5 Conclusion
This paper presents the robustness of model predictive con-trol for an autonomous ground vehicle on a path-followingcontrol with consideration of wind gust effects subjectedto the variation of forward speeds and road surfaces condi-tion in a double-lane change scenario The predictive con-trollers were designed based on two- (lateral-yaw model)and three- (lateral-roll model) degree-of-freedom vehiclemodels The only difference between the models is the rolldynamics factor which is taken into consideration andboth controllers were implemented into a six-degree-of-free-dom vehicle model Based on linear vehicle and tire modelapproximations of a known trajectory we evaluated the effectand impact of roll dynamics at low and high forward vehicle
speeds and at various road frictions (wet concrete to icyroads) in order to follow a desired trajectory as close as pos-sible while rejecting the crosswind effects and maintainingvehicle stability We also evaluated and compared the effi-ciency of the different manoeuvres via two-wheel steeringfour-wheel steering and two-wheel steering with direct yawmoment control
The simulation results showed that with the inclusion ofroll dynamics in the linear vehicle model the vehicle stabilityand adherence to trajectory was considerably improved forthe case of high speed manoeuvres on a higher road surfacefriction coefficient (120583 = 07) and improved even more for thecase of a lower road surface coefficient (120583 = 01) Further-more the obtained results showed and proved that the rearwheels and direct yaw moment are beneficial to help steering
Journal of Control Science and Engineering 17
0 10 20 30
0
5
Time (s)
Ref2WS
4WS2WS + DYC
minus10
minus5
minus15Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0
025
05
Time (s)
Ref2WS
4WS2WS + DYC
minus025
minus05
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
0
500
1000
Time (s)
2WS4WS2WS + DYC
minus500
minus1000
minus1500Fron
t lat
eral
forc
eFyf
(Nm
)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
01
02
03
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS + DYC
minus2
minus4
times10minus4
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 15 3DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
the car by improving the vehicle handling at high speeddecreasing the turning radius at low speeds and reducing thetracking error for lateral position and yaw angle response
The simulation also demonstrated that model predictivecontrol has an ability to reject the side wind effect as adisturbance to the system which is particularly effective at10msminus1 for all cases except for the 2DoF controller duringsimulation of higher speeds on lower road surface adhesionHowever both controllers were not able to provide a betterresponse when the vehicle performed at a high forwardspeedwith a low road adhesion coefficient particularly when2DoF controller was simulated all results showed unstablevehicle response Thus for this scenario we are currentlyinvestigating new approaches with a real-time implementable
control scheme Further improvement for future works oncontrol methodology will also be considered to minimizetracking error and to enhance vehicle stability
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors would like to thank Professor Zidong Wangand Professor Manfred Ploechl for their comments andadvice Special thanks are also extended to Dr A Zahran MKhudzari for his help preparing this paper
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
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Submit your manuscripts athttpwwwhindawicom
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DistributedSensor Networks
International Journal of
8 Journal of Control Science and Engineering
0 50
0
10
20
30
40
50Root locus
Imag
inar
y ax
is (sminus1)
minus10
minus20
minus30
minus40
minus50minus50minus100minus150minus200minus250minus300
Real axis (sminus1)
120595998400
0 20
0
2
4
6
8
10Root locus
Imag
inar
y ax
is (sminus1)
minus2
minus4
minus6
minus8
minus10minus20minus40minus60minus80minus100minus120minus140
Real axis (sminus1)
120573
(a) Without crosswind effect
0 20 40 60
0
2
4
6
8
10Root locus
Imag
inar
y ax
is (sminus1)
Real axis (sminus1)
minus2
minus4
minus6
minus8
minus10
120595998400
minus20minus40minus60minus800 20
0
5
10
15
20
25Root locus
Imag
inar
y ax
is (sminus1)
minus5
minus10
minus15
minus20
minus25
Real axis (sminus1)
120573
minus20minus40minus60minus80minus100minus120minus140
(b) With crosswind effect
Figure 6 Open-loop system in (7)
collisionwith other vehicles)We performed only the double-lane change and roll rate feedback fishhook test for vehiclevalidation purposes in the open-loop simulation as shown inFigures 7 and 8 The vehicle speed was set at 30msminus1 whichis suitable for both tests with a road surface coefficient of 01As illustrated in Figures 7 and 8 the vehicle response in terms
of yaw rate roll rate and lateral acceleration has been provento be validated thus being suitable for other simulations
For easier comparison between controllersrsquo perfor-mances in achieving vehicle stabilization when subjectedto various conditions controllers tuning parameters fromTables 3 and 4 were selected as suitable The weighting gains
Journal of Control Science and Engineering 9
0 5 10 15
0
05
1
Time (s)
minus05
minus10 5 10 15
0
01
02
Time (s)
minus01
minus02
minus030 5 10 15
0
05
1
Time (s)
minus1
minus05
Tire
slip
angl
e120572f
(rad
)
0 5 10 15
0
05
1
Time (s)
minus05Yaw
angl
e res
pons
e120595
(rad
)
0 5 10 15
0
5
10
Time (s)
minus5
minus10
Fron
t ste
erin
g an
gle120575f
(deg
)
5 10 15
0
01
02
Time (s)
minus01
minus02
minus03Roll
angl
e res
pons
e120601
(rad
)
Time (s)0 5 10 15
0
10
20
30
40
50
Late
ral p
ositi
onY
(m)
0 5 10 15
0
2000
4000
Time (s)
minus2000
minus4000
Late
ral f
orce
Fyf
(N)
0 5 10 15Time (s)
10
5
0
minus5
minus10
Late
ral a
ccel
erat
ion
a y(m
s2)
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 7 Vehicle manoeuvre test of single-lane change at 30msminus1 with 120583 = 01
Table 2 Model predictive control parameters
Parameter ValuePredictive horizon 20Control horizon 9Sampling time [s] 005Constraints on max and min steering angles [deg] plusmn20Constraints on max and min changes in steering angles [degs] plusmn10Constraints on max and min changes in moment torque [Nmrad] plusmn2000Constraints on max and min moment torque [Nmrads] plusmn1500Simulation time [s] 15
for controllers input and output were selected using trial-and-error method where selection criteria were based on thebest response output for both input and output for weightinggains
The first simulation revolves around various road surfacefriction coefficients from wet concrete (120583 = 07) to icy sur-face condition (120583 = 01) with a constant forward speed of10msminus1 The MPC weighting tuning parameters are listedin Table 3 We evaluated the controllerrsquos robustness for theoutput responses by comparing the performance of 2WS4WS and 2WS with DYC manoeuvres at a forward speed of
10msminus1 as shown in Figures 9ndash11 The controller trackingerrors based on (17) for lateral position and yaw angleresponses are summarized in Table 5 From Figure 9 thereare not many differences between both controllers underthe conditions of lowest speed and high road adhesioncoefficient both controllers yielded perfect response whenfollowing a given trajectory while maintaining vehicle stabil-ity and rejecting external crosswind effect It can be seen thatfor 4WS and 2WS with DYC manoeuvres the rear steeringangle and the direct yaw moment are almost unused sincethe front steering can provide sufficient control ability This
10 Journal of Control Science and Engineering
0 5 10 15
0
50
100
Time (s)
minus50
minus100
Late
ral a
ccel
erat
ion
a y(m
s2)
Time (s)0 5 10 15
0
1000
2000
3000
minus1000Late
ral p
ositi
onY
(m)
0 5 10 15
0
2
4
Time (s)
minus2
minus4
times104
Late
ral f
orce
Fyf
(N)
0 5 10 15
0
200
400
Time (s)
minus200
minus400
Fron
t ste
erin
g an
gle120575f
(deg
)
0 5 10 15
0
50
100
150
200
Time (s)
minus50
Yaw
angl
e res
pons
e120595
(rad
)
5 10 15
0
1
2
3
Time (s)
minus1
minus2Roll
angl
e res
pons
e120601
(rad
)
0 5 10 15
0
1
2
Time (s)
minus1
minus2Vehi
cle sl
ip an
gle120573
(rad
)
0 5 10 15
0
20
40
60
Time (s)
minus200 5 10 15
0
5
10
Time (s)
minus5
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 8 Vehicle manoeuvre test of roll rate feedback at 30msminus1 with 120583 = 01
Table 3 Model predictive control weighting matrices parameters for V119909
= 10msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 10msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 195 11987622
= 05 11987611
= 525 11987622
= 05
V119909
= 10msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 152 11987622
= 05 11987611
= 395 11987622
= 05
4WS
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 195 11987622
= 05 11987611
= 45 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 153 11987622
= 05 11987611
= 35 11987622
= 05
2WS + DYC
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 193 11987622
= 05 11987611
= 37 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 152 11987622
= 05 11987611
= 35 11987622
= 05
Journal of Control Science and Engineering 11
Time (s)0 5 10 15
0
05
1
4WS
minus05
minus1
minus15
times10minus5Re
ar st
eerin
g an
gle120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
1
2
3
Time (s)
2WS + DYC
minus1
minus2
minus3
times10minus3
Dire
ct y
aw m
omen
tM
z(N
m)
0 5 10 15
0
02
0 5 10 15
0
02
04
Time (s)Time (s)
Ref2WS
4WS2WS + DYC
Ref2WS
4WS2WS + DYC
minus02
minus04
minus02
minus04
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
Yaw
angl
e120595
(rad
)
Yaw
rate
120595998400
(rad
sminus1)
Figure 9 2DoF controller at 10msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
may be partly due to the fact that the rear steering and DYCwere not used in lower speed manoeuvres
Moreover when the road surface friction was set to icythe output responses from both controllers and all controlmanoeuvre techniques did not perfectly follow the desiredtrajectory as shown in Figures 10 and 11 2WS with DYCyielded the best performance output compared to the othersespecially in yaw rate response 4WSwas the next best follow-ing 2WS where the lateral position and yaw angle lookalmost identical It seems that the rear steering and direct yawmoment controls were used in conjunction with front steer-ing for vehicle stabilization along the trajectory Howeverfrom Table 5 the controller with 3DoF gives slightly bettertracking error performance manoeuvres compared to con-troller 2DoF due to the fact that roll dynamics will not havemuch influence during low speed manoeuvres Another factis that at low vehicle speed all control inputs (front steeringangle rear steering angle and direct yawmoment control) areunder constraints
Next we tested the vehicle at various road friction coef-ficients again however this time the vehicle was tested undera high forward speed of 30msminus1 The same MPC design wasused of which the parameter controls are listed in Table 4Wecompared the simulation results for 2WS 4WS and 2WSwithDYC control manoeuvres for both controllers and presentthe comparison in Figures 12ndash15The simulation results show
that for 2DoF controller at 30msminus1 and on wet concrete(120583 = 07) all control manoeuvres give slightly similar out-puts performances Better responses were achieved in lateralpositioning and although the tracking performance of yawangle and yaw rate deteriorated the system still allowedthe vehicle to track and follow the trajectory successfullyrejecting the effects of wind gust that impact the vehicle
As tabulated in Table 6 4WS and 2WS with DYC showslightly better tracking performance compared to 2WS onlyHowever in the case of the 3DoF controller it can be clearlyseen that 2WS with DYC and 4WS control manoeuvresoffered amuch higher performing response especially in yawangle and yaw rate response than 2WS This is shown inFigure 13 In this scenario the rear steering and direct yawmoment control was fully utilized to control and enhancevehicle stability It is therefore very important to consider rolldynamics in order to enhance vehicle stability and to followthe path trajectory From Figures 12 and 13 there are somestrong frequency oscillations in front of some responsesthat is front lateral force vehicle side slip angle and lateralacceleration trace based on authorsrsquo knowledge come fromthe numerical calculation or associated glitches and initialcondition of the system
Furthermore when we simulated vehicle manoeuvre forhigh speed (30msminus1) and icy road condition (120583 = 01) for2DoF controller it can be seen that all manoeuvres control
12 Journal of Control Science and Engineering
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
Time (s)0 5 10 15
0
01
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
Time (s)
2WS + DYC
minus01
minus02Dire
ct y
aw m
omen
tM
z(N
m)
Figure 10 2DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 4 Model predictive control weighting matrices parameters for V119909
= 30msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 30msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 565 11987622
= 05 11987611
= 355 11987622
= 05
V119909
= 30msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 0083 11987622
= 0042 11987611
= 003 11987622
= 045
4WS
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 155 11987622
= 25 11987611
= 135 11987622
= 35
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0005 11987622
= 0001 11987611
= 004 11987622
= 05
2WS + DYC
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 505 11987622
= 05 11987611
= 35 11987622
= 05
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0092 11987622
= 004 11987611
= 05 11987622
= 25
simulation the tracking responses become unstable andimpossible to control especially under the crosswind effectas shown in Figure 14 The most probable cause is the rolldynamic motion neglect in the 2DoF controller design whenthe vehicle itself was modelled by including roll dynamicfactor It can be said that the inclusion of roll dynamic factor
is important for vehicle manoeuvre in terms of stability andcontrollability at high speed and icy road condition Sincehigh speed coupled with icy road condition will render theequation to become highly nonlinear it is impossible fora linear tire model to react positively since the handlingpropertiesmay be significantly different from those generated
Journal of Control Science and Engineering 13
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
01
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
002
004
Time (s)
2WS + DYC
minus002
minus004
minus006Dire
ct y
aw m
omen
tM
z(N
m)
Figure 11 3DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 5 Path following tracking errors with road friction surface 120583 = 07
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00637 00085 00623 000744WS 00548 00054 00526 00051
2WS + DYC 00542 00051 00527 00051
302WS 00639 05348 00616 052154WS 00558 05239 00528 00023
2WS + DYC 00549 05226 00524 00254
by the linear tire model These results show that linear tiremodel is only suitable for analyzing a stable vehicle behaviourunder the assumption of small steering and acceleration
Next for the 3DoF controller simulation 2WS with DYCmanoeuvres provides a much better tracking response incomparison to 4WS particularly in the case of lateral outputOn the contrary 4WS manoeuvre demonstrates a much bet-ter tracking response in the yaw angle and yaw rate responsesas tabulated in Table 6 With appropriate weight tuning gainthe rear steering angle and direct yaw moment control arefully optimized in order to become stable along the given tra-jectory However for the 2WS manoeuvre the vehicleresponses become unstable despite several instances wherethe weighting tuning gains are adjusted for output and inputgains With the inclusion of another input control to the con-troller design we can enhance the vehicle responses for both
