Path tracking and stabilization for a reversing general 2-trailer configuration using a cascaded

control approach

Niclas Evestedt, Oskar Ljungqvist and Daniel Axehill

Linköping University Post Print

N.B.: When citing this work, cite the original article.

Original Publication:

Niclas Evestedt, Oskar Ljungqvist and Daniel Axehill, Path tracking and stabilization for a reversing general 2-trailer configuration using a cascaded control approach, 2016, Intelligent Vehicles Symposium (IV), 2016 IEEE. http://dx.doi.org/10.1109/IVS.2016.7535535 Copyright: http://www.ieee.org Postprint available at: Linköping University Electronic Press

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-130950

Path tracking and stabilization for a reversing general 2-trailerconfiguration using a cascaded control approach

Niclas Evestedt1, Oskar Ljungqvist1, Daniel Axehill1

Abstract— In this paper a cascaded approach for stabilizationand path tracking of a general 2-trailer vehicle configurationwith an off-axle hitching is presented. A low level LinearQuadratic controller is used for stabilization of the internalangles while a pure pursuit path tracking controller is used ona higher level to handle the path tracking. Piecewise linearityis the only requirement on the control reference which makesthe design of reference paths very general. A Graphical UserInterface is designed to make it easy for a user to design controlreferences for complex manoeuvres given some representationof the surroundings. The approach is demonstrated with chal-lenging path following scenarios both in simulation and on asmall scale test platform.

I. INTRODUCTION

Many Advanced Driver Assistance Systems (ADAS) havebeen introduced during the last decade. Historically thefocus has been to increase safety with systems like LaneKeep Assist (LKA), Adaptive Cruise Control (ACC) andAutomatic Breaking, however in recent years systems thathelp the driver to perform complex tasks, such as parallelparking and reversing with a trailer, have been introduced [1].Reversing with a trailer is known to be a task that needs a fairamount of skill and training to perfect and an inexperienceddriver will have problems already performing simple taskssuch as reversing in a straight line or make a simple turnaround an obstacle. To relieve the driver in such situations,trailer assist systems have been developed that stabilize thetrailer around a reference that the driver can specify from acontrol knob. The trailer assist systems have been releasedto the passenger car market but an even greater challengearises when reversing a truck with a dolly steered trailer.This introduces another degree of freedom making it virtuallyimpossible for a driver, without extensive training, to control.In this paper we present a cascaded control scheme usingan LQ-controller, based on the work in [2], to stabilizethe vehicle configuration around an equilibrium point andthen use a pure-pursuit path tracking controller to makethe vehicle configuration follow a piecewise linear referencepath. We also present an intuitive interface where a user canspecify a path from start to goal by hand and then let thesystem execute that path automatically.

A. Related work

The nonlinear dynamics of a standard trailer configurationwith the hitch connection in the center of the rear axle are

*The research leading to these results has been carried out within theiQMatic project funded by FFI/VINNOVA.

1Division of Automatic Control, Linkoping University, Sweden,(e-mail: {niclas.evestedt, oskar.ljungqvist,daniel.axehill}@liu.se)

Fig. 1: Truck and trailer system used as a test platform forevaluation experiments. The truck has been built with LEGONXT and fitted with angle sensor for dolly and hitch angles.

well understood and the derivation of the equations for ann-trailer configuration can be found in [3]. In [4], [5], [6]the flatness property of this system is used and controllersusing feedback linearization are designed, [6] and [5] alsodemonstrate the feasibility of the controllers using some 1-trailer lab experiments. However, the assumption that thehitch connection is longitudinally centered at the rear axlecenter does not hold for passenger cars nor for the truck thatwill be used in this work. The nonlinear dynamics of thegeneral n-trailer system, where no assumptions are made onthe position of the hitching point are derived in [7]. Input-output linearization is used in [1] to derive a controller forthe 1-trailer system with off-axle hitching that stabilize thetrailer’s driving curvature, they also derive a path trackingcontroller and test the controllers with good results on a realcar test platform. Although these results are very encouragingit is shown in [8] and [9] that trailer configurations withmore than one trailer are not input-output linearizable. Toovercome this [10] introduce what they call a ”ghost vehicle”that should be exact linearizable and have similar behaviourto the original model. A controller can then be designedusing the exact linearization techniques for the ghost modeland then apply the controller to the original system. An LQbased approach for reversing the general 2-trailer system withoff-axle hitching is presented in [2]. The system is linearizedaround an equilibrium point and the linearized system is usedto design an LQ control scheme. The work focuses on thestabilization of the internal angles but does not look into pathtracking.

