Research Article Robust PID Steering Control in Parameter...

Research ArticleRobust PID Steering Control in Parameter Space forHighly Automated Driving

Mümin Tolga Emirler,1,2 Esmail Meriç Can Uygan,2,3

Bilin Aksun Güvenç,2,4 and Levent Güvenç2,4

1 Department of Mechanical Engineering, Istanbul Technical University, 34437 Istanbul, Turkey2Mekar Labs, Istanbul Okan University, 34959 Istanbul, Turkey3 Department of Control Engineering, Istanbul Technical University, 34469 Istanbul, Turkey4Department of Mechanical Engineering, Istanbul Okan University, 34959 Istanbul, Turkey

Correspondence should be addressed to Levent Guvenc; [email protected]

Received 24 September 2013; Revised 27 November 2013; Accepted 29 November 2013; Published 4 February 2014

Academic Editor: Luis M. Bergasa

Copyright © 2014 Mumin Tolga Emirler et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

This paper is on the design of a parameter space based robust PID steering controller.This controller is used for automated steeringin automated path following of a midsized sedan. Linear and nonlinear models of this midsized sedan are presented in the paper.Experimental results are used to validate the longitudinal and lateral dynamic models of this vehicle. This paper is on automatedsteering control and concentrates on the lateral direction of motion. The linear model is used to design a PID steering controllerin parameter space that satisfies 𝐷-stability. The PID steering controller that is designed is used in a simulation study to illustratethe effectiveness of the proposed method. Simulation results for a circular trajectory and for a curved trajectory are presented anddiscussed in detail. This study is part of a larger research effort aimed at implementing highly automated driving in a midsizedsedan.

1. Introduction

In recent years, intelligent vehicle systems and highly auto-mated driving technologies have drawn interest amongresearchers. Many research efforts including, for example,the work reported in [1–3] have concentrated on semi-autonomous and fully autonomous vehicles. Autonomousdriving requires coordinated automation of the longitudinaland the lateral driving tasks of speed control and steeringcontrol, respectively. Desired path tracking of an autonomousvehicle requires the proper design and implementation ofsteering and speed controllers at the lower control level. Thispaper concentrates on automated steering control.

The basic automatic steering control algorithms foundin the literature are based on PID (proportional-integral-derivative) type controllers [4, 5]. In these designs, thelateral deviation of the vehicle at a preview distance is fedback for controlling the vehicle’s lateral dynamics. In [6], arobust PIDD controller is designed for automatic bus steering

control as a solution of a benchmark problem.The yaw rate ismeasured in addition to lateral deviation measurements andis fed back for improving the control system performance.For the same benchmark problem, a discrete time add-ondisturbance observer design is realized in [7]. Using the add-on disturbance observer, the performance improvement isachieved without the need for yaw rate feedback. Anotherapproach to automatic steering controller is to design nestedPI and PID controllers [8]. A PI steering controller thatreduces yaw rate tracking error is used to improve the vehiclesteering dynamics and a PID controller is employed to rejectthe lateral deviation from the desired path due to roadcurvature disturbances [8].

In this paper, the parameter space approach based PIDcontroller design is applied to automatic steering control.Thetheoretical background about the parameter space approachand an example of road vehicle yaw stability control can befound in the references [9–11]. The parameters exhibiting the

Hindawi Publishing CorporationInternational Journal of Vehicular TechnologyVolume 2014, Article ID 259465, 8 pageshttp://dx.doi.org/10.1155/2014/259465

2 International Journal of Vehicular Technology

largest variation in automatic steering control are the vehiclemass, the vehicle speed, and the road friction coefficient.In this paper, the controller parameter space for the twofree coefficients of a PID controller which are chosen as theproportional gain 𝑘𝑝 and the derivative gain 𝑘𝑑 is obtainedconsidering 𝐷-stability requirements. An overall solutionregion is calculated by intersecting the solution regions forexemplary points chosen from the boundary of the uncertainrange of parameters. Robust PID coefficients satisfying 𝐷-stability are chosen from the overall calculated parameterspace regions. The designed controller is applied to anexperimentally validated nonlinear simulation model of asedan used in autonomous vehicle tests by the authors.

