IEEE/ASME TRANSACTIONS ON MECHATRONICS 1

Robust Trajectory Tracking Error Model-BasedPredictive Control for Unmanned Ground Vehicles

Erkan Kayacan, Student Member, IEEE, Herman Ramon and Wouter Saeys

Abstract—This paper proposes a new robust trajectory track-ing error-based control approach for unmanned ground vehicles.A trajectory tracking error-based model is used to design a linearmodel predictive controller and its control action is combinedwith feedforward and robust control actions. The experimentalresults show that the proposed control structure is capable to leta tractor-trailer system track both linear and curvilinear targettrajectories with low tracking error.

Index Terms—Autonomous vehicle, unmanned ground vehicle,model predictive control, trajectory tracking, agricultural robot,tractor-trailer system.

I. INTRODUCTION

THANKS to recent developments in satellite technologies,global positioning system (GPS)-based guidance systems

have become very popular. The research on autonomousground vehicles (AGVs), e.g. self-driving cars, has rapidlygrown since the introduction of real-time kinematic (RTK)GPS yielding centimeter precision. As automation of agricul-tural vehicles is essential to remain cost-effective, while theyoperate at relatively low speed in the field, research on au-tonomous agricultural vehicles has been increased dramaticallyafter the first successful results on AGVs [1].Nowadays, GPSguidance systems on agricultural machinery have become verypopular, as they are known to be more accurate than manualnavigation, e.g. visually straight and parallel crop rows. As aresult, the driver no longer has to steer the tractor accuratelywhich is a tiresome task. Moreover, these systems allow toalso perform the field work accurately during night or in foggyweather.

The initial studies on autonomous vehicle guidance allused proportional-integral-derivative (PID) controllers. How-ever, the performance of the currently available machineguidance systems controlled by PID is rather limited due to thecomplex vehicle dynamics which make that the conventional(e.g. PID) controllers for machine guidance have to be tunedguardedly or in an adaptive way [1], [2]. Moreover, PIDcontrol is a convenient choice for single-input single-output(SISO) systems, while autonomous vehicles have multipleinputs ( e.g. speed setting, steering angle setting for the

This work was supported by the IWT-SBO 80032 (LeCoPro) projectfunded by the Institute for the Promotion of Innovation through Science andTechnology in Flanders (IWT-Vlaanderen).

E. Kayacan is with the Delft Center for Systems and Control, DelftUniversity of Technology, 2628 CD Delft, The Netherlands. e-mail:e.kayacan@tudelft.nl

H. Ramon and W. Saeys are with the Division of Mechatronics, Bio-statistics and Sensors, Department of Biosystems, University of Leuven(KU Leuven), Kasteelpark Arenberg 30, B-3001 Leuven, Belgium. e-mail:{herman.ramon, wouter.saeys}@biw.kuleuven.be

tractor, steering angle setting for the trailer) and outputs (e.g.XY-coordinates of the tractor, XY-coordinates of the trailer,longitudinal speed, yaw rates, yaw angles). These multi-inputmulti-output (MIMO) systems are traditionally controlled ina decentralized way by designing a controller for each SISOsubsystem, thus neglecting the interactions. As an alternativemethod to PID controllers, optimal control approaches, suchas the linear quadratic regulator have been proposed, whichare convenient control methods for MIMO systems [3], [4].Furthermore, model predictive control (MPC) has been sug-gested as an evolution of the optimal control approach to dealwith constraints on the states and the inputs.

In mobile robot applications, successful results for linearMPC (LMPC) were reported when mobile robots are closeto the reference. Controllers are generally designed based-onthe derived trajectory tracking error-based model, which is alinearized error dynamics model obtained around the referencetrajectory, and the control inputs are generally obtained bythe combination of feedback and feedforward actions as in[5]–[7]. The LMPC generates the feedback action and thefeedforward action is calculated from the reference trajectory[8]. On the other hand, in vehicle guidance applications, ithas been reported that LMPC worked well for straight linetracking, while no linear control method worked well forcurvilinear trajectories [9]. Moreover, these methods cannotachieve trajectory tracking when the system stays off-trackand also vehicles are not capable of staying on-track whena curvilinear line starts. The reason is that LMPC works finefor processes which stay around fixed operating-points, as thisallows linearization of the process model. However, since theautonomous vehicle has time-varying set points and is subjectto several disturbances (e.g., varying soil conditions, bumpyfields), local linearization is not feasible. Moreover, since themodel mismatch increases when the system is getting far awayfrom the reference trajectory, it can generate large predictionerrors with a consequent instability of the closed-loop system[10]. Therefore, nonlinear MPC (NMPC) was proposed as abetter alternative [9].