This work presents results from the thesis [11], whereresults from [2] are used to stabilize the internal angles ofthe system and then a pure pursuit path tracking controlleris designed for the stabilized system in a similar way as [12]

and [13]. Finally we present an intuitive user interface thatcan be used to perform complex manoeuvres from a user-defined start position to a user-defined goal position. Thesystem is then validated both in simulation and on a small-scale test platform.

The outline of the remainder of the paper is as follows:In Section II the nonlinear equations that are used to modelthe system are presented. In Section III the LQ controllerused will be explained and in Section IV the pure pursuitpath tracking controller will be discussed. Section V presentsthe user interface and Section VI presents the small scaletest platform that was used for experiments. Finally, SectionVII and Section VIII present the results and conclusions,respectively.

II. SYSTEM DYNAMICS

In this section the model used to describe the general 2-trailer system with an off-axle hitching on the pulling vehicleis presented. A schematic overview of the configuration isshown in Fig 2. The generalized coordinates used to modelthe system are, p = [x3,y3,θ3,β3,β2]

T where x3,y3 are theposition of the rear axle center of the trailer, θ3 is the headingof the trailer, β3 is the relative angle between the trailer andthe dolly and β2 is the relative angle between the dolly andthe truck. The parameters L3,L2,L1 are the distances betweenthe axle center for the trailer to the axle center for the dolly,the axle center for the dolly to the off-axle hitch connectionfor the truck and the distance between the axle centers forthe truck, respectively. M1 is the off-axle hitch length for thetruck and α is the steering angle. The dynamic model for ageneral n-trailer system was derived in [7] and the followingequations for the special 2-trailer case was presented in [2]:

x3 = vcosβ3 cosβ2

(1+

M1

L1tanβ2 tanα

)cosθ3 (1)

y3 = vcosβ3 cosβ2

(1+

M1

L1tanβ2 tanα

)sinθ3 (2)

θ3 = vsinβ3 cosβ2

L3

(1+

M1

L1tanβ2 tanα

)(3)

β3 = vcosβ2

(1L2

(tanβ2−

M1

L1tanα

)−

sinβ3

L3

(1+

M1

L1tanβ2 tanα

))(4)

β2 = v(

tanα

L1− sinβ2

L2+

M1

L1L2cosβ2 tanα

)(5)

where v is the longitudinal velocity at the rear axle for thetruck. The model is valid under the no slip condition. Sinceour application only concerns maneuvers at lower speeds thisis a feasible assumption.

A. Linearization

To fit into the LQ-framework used in the next section alinearized system model needs to be derived. When drivingforward, the system is stable but in reverse (v< 0) the systembecome unstable. Since v enters linearly in (1)-(5) it onlyaffect the system as a time scaling [11] and will, theoretically,

α

M1

X

Y

β

θ

2

Z

L 3β

3

3

v

( , )x 3 y3

L 2

L 1

Fig. 2: Schematic view of the configuration used to modelthe general two-trailer system.

have no effect on the controller design. However, in practicebacklash and unmodeled dynamics in the steering mechanismwill limit the maximum speed for stability. Since a cascadedapproach is used the slower states, x3,y3 and θ3 will becontrolled by a high level controller. Hence, only (4) and (5)for the states β2 and β3 are considered in the linearizationand stabilization of the model for the low level controller.

