Maneuvering assistant for truck and trailer combinationswith arbitrary trailer hitching

Yevgen Sklyarenko, Frank Schreiber and Walter Schumacher

Abstract— Stabilizing controllers can assist the driver tosafely maneuver of truck and trailer combinations like theEuroCombis. This paper presents a novel nonlinear controllerfor combinations of a nonholonomic tractor car with twosemi-trailers based upon a instantaneous approximation of thevehicle kinematics, an exact input-state linearization techniqueand a model-reference control scheme. The main advantage ofthe suggested controller is its universality, as it can be appliedto truck and trailer combinations with an arbitrary hitchingoffset as well as in both driving directions. Convergence androbustness properties of the proposed controller have beenvalidated by series of simulations and experiments with acombination, which possesses one off-axle and one on-axle joint.The presented method can also be extended to other truck andtrailer combinations.

Keywords - truck and trailer combination, off-axle hitching,input-state linearization, model-reference control scheme

I. INTRODUCTION

Automatic and autonomous transportation systems are theobjective of numerous research initiatives at present. Amongthose the controllability of articulated multi-body vehiclesplays a prominent role. The major control challenge in thistype of plant arises mainly from the nonlinear nature of theirkinematics and its instability in the case of backward motionin a conjunction with the steering limitations.

The introduction of the European Council Directive96/53/EG in 1996 created a legal basis within the EuropeanUnion for truck and trailer combinations up to lengthsof 25.25 m and up to a total weight of 60 tons. As aresult of the developments in the logistics industry differenttruck and trailer configurations emerged, which are knownas “EuroCombis”, see Fig. 1. Most of those EuroCombis(variants A - E) feature two hitching joints and can beregarded kinematically as a tractor car with two semi-trailers.However, these EuroCombis differ from each other withregard to their trailer hitching offsets. This is a crucial pointwhich leads to partially different behavior of the vehicles.The ability to handle systems with an arbitrary type of trailerhitching is therefore of utmost importance for designing auniversal maneuvering assistant for the most relevant truckand trailer combinations with two semi-trailers (TTC).

Through the many years of international research nu-merous stabilizing controllers for articulated vehicles weredeveloped, depending on the existence and direction of thetrailer hitching offset. The earlier works, based on the methodof exact input-state linearization, were limited to the case,

All authors are with the Technische Universität Braunschweig, Institute ofControl Engineering, Hans-Sommer-Str. 66, 38106 Braunschweig, Germany

Email corresponding author:skliarenko@ifr.ing.tu-bs.de

Fig. 1. Variants of the EuroCombis [1]

that the hitching point was located directly on the centerof the rear axle of the preceding towing unit (so-called “on-axle” hitching) [2], [3], [4], [5]. It was also noted that the ex-act input-state linearization is not applicable for multi-bodyarticulated vehicles with an offset in the trailer hitching (so-called “off-axle” hitching) [6], [7]. In [7] an approximationof the system with a hitching offset at the truck by an on-axlesystem was suggested. The approximated “ghost vehicle”provided the same steady state behavior and allowed an exactinput-state linearization. During path-tracking the modelinginaccuracies were considered as disturbances.

Similar results were later achieved for train-like vehicleswith a hitching offset by applying the method of exact input-output linearization instead of input-state linearization [8],[9]. The peculiarity of these control laws is the dependenceof the choice of the guide-point and therefore the controllerstructure has to be adjusted based upon the driving direction.

It should be noted that most works on the control ofarticulated vehicles are limited to a negative hitching offset(joint behind the rear axle of the preceding unit), while to thepresent time there are relatively few research results availableregarding the control of train-like vehicles with positive (oran arbitrary) hitching offset. The reason are the mentioneddifficulties to apply the method of exact linearization. In[10] this problem was avoided by “conventional” (Jacobian)linearization. Promising results provides the application ofthe Lyapunov method to design a controller. In [11] thisis shown only for a vehicle with a single semi-trailer. Inaddition the application of known approaches to the path-tracking problem for train-like vehicles with off-axle hitchingprovides quite complex control laws and complicates theirpractical application.