All aforementioned studies are related to vehicle guidanceonly. Numerous studies have been reported about the controlof vehicle with towed trailer systems such as a tractor-trailersystem. The mathematical model of a tractor-trailer systemwas studied in [4]. Three different models were obtained anda linear quadratic regulator based on the linearized modelwas designed to control the system. It was reported that ifthe heading angle is more than 10 degrees, the linearizedmodel is not valid anymore. Moreover, no control law wasproposed for the control of the position of the trailer. In [11],

IEEE/ASME TRANSACTIONS ON MECHATRONICS 2

GPS antenna GPS antenna

Internet modem

PXI

Electro-mechanical valve

Potentiometer

Electro-hydraulic valve

Electro-hydraulic valve

Fig. 1. The tractor-trailer system

another linear quadratic regulator was proposed for both thecontrol of the tractor and trailer, and thus the position of thetrailer was controlled actively. The controller gave successfulresults for straight line trajectories. However, it was notedthat a feedforward control action was required for curvedlines tracking. An NMPC implementation for a tractor witha steerable trailer was studied in [12]. It was reported that theNMPC was able to control the tractor and trailer for straightand also curvilinear lines. Moreover, the system model wasmade adaptive to varying soil conditions by adding slip pa-rameters for the tractor-trailer system to take the variability inthe working environment into account. The nonlinear movinghorizon estimator and nonlinear model predictive controllerwere designed based on the adaptive model in a centralizedcase and successful experimental results have been reported[13]. In addition to centralized NMPC, decentralized and dis-tributed NMPC approaches have respectively been proposed in[14], [15] to decrease the computation time. The experimentalresults show that although the trajectory tracking accuracy wasa little bit worse than the one for centralized NMPC, theseapproaches reduce the computational cost significantly. Passivecontrol of vehicles with multiple trailers was studied in [16].

Although tracking performance obtained by NMPC wasquite good, the computational burden of NMPC implementa-tions is expensive. On the other hand, LMPC is not capable totrack curvilinear trajectories accurately although the requiredcomputation time is low. The main motivation of this study isto design a robust trajectory-tracking error-based linear modelpredictive controller for tracking straight and curved lines, andto benchmark its performance in terms of tracking error andcomputation time against the aforementioned NMPC studies.

This paper is organized as follows: The real-time system andthe system model are presented in Section II. The trajectorytracking error-based model is derived in Section III. In SectionIV, the feedback control action as an MPC, the feedforward

and robust control actions are designed, and the control schemeis presented. The experimental results are presented in SectionV. Finally, the main conclusions from this study are presentedin Section VI.

II. AUTONOMOUS TRACTOR-TRAILER SYSTEM ANDKINEMATIC TRICYCLE MODEL

The objective in this study is to track a time-based trajectorywith the small agricultural tractor-trailer system shown in Fig.1. In practice, an accurate trajectory tracking is desired toobtain a constant distance between rows to avoid crop damagewhile difficult and varying soil conditions are faced by abumpy and wet grass field. The experimental set-up is thesame as in [13]–[15], but the target trajectory is a time-basedone instead of a space-based one.

RTK GPS (AsteRx2eH, Septentrio Satellite Navigation NV,Belgium) is used to obtain positional information. For thispurpose, two GPS antennas are located straight up the centerof the tractor rear axle and the center of the trailer. A DigiConnect WAN 3G modem is used to send uncorrected andreceive corrected GPS data from the Flepos network. The non-Gaussian measurement errors of the GPS are 0.03 m accordingto the specifications of the manufacturer.