The organisation of the rest of this paper is as follows. InSection 2, the linear vehicle model used in controller designand the experimental vehicle that it is based on are described.The nonlinear model of this experimental vehicle and modelvalidation results are also presented in Section 2. In Section 3,the robust PID controller design is performed based onmapping 𝐷-stability boundaries into the parameter space.The simulation results in Section 4 illustrate the effectivenessof the designed controller. The paper ends with conclusionsand recommendations for future work in Section 5.

2. Vehicle Models and Experimental Vehicle

2.1. Vehicle Steering Model. The vehicle steering behavioris modeled as single track model that also includes thedynamics of following the reference path as illustrated inFigure 1. The linear vehicle steering model is described instate-space form as

[[[

[

𝛽

𝑟

Δ��

𝑦

]]]

]

=

[[[

[

𝑎11 𝑎12 0 0

𝑎21 𝑎22 0 0

0 1 0 0

𝑉 𝑙𝑠 𝑉 0

]]]

]

[[[

[

𝛽

𝑟

Δ𝜓

𝑦

]]]

]

+

[[[

[

𝑏11 0

𝑏21 0

0 −𝑉

0 0

]]]

]

[𝛿𝑓

𝜌ref] ,

(1)

where 𝛽, 𝑟, 𝑉, ΔΨ, 𝑙𝑠, and 𝑦 are vehicle side slip angle, vehicleyaw rate, vehicle velocity, yaw angle relative to the desiredpath’s tangent, the preview distance, and lateral deviationfrom the desired path at the preview distance, respectively.The control input is the steering angle 𝛿𝑓. 𝜌ref = 1/𝑅 is theroad curvature where 𝑅 is the road radius. The remainingterms are

𝑎11 =

− (𝑐𝑟 + 𝑐𝑓)

��𝑉, 𝑎12 =

−1 + (𝑐𝑟𝑙𝑟 − 𝑐𝑓𝑙𝑓)

��𝑉2,

𝑎21 =

(𝑐𝑟𝑙𝑟 − 𝑐𝑓𝑙𝑓)

𝐽

, 𝑎22 =

− (𝑐𝑟𝑙2

𝑟+ 𝑐𝑓𝑙2

𝑓)

𝐽𝑉2,

𝑏11 =

𝑐𝑓

��𝑉, 𝑏12 =

𝑐𝑓𝑙𝑓

𝐽

,

(2)

where �� = 𝑚/𝜇 is the virtual mass, 𝐽 = 𝐽/𝜇 is the virtualmoment of inertia, 𝜇 is the road friction coefficient, 𝑚 isthe vehicle mass, 𝐽 is the moment of inertia, 𝑐𝑓 and 𝑐𝑟 arethe cornering stiffnesses, 𝑙𝑓 is the distance from the center

Desired path

Fr

Vr𝛼r

FfVf𝛼f

𝛿f

lflr

ls

R

y

y

V

𝛽

x

Δ𝜓

r

CG

Figure 1: Vehicle steering model.

3400

1400

P4 P3

P2P1

1 20

S

V (m/s)

m=m/𝜇

(kg)

Figure 2: Uncertainty box.

of gravity of the vehicle (CG) to the front axle, and 𝑙𝑟 isthe distance from the CG to the rear axle [7]. The values ofthe parameters used in this paper are 𝐽 = 2392 kgm2, 𝑙𝑓 =

1.07m, 𝑙𝑟 = 1.53m, 𝑙𝑠 = 2m, 𝑐𝑓 = 72463N/rad, and 𝑐𝑟 =92492N/rad. The vehicle mass, the vehicle velocity, and theroad friction coefficient are taken as uncertain parameterswithin the ranges of 𝑚 ∈ [1400, 1700] (kg) (the nominalvalue of mass is 1550 kg), 𝜇 ∈ [0.5, 1], and 𝑉 ∈ [1, 20] (m/s),respectively. The virtual mass, then, is within the range �� =

𝑚/𝜇 ∈ [1400, 3400] (kg). The corresponding uncertainty boxof virtual mass and vehicle speed is illustrated in Figure 2.