Given a constant steering angle, αe < αmax, where αmaxis defined in (10), there exists a stationary equilibrium con-figuration where β2 and β3 equals zero. At this equilibriumpoint the configuration will travel along circles with theirradius determined by αe as depicted in Fig. 3. From Fig. 3the angles, β2e and β3e, at this equilibrium can be determinedby basic trigonometry and gives the following relations:

β3e = sign(αe)arctan(

L3

R3

)(6)

β2e = sign(αe)

(arctan

(M1

R1

)+ arctan

(L2

R2

))(7)

where R1 = L1/| tanαe|, R2 =√

R21 +M2

1 −L22, R3 =√

R22−L2

3.

With f (βββ ) = [β3, β2]T and linearizing around this equilib-

rium point and letting βββ = [β3,β2]T we get

βββ = A(βββ −βeβeβe)+ B(α−αe) (8)

where

A =∂ f∂βββ

∣∣∣∣βeβeβe,αe

and B =∂ f∂α

∣∣∣∣βeβeβe,αe

(9)

The set of equilibrium configurations reaches its limit whenthe traveling circle for the rear axle of the trailer collapsesto a point at the center of the rear axle. This happens whenβ3e = π/2 which directly gives R3 = 0 and R2 = L3 at thispoint. Inserting this in the relation for R2 and solving for αewe get the maximum steering angle where the linearizationis valid for a given parameter set L1, L2, L3 and M1.

αmax = arctan

(√L2

1

L23 +L2

2−M21

)(10)

This linearized model can now be used for control designand this is further explained in the next section.

α

R3

e

β2,e

M1

L 3

O

R 2

R1

L 2

L 1

β3,e

α e

β2,e

β3,e

Fig. 3: Stationary equilibrium point for steering angle αe.The system will travel in a circular path with a radiusdetermined by the geometry and αe.

III. STABILIZATION

For the low level stabilization of β2 and β3 a state feedbackcontroller, using LQ-techniques similar to what was donein [2] but extended with gain scheduling, is designed. Thecontroller is designed around an equilibrium point using (8)for the linearized model. To account for model variationdue to the choice of equilibrium point, αe, a gain scheduledapproach depending on αe is used. Using αe as a referenceresults in a controller with the following structure

α = αe−L(αe)(βββ −βββ e(αe)) (11)

where αe is the desired linearization point, L(αe) is thecontroller gain at this point, βββ the current measured anglesand βββ e is the steady state values for β2 and β3 given by (6)and (7) for a given αe.

LQ design is a well known method and thoroughly docu-mented in control literature and we refer to [14] for details.Given a linearization point αe and the linearized model in(8), the LQ design method finds the optimal gain, L(αe)minimizing the cost function

J =∫

∞

0

(βββ

TQβββ + α

2)

dt (12)

where βββ =βββ −βββ e, α = α−αe and Q is a design parameter.By solving the problem for different linearization points,αe, in the range given by the linearization limits given by(10), L(αe) can be obtained as show in Fig. 4. For the highlevel path follower, that will be further explained in thenext section, we want to be able to control β3 instead ofα . To achieve this we introduce β3e as the reference to thecontroller by deriving a pre-compensation link from β3e toαe. From (6) and the definitions for R1, R2 and R3 we get

αe = arctan

L1 sign(β3e)√L2

3

(1+ 1

tan2 β3e

)+L2

2−M21

(13)

IV. PATH TRACKING

In this section the high level path tracking controller isintroduced. When the internal angles β2 and β3 are stabilizedby the LQ controller a pure pursuit path follower is used

αe [rad]

-0.5 0 0.5

Lβ

3

5.2

5.3

5.4

5.5

αe [rad]

-0.5 0 0.5

Lβ

2

-4.6

-4.4

-4.2

-4

Fig. 4: Optimal feedback gains as a function of linearizationpoint αe. Gains calculated for Q = 10I.