The steering mechanisms of the tractor and trailer consistof electro-hydraulic valve actuators (OSPC50-LS/EH-20, Dan-foss, Nordborg, Denmark) and the speed of the tractor-trailersystem is controlled through an electromechanical actuator(LA12, Linak, Nordborg, Denmark) connected to the hydrostatpedal (HP) as shown in Fig. 1. The angle of the front wheelsof the tractor is measured using a potentiometer (533-540-J00A3X0-0, Mobil Elektronik, Langenbeutingen, Germany)mounted on the front axle while the steering angle of the traileris measured by an inductive sensor. The measurements of thesteering angles were found to be perturbed by Gaussian noisewith standard deviations of 1 degree. An encoder mounted on

IEEE/ASME TRANSACTIONS ON MECHATRONICS 3

the rear wheels is used to measure the speed of the systemwith a measurement error (standard deviation) of 0.1 m/s.

The GPS receiver, the internet modem, all actuators and sen-sors are connected to a real time operating system (PXI-8110,National Instruments, Austin, TX, USA) through an RS232serial communication. The PXI system equipped with a 2.26GHz Intel Core 2 Quad Q9100 quad-core processor acquiresall measurements, and controls the tractor-trailer system byapplying voltages to the actuators. A laptop is connected tothe PXI system by WiFi functions as the user interface of theautonomous tractor-trailer. The control algorithms are imple-mented in LabV IEW T M (version 2011, National Instrument,USA). They are executed in real time on the PXI and updatedat a rate of 5-Hz.

The autonomous tractor-trailer system model is a kinematicmodel neglecting the dynamic force balances in the equationsof motion [4]. The yaw angle difference between the tractorand the trailer λ is defined as the measured relative angle. Thetractor and trailer rigid bodies are mechanically linked to eachother by the drawbar. There are two revolute joints (RJs) whichconnect the drawbar to the tractor at RJ1 and the drawbar tothe trailer at RJ2 as illustrated in Fig. 2. The centers of gravityof the tractor and trailer are respectively represented by CGt

and CGi.The equations of motion of the system consisting of the

kinematic and speed models as derived respectively in [13],[17] are written as follows:

xt = vcos(ψ t)

yt = vsin(ψ t)

ψt =

v tan(δ t)

Lt

xi = vcos(ψ i)

yi = vsin(ψ i)

ψi =

vLi

(sin(λ )+

lLt tan(δ t)cos(λ )

)v = − v

τ+

Kτ

HP (1)

where xt and yt represent the position of the tractor, ψ t is theyaw angle of the tractor, xi and yi represent the position of thetrailer, ψ i is the yaw angle of the trailer, v is the longitudinalspeed of the system. Since the tractor and trailer rigid bodiesare linked by two RJs at a hitch point, the tractor and the trailerlongitudinal velocities are coupled to each other. The steeringangle of the front wheel of the tractor is represented by δ t ,β is the angle between the tractor and the drawbar at RJ1, δ i

is the steering angle between the trailer and the drawbar atRJ2, and HP is the hydrostat position. The angle between thetractor and trailer λ is equal to the summation of the anglebetween the tractor and the drawbar at RJ1, and the steeringangle between the trailer and the drawbar at RJ2, (λ = β +δ i).

The physical parameters that can be directly measuredare as follows: The distance between the front axle of thetractor and the rear axle of the tractor Lt(1.4m), the distancebetween RJ2 and the rear axle of the trailer Li(1.3m), and thedistance between the rear axle of the tractor and RJ2 Ld(1.1m),respectively. For an engine speed of 2500 RPM, the identified

Fig. 2. Schematic illustration of tricycle model for an autonomous tractor-trailer system

parameters are as follows [17]: the time-constant τ = 2.05 andthe gain value K = 1.4 for the speed model.

In the rest of the paper, we denote equations (1) as

z = f(z,u)

(2)

where the state, input and output vectors are denoted asfollows:

z =[

xt yt ψ t xi yi ψ i v]T (3)

u =[

δ t λ HP]T (4)

y =[

xt yt xi yi v]T (5)

III. TRAJECTORY TRACKING ERROR-BASED SYSTEMMODEL

The trajectory tracking problem is a nonlinear controlproblem in nature. Therefore, the trajectory tracking control ofan autonomous ground vehicle, e.g. tractor-trailer system, canbe asymptotically stabilized by nonlinear feedback controllers.In case of linearization around the trajectory, a linear time-varying trajectory tracking system is obtained, which can becontrolled by linear controllers [18], [19]. In this section,a new trajectory tracking error-based model is derived. Thetraditional trajectory tracking error-based models were derivedfor mobile robots in [8] and for the trajectory planner of atractor-trailer mobile robot in [20]. The difference betweenthe traditional trajectory tracking error-based model and themethod proposed here is that the speed and yaw rates are theinputs for the traditional one, while the speed and yaw modelsare taken into account to design a controller for the new one.As a result, the gas pedal position and steering angles are theinputs for the new trajectory tracking error-based model.