2.2. Experimental Vehicle. The experimental vehicle is a FiatLineamidsized sedan.Three Fiat Lineamidsized sedans wereused by the authors and their colleagues in lateral dynamicstesting [12], semiautonomous driving in a platoon [2], andautonomous path following experiments [13], respectively.The authors, therefore, have considerable experimental expe-rience with this vehicle. The current paper uses the thirdof these vehicles in [13]. The steering controller designed inthis paper will be tested on that vehicle which is shown inFigure 3. This is the drive-by-wire vehicle of Istanbul OkanUniversity named Okanom [14].Throttle, brake, and steeringactuation signals are provided by a dSpace microautoboxgeneral purpose electronic control unit which is also usedfor all the low level computations. Available signals on thevehicle CAN bus are read by this microautobox. A personalcomputer operating under Linux is used as an upper level

International Journal of Vehicular Technology 3

Figure 3: Experimental vehicle.

Fhcmg

Frr

Fxi

x

𝜃

Faero

Figure 4: The resistive forces acting on the longitudinal dynamicsof the vehicle.

control system. This PC collects data from the GPS receiver,the IMU unit, the Lidar in front of the vehicle, and the IEEE802.11p vehicle to vehicle (V2V) communicationmodem andcommunicates with the low level microautobox controller.The GPS and the IMU signals are used in GPS/INS integra-tion [15]. The GPS position of the vehicle is periodically sentby the V2V system to a nearby road side unit (RSU). Thisinformation is used to track the vehicle position on a remotecomputer and on smartphones. The path to be followed isrecorded as a list of successive GPS waypoints which formthe desired path in Figure 1.

2.3. Nonlinear VehicleModel. Theequations ofmotion for thelongitudinal and the lateral dynamics of the nonlinear vehiclemodel are

𝑚(𝑎𝑥 − 𝑟𝑉𝑦) = ∑

𝑖=𝑓,𝑟

𝐹𝑥𝑖 cos 𝛿𝑖 − 𝐹𝑦𝑖 sin 𝛿𝑖

− (𝐹aero + 𝐹𝑟𝑟 + 𝐹ℎ𝑐) ,

𝑚 (𝑎𝑦 + 𝑟𝑉𝑥) = ∑

𝑖=𝑓,𝑟

𝐹𝑥𝑖 sin 𝛿𝑖 + 𝐹𝑦𝑖 cos 𝛿𝑖

(3)

while the equation of motion around the yaw axis is

𝐼𝑧 𝑟 = 𝑙𝑓𝐹𝑦𝑓 cos 𝛿𝑓 − 𝑙𝑟𝐹𝑦𝑟 cos 𝛿𝑟 + 𝑙𝑓𝐹𝑥𝑓 sin 𝛿𝑓

− 𝑙𝑟𝐹𝑥𝑟 sin 𝛿𝑟,

(4)

where 𝐹𝑥𝑖 and 𝐹𝑦𝑖 are the longitudinal and the lateral tireforces.𝑓 and 𝑟 represent the front and rear tires. 𝑎𝑥, 𝑎𝑦,𝑉𝑥,𝑉𝑦,and 𝐼𝑧 are the longitudinal acceleration at the CG, the lateralacceleration at theCG, the longitudinal velocity at theCG, thelateral velocity at theCG, and themoment of inertia about theyaw axis, respectively. Note that for the front wheel steeredvehicle considered in this paper, the rear wheel steering angle𝛿𝑟 = 0.

z

Tbi𝜔i

TdRw

x

Fxi

Figure 5: The forces and the torques acting on the wheel.