The measures proposed in this paper for TTC witharbitrary trailer hitching differ from previous approachesby the underlying hypothesis that the stabilization and thepath-tracking tasks can be treated separately, resulting ina cascaded control structure. The novelty of the proposedlower-level stabilizing control loop, which forms a ma-neuvering assistant, arises from applying an instantaneousapproximation of the vehicle kinematics, an exact input-state linearization technique and a model-reference controlscheme. This approach offers all the advantages of a cascadecontrol structure and will allow at a later time the designof simpler control laws of a superimposed path-trackingcontroller.

II. STABILIZING CONTROLLER

A. Analysis of possible solutions

The nonlinear control theory provides a variety of methodsfor the analysis and synthesis of nonlinear control systems.The most successful method applied to truck and trailercombinations is still the exact feedback linearization. Thespecial attractiveness of this method is explained by the ele-gant way, in which the requirements of stability and controlperformance can be addressed using the well-known methodsof linear control theory to the linearized system. Thereare two well-known exact linearization techniques, input-output linearization (IOL) and input-state linearization (ISL),applicable for the class of nonlinear affine in control systems.The application of IOL is usually more straightforward thanthat of ISL. However, a control law based on IOL is onlysuitable with a stable internal dynamics of the controlledsystem. In case of IOL of a truck and trailer combination thestability of the internal dynamics as shown in [8] dependsof the direction of motion, the sign of the hitching offsetand a choice of the guide-point of the vehicle. Since theISL produces no internal dynamics in the controlled system,it would allow to design the universal control laws validfor truck and trailer combinations with an arbitrary hitchingoffset as well as independent of the direction of motion.Unfortunately, the analytical determination of the flat outputfor the nonlinear system, like a TTC, can be an unachievabletask. Therefore, irregular approaches to simplify the task maybe required.

It is well-known, that a substitution of variables in non-linear differential equations of the vehicle’s model providesan affine in control system without increasing the orderof the system. Our investigation has shown, that a furtherinstantaneous approximation of a single term in the equationsallows to obtain the solution of a partial differential equationleading to a flat output of the approximated system withrelative ease. Based upon this approach, the state as wellthe input transformation can be found. After such of anapproximate input-state linearization the design of a linearstate regulator for the stabilizing and tracking controller(maneuvering assistant) can be done. To provide robustperformance under the influence of model inaccuracies anddisturbances a model-reference control scheme is used.

B. Kinematic model of the vehicle with two semi-trailers

The design of a controller requires a model of the TTCthat describes the system behavior in forward and reversedirection of movements with sufficient accuracy. A purelykinematic model can be established, which neglects dynamiceffects (e.g. inertia and slip) and reduces to a planar single-track model. The kinematic modeling leads to a muchsimpler and better interpretable system of differential equa-tions, which reproduces the processes in truck and trailercombinations with reasonable accuracy.

Fig. 2 depicts the schematic representation of a TTC.The first joint is assumed to be an off-axle joint with anarbitrary hitching offset (pictured positive) while the secondhitching is on-axle. The following variables and parametersare illustrated in Fig. 2:

• φ0(t) - steering angle• φ1(t) - angle between first semi-trailer and tractor• φ2(t) - angle between second and first semi-trailer• λ - distance traveled along the path• V0(t) - tractors’ front axle speed along the path• V1(t) - speed of first semi-trailer• l0 - distance between tractors’ axles• d1 - hitching offset (positive towards tractors’ front axle)• l1 - length of first semi-trailer• l2 - length of second semi-trailer

f2

f1

f0

d1

V0

l 1

l 2

l 0

l

V1

Fig. 2. Schematic of a vehicle with two semi-trailers

For the design of a stabilizing controller only the behaviorof the internal degrees of freedom φ1(t) and φ2(t) have tobe considered. For ease of analysis and design of the controlsystem, the following notations were introduced:

• state variables: x1 = φ2, x2 = φ1;• control variables: x3 = φ0, V0 ;• model parameters: a1 = V0 1l2 , a2 = V0

d1l0l2

, a3 = V0 1l1 ,a4 = V0

d1l0l1

, a5 = V0 1l0 ;• short form of the functions: ci = cosxi, si = sinxi.