The reference frame represents the inertial reference framefixed to the motion ground. The other reference frames are

IEEE/ASME TRANSACTIONS ON MECHATRONICS 4

moving frames attached to the centers of gravity of thetractor and trailer, which can only translate with respectto the reference frame fixed to the motion ground. Thereference trajectory is described by a reference state vectorzr = (xt

r,ytr,ψ

tr,x

ir,y

ir,ψ

ir,vr)

T and a reference control vectorur = (δ t

r ,λr,HPr)T . The error state ze expressed in the frames

on the tractor and trailer is written as follows:

ze = T × [zr− z] (6)

where T is the transformation matrix between reference framesas follows:

cos(ψ t) sin(ψ t) 0 0 0 0 0−sin(ψ t) cos(ψ t) 0 0 0 0 0

0 0 1 0 0 0 00 0 0 cos(ψ i) sin(ψ i) 0 00 0 0 −sin(ψ i) cos(ψ i) 0 00 0 0 0 0 1 00 0 0 0 0 0 1

The trajectory tracking error-based model is derived by

taking the derivative of the error state in (6) and taking thesystem model in (1) into account as follows:

xte = γ

tyte− v+ vr cos(ψ t

e)

yte = −γ

txte + vr sin(ψ t

e)

ψte =

vr tan(δ tr )− v tan(δ t)

Lt

xie = γ

iyte− v+ vr cos(ψ i

e)

yie = −γ

ixte + vr sin(ψ i

e)

ψie =

vr

Li

(sin(λr)+

Ld

Lt tan(δ tr )cos(λr)

)− v

Li

(sin(λ )+

Ld

Lt tan(δ t)cos(λ ))

ve = −ve

τ+

Kτ(HPr−HP) (7)

where γ t and γ i are the yaw rates of the tractor and trailer,respectively.

The trajectory tracking error-based model in the state-spaceform is written by linearizing the error model in (7) aroundthe reference trajectory (xt

e = yte = ψ t

e = xie = yi

e = ψ ie = ve =

δ te = λe = HPe = 0) as follows:

ze = Aze +Bue

ze =

0 γ tr 0 0 0 0 1

−γ tr 0 vr 0 0 0 0

0 0 0 0 0 0 00 0 0 0 γ i

r 0 10 0 0 −γ i

r 0 vr 00 0 0 0 0 0 00 0 0 0 0 0 − 1

τ

ze

+

0 0 00 0 0vrLt 0 00 0 00 0 0

vrLd

Lt LivrLi 0

0 0 Kτ

ue (8)

where the state and control vectors for the trajectory trackingerror-based model are denoted as

ze =[

xte yt

e ψ te xi

e yie ψ i

e ve]T (9)

ue =[

δ te λe HPe

]T (10)

Remark 1: The trajectory tracking error-based model iscontrollable when either the reference longitudinal velocity vror the reference yaw rates γ t

r and γ ir are nonzero, which is a

sufficient condition.

IV. DESIGN OF THE ROBUST TRAJECTORY TRACKINGERROR-BASED CONTROLLER

The control scheme is illustrated in Fig. 3. The control inputapplied to the real-time system is calculated as the differencebetween the summation of the feedforward u f and robust umcontrol actions, and the feedback ub control action:

u = u f −ub +um (11)

In following subsections, the feedback, feedforward androbust control actions are formulated.

A. Feedback Control Action: Model Predictive Control

The system to be controlled is described by the followinglinear discrete-time model:

ze(k+1) = Adze(k)+Bdue(k) (12)

where ze(k) ∈ Rnz is the state vector and ue(k) ∈ Rnu isthe control input. The matrices Ad and Bd are calculatedconsidering the sampling time of the real-time system by usingthe continuous-time version of the trajectory tracking error-based model in (8).