The resistive forces which affect the longitudinal dynam-ics of the vehicle are shown in Figure 4.The aerodynamic dragforce 𝐹aero is given by

𝐹aero =1

2𝐴𝜌𝐶𝑑𝑉

2, (5)

where 𝐴 is the effective frontal area of the vehicle, 𝜌 is themass density of air, 𝐶𝑑 is the drag coefficient, and 𝑉 is thevelocity of the vehicle. The rolling resistance force 𝐹𝑟𝑟 isdetermined as

𝐹𝑟𝑟 = 𝐶𝑟𝑟𝑚𝑔 cos (𝜃) , (6)

where𝐶𝑟𝑟 is the rolling resistance coefficient and 𝜃 is the roadinclination angle. The gravitational slope resistance force 𝐹ℎ𝑐is modeled as

𝐹ℎ𝑐 = 𝑚𝑔 cos (𝜃) . (7)

The internal combustion engine (ICE) is modeled using astatic engine map that defines the relationship between theinputs of throttle position 𝛼, the engine speed 𝜔, and theoutput engine torque 𝑇ICE(𝜔, 𝛼). The engine torque output istransmitted to the wheels through the driveline as torque 𝑇𝑑according to

𝑇𝑑 = 𝜂𝑡𝑖𝑡𝑇ice (𝜔, 𝛼) , (8)

where 𝜂𝑡 is a static efficiency factor used tomodel mechanicallosses and 𝑖𝑡 is the transmission ratio. These parameters areused to model the transmission of the vehicle.

The forces and torques acting on the wheel are shown inFigure 5. The moment balance at the center of the wheel isgiven by

𝐼𝑤��𝑖 = 𝑇𝑑 − 𝑇𝑏𝑖 − 𝐹𝑥𝑖𝑅𝑤, (9)

where 𝐼𝑤 is the moment of inertia of the wheel, 𝜔 is theangular velocity of the 𝑖th wheel, 𝑇𝑏 is the braking torqueon the 𝑖th wheel applied through the brake system, 𝐹𝑥𝑖 is thelongitudinal tire force of the 𝑖th wheel, and 𝑅𝑤 is the effectivewheel radius.


−0.1

0.1

0

0

30

20

10

0.5

0

−0.5

5 10 15 20 25 30 35 40

𝛿f

(rad

)V

(m/s

)r

(rad

/s)

Time (s)

0 5 10 15 20 25 30 35 40

Time (s)

Experimental result

Experimental result

Simulation result

0 5 10 15 20 25 30 35 40

Time (s)

Simulation result

Figure 6: Comparison of test data and nonlinear vehicle simulationresults for lateral dynamics.

The longitudinal velocities of the front and rear wheelscan be determined as follows:

𝑉𝑓𝑥 =√𝑉2𝑥+ (𝑉𝑦 + 𝑙𝑓𝑟)

2

cos𝛼𝑓,

𝑉𝑟𝑥 =√𝑉2𝑥+ (𝑉𝑦 − 𝑙𝑟𝑟)

2

cos𝛼𝑟,

(10)

where the tire slip angles are

𝛼𝑓 = 𝛿𝑓 − 𝑎 tan(tan𝛽 +

𝑙𝑓𝑟

𝑉𝑥

) ,

𝛼𝑟 = 𝛿𝑟 − 𝑎 tan(tan𝛽 −𝑙𝑟𝑟

𝑉𝑥

) .

(11)

The longitudinal wheel slip ratio is defined as

𝑠𝑖 =

{{{

{{{

{

𝑅𝑤𝜔𝑖 − 𝑉𝑖𝑥

𝑉𝑖𝑥

, 𝑅𝑤𝜔𝑖 < 𝑉𝑖𝑥 (braking)𝑅𝑤𝜔𝑖 − 𝑉𝑖𝑥

𝑅𝑤𝜔𝑖

, 𝑅𝑤𝜔𝑖 > 𝑉𝑖𝑥 (traction) , (𝑖 = 𝑓, 𝑟) .