2WS with DYC and 4WS control manoeuvres show Thismeans that for 2WS controller design it may have to usemore than just front steering to stabilize the vehicle under theeffects of wind gusts Figure 15 shows that rear steering angleand direct yaw moment are fully utilized in order to stabilizethe vehicle for 2WS with DYC and 4WSmanoeuvres control
In contrast to the 2DoF controller as shown in Figure 14in the 3DoF controllers the 2WS control manoeuvre canstill be used to track a given trajectory despite not perfectlyfollowing the trajectory in a satisfactory manner especiallyunder crosswind effects In Figure 15 we can see some oscil-lations in few traces responses at the end of the responsesBased on the authorsrsquo knowledge these oscillationsmay comefrom the vehicle model and some initial conditions of thesystem with imperfect controllers tuning weighting gains Inthese conditions neither 2WS nor 4WS control manoeuvres
14 Journal of Control Science and Engineering
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
Time (s)
2WS4WS2WS + DYC
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
001
002
Time (s)
2WS4WS2WS + DYC
minus001Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
1
2
Time (s)
2WS4WS2WS + DYC
minus1
minus2
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
001
002
003
Time (s)
2WS4WS2WS + DYC
minus001
minus002
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
00005
0001
4WS
Time (s)
minus00005
minus0001
minus00015
Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
001
002
Time (s)
2WS + DYC
minus001
minus002
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 12 2DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
for lateral response and neither 2WS nor 2WS with DYCcontrol manoeuvres for yaw angle or yaw rate responseperformed well causing an increase in vehicle instabilityincreased vibration and tracking responses deteriorationWewill next focus on how to enhance the controller in order tostabilize vehicle manoeuvrability and handling stability
We can therefore conclude that for 4WS the rear wheelswere helping the car to steer by improving the vehiclehandling at high speed while decreasing the turning radiusat low speed as shown by the control signal in Figures 12 and13 Meanwhile for 2WS with DYC active front steering wasused in low speed manoeuvres for lateral acceleration whileinclusion of DYC was adopted for high lateral accelerationwhen the tires were saturated and could not produce enoughlateral force for vehicle control and stability as intendedIn 2WS vehicles the rear set of wheels do not play an
active role in controlling the steering From Table 5 it canbe seen that among the three controllers 4WS gave the bestperformance by reducing the tracking error for lateral andyaw angle responses when compared with 2WS with DYCand 2WS only Tables 5 and 6 tabulated the MPC robustnessand the tracking errors in (17) for all types of manoeuvre at10msminus1 and 30msminus1 vehicle speeds and under various roadsurfaces Furthermore all controlmanoeuvre signals for bothcontrollers were within the input constraints We would liketo highlight that in MPC approaches although there was anadvantage in multivariable systems in this study there was atrade-off between theweighting tuning parameter to focus oneither lateral position or yaw angle trajectory In this paper wehave chosen lateral position as the weighting tuning priorityso we may get a better response on yaw angle and yaw rateresponse by sacrificing lateral position precision
Journal of Control Science and Engineering 15
0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
Time (s)
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
2WS4WS2WS + DYC
Time (s)
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02
Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
5
10
2WS4WS2WS + DYC
Time (s)
minus5
minus10Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
03
4WS
Time (s)
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
2WS + DYC
Time (s)
minus005
minus01
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 13 3DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
Table 6 Path following tracking errors with road friction surface 120583 = 01
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00874 00115 00841 001044WS 00838 00109 00832 00094
2WS + DYC 00838 00110 00835 00083
302WS Uncontrolled Uncontrolled 21352 052644WS Uncontrolled Uncontrolled 08824 02982
2WS + DYC Uncontrolled Uncontrolled 09964 04580
16 Journal of Control Science and Engineering
0 10 20 30
02468
10
Time (s)
Ref2WS
4WS2WS + DYC
minus2
minus4Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0020406
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
minus06
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15
Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
0020406
Time (s)
2WS4WS2WS + DYC
minus02
minus04
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
02
04
06
Time (s)
4WS
minus02
Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
005
01
Time (s)
2WS + DYC
minus005
Dire
ct y
aw m
omen
tM
z(N
m)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15Fron
t lat
eral
forc
eFyf
(Nm
) times104
Figure 14 2DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
5 Conclusion
This paper presents the robustness of model predictive con-trol for an autonomous ground vehicle on a path-followingcontrol with consideration of wind gust effects subjectedto the variation of forward speeds and road surfaces condi-tion in a double-lane change scenario The predictive con-trollers were designed based on two- (lateral-yaw model)and three- (lateral-roll model) degree-of-freedom vehiclemodels The only difference between the models is the rolldynamics factor which is taken into consideration andboth controllers were implemented into a six-degree-of-free-dom vehicle model Based on linear vehicle and tire modelapproximations of a known trajectory we evaluated the effectand impact of roll dynamics at low and high forward vehicle
speeds and at various road frictions (wet concrete to icyroads) in order to follow a desired trajectory as close as pos-sible while rejecting the crosswind effects and maintainingvehicle stability We also evaluated and compared the effi-ciency of the different manoeuvres via two-wheel steeringfour-wheel steering and two-wheel steering with direct yawmoment control
The simulation results showed that with the inclusion ofroll dynamics in the linear vehicle model the vehicle stabilityand adherence to trajectory was considerably improved forthe case of high speed manoeuvres on a higher road surfacefriction coefficient (120583 = 07) and improved even more for thecase of a lower road surface coefficient (120583 = 01) Further-more the obtained results showed and proved that the rearwheels and direct yaw moment are beneficial to help steering
Journal of Control Science and Engineering 17
0 10 20 30
0
5
Time (s)
Ref2WS
4WS2WS + DYC
minus10
minus5
minus15Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0
025
05
Time (s)
Ref2WS
4WS2WS + DYC
minus025
minus05
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
0
500
1000
Time (s)
2WS4WS2WS + DYC
minus500
minus1000
minus1500Fron
t lat
eral
forc
eFyf
(Nm
)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
01
02
03
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS + DYC
minus2
minus4
times10minus4
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 15 3DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
the car by improving the vehicle handling at high speeddecreasing the turning radius at low speeds and reducing thetracking error for lateral position and yaw angle response
The simulation also demonstrated that model predictivecontrol has an ability to reject the side wind effect as adisturbance to the system which is particularly effective at10msminus1 for all cases except for the 2DoF controller duringsimulation of higher speeds on lower road surface adhesionHowever both controllers were not able to provide a betterresponse when the vehicle performed at a high forwardspeedwith a low road adhesion coefficient particularly when2DoF controller was simulated all results showed unstablevehicle response Thus for this scenario we are currentlyinvestigating new approaches with a real-time implementable
control scheme Further improvement for future works oncontrol methodology will also be considered to minimizetracking error and to enhance vehicle stability
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors would like to thank Professor Zidong Wangand Professor Manfred Ploechl for their comments andadvice Special thanks are also extended to Dr A Zahran MKhudzari for his help preparing this paper
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
International Journal of
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Navigation and Observation
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DistributedSensor Networks
International Journal of
Journal of Control Science and Engineering 9
0 5 10 15
0
05
1
Time (s)
minus05
minus10 5 10 15
0
01
02
Time (s)
minus01
minus02
minus030 5 10 15
0
05
1
Time (s)
minus1
minus05
Tire
slip
angl
e120572f
(rad
)
0 5 10 15
0
05
1
Time (s)
minus05Yaw
angl
e res
pons
e120595
(rad
)
0 5 10 15
0
5
10
Time (s)
minus5
minus10
Fron
t ste
erin
g an
gle120575f
(deg
)
5 10 15
0
01
02
Time (s)
minus01
minus02
minus03Roll
angl
e res
pons
e120601
(rad
)
Time (s)0 5 10 15
0
10
20
30
40
50
Late
ral p
ositi
onY
(m)
0 5 10 15
0
2000
4000
Time (s)
minus2000
minus4000
Late
ral f
orce
Fyf
(N)
0 5 10 15Time (s)
10
5
0
minus5
minus10
Late
ral a
ccel
erat
ion
a y(m
s2)
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 7 Vehicle manoeuvre test of single-lane change at 30msminus1 with 120583 = 01
Table 2 Model predictive control parameters
Parameter ValuePredictive horizon 20Control horizon 9Sampling time [s] 005Constraints on max and min steering angles [deg] plusmn20Constraints on max and min changes in steering angles [degs] plusmn10Constraints on max and min changes in moment torque [Nmrad] plusmn2000Constraints on max and min moment torque [Nmrads] plusmn1500Simulation time [s] 15
for controllers input and output were selected using trial-and-error method where selection criteria were based on thebest response output for both input and output for weightinggains
The first simulation revolves around various road surfacefriction coefficients from wet concrete (120583 = 07) to icy sur-face condition (120583 = 01) with a constant forward speed of10msminus1 The MPC weighting tuning parameters are listedin Table 3 We evaluated the controllerrsquos robustness for theoutput responses by comparing the performance of 2WS4WS and 2WS with DYC manoeuvres at a forward speed of
10msminus1 as shown in Figures 9ndash11 The controller trackingerrors based on (17) for lateral position and yaw angleresponses are summarized in Table 5 From Figure 9 thereare not many differences between both controllers underthe conditions of lowest speed and high road adhesioncoefficient both controllers yielded perfect response whenfollowing a given trajectory while maintaining vehicle stabil-ity and rejecting external crosswind effect It can be seen thatfor 4WS and 2WS with DYC manoeuvres the rear steeringangle and the direct yaw moment are almost unused sincethe front steering can provide sufficient control ability This
10 Journal of Control Science and Engineering
0 5 10 15
0
50
100
Time (s)
minus50
minus100
Late
ral a
ccel
erat
ion
a y(m
s2)
Time (s)0 5 10 15
0
1000
2000
3000
minus1000Late
ral p
ositi
onY
(m)
0 5 10 15
0
2
4
Time (s)
minus2
minus4
times104
Late
ral f
orce
Fyf
(N)
0 5 10 15
0
200
400
Time (s)
minus200
minus400
Fron
t ste
erin
g an
gle120575f
(deg
)
0 5 10 15
0
50
100
150
200
Time (s)
minus50
Yaw
angl
e res
pons
e120595
(rad
)
5 10 15
0
1
2
3
Time (s)
minus1
minus2Roll
angl
e res
pons
e120601
(rad
)
0 5 10 15
0
1
2
Time (s)
minus1
minus2Vehi
cle sl
ip an
gle120573
(rad
)
0 5 10 15
0
20
40
60
Time (s)
minus200 5 10 15
0
5
10
Time (s)
minus5
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 8 Vehicle manoeuvre test of roll rate feedback at 30msminus1 with 120583 = 01
Table 3 Model predictive control weighting matrices parameters for V119909
= 10msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 10msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 195 11987622
= 05 11987611
= 525 11987622
= 05
V119909
= 10msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 152 11987622
= 05 11987611
= 395 11987622
= 05
4WS
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 195 11987622
= 05 11987611
= 45 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 153 11987622
= 05 11987611
= 35 11987622
= 05
2WS + DYC
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 193 11987622
= 05 11987611
= 37 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 152 11987622
= 05 11987611
= 35 11987622
= 05
Journal of Control Science and Engineering 11
Time (s)0 5 10 15
0
05
1
4WS
minus05
minus1
minus15
times10minus5Re
ar st
eerin
g an
gle120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
1
2
3
Time (s)
2WS + DYC
minus1
minus2
minus3
times10minus3
Dire
ct y
aw m
omen
tM
z(N
m)
0 5 10 15
0
02
0 5 10 15
0
02
04
Time (s)Time (s)
Ref2WS
4WS2WS + DYC
Ref2WS
4WS2WS + DYC
minus02
minus04
minus02
minus04
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
Yaw
angl
e120595
(rad
)
Yaw
rate
120595998400
(rad
sminus1)
Figure 9 2DoF controller at 10msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
may be partly due to the fact that the rear steering and DYCwere not used in lower speed manoeuvres
Moreover when the road surface friction was set to icythe output responses from both controllers and all controlmanoeuvre techniques did not perfectly follow the desiredtrajectory as shown in Figures 10 and 11 2WS with DYCyielded the best performance output compared to the othersespecially in yaw rate response 4WSwas the next best follow-ing 2WS where the lateral position and yaw angle lookalmost identical It seems that the rear steering and direct yawmoment controls were used in conjunction with front steer-ing for vehicle stabilization along the trajectory Howeverfrom Table 5 the controller with 3DoF gives slightly bettertracking error performance manoeuvres compared to con-troller 2DoF due to the fact that roll dynamics will not havemuch influence during low speed manoeuvres Another factis that at low vehicle speed all control inputs (front steeringangle rear steering angle and direct yawmoment control) areunder constraints
Next we tested the vehicle at various road friction coef-ficients again however this time the vehicle was tested undera high forward speed of 30msminus1 The same MPC design wasused of which the parameter controls are listed in Table 4Wecompared the simulation results for 2WS 4WS and 2WSwithDYC control manoeuvres for both controllers and presentthe comparison in Figures 12ndash15The simulation results show
that for 2DoF controller at 30msminus1 and on wet concrete(120583 = 07) all control manoeuvres give slightly similar out-puts performances Better responses were achieved in lateralpositioning and although the tracking performance of yawangle and yaw rate deteriorated the system still allowedthe vehicle to track and follow the trajectory successfullyrejecting the effects of wind gust that impact the vehicle