The constraints are written for all k ≥ 0 as follows:

−55 degrees/s≤ ∆δ te(k) ≤ 55 degrees/s

−35 degrees/s≤ ∆λe(k) ≤ 35 degrees/s

−30 %/s≤ ∆HPe(k) ≤ 30 %/s

−12 degrees≤ δ te(k) ≤ 12 degrees

−6 degrees≤ λe(k) ≤ 6 degrees

−10 %≤ HPe(k) ≤ 10 % (13)

The cost function in its general form is written as follows:

J(∆U,ze(k)

)=

Np

∑i=0

zTek+i|k

Qzek+i|k +Nc−1

∑i=0

∆uTek+i

R∆uek+i (14)

where Np = 8 and Nc = 3 represent the prediction and con-trol horizons, 4ue is the change of the input, and ∆U =[∆uT

ek, ...,∆uT

ek+Nc−1]T is the vector of the input steps from

sampling instant k to sampling instant k+Nc− 1. Since thesampling time of the real-time experiments has been equal to200 ms, the prediction and control horizons are respectivelyequal to 1.6 s and 0.6 s. The positive-definite weightingmatrices Qnz×nz and Rnu×nu are defined as follows:

Q = diag(1,1,0,1,1,0,0) , R = diag(1,1,1) (15)

IEEE/ASME TRANSACTIONS ON MECHATRONICS 5

TransformationTrajectory

Planner

Calculated

Measured

MPC

Feedforward

Control Action

Robust

Control Action

Nominal Model

Fig. 3. Block diagram of the control scheme combining feedback MPC, feedforward and robust control actions.

Since the aim is trajectory tracking control of the tractor-trailer system, we try to minimize the tractor and trailertracking errors on x- and y-axes. If oscillatory behavior isobserved, then the yaw angle error for the lateral motion andthe speed error for the longitudinal motion might be needed tominimize. On the other hand, if the values for the yaw angleand speed errors are set to very large values, the system maynot be able to track the reference trajectory. Since we havenot observed any oscillatory behavior, we have not needed tominimize the yaw angle and speed errors.

The following plant objective function is solved at eachsampling time for the LMPC:

minze(.),ue(.)

Np

∑i=0

zTe k+i|kQzek+i|k +

Nc−1

∑i=0

∆uTek+i

R∆uek+i

subject to ze(k+1) = Adze(k)+Bdue(k)

−55 degrees/s≤ ∆δte(k)≤ 55 degrees/s

−35 degrees/s≤ ∆λe(k)≤ 35 degrees/s

−30≤ ∆HPe(k)≤ 30−12 degrees≤ δ

te(k)≤ 12 degrees

−6 degrees≤ λe(k)≤ 6 degrees

−10 %≤ HPe(k)≤ 10 %

(16)

In the real-time implementation of the LMPC, the linearoptimization problem in (16) is solved online for a givenze(k) in a receding horizon fashion. In this approach, thefirst element of the input sequence is applied to the system,while the rest is discarded. For the next time step, the entireprocedure is repeated for the new measured or estimatedoutput. The online MPC algorithm can be implemented in thefollowing steps:

1) Measure or estimate the current system states ze(k)2) Solve the optimization problem in (16) to obtain 4U∗ =

[4u∗e(k), . . . ,4u∗e(k+Nc−1)]T

3) Apply u∗e(k) =4u∗e(k)+u∗e(k−1)The optimization problem is then solved over a shifted horizonfor the next sampling time.

In our case, the designed LMPC minimizes the differencesbetween the reference trajectory and the measured positions

of the tractor and trailer in x- and y-axes, and finds thedifferences between the reference and actual control inputs.For this reason, the generated inputs by the LMPC are not theactual control inputs to the real-time system. Therefore, wewill define feedforward control actions in the next subsectionIV-B to calculate the control inputs applied to the real-timesystem. The input calculated by the LMPC u∗e contributes tothe input signal to the system u as a feedback control actionub:

ub = u∗e (17)

Remark 2: As can be seen from (16), the equality andinequality constraints are linear such that the formulation is aconvex optimization problem. A quadratic programming solvercan be used for this optimization problem. The formulationfor NMPC is the constrained nonlinear optimization problemwhich is non-convex. For this reason, it is to be noted that thecomputational burden of the optimization problem for LMPCin (16) is significantly lower than the one for NMPC.