(12)

The Dugoff tire model is used for the calculations of thetire forces as

𝐹𝑥𝑖 = 𝑓𝑖𝐶𝑥𝑖𝑠𝑖,

𝐹𝑦𝑖 = 𝑓𝑖𝐶𝑦𝑖𝛼𝑖,

(13)

Time (s)

Experimental resultSimulation result

25

20

15

10

5

00

V(m

/s)

50 100 150 200

Figure 7: Comparison of test data and nonlinear vehicle simulationresults for longitudinal dynamics.

where𝐶𝑥𝑖 and𝐶𝑦𝑖 are the longitudinal stiffness and the lateralcornering stiffness of the 𝑖th wheel. The coefficients 𝑓𝑖 aredetermined using [2]

𝑓𝑖 =

{{{{

{{{{

{

1, 𝐹𝑅𝑖 ≤𝜇𝐹𝑧𝑖

2

(2 −𝜇𝐹𝑧𝑖

2𝐹𝑅𝑖

)𝜇𝐹𝑧𝑖

2𝐹𝑅𝑖

, 𝐹𝑅𝑖 >𝜇𝐹𝑧𝑖

2,

(14)

𝐹𝑅𝑖 =√(𝐶𝑥𝑖𝑠𝑖)

2+ (𝐶𝑦𝑖𝛼𝑖)

2

. (15)

Model validation studies were performed using the dataobtained from the experimental vehicle test runs. The mea-sured steering wheel and vehicle velocity were used asinputs to the nonlinear vehicle model. The simulated outputsfor vehicle velocity and yaw rate were compared with theobtained experimental data. A comparison result from atest run is shown in Figure 6. A 𝐽-turn-like maneuver wasapplied to the vehicle. The nonlinear vehicle model resultsof the vehicle velocity and yaw rate were consistent with theobtained data from the experimental vehicle. In the test runshown in Figure 7, the experimental vehicle followed a veloc-ity profile from the Grand Cooperative Driving Challenge[2]. The steering wheel input was zero. It can be seen fromFigure 7 that the velocity obtained from the nonlinear vehiclemodel coincides with the experimental test result closely.

3. Robust PID Steering Controller DesignUsing the Parameter Space Approach

3.1. Mapping D-Stability Requirements into the ParameterSpace. Similar to the approach in [11], 𝐷-stability require-ments can be mapped into the parameter space.

Consider the plant given by

𝐺 (𝑠) =𝑁 (𝑠)

𝐷 (𝑠), (16)

where 𝑁(𝑠) represents the numerator of the plant and 𝐷(𝑠)

represents its denominator. The real and imaginary parts of


Rregion

Im

Re𝜃

𝜎

𝜕1

𝜕2𝜕3

D-stable

Figure 8:𝐷-stable region in the complex plane.

the numerator and denominator can be defined as 𝑁(𝑗𝜔) =

𝑁𝑅(𝜔) + 𝑗𝑁𝐼(𝜔) and𝐷(𝑗𝜔) = 𝐷𝑅(𝜔) + 𝑗𝐷𝐼(𝜔).The PID controlled closed loop system characteristic

equation can be written as

𝑝𝑐 (𝑠) = 𝑠𝐷 (𝑠) + (𝑘𝑝𝑠 + 𝑘𝑖 + 𝑘𝑑𝑠2)𝑁 (𝑠)

= 𝑎𝑛+1𝑠𝑛+1

+ 𝑎𝑛𝑠𝑛+ ⋅ ⋅ ⋅ + 𝑎1𝑠 + 𝑎0 = 0,

(17)

where 𝑛 is the degree of the plant 𝐺(𝑠).The Hurwitz stability boundary crossed by a pair of

complex conjugate roots is characterized by the followingequations:

Re [𝑝𝑐 (𝑗𝜔)] = 0,

Im [𝑝𝑐 (𝑗𝜔)] = 0, ∀𝜔 ∈ (0,∞] .

(18)

This is called the complex root boundary (CRB).There may be a real root boundary such that a single real

root crosses the boundary at frequency𝜔 = 0 as characterizedby

𝑝𝑐 (0) = 0 or 𝑎0 = 0. (19)

This is called the real root boundary (RRB).There may exist an infinite root boundary (IRB) which

is characterized by a degree drop in the characteristic poly-nomial at 𝜔 = ∞. This degree drop in the characteristicpolynomial is characterized as

𝑎𝑛+1 = 0. (20)

CRB, RRB, and IRB solutions parameterized by frequency𝜔 can be plotted in the parameter plane of two free designparameters to show the Hurwitz stability regions of the givenclosed loop system. The free parameter pairs which provideHurwitz stability can be chosen visually from the stableregion of the parameter plane.