As tabulated in Table 6 4WS and 2WS with DYC showslightly better tracking performance compared to 2WS onlyHowever in the case of the 3DoF controller it can be clearlyseen that 2WS with DYC and 4WS control manoeuvresoffered amuch higher performing response especially in yawangle and yaw rate response than 2WS This is shown inFigure 13 In this scenario the rear steering and direct yawmoment control was fully utilized to control and enhancevehicle stability It is therefore very important to consider rolldynamics in order to enhance vehicle stability and to followthe path trajectory From Figures 12 and 13 there are somestrong frequency oscillations in front of some responsesthat is front lateral force vehicle side slip angle and lateralacceleration trace based on authorsrsquo knowledge come fromthe numerical calculation or associated glitches and initialcondition of the system
Furthermore when we simulated vehicle manoeuvre forhigh speed (30msminus1) and icy road condition (120583 = 01) for2DoF controller it can be seen that all manoeuvres control
12 Journal of Control Science and Engineering
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
Time (s)0 5 10 15
0
01
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
Time (s)
2WS + DYC
minus01
minus02Dire
ct y
aw m
omen
tM
z(N
m)
Figure 10 2DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 4 Model predictive control weighting matrices parameters for V119909
= 30msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 30msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 565 11987622
= 05 11987611
= 355 11987622
= 05
V119909
= 30msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 0083 11987622
= 0042 11987611
= 003 11987622
= 045
4WS
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 155 11987622
= 25 11987611
= 135 11987622
= 35
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0005 11987622
= 0001 11987611
= 004 11987622
= 05
2WS + DYC
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 505 11987622
= 05 11987611
= 35 11987622
= 05
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0092 11987622
= 004 11987611
= 05 11987622
= 25
simulation the tracking responses become unstable andimpossible to control especially under the crosswind effectas shown in Figure 14 The most probable cause is the rolldynamic motion neglect in the 2DoF controller design whenthe vehicle itself was modelled by including roll dynamicfactor It can be said that the inclusion of roll dynamic factor
is important for vehicle manoeuvre in terms of stability andcontrollability at high speed and icy road condition Sincehigh speed coupled with icy road condition will render theequation to become highly nonlinear it is impossible fora linear tire model to react positively since the handlingpropertiesmay be significantly different from those generated
Journal of Control Science and Engineering 13
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
01
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
002
004
Time (s)
2WS + DYC
minus002
minus004
minus006Dire
ct y
aw m
omen
tM
z(N
m)
Figure 11 3DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 5 Path following tracking errors with road friction surface 120583 = 07
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00637 00085 00623 000744WS 00548 00054 00526 00051
2WS + DYC 00542 00051 00527 00051
302WS 00639 05348 00616 052154WS 00558 05239 00528 00023
2WS + DYC 00549 05226 00524 00254
by the linear tire model These results show that linear tiremodel is only suitable for analyzing a stable vehicle behaviourunder the assumption of small steering and acceleration
Next for the 3DoF controller simulation 2WS with DYCmanoeuvres provides a much better tracking response incomparison to 4WS particularly in the case of lateral outputOn the contrary 4WS manoeuvre demonstrates a much bet-ter tracking response in the yaw angle and yaw rate responsesas tabulated in Table 6 With appropriate weight tuning gainthe rear steering angle and direct yaw moment control arefully optimized in order to become stable along the given tra-jectory However for the 2WS manoeuvre the vehicleresponses become unstable despite several instances wherethe weighting tuning gains are adjusted for output and inputgains With the inclusion of another input control to the con-troller design we can enhance the vehicle responses for both
2WS with DYC and 4WS control manoeuvres show Thismeans that for 2WS controller design it may have to usemore than just front steering to stabilize the vehicle under theeffects of wind gusts Figure 15 shows that rear steering angleand direct yaw moment are fully utilized in order to stabilizethe vehicle for 2WS with DYC and 4WSmanoeuvres control
In contrast to the 2DoF controller as shown in Figure 14in the 3DoF controllers the 2WS control manoeuvre canstill be used to track a given trajectory despite not perfectlyfollowing the trajectory in a satisfactory manner especiallyunder crosswind effects In Figure 15 we can see some oscil-lations in few traces responses at the end of the responsesBased on the authorsrsquo knowledge these oscillationsmay comefrom the vehicle model and some initial conditions of thesystem with imperfect controllers tuning weighting gains Inthese conditions neither 2WS nor 4WS control manoeuvres
14 Journal of Control Science and Engineering
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
Time (s)
2WS4WS2WS + DYC
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
001
002
Time (s)
2WS4WS2WS + DYC
minus001Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
1
2
Time (s)
2WS4WS2WS + DYC
minus1
minus2
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
001
002
003
Time (s)
2WS4WS2WS + DYC
minus001
minus002
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
00005
0001
4WS
Time (s)
minus00005
minus0001
minus00015
Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
001
002
Time (s)
2WS + DYC
minus001
minus002
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 12 2DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
for lateral response and neither 2WS nor 2WS with DYCcontrol manoeuvres for yaw angle or yaw rate responseperformed well causing an increase in vehicle instabilityincreased vibration and tracking responses deteriorationWewill next focus on how to enhance the controller in order tostabilize vehicle manoeuvrability and handling stability
We can therefore conclude that for 4WS the rear wheelswere helping the car to steer by improving the vehiclehandling at high speed while decreasing the turning radiusat low speed as shown by the control signal in Figures 12 and13 Meanwhile for 2WS with DYC active front steering wasused in low speed manoeuvres for lateral acceleration whileinclusion of DYC was adopted for high lateral accelerationwhen the tires were saturated and could not produce enoughlateral force for vehicle control and stability as intendedIn 2WS vehicles the rear set of wheels do not play an
active role in controlling the steering From Table 5 it canbe seen that among the three controllers 4WS gave the bestperformance by reducing the tracking error for lateral andyaw angle responses when compared with 2WS with DYCand 2WS only Tables 5 and 6 tabulated the MPC robustnessand the tracking errors in (17) for all types of manoeuvre at10msminus1 and 30msminus1 vehicle speeds and under various roadsurfaces Furthermore all controlmanoeuvre signals for bothcontrollers were within the input constraints We would liketo highlight that in MPC approaches although there was anadvantage in multivariable systems in this study there was atrade-off between theweighting tuning parameter to focus oneither lateral position or yaw angle trajectory In this paper wehave chosen lateral position as the weighting tuning priorityso we may get a better response on yaw angle and yaw rateresponse by sacrificing lateral position precision
Journal of Control Science and Engineering 15
0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
Time (s)
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
2WS4WS2WS + DYC
Time (s)
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02
Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
5
10
2WS4WS2WS + DYC
Time (s)
minus5
minus10Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
03
4WS
Time (s)
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
2WS + DYC
Time (s)
minus005
minus01
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 13 3DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
Table 6 Path following tracking errors with road friction surface 120583 = 01
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00874 00115 00841 001044WS 00838 00109 00832 00094
2WS + DYC 00838 00110 00835 00083
302WS Uncontrolled Uncontrolled 21352 052644WS Uncontrolled Uncontrolled 08824 02982
2WS + DYC Uncontrolled Uncontrolled 09964 04580
16 Journal of Control Science and Engineering
0 10 20 30
02468
10
Time (s)
Ref2WS
4WS2WS + DYC
minus2
minus4Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0020406
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
minus06
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15
Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
0020406
Time (s)
2WS4WS2WS + DYC
minus02
minus04
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
02
04
06
Time (s)
4WS
minus02
Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
005
01
Time (s)
2WS + DYC
minus005
Dire
ct y
aw m
omen
tM
z(N
m)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15Fron
t lat
eral
forc
eFyf
(Nm
) times104
Figure 14 2DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
5 Conclusion
This paper presents the robustness of model predictive con-trol for an autonomous ground vehicle on a path-followingcontrol with consideration of wind gust effects subjectedto the variation of forward speeds and road surfaces condi-tion in a double-lane change scenario The predictive con-trollers were designed based on two- (lateral-yaw model)and three- (lateral-roll model) degree-of-freedom vehiclemodels The only difference between the models is the rolldynamics factor which is taken into consideration andboth controllers were implemented into a six-degree-of-free-dom vehicle model Based on linear vehicle and tire modelapproximations of a known trajectory we evaluated the effectand impact of roll dynamics at low and high forward vehicle
speeds and at various road frictions (wet concrete to icyroads) in order to follow a desired trajectory as close as pos-sible while rejecting the crosswind effects and maintainingvehicle stability We also evaluated and compared the effi-ciency of the different manoeuvres via two-wheel steeringfour-wheel steering and two-wheel steering with direct yawmoment control
The simulation results showed that with the inclusion ofroll dynamics in the linear vehicle model the vehicle stabilityand adherence to trajectory was considerably improved forthe case of high speed manoeuvres on a higher road surfacefriction coefficient (120583 = 07) and improved even more for thecase of a lower road surface coefficient (120583 = 01) Further-more the obtained results showed and proved that the rearwheels and direct yaw moment are beneficial to help steering
Journal of Control Science and Engineering 17
0 10 20 30
0
5
Time (s)
Ref2WS
4WS2WS + DYC
minus10
minus5
minus15Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0
025
05
Time (s)
Ref2WS
4WS2WS + DYC
minus025
minus05
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
0
500
1000
Time (s)
2WS4WS2WS + DYC
minus500
minus1000
minus1500Fron
t lat
eral
forc
eFyf
(Nm
)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
01
02
03
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS + DYC
minus2
minus4
times10minus4
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 15 3DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
the car by improving the vehicle handling at high speeddecreasing the turning radius at low speeds and reducing thetracking error for lateral position and yaw angle response
The simulation also demonstrated that model predictivecontrol has an ability to reject the side wind effect as adisturbance to the system which is particularly effective at10msminus1 for all cases except for the 2DoF controller duringsimulation of higher speeds on lower road surface adhesionHowever both controllers were not able to provide a betterresponse when the vehicle performed at a high forwardspeedwith a low road adhesion coefficient particularly when2DoF controller was simulated all results showed unstablevehicle response Thus for this scenario we are currentlyinvestigating new approaches with a real-time implementable
control scheme Further improvement for future works oncontrol methodology will also be considered to minimizetracking error and to enhance vehicle stability
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors would like to thank Professor Zidong Wangand Professor Manfred Ploechl for their comments andadvice Special thanks are also extended to Dr A Zahran MKhudzari for his help preparing this paper
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
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DistributedSensor Networks
International Journal of
10 Journal of Control Science and Engineering
0 5 10 15
0
50
100
Time (s)
minus50
minus100
Late
ral a
ccel
erat
ion
a y(m
s2)
Time (s)0 5 10 15
0
1000
2000
3000
minus1000Late
ral p
ositi
onY
(m)
0 5 10 15
0
2
4
Time (s)
minus2
minus4
times104
Late
ral f
orce
Fyf
(N)
0 5 10 15
0
200
400
Time (s)
minus200
minus400
Fron
t ste
erin
g an
gle120575f
(deg
)
0 5 10 15
0
50
100
150
200
Time (s)
minus50
Yaw
angl
e res
pons
e120595
(rad
)
5 10 15
0
1
2
3
Time (s)
minus1
minus2Roll
angl
e res
pons
e120601
(rad
)
0 5 10 15
0
1
2
Time (s)
minus1
minus2Vehi
cle sl
ip an
gle120573
(rad
)
0 5 10 15
0
20
40
60
Time (s)
minus200 5 10 15
0
5
10
Time (s)
minus5
(rad
s)
Roll
rate
resp
onse
120601(rad
s)
Yaw
rate
resp
onse
120595
Figure 8 Vehicle manoeuvre test of roll rate feedback at 30msminus1 with 120583 = 01
Table 3 Model predictive control weighting matrices parameters for V119909
= 10msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 10msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 195 11987622
= 05 11987611
= 525 11987622
= 05
V119909
= 10msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 152 11987622
= 05 11987611
= 395 11987622
= 05
4WS
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 195 11987622
= 05 11987611
= 45 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 153 11987622
= 05 11987611
= 35 11987622
= 05
2WS + DYC
V119909
= 10msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 193 11987622
= 05 11987611
= 37 11987622
= 05
V119909
= 10msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 152 11987622
= 05 11987611
= 35 11987622
= 05
Journal of Control Science and Engineering 11
Time (s)0 5 10 15
0
05
1
4WS
minus05
minus1
minus15
times10minus5Re
ar st
eerin
g an
gle120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
1
2
3
Time (s)
2WS + DYC
minus1
minus2
minus3
times10minus3
Dire
ct y
aw m
omen
tM
z(N
m)
0 5 10 15
0
02
0 5 10 15
0
02
04
Time (s)Time (s)
Ref2WS
4WS2WS + DYC
Ref2WS
4WS2WS + DYC
minus02
minus04
minus02
minus04
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
Yaw
angl
e120595
(rad
)
Yaw
rate
120595998400
(rad
sminus1)
Figure 9 2DoF controller at 10msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
may be partly due to the fact that the rear steering and DYCwere not used in lower speed manoeuvres
Moreover when the road surface friction was set to icythe output responses from both controllers and all controlmanoeuvre techniques did not perfectly follow the desiredtrajectory as shown in Figures 10 and 11 2WS with DYCyielded the best performance output compared to the othersespecially in yaw rate response 4WSwas the next best follow-ing 2WS where the lateral position and yaw angle lookalmost identical It seems that the rear steering and direct yawmoment controls were used in conjunction with front steer-ing for vehicle stabilization along the trajectory Howeverfrom Table 5 the controller with 3DoF gives slightly bettertracking error performance manoeuvres compared to con-troller 2DoF due to the fact that roll dynamics will not havemuch influence during low speed manoeuvres Another factis that at low vehicle speed all control inputs (front steeringangle rear steering angle and direct yawmoment control) areunder constraints