Once the control input applied to the real-time system hasbeen found, a modification is required to find the steering anglefor the trailer δ i. This is found by subtracting the angle β

between the tractor and the drawbar at RJ1 from the angle λ

between the tractor and the trailer as follows:

δi = λ −β (18)

B. Feedforward Control Action

As the LMPC generates the differences between the ref-erence and actual control variables, the outputs of the MPChave to be combined with a feedforward control action tocalculate the actual control inputs to be applied to the real-time system. Feedforward control inputs δ t

r , λr and HPrare derived for given reference trajectories (xt

r, ytr, xi

r, yir)

by using the system model in (1). The feedforward controlactions for the tractor-trailer system are the reference steeringangles and the reference hydrostat position. The referencelongitudinal velocity vr, and the reference yaw rates γ t

r and γ ir

for the tractor-trailer system are derived for a given referencetrajectory (xt

r,ytr,x

ir,y

ir) defined in a time interval t ∈ [0,T ] as

IEEE/ASME TRANSACTIONS ON MECHATRONICS 6

follows:

vr = ±√(xt

r)2 +(yt

r)2

γtr =

xtr y

tr− yt

r xtr

(xtr)

2 +(ytr)

2

γir =

xir y

ir− yi

r xir

(xir)

2 +(yir)

2 (19)

where the sign ± indicates the desired driving direction of thesystem (+ for forward, − for reverse).

To calculate the feedforward control action for the tractor-trailer system, the steering angles are assumed to be small,and the steady-state behaviour of the relation between thelongitudinal velocity and the hydrostat position is taken intoaccount. Under these assumptions, the feedforward controlactions can be derived from (1) as:

δtr =

γ trLt

vr

δir =

γ irL

i− γ trLd

vr−β

HPr =vr

K(20)

The defined feedforward control action uF = [δ tr ,δ

ir ,HPr]

T

provides the calculated references for the control inputs. Thecalculated feedforward control action will only be able to drivethe tractor-trailer system on the reference trajectory if there areno disturbances, uncertainties and initial state errors.

Remark 3: The necessary condition in the trajectory designis that the trajectory is twice-differentiable, and the velocityreference vr 6= 0 and the gain of the speed model K 6= 0 arenonzero.

C. Robust Control ActionSince the trajectory tracking error-based model has been

obtained by linearizing the system around the reference tra-jectory, the mismatch between the trajectory tracking error-based model and the real system can result in poor controlperformance when the system is not close to the reference.Therefore, a robust control action is required to bring and tokeep the system close to the reference trajectory.

In [21]–[23], a tube-based approach for (N)MPC was pro-posed to obtain robust and better control performance of thesystem. The robust control law is written as follows:

um = K(ze(t)− ze(t)

)(21)

where K ∈ Rnu×nz is the feedback gain and ze(t)− ze(t) isthe modeling error between the nominal model in (8) and thereal system.

The modeling error term is calculated as the differencebetween the linearized model and the real system as follows:

zm = g(ze(t),u(t)

)(22)

where zm ∈Zm is a robust positively invariant set. It is assumedthat Zm ⊂Ze and KZm ⊂U. The nominal state and input haveto satisfy:

ze ∈ Ze = ZeZm

u ∈ U= UKZm (23)

where they are in the neighborhoods of the origin.Since only measurable states must be considered in the

robust control law due to the estimation error, the uncertaintyvector is written as follows:

zm =

xt

myt

mxi

myi

m

=

xt

e− xte

yte− yt

exi

e− xie

yie− yi

e

(24)

where zm vector is defined as −1≤ zm ≤ 1 in this paper.In most tractor-trailer systems, there are actuator limits. For

this reason, the constraints on the actuators must be taken intoaccount. Therefore, we propose a tanh function to place asaturation for the robust control action [24], [25]. Moreover,during the real-time experiments, it was observed that if onlythe uncertainty vector was considered, the system exhibitedoscillatory behaviour. For this reason, the derivative of theuncertainty vector is also considered to reduce overshoots.This results in the following robust control law for the tractorand trailer:

um =

δ tm

δ im

HPm

=

ks1 tanh(kp1ytm + kd1 yt

m)ks2 tanh(kp2yi

m + kd2 yim)

ks3 tanh(kp3xtm + kd3 xt

m)