The aforementioned parameter space computationmethod to determine Hurwitz stability regions can beextended to specify relative stability regions such as 𝐷-stability. A closed loop system is 𝐷-stable when the roots ofthe closed loop characteristic equation lie in the 𝐷-stableregion in the complex plane as depicted in Figure 8.

𝜌ref

𝛿f + +

−1

y

Robust PID controller

Gy𝜌ref(s)

Gy𝛿𝑓(s)

Figure 9: Control system structure.

25

25

20

20

15

15

10

10

5

5

0

0−5

D-stable region

kd

kp

𝜕3CRB 𝜕3RRB

𝜕2CRB

𝜕1CRB

𝜕1RRB

Figure 10: Detailed view of𝐷-stability in parameter space for P1.

25

20

15

10

5

0

−50 5 10 15 20 25

kd

kp

Figure 11: Overall𝐷-stability solution region.

The boundary 𝜕1 in Figure 8 can be mapped into theparameter space by using 𝑠-𝜎 instead of 𝑠 in (17) in orderto shift the stability boundary to 𝜕1 in the complex plane.Solving for two free parameters in (18) for CRB and (19) forRRB and then plotting results will result in the 𝜕1 boundaryin the parameter space. For the 𝜕1 boundary, there is no IRBbecause 𝑠 is never equal to infinity in the 𝐷-shaped region.For mapping the 𝜕2 boundary, use 𝑟𝑒

𝑗𝜃 for 𝑠 with constant 𝜃in (17) and parameterize 𝑟 in (18) to obtain the CRB of 𝜕2. NoRRB and IRB solution exists because 𝑟 is never equal to zeroor infinity. Lastly, the 𝜕3 boundary maps into the parameterspace by substituting 𝑠 with 𝑅𝑒

𝑗𝜃 where 𝑅 is constant and themap is parameterized over 𝜃 in (17). This results in CRB forchanging 𝜃 and RRB for 𝜃 = 0∘.


DesiredActual

200

200

150

100

100

50

00

Y(m

)

300

X (m)

Start Finish

(a)

Late

ral d

evia

tion,y

(m)

2

1

0

−1

−2

×10−3

10 20 30 40 50 60

Time (s)0

DesiredActual

(b)

Vehi

cle v

eloci

ty,V

(m/s

)

15.1

15.05

15

14.95

14.910 20 30 40 50 60

Time (s)0

DesiredActual

(c)

0.3

0.2

0.1

0

−0.110 20 30 40 50 60

Time (s)0

Yaw

rate

,r(r

ad/s

)

(d)

Figure 12: Simulation results for circular trajectory.

3.2. Application to Automatic Steering Control. The vehiclesteering dynamics state-spacemodel corresponding to (1) canbe expressed in standard form as

�� = 𝐴𝑥 + 𝐵𝑢. (21)

Using the state-space form, the transfer function 𝐺𝑦𝛿𝑓 fromthe steering angle 𝛿𝑓 to the lateral deviation 𝑦 is written as

𝐺𝑦𝛿𝑓= [0 0 0 1] (𝑠𝐼 − 𝐴)

−1[[[

[

𝑏11

𝑏21

0

0

]]]

]

(22)

and the transfer function from the road curvature 𝜌ref to thelateral deviation 𝑦 is described as

𝐺𝑦𝜌ref= [0 0 0 1] (𝑠𝐼 − 𝐴)

−1[[[

[

0

0

−𝑉

0

]]]

]

. (23)

These transfer functions are used in designing the robustPID controller for the automatic steering system.The controlsystem structure is illustrated in Figure 9.

The robust PID controller is designed based on theparameter space approach. The 𝐷-stability requirements aretaken into consideration. The𝐷-stability boundaries (shownin Figure 8) are formed by assuming roots no closer than 0.5to the imaginary axis and no further in magnitude than 2.7from the imaginary axis (𝜎 = 0.5 and 𝑅 = 2.7). A minimumdamping ratio corresponding to 𝜃 = 45∘ is determined as0.707.