Next we tested the vehicle at various road friction coef-ficients again however this time the vehicle was tested undera high forward speed of 30msminus1 The same MPC design wasused of which the parameter controls are listed in Table 4Wecompared the simulation results for 2WS 4WS and 2WSwithDYC control manoeuvres for both controllers and presentthe comparison in Figures 12ndash15The simulation results show
that for 2DoF controller at 30msminus1 and on wet concrete(120583 = 07) all control manoeuvres give slightly similar out-puts performances Better responses were achieved in lateralpositioning and although the tracking performance of yawangle and yaw rate deteriorated the system still allowedthe vehicle to track and follow the trajectory successfullyrejecting the effects of wind gust that impact the vehicle
As tabulated in Table 6 4WS and 2WS with DYC showslightly better tracking performance compared to 2WS onlyHowever in the case of the 3DoF controller it can be clearlyseen that 2WS with DYC and 4WS control manoeuvresoffered amuch higher performing response especially in yawangle and yaw rate response than 2WS This is shown inFigure 13 In this scenario the rear steering and direct yawmoment control was fully utilized to control and enhancevehicle stability It is therefore very important to consider rolldynamics in order to enhance vehicle stability and to followthe path trajectory From Figures 12 and 13 there are somestrong frequency oscillations in front of some responsesthat is front lateral force vehicle side slip angle and lateralacceleration trace based on authorsrsquo knowledge come fromthe numerical calculation or associated glitches and initialcondition of the system
Furthermore when we simulated vehicle manoeuvre forhigh speed (30msminus1) and icy road condition (120583 = 01) for2DoF controller it can be seen that all manoeuvres control
12 Journal of Control Science and Engineering
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
Time (s)0 5 10 15
0
01
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
Time (s)
2WS + DYC
minus01
minus02Dire
ct y
aw m
omen
tM
z(N
m)
Figure 10 2DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 4 Model predictive control weighting matrices parameters for V119909
= 30msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 30msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 565 11987622
= 05 11987611
= 355 11987622
= 05
V119909
= 30msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 0083 11987622
= 0042 11987611
= 003 11987622
= 045
4WS
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 155 11987622
= 25 11987611
= 135 11987622
= 35
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0005 11987622
= 0001 11987611
= 004 11987622
= 05
2WS + DYC
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 505 11987622
= 05 11987611
= 35 11987622
= 05
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0092 11987622
= 004 11987611
= 05 11987622
= 25
simulation the tracking responses become unstable andimpossible to control especially under the crosswind effectas shown in Figure 14 The most probable cause is the rolldynamic motion neglect in the 2DoF controller design whenthe vehicle itself was modelled by including roll dynamicfactor It can be said that the inclusion of roll dynamic factor
is important for vehicle manoeuvre in terms of stability andcontrollability at high speed and icy road condition Sincehigh speed coupled with icy road condition will render theequation to become highly nonlinear it is impossible fora linear tire model to react positively since the handlingpropertiesmay be significantly different from those generated
Journal of Control Science and Engineering 13
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
01
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
002
004
Time (s)
2WS + DYC
minus002
minus004
minus006Dire
ct y
aw m
omen
tM
z(N
m)
Figure 11 3DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 5 Path following tracking errors with road friction surface 120583 = 07
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00637 00085 00623 000744WS 00548 00054 00526 00051
2WS + DYC 00542 00051 00527 00051
302WS 00639 05348 00616 052154WS 00558 05239 00528 00023
2WS + DYC 00549 05226 00524 00254
by the linear tire model These results show that linear tiremodel is only suitable for analyzing a stable vehicle behaviourunder the assumption of small steering and acceleration
Next for the 3DoF controller simulation 2WS with DYCmanoeuvres provides a much better tracking response incomparison to 4WS particularly in the case of lateral outputOn the contrary 4WS manoeuvre demonstrates a much bet-ter tracking response in the yaw angle and yaw rate responsesas tabulated in Table 6 With appropriate weight tuning gainthe rear steering angle and direct yaw moment control arefully optimized in order to become stable along the given tra-jectory However for the 2WS manoeuvre the vehicleresponses become unstable despite several instances wherethe weighting tuning gains are adjusted for output and inputgains With the inclusion of another input control to the con-troller design we can enhance the vehicle responses for both
2WS with DYC and 4WS control manoeuvres show Thismeans that for 2WS controller design it may have to usemore than just front steering to stabilize the vehicle under theeffects of wind gusts Figure 15 shows that rear steering angleand direct yaw moment are fully utilized in order to stabilizethe vehicle for 2WS with DYC and 4WSmanoeuvres control
In contrast to the 2DoF controller as shown in Figure 14in the 3DoF controllers the 2WS control manoeuvre canstill be used to track a given trajectory despite not perfectlyfollowing the trajectory in a satisfactory manner especiallyunder crosswind effects In Figure 15 we can see some oscil-lations in few traces responses at the end of the responsesBased on the authorsrsquo knowledge these oscillationsmay comefrom the vehicle model and some initial conditions of thesystem with imperfect controllers tuning weighting gains Inthese conditions neither 2WS nor 4WS control manoeuvres
14 Journal of Control Science and Engineering
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
Time (s)
2WS4WS2WS + DYC
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
001
002
Time (s)
2WS4WS2WS + DYC
minus001Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
1
2
Time (s)
2WS4WS2WS + DYC
minus1
minus2
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
001
002
003
Time (s)
2WS4WS2WS + DYC
minus001
minus002
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
00005
0001
4WS
Time (s)
minus00005
minus0001
minus00015
Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
001
002
Time (s)
2WS + DYC
minus001
minus002
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 12 2DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
for lateral response and neither 2WS nor 2WS with DYCcontrol manoeuvres for yaw angle or yaw rate responseperformed well causing an increase in vehicle instabilityincreased vibration and tracking responses deteriorationWewill next focus on how to enhance the controller in order tostabilize vehicle manoeuvrability and handling stability
We can therefore conclude that for 4WS the rear wheelswere helping the car to steer by improving the vehiclehandling at high speed while decreasing the turning radiusat low speed as shown by the control signal in Figures 12 and13 Meanwhile for 2WS with DYC active front steering wasused in low speed manoeuvres for lateral acceleration whileinclusion of DYC was adopted for high lateral accelerationwhen the tires were saturated and could not produce enoughlateral force for vehicle control and stability as intendedIn 2WS vehicles the rear set of wheels do not play an
active role in controlling the steering From Table 5 it canbe seen that among the three controllers 4WS gave the bestperformance by reducing the tracking error for lateral andyaw angle responses when compared with 2WS with DYCand 2WS only Tables 5 and 6 tabulated the MPC robustnessand the tracking errors in (17) for all types of manoeuvre at10msminus1 and 30msminus1 vehicle speeds and under various roadsurfaces Furthermore all controlmanoeuvre signals for bothcontrollers were within the input constraints We would liketo highlight that in MPC approaches although there was anadvantage in multivariable systems in this study there was atrade-off between theweighting tuning parameter to focus oneither lateral position or yaw angle trajectory In this paper wehave chosen lateral position as the weighting tuning priorityso we may get a better response on yaw angle and yaw rateresponse by sacrificing lateral position precision
Journal of Control Science and Engineering 15
0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
Time (s)
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
2WS4WS2WS + DYC
Time (s)
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02
Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
5
10
2WS4WS2WS + DYC
Time (s)
minus5
minus10Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
03
4WS
Time (s)
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
2WS + DYC
Time (s)
minus005
minus01
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 13 3DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
Table 6 Path following tracking errors with road friction surface 120583 = 01
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00874 00115 00841 001044WS 00838 00109 00832 00094
2WS + DYC 00838 00110 00835 00083
302WS Uncontrolled Uncontrolled 21352 052644WS Uncontrolled Uncontrolled 08824 02982
2WS + DYC Uncontrolled Uncontrolled 09964 04580
16 Journal of Control Science and Engineering
0 10 20 30
02468
10
Time (s)
Ref2WS
4WS2WS + DYC
minus2
minus4Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0020406
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
minus06
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15
Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
0020406
Time (s)
2WS4WS2WS + DYC
minus02
minus04
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
02
04
06
Time (s)
4WS
minus02
Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
005
01
Time (s)
2WS + DYC
minus005
Dire
ct y
aw m
omen
tM
z(N
m)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15Fron
t lat
eral
forc
eFyf
(Nm
) times104
Figure 14 2DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
5 Conclusion
This paper presents the robustness of model predictive con-trol for an autonomous ground vehicle on a path-followingcontrol with consideration of wind gust effects subjectedto the variation of forward speeds and road surfaces condi-tion in a double-lane change scenario The predictive con-trollers were designed based on two- (lateral-yaw model)and three- (lateral-roll model) degree-of-freedom vehiclemodels The only difference between the models is the rolldynamics factor which is taken into consideration andboth controllers were implemented into a six-degree-of-free-dom vehicle model Based on linear vehicle and tire modelapproximations of a known trajectory we evaluated the effectand impact of roll dynamics at low and high forward vehicle
speeds and at various road frictions (wet concrete to icyroads) in order to follow a desired trajectory as close as pos-sible while rejecting the crosswind effects and maintainingvehicle stability We also evaluated and compared the effi-ciency of the different manoeuvres via two-wheel steeringfour-wheel steering and two-wheel steering with direct yawmoment control
The simulation results showed that with the inclusion ofroll dynamics in the linear vehicle model the vehicle stabilityand adherence to trajectory was considerably improved forthe case of high speed manoeuvres on a higher road surfacefriction coefficient (120583 = 07) and improved even more for thecase of a lower road surface coefficient (120583 = 01) Further-more the obtained results showed and proved that the rearwheels and direct yaw moment are beneficial to help steering
Journal of Control Science and Engineering 17
0 10 20 30
0
5
Time (s)
Ref2WS
4WS2WS + DYC
minus10
minus5
minus15Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0
025
05
Time (s)
Ref2WS
4WS2WS + DYC
minus025
minus05
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
0
500
1000
Time (s)
2WS4WS2WS + DYC
minus500
minus1000
minus1500Fron
t lat
eral
forc
eFyf
(Nm
)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
01
02
03
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS + DYC
minus2
minus4
times10minus4
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 15 3DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
the car by improving the vehicle handling at high speeddecreasing the turning radius at low speeds and reducing thetracking error for lateral position and yaw angle response
The simulation also demonstrated that model predictivecontrol has an ability to reject the side wind effect as adisturbance to the system which is particularly effective at10msminus1 for all cases except for the 2DoF controller duringsimulation of higher speeds on lower road surface adhesionHowever both controllers were not able to provide a betterresponse when the vehicle performed at a high forwardspeedwith a low road adhesion coefficient particularly when2DoF controller was simulated all results showed unstablevehicle response Thus for this scenario we are currentlyinvestigating new approaches with a real-time implementable
control scheme Further improvement for future works oncontrol methodology will also be considered to minimizetracking error and to enhance vehicle stability
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors would like to thank Professor Zidong Wangand Professor Manfred Ploechl for their comments andadvice Special thanks are also extended to Dr A Zahran MKhudzari for his help preparing this paper
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
International Journal of
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DistributedSensor Networks
International Journal of
Journal of Control Science and Engineering 11
Time (s)0 5 10 15
0
05
1
4WS
minus05
minus1
minus15
times10minus5Re
ar st
eerin
g an
gle120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
1
2
3
Time (s)
2WS + DYC
minus1
minus2
minus3
times10minus3
Dire
ct y
aw m
omen
tM
z(N
m)
0 5 10 15
0
02
0 5 10 15
0
02
04
Time (s)Time (s)
Ref2WS
4WS2WS + DYC
Ref2WS
4WS2WS + DYC
minus02
minus04
minus02
minus04
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
Yaw
angl
e120595
(rad
)
Yaw
rate
120595998400
(rad
sminus1)
Figure 9 2DoF controller at 10msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
may be partly due to the fact that the rear steering and DYCwere not used in lower speed manoeuvres
Moreover when the road surface friction was set to icythe output responses from both controllers and all controlmanoeuvre techniques did not perfectly follow the desiredtrajectory as shown in Figures 10 and 11 2WS with DYCyielded the best performance output compared to the othersespecially in yaw rate response 4WSwas the next best follow-ing 2WS where the lateral position and yaw angle lookalmost identical It seems that the rear steering and direct yawmoment controls were used in conjunction with front steer-ing for vehicle stabilization along the trajectory Howeverfrom Table 5 the controller with 3DoF gives slightly bettertracking error performance manoeuvres compared to con-troller 2DoF due to the fact that roll dynamics will not havemuch influence during low speed manoeuvres Another factis that at low vehicle speed all control inputs (front steeringangle rear steering angle and direct yawmoment control) areunder constraints