as a high level controller to stabilize the system around areference path. The pure pursuit controller has successfullybeen used for path tracking for mobile platforms before, e.g[15]. In forward motion the controller has direct control ofthe steering angle α and the anchor point of the look-aheadcircle is set at the rear axle center of the pulling vehicle.In backward motion however the look-ahead circle anchorpoint, P∗, is set in the center of the rear axle of the lasttrailer and the pure pursuit controller gives the reference β3dto the low level stabilizing controller as depicted in Fig.5. A piecewise linear reference path is given as input andby calculating the intersection of the look-ahead circle withradius, Lr, and the line segment between two points on thereference path, the error heading, θe, can be calculated. Thecontrol law for β3d that will drive the vehicle along a circle tothe look-ahead point can now be found using basic geometrygiving

β3d =−arctan(

2L3 sinθe

Lr

)(14)

The only tunable parameter is the look-ahead distance, Lr,where a shorter look-ahead gives a more aggressive trackingbut can cause instability while a longer look-ahead givesa smoother tracking but causes bigger tracking offset. Byintroducing a proportionality term

β3e = β3d +Kp(β3d−β3) (15)

a more aggressive control can be achieved and experimentalresults have shown an increase in tracking performance.

α

β 3

β2

L 3

P*

Pr

Reference path

L

θ e

Fig. 5: Geometry of the pure pursuit control law: Lr look-ahead distance, θe angle to look-ahead point and α steerangle.

Fig. 6: Simulated region of attraction for different initialconfigurations of β2 and β3 for a straight line lineariza-tion reference. Light gray region corresponds to unstableinitializations and dark gray region corresponds to stableinitializations where the truck will converge to the straightline reference.

A. Stability

Due to the input constraints it is not possible to globallystabilize the system which might reach a jack-knife state thatis impossible to get out of by only reversing. A numericalsimulation approach have been used to evaluate convergencefrom different initial states for β2 and β3. Given a straightline reference, the parameters for our test platform, Q = 10I,Lr = 50 cm, Kp = 0.3 and θe = 0 the system is simulated fromrest and is checked for convergence to produce a region ofattraction map as shown in Fig. 6. When the configurationhave initial angles with the same sign and is close to anequilibrium point the stability region is quite large but whenthe angles have opposite sign and is closer to a jack-knifeconfiguration the region shrinks giving the elliptical shapeof the region of attraction.

V. USER INTERFACE

To make the control system useful for drivers, the piece-wise linear reference path that is supposed to be followedby the path tracker need to be created. A path plannercould be used to supply the reference but we argue that theavailable computational resources that can be dedicated tosuch a task is not enough in the current hardware setup ina modern truck. Instead we present a fast and user friendlyapproach that could easily be implemented on a touch screensystem inside the driver cabin. Assuming we have accuratepositioning and some knowledge of the surrounding, e.g.a photographic underlay of the scene or a map createdusing e.g. ultrasonic sensors, we let the user interactivelychange the piecewise linear reference path by changingthe connection points between the linear elements using aGraphical User Interface (GUI). The system with controllerand vehicle model is then simulated along the reference,creating a feasible drivable path that is displayed in real-timeon the screen1. Since the only requirement on the referenceis for it to be piecewise linear, complicated manoeuvres can

1Video demonstration of GUI and lab experiments can be found at:https://youtu.be/4EU-t5_mVmA

Fig. 7: GUI for creating reference paths for a parkingscenario where the driver starts in the lower part of thefigure and wants to park the trailer in between the other twotrailers next to the end position. The yellow dots connectedby the yellow lines represents the reference path. For thismanoeuvre the user has only used two control points to createthe simulated manoeuvre shown.

easily be created using only a few points as seen in Fig. 7.The simulated path can then be sent to the real platformfor execution and its feasibility has been ensured by thesimulation of the model.