(25)

where kp > 0, kd > 0 and ks > 0 are respectively the propor-tional, derivative and saturation gains of the robust controlterm. The saturation coefficients have been determined byconsidering the constraints on the actuators, the feedback con-troller and the feedforward control action. For this reason, thesaturation coefficients ks1 , ks2 and ks3 have been respectivelyset to 0.2, 0.1 and 10. Moreover, the system shows oscillatorybehaviour when the derivative coefficients are not larger thanthe proportional ones. Therefore, the proportional coefficientskp1 , kp2 and kp3 have been respectively set to 2, 1 and 10while the derivative coefficients kd1 , kd2 and kd3 have beenrespectively set to 4, 2 and 20. The robust control actions δ t

m,δ i

m and HPm are the steering angles of the tractor and trailer,and the hydrostat position.

The nominal controller ue(t) in (17) is calculated online,while the ancillary control law kp, kd and ks obtained offlinekeeps the trajectories of the system error in the robust controlinvariant set zm centered along the nominal trajectory [21]. Thestability issue of the robust tube-based MPC of constrainedlinear system with disturbances was clarified in [21], [26].

V. EXPERIMENTAL RESULTS

The time-based, 8-shaped trajectory illustrated in Fig. 4 hasbeen used as the reference signal. The 8-shaped trajectoryconsists of two straight lines and two smooth curves. Sincethe radius of the curves is equal to 10 m, the curvature of thesmooth curves is equal to 0.1. (The curvature of a circle is theinverse of its radius).

The actual trajectories of the tractor and trailer are shownin Fig. 4, while close ups are shown in Fig. 5. Thanks tothe robust control action, the new robust trajectory tracking-based model predictive controller is capable to navigate the

IEEE/ASME TRANSACTIONS ON MECHATRONICS 7

Y axis (m)25 30 35 40 45 50 55 60 65 70 75

X a

xis

(m)

10

20

30

40

50

60

70

80

← Initial point

↑ Movement direction

Reference trajectoryActual trajectory of the tractorActual trajectory of the trailer

Fig. 4. Reference and actual trajectories

Y axis (m)38 40 42 44 46 48

X a

xis

(m)

50

51

52

53

54

55

Y axis (m)52 54 56 58 60 62

X a

xis

(m)

45

46

47

48

49

50

Y axis (m)30 35 40 45

X a

xis

(m)

65

70

75

Y axis (m)55 60 65 70

X a

xis

(m)

25

30

35

Fig. 5. Zoom versions of trajectories

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

Err

or (

m)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Tractor Trailer

Fig. 6. Euclidian distance error to the reference trajectory

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

δt (

rad)

-0.5

0

0.5

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

δi (

rad)

-0.5

0

0.5

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

HP

(%

)

0

50

100

Fig. 7. Control Signals: dashed line: bound for the total control action, blackline: total control action, cyan line: feedforward control, red line: feedbackcontrol, blue line: robust control.

autonomous tractor-trailer system close to the target trajectory.Moreover, the system did not exhibit oscillatory behaviour.

The Euclidian distance errors to the time-based referencetrajectory for both the tractor and the trailer are shown in Fig.6. The mean values of the Euclidian distance errors of thetractor and the trailer for the straight lines are respectively23.49 cm and 21.21 cm. Besides, the mean values of theEuclidian distance errors of the tractor and the trailer for thecurved lines are respectively 39.82 cm and 36.21 cm. As canbe observed, the trajectory tracking error of the tractor-trailersystem for straight lines is lower than for the curved lines.NMPC was used for the space-based trajectory approach in[13]. It was reported that the Euclidean error values of thetractor and the trailer for the straight lines are respectively6.44 cm and 3.61 cm, while the Euclidean error values ofthe tractor and the trailer for the curved lines are respectively49.78 cm and 41.52 cm. As can be observed, the tracking errorto the space-based trajectory was less than the one to the time-based trajectory for straight lines, while it was more than theone to the time-based trajectory for curved lines. Therefore, itcan be concluded that the preferred approach depends on theshape of the trajectory.