Under these notations the behavior of the internal coor-dinates φ2(t) und φ1(t) without consideration of externaldegrees of freedom leads to a system of two nonlineardifferential equations:

ẋ1 = −a1s1c2c3 − a2s1s2s3 + a3s2c3 − a4c2s3ẋ2 = −a3s2c3 + a4c2s3 − a5s3

(1)

ISL is applicable to a class of nonlinear affine in controlsystems, whose origin is an equilibrium with zero control:

~̇x = ~f(~x) + ~g(~x) · u, ~x ∈ Rn, u ∈ R, (2)

with ~f(~x) and ~g(~x) smooth n× 1 vector functions.To convert the model (1) into (2), the speed of the first

semi-trailer has to be used instead of the tractors’ frontaxle speed. This is equivalent to introducing the followingvariable substitution:

V1 = V0 · c2 · c3. (3)

Further, after another substitution

u = tanx3 (4)

the functions of the affine in control plant model (2) become:

~f =

[f1(~x)f2(~x)

]=

[−q1s1 + q3 tanx2−q3 tanx2

],

~g =

[g1(~x)g2(~x)

]=

[−q2s1 tanx2 − q4

q4 −q5c2

]. (5)

The model parameters qi correspond to the parameters ai,if the speed V1 replaces V0. This means, that the speed V1must be kept constant by the tractor’s speed control accordingto (3). The speed V0 can be completely eliminated from thedifferential equations to avoid singularities at V0 = 0, if thetime derivatives are replaced by the derivatives with respectto a new independent variable λ, e.g. by means of the so-called time scaling [12].

C. Approximation and input-state linearization

ISL is achieved by a conversion from a system of non-linear differential equations (2) to the canonical Brunovskyform [13]:

ż1 = z2, ż2 = z3, . . . żn−1 = znżn = v

(6)

by a linearizing feedback, consisting of:• diffeomorphic state transformation~z(~x) = [z1(~x) . . . zn(~x)]

T and• input transformation u = 1β(~z) (v − α(~z)), with v the

new linearized input.The plant in this case is covered by the feedback on

theoretically all state variables. From this follows the def-inition of the feedback linearization. There are no internaldynamics, because all state variables are linearized by ISL.The principal difference from the ”conventional” (Jacobian)linearization is that the exact linearization is not approximatebut an equivalent transformation.

The plant (2), (5) is controllable in the vicinity of the ori-gin, where controllability matrix Qc =

[ad0f~g . . . ad

n−1f ~g

]has full rank. It is easy to obtain the controllability conditionsin the origin: q4 6= q5 and q1q4 6= q3q5, which canbe summarized into constraints on the vehicle’s geometry:d1 6= l1 and d1 6= l2. In truck and trailer combinationsthe hitching offset is almost always less than the length oftrailers, therefore the plant is almost always controllable inthe neighborhood of the origin.