VI. EXPERIMENTAL PLATFORM

The test platform shown in Fig. 1 is used to evaluate thesystem performance. The platform consist of a small scaletruck with Ackerman steering and an off-axle hitching whichis connected to a trailer with a turntable dolly. A LEGONXT control brick is used as the on board computer and theangles β2 and β3 are measured using two HiTechnic anglesensors. The LQ-controller and the pure pursuit controlleris running onboard the NXT control brick with an updatefrequency of 100 Hz and 10 Hz respectively. A high accuracyQualisys Oqus motion capture system is used for positioningand the information is relayed to the NXT via a Bluetoothconnection. Overall control of the system is done through alaptop computer running the GUI where reference paths canbe created and then sent via Bluetooth to the vehicle.

A. Parameters

By measuring the distances between the wheel axles thefollowing parameters can be found for our vehicle, L1 =19.0 cm, L2 = 14.0 cm, L3 = 34.5 cm, M1 = 3.6 cm and thesteering angle is limited to ±44 degrees.

VII. RESULTS

In this section the tracking performance results from bothsimulation and real experiments are presented. A parkingscenario where the GUI is used to plan a path is also studied.The LQ-controller gains used have been found throughexperiments and the final parameters used for both simulationand real world experiments were Q = 10I.

A. Simulation experiments

The tracking performance is first evaluated in a simulationenvironment where the system can be tested without anyunknown disturbances. To challenge the control system areference in the shape of an eight is constructed to make the

-1 -0.5 0 0.5x[m]

-0.2

0

0.2

0.4

0.6

0.8

1

y[m

]

Trailer position

Reference path

Fig. 8: Simulation result when driving the eight shapedreference path with Lr = 0.4 m and Kp = 0.3. Blue line is therecorded trailer position and the black dotted line representsthe reference path.

vehicle shift between hard left and right turns. The systemis then simulated along the reference for five laps usingthe same specifications as for our test platform and purepursuit parameters Lr = 0.4 m and Kp = 0.3. The resultingmaximum and mean tracking error in this simulation were2.81 cm and 0.45 cm, respectively. From Fig. 8 it is seen thatthe reference is closely tracked and only diverging from thereference a little at the shift between the turns and the straightsection. The results are encouraging but measurement errorsand the steering backlash is not included in the simulationmodel, so worse results should be expected for the real worldexperiments.

B. Lab experiments

1) Eight shaped reference: In the first lab experiment thesame reference path as in the simulation experiment is used.In the lab experiments it was found that the pure pursuitparameters that were used for the simulations gave unstablebehaviour and the look-ahead, Lr, had to be increased to0.5 m to have reliable operation. The vehicle was drivenfive laps around the course while logging onboard sensorsand position from the Qualisys motion capture system. Theresulting path is shown in Fig. 9 and the angle measurementsfor α , β2 and β3 during the run are shown in Fig. 10.From Fig. 9 it is seen that the tracking performance is veryconsistent and only differ by a few centimeters betweenthe laps and the trailer also tracks the reference quite wellbut gives a larger error when entering the turns. It can beseen from Fig. 11 that the average tracking error duringthe five laps is 1.67 cm and the maximum tracking errorat any time is 4.15 cm. As expected the tracking error isworse when compared to the simulations and the look-aheaddistance even had to be increased to maintain stability, evenso a average tracking error of 1.67 cm is acceptable inmost situations. The steering mechanism on the test vehiclehas a quite significant backlash in the gearing between thesteering servo and the wheels which makes the LQ-controllerto constantly adjust as can be seen in the plot of the steeringangle, α , in Fig. 10. A small delay between the β3 referenceand the β3 measurement is seen in Fig. 10 due to thetime it takes the LQ-controller to move the system into a

-0.5 0 0.5 1 1.5x[m]

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

y[m

]

Trailer position

Reference path

Fig. 9: Tracking results when using the small scale testplatform when driving the eight shaped reference path withLr = 0.5 m and Kp = 0.3. Blue line is the recorded trailerposition and the black dotted line represents the referencepath.