Two parameters of the PID controller are selected as freedesign parameters. In this paper, these free parameters arechosen as the proportional gain 𝑘𝑝 and the derivative gain 𝑘𝑑

of the PID controller.The integral gain 𝑘𝑖 of the PID controlleris determined as a fixed parameter by the designer. Here, 𝑘𝑖 isselected as 5.

Figure 10 shows the solution region for the P1 vertex ofthe uncertainty box shown in Figure 2. 𝜕1 CRB and RRB,𝜕2 CRB, and 𝜕3 CRB and RRB of the 𝐷-stability boundsare depicted with different colors. The intersection of thesebounds determines the boundary of the 𝐷-stable region inthe 𝑘𝑝-𝑘𝑑 plane.

The overall solution region which combines all the solu-tions for the vertices of the uncertainty box in Figure 2 isshown in Figure 11. The design point for 𝑘𝑝 and 𝑘𝑑 is selected


−0.02

0.02

0

0 20 40 60 80

Time (s)Road

curv

atur

e,𝜌

ref

(1/m

)

(a)

1

0.5

0 20 40 60 80

Time (s)

Fric

tion

coef

.,𝜇(—

)

Desired

(b)

400

200

0

Y(m

)

−200 0 200 400 600 800 1000

X (m)

Start

Finish

(c)

0 20 40 60 80

Time (s)

5

0

−5

Late

ral d

evia

tion,y

(m)

×10−3

DesiredActual

(d)

DesiredActual

00

20

20

40 60 80

Time (s)

10

Vehi

cle v

eloci

ty (m

/s)

(e)

0 20 40 60 80

Time (s)

Yaw

rate

,r(r

ad/s

)

0.5

0

−0.5

(f)

Figure 13: Simulation results for curved trajectory.

as (15, 12.5) from the shaded area in Figure 11 which satisfiesthe design requirements for all operating points.

4. Simulation Study

The simulation studies are performed to test the effectivenessof the designed robust PID controller. In the first simulation,vehiclemass, vehicle velocity, and road friction coefficient aretaken as 1500 kg, 15m/s, and 1, respectively. These parametervalues correspond to the point marked with 𝑆 in the uncer-tainty box of Figure 2.

In the first simulation of Figure 12, the vehicle tries tofollow a path consisting of a straight track of 150m followedby a full turn in a circle of radius 100m followed by a150m straight track. The vehicle velocity is kept constantat 15m/s along the way. In the nonlinear vehicle model, aPI-based cruise control algorithm keeps the vehicle velocityconstant. Also, the steering angle saturation is taken intoconsideration in the simulations. The front wheel steeringangle 𝛿𝑓 is limited to 40 degrees.The simulation results givenin Figure 12 show the vehicle trajectory, the lateral deviation,the vehicle velocity, and the yaw rate. It is seen that the

vehicle follows the desired trajectory successfully.The vehiclevelocity is kept around 15m/s by the cruise control algorithmand the vehicle yaw rate is at acceptable values.

In the second simulation, the vehicle tries to follow acurved path with different road curvature values. The tire-road friction coefficient alters between 1 and 0.5 to simulatedifferent road conditions such as dry asphalt and slipperyroad surface.The vehicle tries to track a velocity profile whichchanges between 5 and 18m/sec.The followed road curvature,the variable tire-road friction coefficient profile, the vehicletrajectory, the lateral deviation of the vehicle, the velocityprofile followed, the vehicle velocity, and also the vehicle yawrate change are shown in Figure 13. It is seen that the vehiclefollows the given trajectory with very small lateral deviationand that the velocity profile is followed successfully by thePI-based cruise controller. The vehicle yaw rate is also atacceptable values.

5. Conclusions

A parameter space based robust PID steering controllerdesign for automated steering was developed and tested in


a simulation environment in this paper. A validated modelof a midsized sedan was used in the design and simulations.The simulation results showed the success of this controllerin path following.This study is part of a larger research effortaimed at highly automated driving. The steering controllerdesigned here will be implemented and used in the experi-mental vehicle Okanom presented in this paper.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors would like to thank the Istanbul Okan Univer-sity Transportation Technologies and Intelligent AutomotiveSystems Application and Research Center (UTAS) and Tofas-Fiat. Also, the first and the second authors would like tothank the support of TUBITAK (The Scientific and Tech-nological Research Council of Turkey) National ScholarshipProgramme for PhD Students.