Next we tested the vehicle at various road friction coef-ficients again however this time the vehicle was tested undera high forward speed of 30msminus1 The same MPC design wasused of which the parameter controls are listed in Table 4Wecompared the simulation results for 2WS 4WS and 2WSwithDYC control manoeuvres for both controllers and presentthe comparison in Figures 12ndash15The simulation results show
that for 2DoF controller at 30msminus1 and on wet concrete(120583 = 07) all control manoeuvres give slightly similar out-puts performances Better responses were achieved in lateralpositioning and although the tracking performance of yawangle and yaw rate deteriorated the system still allowedthe vehicle to track and follow the trajectory successfullyrejecting the effects of wind gust that impact the vehicle
As tabulated in Table 6 4WS and 2WS with DYC showslightly better tracking performance compared to 2WS onlyHowever in the case of the 3DoF controller it can be clearlyseen that 2WS with DYC and 4WS control manoeuvresoffered amuch higher performing response especially in yawangle and yaw rate response than 2WS This is shown inFigure 13 In this scenario the rear steering and direct yawmoment control was fully utilized to control and enhancevehicle stability It is therefore very important to consider rolldynamics in order to enhance vehicle stability and to followthe path trajectory From Figures 12 and 13 there are somestrong frequency oscillations in front of some responsesthat is front lateral force vehicle side slip angle and lateralacceleration trace based on authorsrsquo knowledge come fromthe numerical calculation or associated glitches and initialcondition of the system
Furthermore when we simulated vehicle manoeuvre forhigh speed (30msminus1) and icy road condition (120583 = 01) for2DoF controller it can be seen that all manoeuvres control
12 Journal of Control Science and Engineering
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
Time (s)0 5 10 15
0
01
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
Time (s)
2WS + DYC
minus01
minus02Dire
ct y
aw m
omen
tM
z(N
m)
Figure 10 2DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 4 Model predictive control weighting matrices parameters for V119909
= 30msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 30msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 565 11987622
= 05 11987611
= 355 11987622
= 05
V119909
= 30msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 0083 11987622
= 0042 11987611
= 003 11987622
= 045
4WS
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 155 11987622
= 25 11987611
= 135 11987622
= 35
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0005 11987622
= 0001 11987611
= 004 11987622
= 05
2WS + DYC
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 505 11987622
= 05 11987611
= 35 11987622
= 05
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0092 11987622
= 004 11987611
= 05 11987622
= 25
simulation the tracking responses become unstable andimpossible to control especially under the crosswind effectas shown in Figure 14 The most probable cause is the rolldynamic motion neglect in the 2DoF controller design whenthe vehicle itself was modelled by including roll dynamicfactor It can be said that the inclusion of roll dynamic factor
is important for vehicle manoeuvre in terms of stability andcontrollability at high speed and icy road condition Sincehigh speed coupled with icy road condition will render theequation to become highly nonlinear it is impossible fora linear tire model to react positively since the handlingpropertiesmay be significantly different from those generated
Journal of Control Science and Engineering 13
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
01
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
002
004
Time (s)
2WS + DYC
minus002
minus004
minus006Dire
ct y
aw m
omen
tM
z(N
m)
Figure 11 3DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 5 Path following tracking errors with road friction surface 120583 = 07
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00637 00085 00623 000744WS 00548 00054 00526 00051
2WS + DYC 00542 00051 00527 00051
302WS 00639 05348 00616 052154WS 00558 05239 00528 00023
2WS + DYC 00549 05226 00524 00254
by the linear tire model These results show that linear tiremodel is only suitable for analyzing a stable vehicle behaviourunder the assumption of small steering and acceleration
Next for the 3DoF controller simulation 2WS with DYCmanoeuvres provides a much better tracking response incomparison to 4WS particularly in the case of lateral outputOn the contrary 4WS manoeuvre demonstrates a much bet-ter tracking response in the yaw angle and yaw rate responsesas tabulated in Table 6 With appropriate weight tuning gainthe rear steering angle and direct yaw moment control arefully optimized in order to become stable along the given tra-jectory However for the 2WS manoeuvre the vehicleresponses become unstable despite several instances wherethe weighting tuning gains are adjusted for output and inputgains With the inclusion of another input control to the con-troller design we can enhance the vehicle responses for both
2WS with DYC and 4WS control manoeuvres show Thismeans that for 2WS controller design it may have to usemore than just front steering to stabilize the vehicle under theeffects of wind gusts Figure 15 shows that rear steering angleand direct yaw moment are fully utilized in order to stabilizethe vehicle for 2WS with DYC and 4WSmanoeuvres control
In contrast to the 2DoF controller as shown in Figure 14in the 3DoF controllers the 2WS control manoeuvre canstill be used to track a given trajectory despite not perfectlyfollowing the trajectory in a satisfactory manner especiallyunder crosswind effects In Figure 15 we can see some oscil-lations in few traces responses at the end of the responsesBased on the authorsrsquo knowledge these oscillationsmay comefrom the vehicle model and some initial conditions of thesystem with imperfect controllers tuning weighting gains Inthese conditions neither 2WS nor 4WS control manoeuvres
14 Journal of Control Science and Engineering
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
Time (s)
2WS4WS2WS + DYC
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
001
002
Time (s)
2WS4WS2WS + DYC
minus001Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
1
2
Time (s)
2WS4WS2WS + DYC
minus1
minus2
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
001
002
003
Time (s)
2WS4WS2WS + DYC
minus001
minus002
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
00005
0001
4WS
Time (s)
minus00005
minus0001
minus00015
Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
001
002
Time (s)
2WS + DYC
minus001
minus002
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 12 2DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
for lateral response and neither 2WS nor 2WS with DYCcontrol manoeuvres for yaw angle or yaw rate responseperformed well causing an increase in vehicle instabilityincreased vibration and tracking responses deteriorationWewill next focus on how to enhance the controller in order tostabilize vehicle manoeuvrability and handling stability
We can therefore conclude that for 4WS the rear wheelswere helping the car to steer by improving the vehiclehandling at high speed while decreasing the turning radiusat low speed as shown by the control signal in Figures 12 and13 Meanwhile for 2WS with DYC active front steering wasused in low speed manoeuvres for lateral acceleration whileinclusion of DYC was adopted for high lateral accelerationwhen the tires were saturated and could not produce enoughlateral force for vehicle control and stability as intendedIn 2WS vehicles the rear set of wheels do not play an
active role in controlling the steering From Table 5 it canbe seen that among the three controllers 4WS gave the bestperformance by reducing the tracking error for lateral andyaw angle responses when compared with 2WS with DYCand 2WS only Tables 5 and 6 tabulated the MPC robustnessand the tracking errors in (17) for all types of manoeuvre at10msminus1 and 30msminus1 vehicle speeds and under various roadsurfaces Furthermore all controlmanoeuvre signals for bothcontrollers were within the input constraints We would liketo highlight that in MPC approaches although there was anadvantage in multivariable systems in this study there was atrade-off between theweighting tuning parameter to focus oneither lateral position or yaw angle trajectory In this paper wehave chosen lateral position as the weighting tuning priorityso we may get a better response on yaw angle and yaw rateresponse by sacrificing lateral position precision
Journal of Control Science and Engineering 15
0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
Time (s)
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
2WS4WS2WS + DYC
Time (s)
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02
Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
5
10
2WS4WS2WS + DYC
Time (s)
minus5
minus10Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
03
4WS
Time (s)
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
2WS + DYC
Time (s)
minus005
minus01
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 13 3DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
Table 6 Path following tracking errors with road friction surface 120583 = 01
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00874 00115 00841 001044WS 00838 00109 00832 00094
2WS + DYC 00838 00110 00835 00083
302WS Uncontrolled Uncontrolled 21352 052644WS Uncontrolled Uncontrolled 08824 02982
2WS + DYC Uncontrolled Uncontrolled 09964 04580
16 Journal of Control Science and Engineering
0 10 20 30
02468
10
Time (s)
Ref2WS
4WS2WS + DYC
minus2
minus4Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0020406
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
minus06
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15
Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
0020406
Time (s)
2WS4WS2WS + DYC
minus02
minus04
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
02
04
06
Time (s)
4WS
minus02
Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
005
01
Time (s)
2WS + DYC
minus005
Dire
ct y
aw m
omen
tM
z(N
m)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15Fron
t lat
eral
forc
eFyf
(Nm
) times104
Figure 14 2DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
5 Conclusion
This paper presents the robustness of model predictive con-trol for an autonomous ground vehicle on a path-followingcontrol with consideration of wind gust effects subjectedto the variation of forward speeds and road surfaces condi-tion in a double-lane change scenario The predictive con-trollers were designed based on two- (lateral-yaw model)and three- (lateral-roll model) degree-of-freedom vehiclemodels The only difference between the models is the rolldynamics factor which is taken into consideration andboth controllers were implemented into a six-degree-of-free-dom vehicle model Based on linear vehicle and tire modelapproximations of a known trajectory we evaluated the effectand impact of roll dynamics at low and high forward vehicle
speeds and at various road frictions (wet concrete to icyroads) in order to follow a desired trajectory as close as pos-sible while rejecting the crosswind effects and maintainingvehicle stability We also evaluated and compared the effi-ciency of the different manoeuvres via two-wheel steeringfour-wheel steering and two-wheel steering with direct yawmoment control
The simulation results showed that with the inclusion ofroll dynamics in the linear vehicle model the vehicle stabilityand adherence to trajectory was considerably improved forthe case of high speed manoeuvres on a higher road surfacefriction coefficient (120583 = 07) and improved even more for thecase of a lower road surface coefficient (120583 = 01) Further-more the obtained results showed and proved that the rearwheels and direct yaw moment are beneficial to help steering
Journal of Control Science and Engineering 17
0 10 20 30
0
5
Time (s)
Ref2WS
4WS2WS + DYC
minus10
minus5
minus15Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0
025
05
Time (s)
Ref2WS
4WS2WS + DYC
minus025
minus05
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
0
500
1000
Time (s)
2WS4WS2WS + DYC
minus500
minus1000
minus1500Fron
t lat
eral
forc
eFyf
(Nm
)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
01
02
03
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS + DYC
minus2
minus4
times10minus4
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 15 3DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
the car by improving the vehicle handling at high speeddecreasing the turning radius at low speeds and reducing thetracking error for lateral position and yaw angle response
The simulation also demonstrated that model predictivecontrol has an ability to reject the side wind effect as adisturbance to the system which is particularly effective at10msminus1 for all cases except for the 2DoF controller duringsimulation of higher speeds on lower road surface adhesionHowever both controllers were not able to provide a betterresponse when the vehicle performed at a high forwardspeedwith a low road adhesion coefficient particularly when2DoF controller was simulated all results showed unstablevehicle response Thus for this scenario we are currentlyinvestigating new approaches with a real-time implementable
control scheme Further improvement for future works oncontrol methodology will also be considered to minimizetracking error and to enhance vehicle stability
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors would like to thank Professor Zidong Wangand Professor Manfred Ploechl for their comments andadvice Special thanks are also extended to Dr A Zahran MKhudzari for his help preparing this paper
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
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DistributedSensor Networks
International Journal of
12 Journal of Control Science and Engineering
Time (s)0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
Time (s)0 5 10 15
0
01
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
Time (s)
2WS + DYC
minus01
minus02Dire
ct y
aw m
omen
tM
z(N
m)
Figure 10 2DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 4 Model predictive control weighting matrices parameters for V119909
= 30msminus1
Control manoeuvres 2DoF 3DoF
2WSV119909
= 30msminus1 120583 = 07 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 565 11987622
= 05 11987611
= 355 11987622
= 05
V119909
= 30msminus1 120583 = 01 1198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 00311987611
= 0083 11987622
= 0042 11987611
= 003 11987622
= 045
4WS
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 155 11987622
= 25 11987611
= 135 11987622
= 35
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0005 11987622
= 0001 11987611
= 004 11987622
= 05
2WS + DYC
V119909
= 30msminus1 120583 = 071198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 505 11987622
= 05 11987611
= 35 11987622
= 05
V119909
= 30msminus1 120583 = 011198771
= 01 Δ1198771
= 003 1198771
= 01 Δ1198771
= 0031198772
= 01 Δ1198772
= 003 1198772
= 01 Δ1198772
= 00311987611
= 0092 11987622
= 004 11987611
= 05 11987622
= 25
simulation the tracking responses become unstable andimpossible to control especially under the crosswind effectas shown in Figure 14 The most probable cause is the rolldynamic motion neglect in the 2DoF controller design whenthe vehicle itself was modelled by including roll dynamicfactor It can be said that the inclusion of roll dynamic factor