100 150 200 250 300 350 400-50

0

50

α [d

eg]

100 150 200 250 300 350 400-50

0

50

β2

[deg

]

100 150 200 250 300 350 400Time[s]

-50

0

50

β3

[deg

]

Fig. 10: Top figure: measured steering angle α . Middlefigure: measured angle β2. Lower figure: Blue line showsmeasured angle β3, red dotted line shows β3 reference.

new equilibrium configuration. Due to the cascaded natureof the approach this delay can be problematic since it isnot explicitly modeled and is one of the drawbacks whenpath tracking and β3 tracking is separated into individualcontrollers.

2) GUI reference: To demonstrate the usefulness of theGUI, the same parking scenario as in Section V is used.In the previous experiments the reference path consistedof only position waypoints for the trailer, but when thesimulation model in the reference generation is used thestate for the whole configuration is returned. The pure pursuitcontroller can still only use the simulated path of the traileras reference but in a parking scenario where we expect thevehicle to behave in the same way as the simulation andother objects should be avoided, it is also important to checkthe deviation of the truck and dolly against the simulatedpath. The path shown in Fig. 7 is easily created in theGUI and can then be sent to the platform for execution.The reference for trailer, dolly and truck together with themeasured positions are shown in Fig. 12. The mean and maxdeviation from the reference while performing the parkingmaneuver were emean = 2.11 cm and emax = 5.10 cm for thetruck, emean = 2.13 cm and emax = 5.11 cm for the dolly andemean = 1.75 cm and emax = 4.25 cm for the trailer.

50 100 150 200 250 300 350 400Time[s]

0

0.01

0.02

0.03

0.04E

orro

r [m

]

Fig. 11: Blue line represents absolute value of tracking errorand red dotted lines shows mean tracking error when drivingthe eight shaped path.

-1 -0.5 0 0.5 1 1.5x[m]

-1

-0.5

0

0.5

y[m

] Trailer position

Dolly position

Truck position

Trailer reference

Dolly reference

Truck reference

Fig. 12: Path deviation when using the small scale testplatform when driving the eight shaped reference path. Blue,red and green lines are the recorded trailer, dolly and truckpositions respectively and the blue, red and green dotted linesrepresents the reference paths.

VIII. CONCLUSIONS AND FUTURE WORK

This paper presents a cascaded path tracking and stabi-lization scheme for a reversing 2-trailer vehicle configurationwith off-axle hitching and evaluates the tracking performancein simulation and real world experiments on a small scaletest platform. The low level stabilization of the configurationis handled by a gain scheduled LQ-controller that is designedby linearizing the general 2-trailer equations around circularequilibrium configurations, while the path tracking is handledby the well known pure pursuit path tracking algorithm. Amaximum and mean tracking error of 2.81 cm and 0.45 cm,respectively, was achieved in the simulations while 4.15 cmand 1.67 cm on the test platform when the system was testedwith an eight shaped reference. A GUI was also presentedthat, given a representation of the surroundings, makes iteasy for an operator to manually plan a path for otherwisechallenging tasks such as reverse parking manoeuvres. Whenperforming such a manoeuvre it is important to have a smallpath deviation, not only for the trailer but also for the pushingtruck, to avoid collisions with nearby objects. A reversingparking scenario was created in the GUI and then executedby the test platform giving a maximum deviation error of afew centimeters for both the trailer and the truck.

Very general piecewise linear references can be handled bythe cascaded approach with a pure pursuit path tracking con-troller. However, since the controller only concerns the path

tracking for the trailer and no explicit tracking is done for thetruck and the dolly it could be risky when performing tightmanoeuvres. Since the path generation returns the states forthe whole configuration, future work will include controllerdesigns that exploit this and minimizes the tracking error forthe whole configuration. We will also look into the possibilityof using a path planner instead of the manual planning withthe GUI and also perform full scale experiments with a fullsize trailer configuration.

ACKNOWLEDGMENT

We gratefully acknowledge the Royal Institute of Tech-nology for providing us with the opportunity to performexperiments at their facilities at the Smart Mobility Lablocated in Stockholm, Sweden.

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