In Fig. 7, the outputs, the steering angle (δ t ) reference forthe tractor, the steering angle (δ i) reference for the trailer, and

IEEE/ASME TRANSACTIONS ON MECHATRONICS 8

the hydrostat position (HP) reference, of the controller areillustrated. As can be seen from this figure, the total controlinputs are within the bounds, and the feedback, feedforwardand robust control actions can be observed. The contribution ofthe robust control action is more than the one of the feedbackcontrol action for curved lines, while the one of feedbackcontrol action is dominant for straight lines. The reason is thatthe yaw angle is time-invariant for straight lines, while it istime-varying for curved lines. This results in larger mismatchproblem during tracking curved lines. Moreover, the steeringangle reference for the trailer has some oscillation. As canbe seen in (20), it is calculated considering the referencetrajectory and the measured hitch point angle β . Thus, thisoscillatory behaviour is caused by the measured hitch pointangle β .

The average computation time for LMPC was equal to 1.1ms and feasible in real-time. As reported in [13], while thecomputation time for NMPC was still acceptable for real-timeapplications, the average computation time for NMPC was 6times larger with 6.8 ms. It is to be noted that the computationtime increases exponentially when the number of the state andinput increases.

VI. CONCLUSIONS

A new robust trajectory tracking error-based model pre-dictive controller has been elaborated for the control of anautonomous tractor-trailer system. To increase the robustnessof the algorithm, the tube-based approach has been used, and itwas evaluated in real-time with respect to its computation timeand tracking accuracy. The experimental results in the fieldhave shown that the designed controller is able to control thesystem with a reasonable accuracy due to the modeling errorsand disturbances. The mean values of the Euclidian distanceerrors on the straight lines for the tractor and the trailer wererespectively equal to 23.49 cm and 21.21 cm, while the onesfor the curved lines are respectively 39.82 cm and 36.21 cm.The computation time for LMPC was around 1.1 ms and issignificantly smaller than for NMPC.

ACKNOWLEDGMENT

We would like to thank Mr. Soner Akpinar for his technicalsupport for the preparation of the experimental set up.

REFERENCES

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IEEE/ASME TRANSACTIONS ON MECHATRONICS 9

Erkan Kayacan (S’12) was born in Istanbul,Turkey, on April 17, 1985. He received the B.Sc. andthe M.Sc. degrees in mechanical engineering fromIstanbul Technical University, Istanbul, Turkey in2008 and 2010, respectively. He received the Ph.D.degree in Mechatronics, Biostatistics and Sensorsfrom University of Leuven (KU Leuven), Leuven,Belgium in 2014.

He is currently a Postdoctoral Researcher with theDelft Center for Systems and Control, Delft Uni-versity of Technology, Delft, The Netherlands. His

current research interests include model predictive control, state estimation,unmanned vehicles and autonomous systems.

Herman Ramon received the M.Sc. degree inbioscience engineering from Gent University, Gent,Belgium and the Ph.D. degree in biological sciencesfrom the University of Leuven (KU Leuven), Leu-ven, Belgium, in 1993.

He is currently a Professor with the Faculty ofBioscience Engineering, KU Leuven, lecturing onfield robotics, system dynamics, applied mechanicsand mathematical biology. His current research in-terests include precision technologies and advancedmechatronic systems for processes involved in the

production chain of food and non-food materials, from the field to the enduser. He has authored and co-authored more than 200 peer reviewed journalarticles (ISI).

Wouter Saeys received the M.Sc degree in Bio-science Engineering from University of Leuven (KULeuven), Leuven, Belgium in 2002. On the basis ofhis Masters thesis, he was awarded the engineeringprize by the Royal Flemish Society of Engineers(KVIV). In 2006, he received the Ph.D. in Bio-science Engineering from KU Leuven, Leuven, Bel-gium. under the supervision of Professors HermanRamon and Josse De Baerdemaeker.

Since 2010 he is an Assistant Professor at theBiosystems Department of KU Leuven, where he

leads a group focusing on technology for the AgroFood chain. His mainresearch interests include agricultural automation and robotics, chemometrics,light transport modelling and optical characterisation of biological materials.He has supervised more than 10 PhDs and is (co-)author of over 110 peerreviewed journal articles (ISI).”