is important for vehicle manoeuvre in terms of stability andcontrollability at high speed and icy road condition Sincehigh speed coupled with icy road condition will render theequation to become highly nonlinear it is impossible fora linear tire model to react positively since the handlingpropertiesmay be significantly different from those generated
Journal of Control Science and Engineering 13
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
01
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
002
004
Time (s)
2WS + DYC
minus002
minus004
minus006Dire
ct y
aw m
omen
tM
z(N
m)
Figure 11 3DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 5 Path following tracking errors with road friction surface 120583 = 07
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00637 00085 00623 000744WS 00548 00054 00526 00051
2WS + DYC 00542 00051 00527 00051
302WS 00639 05348 00616 052154WS 00558 05239 00528 00023
2WS + DYC 00549 05226 00524 00254
by the linear tire model These results show that linear tiremodel is only suitable for analyzing a stable vehicle behaviourunder the assumption of small steering and acceleration
Next for the 3DoF controller simulation 2WS with DYCmanoeuvres provides a much better tracking response incomparison to 4WS particularly in the case of lateral outputOn the contrary 4WS manoeuvre demonstrates a much bet-ter tracking response in the yaw angle and yaw rate responsesas tabulated in Table 6 With appropriate weight tuning gainthe rear steering angle and direct yaw moment control arefully optimized in order to become stable along the given tra-jectory However for the 2WS manoeuvre the vehicleresponses become unstable despite several instances wherethe weighting tuning gains are adjusted for output and inputgains With the inclusion of another input control to the con-troller design we can enhance the vehicle responses for both
2WS with DYC and 4WS control manoeuvres show Thismeans that for 2WS controller design it may have to usemore than just front steering to stabilize the vehicle under theeffects of wind gusts Figure 15 shows that rear steering angleand direct yaw moment are fully utilized in order to stabilizethe vehicle for 2WS with DYC and 4WSmanoeuvres control
In contrast to the 2DoF controller as shown in Figure 14in the 3DoF controllers the 2WS control manoeuvre canstill be used to track a given trajectory despite not perfectlyfollowing the trajectory in a satisfactory manner especiallyunder crosswind effects In Figure 15 we can see some oscil-lations in few traces responses at the end of the responsesBased on the authorsrsquo knowledge these oscillationsmay comefrom the vehicle model and some initial conditions of thesystem with imperfect controllers tuning weighting gains Inthese conditions neither 2WS nor 4WS control manoeuvres
14 Journal of Control Science and Engineering
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
Time (s)
2WS4WS2WS + DYC
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
001
002
Time (s)
2WS4WS2WS + DYC
minus001Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
1
2
Time (s)
2WS4WS2WS + DYC
minus1
minus2
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
001
002
003
Time (s)
2WS4WS2WS + DYC
minus001
minus002
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
00005
0001
4WS
Time (s)
minus00005
minus0001
minus00015
Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
001
002
Time (s)
2WS + DYC
minus001
minus002
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 12 2DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
for lateral response and neither 2WS nor 2WS with DYCcontrol manoeuvres for yaw angle or yaw rate responseperformed well causing an increase in vehicle instabilityincreased vibration and tracking responses deteriorationWewill next focus on how to enhance the controller in order tostabilize vehicle manoeuvrability and handling stability
We can therefore conclude that for 4WS the rear wheelswere helping the car to steer by improving the vehiclehandling at high speed while decreasing the turning radiusat low speed as shown by the control signal in Figures 12 and13 Meanwhile for 2WS with DYC active front steering wasused in low speed manoeuvres for lateral acceleration whileinclusion of DYC was adopted for high lateral accelerationwhen the tires were saturated and could not produce enoughlateral force for vehicle control and stability as intendedIn 2WS vehicles the rear set of wheels do not play an
active role in controlling the steering From Table 5 it canbe seen that among the three controllers 4WS gave the bestperformance by reducing the tracking error for lateral andyaw angle responses when compared with 2WS with DYCand 2WS only Tables 5 and 6 tabulated the MPC robustnessand the tracking errors in (17) for all types of manoeuvre at10msminus1 and 30msminus1 vehicle speeds and under various roadsurfaces Furthermore all controlmanoeuvre signals for bothcontrollers were within the input constraints We would liketo highlight that in MPC approaches although there was anadvantage in multivariable systems in this study there was atrade-off between theweighting tuning parameter to focus oneither lateral position or yaw angle trajectory In this paper wehave chosen lateral position as the weighting tuning priorityso we may get a better response on yaw angle and yaw rateresponse by sacrificing lateral position precision
Journal of Control Science and Engineering 15
0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
Time (s)
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
2WS4WS2WS + DYC
Time (s)
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02
Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
5
10
2WS4WS2WS + DYC
Time (s)
minus5
minus10Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
03
4WS
Time (s)
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
2WS + DYC
Time (s)
minus005
minus01
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 13 3DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
Table 6 Path following tracking errors with road friction surface 120583 = 01
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00874 00115 00841 001044WS 00838 00109 00832 00094
2WS + DYC 00838 00110 00835 00083
302WS Uncontrolled Uncontrolled 21352 052644WS Uncontrolled Uncontrolled 08824 02982
2WS + DYC Uncontrolled Uncontrolled 09964 04580
16 Journal of Control Science and Engineering
0 10 20 30
02468
10
Time (s)
Ref2WS
4WS2WS + DYC
minus2
minus4Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0020406
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
minus06
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15
Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
0020406
Time (s)
2WS4WS2WS + DYC
minus02
minus04
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
02
04
06
Time (s)
4WS
minus02
Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
005
01
Time (s)
2WS + DYC
minus005
Dire
ct y
aw m
omen
tM
z(N
m)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15Fron
t lat
eral
forc
eFyf
(Nm
) times104
Figure 14 2DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
5 Conclusion
This paper presents the robustness of model predictive con-trol for an autonomous ground vehicle on a path-followingcontrol with consideration of wind gust effects subjectedto the variation of forward speeds and road surfaces condi-tion in a double-lane change scenario The predictive con-trollers were designed based on two- (lateral-yaw model)and three- (lateral-roll model) degree-of-freedom vehiclemodels The only difference between the models is the rolldynamics factor which is taken into consideration andboth controllers were implemented into a six-degree-of-free-dom vehicle model Based on linear vehicle and tire modelapproximations of a known trajectory we evaluated the effectand impact of roll dynamics at low and high forward vehicle
speeds and at various road frictions (wet concrete to icyroads) in order to follow a desired trajectory as close as pos-sible while rejecting the crosswind effects and maintainingvehicle stability We also evaluated and compared the effi-ciency of the different manoeuvres via two-wheel steeringfour-wheel steering and two-wheel steering with direct yawmoment control
The simulation results showed that with the inclusion ofroll dynamics in the linear vehicle model the vehicle stabilityand adherence to trajectory was considerably improved forthe case of high speed manoeuvres on a higher road surfacefriction coefficient (120583 = 07) and improved even more for thecase of a lower road surface coefficient (120583 = 01) Further-more the obtained results showed and proved that the rearwheels and direct yaw moment are beneficial to help steering
Journal of Control Science and Engineering 17
0 10 20 30
0
5
Time (s)
Ref2WS
4WS2WS + DYC
minus10
minus5
minus15Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0
025
05
Time (s)
Ref2WS
4WS2WS + DYC
minus025
minus05
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
0
500
1000
Time (s)
2WS4WS2WS + DYC
minus500
minus1000
minus1500Fron
t lat
eral
forc
eFyf
(Nm
)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
01
02
03
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS + DYC
minus2
minus4
times10minus4
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 15 3DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
the car by improving the vehicle handling at high speeddecreasing the turning radius at low speeds and reducing thetracking error for lateral position and yaw angle response
The simulation also demonstrated that model predictivecontrol has an ability to reject the side wind effect as adisturbance to the system which is particularly effective at10msminus1 for all cases except for the 2DoF controller duringsimulation of higher speeds on lower road surface adhesionHowever both controllers were not able to provide a betterresponse when the vehicle performed at a high forwardspeedwith a low road adhesion coefficient particularly when2DoF controller was simulated all results showed unstablevehicle response Thus for this scenario we are currentlyinvestigating new approaches with a real-time implementable
control scheme Further improvement for future works oncontrol methodology will also be considered to minimizetracking error and to enhance vehicle stability
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors would like to thank Professor Zidong Wangand Professor Manfred Ploechl for their comments andadvice Special thanks are also extended to Dr A Zahran MKhudzari for his help preparing this paper
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
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Navigation and Observation
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DistributedSensor Networks
International Journal of
Journal of Control Science and Engineering 13
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
01
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
Time (s)
2WS4WS2WS + DYC
minus005
minus01
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
002
004
Time (s)
2WS + DYC
minus002
minus004
minus006Dire
ct y
aw m
omen
tM
z(N
m)
Figure 11 3DoF controller at 10msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
Table 5 Path following tracking errors with road friction surface 120583 = 07
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00637 00085 00623 000744WS 00548 00054 00526 00051
2WS + DYC 00542 00051 00527 00051
302WS 00639 05348 00616 052154WS 00558 05239 00528 00023
2WS + DYC 00549 05226 00524 00254
by the linear tire model These results show that linear tiremodel is only suitable for analyzing a stable vehicle behaviourunder the assumption of small steering and acceleration
Next for the 3DoF controller simulation 2WS with DYCmanoeuvres provides a much better tracking response incomparison to 4WS particularly in the case of lateral outputOn the contrary 4WS manoeuvre demonstrates a much bet-ter tracking response in the yaw angle and yaw rate responsesas tabulated in Table 6 With appropriate weight tuning gainthe rear steering angle and direct yaw moment control arefully optimized in order to become stable along the given tra-jectory However for the 2WS manoeuvre the vehicleresponses become unstable despite several instances wherethe weighting tuning gains are adjusted for output and inputgains With the inclusion of another input control to the con-troller design we can enhance the vehicle responses for both
2WS with DYC and 4WS control manoeuvres show Thismeans that for 2WS controller design it may have to usemore than just front steering to stabilize the vehicle under theeffects of wind gusts Figure 15 shows that rear steering angleand direct yaw moment are fully utilized in order to stabilizethe vehicle for 2WS with DYC and 4WSmanoeuvres control
In contrast to the 2DoF controller as shown in Figure 14in the 3DoF controllers the 2WS control manoeuvre canstill be used to track a given trajectory despite not perfectlyfollowing the trajectory in a satisfactory manner especiallyunder crosswind effects In Figure 15 we can see some oscil-lations in few traces responses at the end of the responsesBased on the authorsrsquo knowledge these oscillationsmay comefrom the vehicle model and some initial conditions of thesystem with imperfect controllers tuning weighting gains Inthese conditions neither 2WS nor 4WS control manoeuvres
14 Journal of Control Science and Engineering
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
Time (s)
2WS4WS2WS + DYC
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
001
002
Time (s)
2WS4WS2WS + DYC
minus001Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
1
2
Time (s)
2WS4WS2WS + DYC
minus1
minus2
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
001
002
003
Time (s)
2WS4WS2WS + DYC
minus001
minus002
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
00005
0001
4WS
Time (s)
minus00005
minus0001
minus00015
Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
001
002
Time (s)
2WS + DYC
minus001
minus002
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 12 2DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
for lateral response and neither 2WS nor 2WS with DYCcontrol manoeuvres for yaw angle or yaw rate responseperformed well causing an increase in vehicle instabilityincreased vibration and tracking responses deteriorationWewill next focus on how to enhance the controller in order tostabilize vehicle manoeuvrability and handling stability
We can therefore conclude that for 4WS the rear wheelswere helping the car to steer by improving the vehiclehandling at high speed while decreasing the turning radiusat low speed as shown by the control signal in Figures 12 and13 Meanwhile for 2WS with DYC active front steering wasused in low speed manoeuvres for lateral acceleration whileinclusion of DYC was adopted for high lateral accelerationwhen the tires were saturated and could not produce enoughlateral force for vehicle control and stability as intendedIn 2WS vehicles the rear set of wheels do not play an
active role in controlling the steering From Table 5 it canbe seen that among the three controllers 4WS gave the bestperformance by reducing the tracking error for lateral andyaw angle responses when compared with 2WS with DYCand 2WS only Tables 5 and 6 tabulated the MPC robustnessand the tracking errors in (17) for all types of manoeuvre at10msminus1 and 30msminus1 vehicle speeds and under various roadsurfaces Furthermore all controlmanoeuvre signals for bothcontrollers were within the input constraints We would liketo highlight that in MPC approaches although there was anadvantage in multivariable systems in this study there was atrade-off between theweighting tuning parameter to focus oneither lateral position or yaw angle trajectory In this paper wehave chosen lateral position as the weighting tuning priorityso we may get a better response on yaw angle and yaw rateresponse by sacrificing lateral position precision