References

[1] K. L. R. Talvala, K. Kritayakirana, and J. C. Gerdes, “Pushingthe limits: from lanekeeping to autonomous racing,” AnnualReviews in Control, vol. 35, no. 1, pp. 137–148, 2011.

[2] L. Guvenc, I. M. C. Uygan, K. Kahraman et al., “Cooperativeadaptive cruise control implementation of team mekar at thegrand cooperative driving challenge,” IEEE Transactions onIntelligent Transportation Systems, vol. 13, no. 3, pp. 1062–1074,2012.

[3] A. Broggi, P. Medici, P. Zani, A. Coati, and M. Panciroli,“Autonomous vehicles control in the VisLab intercontinentalautonomous challenge,” Annual Reviews in Control, vol. 36, no.1, pp. 161–171, 2012.

[4] K. A. Unyelioglu, C. Hatipoglu, and U. Ozguner, “Designand stability analysis of a lane following controller,” IEEETransactions onControl SystemsTechnology, vol. 5, no. 1, pp. 127–134, 1997.

[5] A. Broggi, M. Bertozzi, A. Fascioli, C. G. LoBianco, and A.Piazzi, “The ARGO autonomous vehicle’s vision and controlsystems,” International Journal of Intelligent Control and Sys-tems, vol. 3, no. 4, pp. 409–441, 1999.

[6] J. Ackermann, J. Guldner, W. Sienel, R. Steinhauser, and V.I. Utkin, “Linear and nonlinear controller design for robustautomatic steering,” IEEE Transactions on Control SystemsTechnology, vol. 3, no. 1, pp. 132–143, 1995.

[7] B. Aksun Guvenc and L. Guvenc, “Robust two degree-of-freedom add-on controller design for automatic steering,” IEEETransactions on Control Systems Technology, vol. 10, no. 1, pp.137–148, 2002.

[8] R.Marino, S. Scalzi, andM. Netto, “Nested PID steering controlfor lane keeping in autonomous vehicles,” Control EngineeringPractice, vol. 19, no. 12, pp. 1459–1467, 2011.

[9] J. Ackermann, P. Blue, T. Bunte et al., Robust Control: TheParameter Space Approach, Springer, London, UK, 2002.

[10] B. Aksun Guenc, L. Guvenc, and S. Karaman, “Robust yawstability controller design and hardware-in-the-loop testing for

a road vehicle,” IEEE Transactions on Vehicular Technology, vol.58, no. 2, pp. 555–571, 2009.

[11] B. Aksun Guvenc, L. Guvenc, and S. Karaman, “Robust MIMOdisturbance observer analysis and design with applicationto active car steering,” International Journal of Robust andNonlinear Control, vol. 20, no. 8, pp. 873–891, 2010.

[12] K. Kahraman, M. T. Emirler, M. Senturk, B. Aksun Guvenc,L. Guvenc, and B. Efendioglu, “Estimation of vehicle yaw rateusing a virtual sensor with a speed scheduled observer,” inProceedings of the IFAC Symposium on Advances in AutomotiveControl, pp. 632–637, Munich, Germany, July 2010.

[13] AutonomousDrivingDemo at IstanbulOkanUniversity, http://www.youtube.com/watch?v=6MQdX4 dx4U.

[14] Okan Autonomous Vehicle Project Presentation, http://arastir-ma.okan.edu.tr/sayfa/okanom-otonom-arac-projesi-tanitimi.

[15] I. Altay, B. Aksun Guvenc, and L. Guvenc, “A simulation studyof GPS/INS integration for use in ACC/CACC and HAD,”in Proceedings of the 9th Asian Control Conference, Istanbul,Turkey, June 2013.

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


Research Article Robust PID Steering Control in Parameter...

Documents

Transcript of Research Article Robust PID Steering Control in Parameter...