Journal of Control Science and Engineering 15
0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
Time (s)
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
2WS4WS2WS + DYC
Time (s)
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02
Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
5
10
2WS4WS2WS + DYC
Time (s)
minus5
minus10Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
03
4WS
Time (s)
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
2WS + DYC
Time (s)
minus005
minus01
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 13 3DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
Table 6 Path following tracking errors with road friction surface 120583 = 01
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00874 00115 00841 001044WS 00838 00109 00832 00094
2WS + DYC 00838 00110 00835 00083
302WS Uncontrolled Uncontrolled 21352 052644WS Uncontrolled Uncontrolled 08824 02982
2WS + DYC Uncontrolled Uncontrolled 09964 04580
16 Journal of Control Science and Engineering
0 10 20 30
02468
10
Time (s)
Ref2WS
4WS2WS + DYC
minus2
minus4Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0020406
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
minus06
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15
Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
0020406
Time (s)
2WS4WS2WS + DYC
minus02
minus04
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
02
04
06
Time (s)
4WS
minus02
Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
005
01
Time (s)
2WS + DYC
minus005
Dire
ct y
aw m
omen
tM
z(N
m)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15Fron
t lat
eral
forc
eFyf
(Nm
) times104
Figure 14 2DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
5 Conclusion
This paper presents the robustness of model predictive con-trol for an autonomous ground vehicle on a path-followingcontrol with consideration of wind gust effects subjectedto the variation of forward speeds and road surfaces condi-tion in a double-lane change scenario The predictive con-trollers were designed based on two- (lateral-yaw model)and three- (lateral-roll model) degree-of-freedom vehiclemodels The only difference between the models is the rolldynamics factor which is taken into consideration andboth controllers were implemented into a six-degree-of-free-dom vehicle model Based on linear vehicle and tire modelapproximations of a known trajectory we evaluated the effectand impact of roll dynamics at low and high forward vehicle
speeds and at various road frictions (wet concrete to icyroads) in order to follow a desired trajectory as close as pos-sible while rejecting the crosswind effects and maintainingvehicle stability We also evaluated and compared the effi-ciency of the different manoeuvres via two-wheel steeringfour-wheel steering and two-wheel steering with direct yawmoment control
The simulation results showed that with the inclusion ofroll dynamics in the linear vehicle model the vehicle stabilityand adherence to trajectory was considerably improved forthe case of high speed manoeuvres on a higher road surfacefriction coefficient (120583 = 07) and improved even more for thecase of a lower road surface coefficient (120583 = 01) Further-more the obtained results showed and proved that the rearwheels and direct yaw moment are beneficial to help steering
Journal of Control Science and Engineering 17
0 10 20 30
0
5
Time (s)
Ref2WS
4WS2WS + DYC
minus10
minus5
minus15Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0
025
05
Time (s)
Ref2WS
4WS2WS + DYC
minus025
minus05
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
0
500
1000
Time (s)
2WS4WS2WS + DYC
minus500
minus1000
minus1500Fron
t lat
eral
forc
eFyf
(Nm
)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
01
02
03
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS + DYC
minus2
minus4
times10minus4
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 15 3DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
the car by improving the vehicle handling at high speeddecreasing the turning radius at low speeds and reducing thetracking error for lateral position and yaw angle response
The simulation also demonstrated that model predictivecontrol has an ability to reject the side wind effect as adisturbance to the system which is particularly effective at10msminus1 for all cases except for the 2DoF controller duringsimulation of higher speeds on lower road surface adhesionHowever both controllers were not able to provide a betterresponse when the vehicle performed at a high forwardspeedwith a low road adhesion coefficient particularly when2DoF controller was simulated all results showed unstablevehicle response Thus for this scenario we are currentlyinvestigating new approaches with a real-time implementable
control scheme Further improvement for future works oncontrol methodology will also be considered to minimizetracking error and to enhance vehicle stability
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors would like to thank Professor Zidong Wangand Professor Manfred Ploechl for their comments andadvice Special thanks are also extended to Dr A Zahran MKhudzari for his help preparing this paper
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Active and Passive Electronic Components
Control Scienceand Engineering
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RotatingMachinery
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Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Electrical and Computer Engineering
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Advances inOptoElectronics
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Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
14 Journal of Control Science and Engineering
0 5 10 15
0
2
4
Time (s)
Ref2WS
4WS2WS + DYC
minus2Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
Time (s)
2WS4WS2WS + DYC
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
001
002
Time (s)
2WS4WS2WS + DYC
minus001Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
1
2
Time (s)
2WS4WS2WS + DYC
minus1
minus2
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
001
002
003
Time (s)
2WS4WS2WS + DYC
minus001
minus002
Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
00005
0001
4WS
Time (s)
minus00005
minus0001
minus00015
Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
001
002
Time (s)
2WS + DYC
minus001
minus002
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 12 2DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
for lateral response and neither 2WS nor 2WS with DYCcontrol manoeuvres for yaw angle or yaw rate responseperformed well causing an increase in vehicle instabilityincreased vibration and tracking responses deteriorationWewill next focus on how to enhance the controller in order tostabilize vehicle manoeuvrability and handling stability
We can therefore conclude that for 4WS the rear wheelswere helping the car to steer by improving the vehiclehandling at high speed while decreasing the turning radiusat low speed as shown by the control signal in Figures 12 and13 Meanwhile for 2WS with DYC active front steering wasused in low speed manoeuvres for lateral acceleration whileinclusion of DYC was adopted for high lateral accelerationwhen the tires were saturated and could not produce enoughlateral force for vehicle control and stability as intendedIn 2WS vehicles the rear set of wheels do not play an
active role in controlling the steering From Table 5 it canbe seen that among the three controllers 4WS gave the bestperformance by reducing the tracking error for lateral andyaw angle responses when compared with 2WS with DYCand 2WS only Tables 5 and 6 tabulated the MPC robustnessand the tracking errors in (17) for all types of manoeuvre at10msminus1 and 30msminus1 vehicle speeds and under various roadsurfaces Furthermore all controlmanoeuvre signals for bothcontrollers were within the input constraints We would liketo highlight that in MPC approaches although there was anadvantage in multivariable systems in this study there was atrade-off between theweighting tuning parameter to focus oneither lateral position or yaw angle trajectory In this paper wehave chosen lateral position as the weighting tuning priorityso we may get a better response on yaw angle and yaw rateresponse by sacrificing lateral position precision
Journal of Control Science and Engineering 15
0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
Time (s)
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
2WS4WS2WS + DYC
Time (s)
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02
Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
5
10
2WS4WS2WS + DYC
Time (s)
minus5
minus10Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
03
4WS
Time (s)
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
2WS + DYC
Time (s)
minus005
minus01
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 13 3DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
Table 6 Path following tracking errors with road friction surface 120583 = 01
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00874 00115 00841 001044WS 00838 00109 00832 00094
2WS + DYC 00838 00110 00835 00083
302WS Uncontrolled Uncontrolled 21352 052644WS Uncontrolled Uncontrolled 08824 02982
2WS + DYC Uncontrolled Uncontrolled 09964 04580
16 Journal of Control Science and Engineering
0 10 20 30
02468
10
Time (s)
Ref2WS
4WS2WS + DYC
minus2
minus4Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0020406
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
minus06
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15
Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
0020406
Time (s)
2WS4WS2WS + DYC
minus02
minus04
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
02
04
06
Time (s)
4WS
minus02
Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
005
01
Time (s)
2WS + DYC
minus005
Dire
ct y
aw m
omen
tM
z(N
m)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15Fron
t lat
eral
forc
eFyf
(Nm
) times104
Figure 14 2DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
5 Conclusion
This paper presents the robustness of model predictive con-trol for an autonomous ground vehicle on a path-followingcontrol with consideration of wind gust effects subjectedto the variation of forward speeds and road surfaces condi-tion in a double-lane change scenario The predictive con-trollers were designed based on two- (lateral-yaw model)and three- (lateral-roll model) degree-of-freedom vehiclemodels The only difference between the models is the rolldynamics factor which is taken into consideration andboth controllers were implemented into a six-degree-of-free-dom vehicle model Based on linear vehicle and tire modelapproximations of a known trajectory we evaluated the effectand impact of roll dynamics at low and high forward vehicle
speeds and at various road frictions (wet concrete to icyroads) in order to follow a desired trajectory as close as pos-sible while rejecting the crosswind effects and maintainingvehicle stability We also evaluated and compared the effi-ciency of the different manoeuvres via two-wheel steeringfour-wheel steering and two-wheel steering with direct yawmoment control
The simulation results showed that with the inclusion ofroll dynamics in the linear vehicle model the vehicle stabilityand adherence to trajectory was considerably improved forthe case of high speed manoeuvres on a higher road surfacefriction coefficient (120583 = 07) and improved even more for thecase of a lower road surface coefficient (120583 = 01) Further-more the obtained results showed and proved that the rearwheels and direct yaw moment are beneficial to help steering
Journal of Control Science and Engineering 17
0 10 20 30
0
5
Time (s)
Ref2WS
4WS2WS + DYC
minus10
minus5
minus15Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0
025
05
Time (s)
Ref2WS
4WS2WS + DYC
minus025
minus05
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
0
500
1000
Time (s)
2WS4WS2WS + DYC
minus500
minus1000
minus1500Fron
t lat
eral
forc
eFyf
(Nm
)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
01
02
03
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS + DYC
minus2
minus4
times10minus4
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 15 3DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
the car by improving the vehicle handling at high speeddecreasing the turning radius at low speeds and reducing thetracking error for lateral position and yaw angle response
The simulation also demonstrated that model predictivecontrol has an ability to reject the side wind effect as adisturbance to the system which is particularly effective at10msminus1 for all cases except for the 2DoF controller duringsimulation of higher speeds on lower road surface adhesionHowever both controllers were not able to provide a betterresponse when the vehicle performed at a high forwardspeedwith a low road adhesion coefficient particularly when2DoF controller was simulated all results showed unstablevehicle response Thus for this scenario we are currentlyinvestigating new approaches with a real-time implementable
control scheme Further improvement for future works oncontrol methodology will also be considered to minimizetracking error and to enhance vehicle stability
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors would like to thank Professor Zidong Wangand Professor Manfred Ploechl for their comments andadvice Special thanks are also extended to Dr A Zahran MKhudzari for his help preparing this paper
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Journal of Control Science and Engineering 15
0 5 10 15
0
2
4
Ref2WS
4WS2WS + DYC
Time (s)
minus2
Late
ral p
ositi
onY
(m)
0 5 10 15
0
02
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
angl
e120595
(rad
)
0 5 10 15
0
02
04
Ref2WS
4WS2WS + DYC
Time (s)
minus02
minus04
Yaw
rate
120595998400
(rad
sminus1)
0 5 10 15
0
1000
2000
2WS4WS2WS + DYC
Time (s)
minus1000
minus2000Fron
t lat
eral
forc
eFyf
(Nm
)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02
Vehi
cle si
de sl
ip120573
(rad
)
0 5 10 15
0
5
10
2WS4WS2WS + DYC
Time (s)
minus5
minus10Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 5 10 15
0
01
02
03
2WS4WS2WS + DYC
Time (s)
minus01
minus02Fron
t ste
erin
g an
gle120575f
(rad
)
0 5 10 15
0
01
02
03
4WS
Time (s)
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 5 10 15
0
005
01
2WS + DYC
Time (s)
minus005
minus01
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 13 3DoF controller at 30msminus1 and 120583 = 07 by 2WS 4WS and 2WS + DYC
Table 6 Path following tracking errors with road friction surface 120583 = 01
Vehicle speed Manoeuvre control Controller 2DoF Controller 3DoF119884 [m] Ψ [rad] 119884 [m] Ψ [rad]
102WS 00874 00115 00841 001044WS 00838 00109 00832 00094
2WS + DYC 00838 00110 00835 00083
302WS Uncontrolled Uncontrolled 21352 052644WS Uncontrolled Uncontrolled 08824 02982
2WS + DYC Uncontrolled Uncontrolled 09964 04580
16 Journal of Control Science and Engineering
0 10 20 30
02468
10
Time (s)
Ref2WS
4WS2WS + DYC
minus2
minus4Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0020406
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
minus06
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15
Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
0020406
Time (s)
2WS4WS2WS + DYC
minus02
minus04
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
02
04
06
Time (s)
4WS
minus02
Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
005
01
Time (s)
2WS + DYC
minus005
Dire
ct y
aw m
omen
tM
z(N
m)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15Fron
t lat
eral
forc
eFyf
(Nm
) times104
Figure 14 2DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
5 Conclusion
This paper presents the robustness of model predictive con-trol for an autonomous ground vehicle on a path-followingcontrol with consideration of wind gust effects subjectedto the variation of forward speeds and road surfaces condi-tion in a double-lane change scenario The predictive con-trollers were designed based on two- (lateral-yaw model)and three- (lateral-roll model) degree-of-freedom vehiclemodels The only difference between the models is the rolldynamics factor which is taken into consideration andboth controllers were implemented into a six-degree-of-free-dom vehicle model Based on linear vehicle and tire modelapproximations of a known trajectory we evaluated the effectand impact of roll dynamics at low and high forward vehicle
speeds and at various road frictions (wet concrete to icyroads) in order to follow a desired trajectory as close as pos-sible while rejecting the crosswind effects and maintainingvehicle stability We also evaluated and compared the effi-ciency of the different manoeuvres via two-wheel steeringfour-wheel steering and two-wheel steering with direct yawmoment control
The simulation results showed that with the inclusion ofroll dynamics in the linear vehicle model the vehicle stabilityand adherence to trajectory was considerably improved forthe case of high speed manoeuvres on a higher road surfacefriction coefficient (120583 = 07) and improved even more for thecase of a lower road surface coefficient (120583 = 01) Further-more the obtained results showed and proved that the rearwheels and direct yaw moment are beneficial to help steering
Journal of Control Science and Engineering 17
0 10 20 30
0
5
Time (s)
Ref2WS
4WS2WS + DYC
minus10
minus5
minus15Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0
025
05
Time (s)
Ref2WS
4WS2WS + DYC
minus025
minus05
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
0
500
1000
Time (s)
2WS4WS2WS + DYC
minus500
minus1000
minus1500Fron
t lat
eral
forc
eFyf
(Nm
)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
01
02
03
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS + DYC
minus2
minus4
times10minus4
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 15 3DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
the car by improving the vehicle handling at high speeddecreasing the turning radius at low speeds and reducing thetracking error for lateral position and yaw angle response
The simulation also demonstrated that model predictivecontrol has an ability to reject the side wind effect as adisturbance to the system which is particularly effective at10msminus1 for all cases except for the 2DoF controller duringsimulation of higher speeds on lower road surface adhesionHowever both controllers were not able to provide a betterresponse when the vehicle performed at a high forwardspeedwith a low road adhesion coefficient particularly when2DoF controller was simulated all results showed unstablevehicle response Thus for this scenario we are currentlyinvestigating new approaches with a real-time implementable
control scheme Further improvement for future works oncontrol methodology will also be considered to minimizetracking error and to enhance vehicle stability
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors would like to thank Professor Zidong Wangand Professor Manfred Ploechl for their comments andadvice Special thanks are also extended to Dr A Zahran MKhudzari for his help preparing this paper
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
16 Journal of Control Science and Engineering
0 10 20 30
02468
10
Time (s)
Ref2WS
4WS2WS + DYC
minus2
minus4Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
04
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0020406
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
minus06
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15
Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
0020406
Time (s)
2WS4WS2WS + DYC
minus02
minus04
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
02
04
06
Time (s)
4WS
minus02
Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
005
01
Time (s)
2WS + DYC
minus005
Dire
ct y
aw m
omen
tM
z(N
m)
0 10 20 30
005
115
Time (s)
2WS4WS2WS + DYC
minus05
minus1
minus15Fron
t lat
eral
forc
eFyf
(Nm
) times104
Figure 14 2DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
5 Conclusion
This paper presents the robustness of model predictive con-trol for an autonomous ground vehicle on a path-followingcontrol with consideration of wind gust effects subjectedto the variation of forward speeds and road surfaces condi-tion in a double-lane change scenario The predictive con-trollers were designed based on two- (lateral-yaw model)and three- (lateral-roll model) degree-of-freedom vehiclemodels The only difference between the models is the rolldynamics factor which is taken into consideration andboth controllers were implemented into a six-degree-of-free-dom vehicle model Based on linear vehicle and tire modelapproximations of a known trajectory we evaluated the effectand impact of roll dynamics at low and high forward vehicle
speeds and at various road frictions (wet concrete to icyroads) in order to follow a desired trajectory as close as pos-sible while rejecting the crosswind effects and maintainingvehicle stability We also evaluated and compared the effi-ciency of the different manoeuvres via two-wheel steeringfour-wheel steering and two-wheel steering with direct yawmoment control
The simulation results showed that with the inclusion ofroll dynamics in the linear vehicle model the vehicle stabilityand adherence to trajectory was considerably improved forthe case of high speed manoeuvres on a higher road surfacefriction coefficient (120583 = 07) and improved even more for thecase of a lower road surface coefficient (120583 = 01) Further-more the obtained results showed and proved that the rearwheels and direct yaw moment are beneficial to help steering
Journal of Control Science and Engineering 17
0 10 20 30
0
5
Time (s)
Ref2WS
4WS2WS + DYC
minus10
minus5
minus15Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0
025
05
Time (s)
Ref2WS
4WS2WS + DYC
minus025
minus05
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
0
500
1000
Time (s)
2WS4WS2WS + DYC
minus500
minus1000
minus1500Fron
t lat
eral
forc
eFyf
(Nm
)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
01
02
03
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS + DYC
minus2
minus4
times10minus4
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 15 3DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
the car by improving the vehicle handling at high speeddecreasing the turning radius at low speeds and reducing thetracking error for lateral position and yaw angle response
The simulation also demonstrated that model predictivecontrol has an ability to reject the side wind effect as adisturbance to the system which is particularly effective at10msminus1 for all cases except for the 2DoF controller duringsimulation of higher speeds on lower road surface adhesionHowever both controllers were not able to provide a betterresponse when the vehicle performed at a high forwardspeedwith a low road adhesion coefficient particularly when2DoF controller was simulated all results showed unstablevehicle response Thus for this scenario we are currentlyinvestigating new approaches with a real-time implementable
control scheme Further improvement for future works oncontrol methodology will also be considered to minimizetracking error and to enhance vehicle stability
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors would like to thank Professor Zidong Wangand Professor Manfred Ploechl for their comments andadvice Special thanks are also extended to Dr A Zahran MKhudzari for his help preparing this paper
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Journal of Control Science and Engineering 17
0 10 20 30
0
5
Time (s)
Ref2WS
4WS2WS + DYC
minus10
minus5
minus15Late
ral p
ositi
onY
(m)
0 10 20 30
0
02
Time (s)
Ref2WS
4WS2WS + DYC
minus02
minus04
Yaw
angl
e120595
(rad
)
0 10 20 30
0
025
05
Time (s)
Ref2WS
4WS2WS + DYC
minus025
minus05
Yaw
rate
120595998400
(rad
sminus1)
0 10 20 30
0
500
1000
Time (s)
2WS4WS2WS + DYC
minus500
minus1000
minus1500Fron
t lat
eral
forc
eFyf
(Nm
)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02Vehi
cle si
de sl
ip120573
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS4WS2WS + DYC
minus2
minus4
Late
ral a
ccel
erat
ion
a y(m
sminus2)
0 10 20 30
001020304
Time (s)
2WS4WS2WS + DYC
minus01
minus02
Fron
t ste
erin
g an
gle120575f
(rad
)
0 10 20 30
0
01
02
03
Time (s)
4WS
minus01
minus02Rear
stee
ring
angl
e120575r
(rad
)
0 10 20 30
0
2
4
Time (s)
2WS + DYC
minus2
minus4
times10minus4
Dire
ct y
aw m
omen
tM
z(N
m)
Figure 15 3DoF controller at 30msminus1 and 120583 = 01 by 2WS 4WS and 2WS + DYC
the car by improving the vehicle handling at high speeddecreasing the turning radius at low speeds and reducing thetracking error for lateral position and yaw angle response
The simulation also demonstrated that model predictivecontrol has an ability to reject the side wind effect as adisturbance to the system which is particularly effective at10msminus1 for all cases except for the 2DoF controller duringsimulation of higher speeds on lower road surface adhesionHowever both controllers were not able to provide a betterresponse when the vehicle performed at a high forwardspeedwith a low road adhesion coefficient particularly when2DoF controller was simulated all results showed unstablevehicle response Thus for this scenario we are currentlyinvestigating new approaches with a real-time implementable
control scheme Further improvement for future works oncontrol methodology will also be considered to minimizetracking error and to enhance vehicle stability
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors would like to thank Professor Zidong Wangand Professor Manfred Ploechl for their comments andadvice Special thanks are also extended to Dr A Zahran MKhudzari for his help preparing this paper
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
18 Journal of Control Science and Engineering
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] C E Garcıa DM Prett andMMorari ldquoModel predictive con-trol theory and practicemdasha surveyrdquo Automatica vol 25 no 3pp 335ndash348 1989
[3] P Mhaskar ldquoRobust model predictive control design for fault-tolerant control of process systemsrdquo Industrial amp EngineeringChemistry Research vol 45 no 25 pp 8565ndash8574 2006
[4] V Vesely and D Rosinova ldquoRobust model predictive controldesignrdquo in Model Predictive Control T Zheng Ed pp 1ndash24InTech Rijeka Croatia 2010
[5] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002
[6] F Yakub and Y Mori ldquoComparative study of autonomous path-following vehicle control viamodel predictive control and linearquadratic controlrdquo Proceedings of the Institution of MechanicalEngineers Part D vol 229 no 12 pp 1695ndash1714 2015
[7] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers Part D Journal of Auto-mobile Engineering vol 221 no 4 pp 377ndash391 2007
[8] J Wang D A Wilson W Xu and D A Crolla ldquoIntegratedvehicle ride and steady-state handling control via active suspen-sionsrdquo International Journal of Vehicle Design vol 42 no 3-4pp 306ndash327 2006
[9] M J L Boada B L Boada A Munoz and V Dıaz ldquoIntegratedcontrol of front-wheel steering and front braking forces on thebasis of fuzzy logicrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 220no 3 pp 253ndash267 2006
[10] A Balluchi A Bicchi and P Soueres ldquoPath-following with abounded-curvature vehicle a hybrid control approachrdquo Inter-national Journal of Control vol 78 no 15 pp 1228ndash1247 2005
[11] KMoriwaki ldquoAutonomous steering control for electric vehiclesusing nonlinear state feedback119867
infin
controlrdquoNonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e2257ndashe2268 2005
[12] B Mashadi P Ahmadizadeh and M Majidi ldquoIntegrated con-troller design for path following in autonomous vehiclesrdquo SAETechnical Paper 2011-01-1032 SAE International 2011
[13] J-M Park D-W Kim Y-S Yoon H J Kim and K-S YildquoObstacle avoidance of autonomous vehicles based on modelpredictive controlrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 223no 12 pp 1499ndash1516 2009
[14] Y Yoon J Shin H J Kim Y Park and S Sastry ldquoModel-pre-dictive active steering and obstacle avoidance for autonomousground vehiclesrdquo Control Engineering Practice vol 17 no 7 pp741ndash750 2009
[15] P Falcone F Borrelli J Asgari H E Tseng andDHrovat ldquoPre-dictive active steering control for autonomous vehicle systemsrdquoIEEE Transactions on Control Systems Technology vol 15 no 3pp 566ndash580 2007
[16] T Keviczky P Falcone F Borrelli J Asgari and D HrovatldquoPredictive control approach to autonomous vehicle steeringrdquoin Proceedings of the American Control Conference pp 4670ndash4675 IEEE Minneapolis Minn USA June 2006
[17] F Borrelli P Falcone T Keviczky J Asgari and D HrovatldquoMPC-based approach to active steering for autonomous
vehicle systemsrdquo International Journal of Vehicle AutonomousSystems vol 3 no 2ndash4 pp 265ndash291 2005
[18] K Nam S Oh H Fujimoto and Y Hori ldquoRobust yaw stabilitycontrol for electric vehicles based on active front steering con-trol through a steer-by-wire systemrdquo International Journal ofAutomotive Technology vol 13 no 7 pp 1169ndash1176 2012
[19] H Yoshida S Shinohara and M Nagai ldquoLane change steeringmanoeuvre using model predictive control theoryrdquo VehicleSystem Dynamics vol 46 no 1 pp 669ndash681 2008
[20] P Falcone H Eric Tseng F Borrelli J Asgari and D HrovatldquoMPC-based yaw and lateral stabilisation via active frontsteering and brakingrdquo Vehicle System Dynamics vol 46 no 1pp 611ndash628 2008
[21] W Kim D Kim K Yi and H J Kim ldquoDevelopment of apath-tracking control system based onmodel predictive controlusing infrastructure sensorsrdquo Vehicle System Dynamics vol 50no 6 pp 1001ndash1023 2012
[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001
[23] G Palmieri P Falcone H E Tseng and L Glielmo ldquoA prelim-inary study on the effects of roll dynamics in predictive vehiclestability controlrdquo in Proceedings of the 47th IEEE Conference onDecision and Control (CDC rsquo08) pp 5354ndash5359 IEEE CancunMexico December 2008
[24] H PacejkaTire andVehicleDynamics ElsevierOxfordUK 3rdedition 2012
[25] R N Jazar Vehicle Dynamics Theory and Application SpringerNew York NY USA 3rd edition 2009
[26] C R Carlson and J C Gerdes ldquoOptimal rollover preventionwith steer by wire and differential brakingrdquo in Proceedings of theASME InternationalMechanical Engineering Congress vol 1 no1 pp 345ndash354 Washington Wash USA November 2003
[27] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[28] M Mahmood and P Mhaskar ldquoOn constructing constrainedcontrol Lyapunov functions for linear systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 5 pp 1136ndash1140 2011
[29] M Mahmood and P Mhaskar ldquoEnhanced stability regions formodel predictive control of nonlinear process systemsrdquo AIChEJournal vol 54 no 6 pp 1487ndash1498 2008
[30] F Yakub and Y Mori ldquoHeavy vehicle stability and rollover pre-vention via switching model predictive controlrdquo InternationalJournal of Innovative Computing Information and Control vol11 no 5 pp 1751ndash1764 2015
[31] S Solmaz M Corless and R Shorten ldquoA methodology for thedesign of robust rollover prevention controllers for automotivevehicles with active steeringrdquo International Journal of Controlvol 80 no 11 pp 1763ndash1779 2007
[32] D C Viano and C Parenteau ldquoCase study of vehicle maneuversleading to rollovers need for a vehicle test simulating off roadexcursions recovery and handlingrdquo SAE Technical Paper 2003-01-0169 SAE International 2003
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Recommended