For a second order system the involutivity condition of theset{ad0f~g . . . ad

n−2f ~g

}is always satisfied. This means

that a diffeomorphic state transformation ~z(~x) for the sys-tem (2), (5) theoretically exists. The first element of the statetransformation vector is found by solving the system of n−1homogeneous partial differential equations dz1d~x · ad

if~g = 0,

i = 0 . . . n − 2. Unfortunately, the analytic solution of thesingle partial differential equation for the system (2), (5)∂z1∂x1

(−q2 sinx1 tanx2 − q4) +∂z1∂x2

(q4 −

q5cosx2

)= 0

and the flat output can’t be found. If we assume thatq24 is the momentary value of q2 sinx1 tanx2 + q4, thenthe instantaneously approximated equation ∂z1∂x1 (−q24) +∂z1∂x2

(q4 −

q5cos x2

)= 0 results, for which the expression of

the flat output can easily be found:

z1(~x) =2q24q5

q4√q25 − q24

arctan

(q5 + q4√q25 − q24

tanx22

)+

−q24q4x2 − x1 for |q5| > |q4| or l1 > d1 (7)

Based upon this, the next elements of the state transformationcan be derived as zi = Li−1f z1 = Lfzi−1, i = 2 . . . n [13]:

z2(~x) = q1s1 − q3 tanx2 +q3q24s2q4c2 − q5

. (8)

The calculation of the input transformation also does notcause too many difficulties α(~x) = Lnf z1 = Lfzn,β(~x) = LgL

n−1f z1 = Lgzn [13]:

α(~x) = q1c1(q3 tanx2 − q1s1) +

−

(q24 (q4 − q5c2)(q4c2 − q5)2

− 1c22

)q23 tanx2, (9)

β(~x) = −q1q24c1 +

+

(q24 (q4 − q5c2)(q4c2 − q5)2

− 1c22

)q3q4c2 − q5

c2.(10)

With |q5| < |q4| or l1 < |d1|, although this is not typicalfor truck and trailer combinations, the flat output for theapproximated system also exists

z1(~x) =−2q24q5

q4√q24 − q25

arctanh

(q5 + q4√q24 − q25

tanx22

)+

−q24q4x2 − x1. (11)

This leads to the same results for the functions z2(~x) , α(~x)and β(~x).

D. Control law

Considering the substitution (4) the nonlinear controllaw [13]

u =1

β(~x)(v − α(~x)) (12)

=1

β(~x)

(z̈1d +~b

T (~zd − ~z)− α(~x))

=1

β(~x)(z̈1d + b1 (z2d − z2) + b0 (z1d − z1)− α(~x))

φ0 = arctan ( u )

can be represented in a form of two loops, comprising theplant, see Fig. 3:

1) linearization loop with input v (according to (6)),nonlinear functions β−1(~x) and arctan in the feed-forward path, as well as α(~x) in the feedback path;

2) pole-placement loop with the linear gain ~b = [b0 b1]T

according to the state ~z = [z1(~x) z2(~x)]T withreferences ~zd = [z1d(~xd) z2d(~xd)]T and z̈1d.

When choosing a stable characteristic polynomial N(s) =s2+ b1s+ b0 the control law (12) will provide local stabilityof the closed system, asymptotic convergence to zero of thetracking error ~ε(t) = ~zd(t)−~z(t) and the boundedness of allthe state variables [13]. In this case transient processes of theflat output of the plant will be described by a linear transferfunction G(s) = 1N(s) [13]. The coefficients b0 and b1 ofthe polynomial N(s) must be chosen based on the desiredperformance of the closed system as well on the performanceof the steering servo.

E. Control scheme

The tracking controller requires smooth reference signalszd, żd, . . . , z

(n)d . By a common flatness-based technique they

are generated in ~z after transformation of the referencesignals ~xd to ~zd. Additionally, to compensate tracking er-rors occurring under the influence of model uncertaintiesand disturbances the measured system output ~xs has to becompared with the output of the nonlinear system model ~xm,calculated from ~zm by the inverse transformation. Here twoproblems arise: the state transformation is only known forthe approximated system, additionally the analytic solutionof (7) and (8) is necessary whose derivation is a hard task.

A model-reference control scheme was chosen for thisapplication, comprising a reference-model control loop anda disturbance control loop, see Fig. 4. The reference-modelcontrol loop contains the instantaneously approximated non-linear system model, which is exactly feedback linearizedby (9) and (10). The output of the model-reference controlloop is ~xm, which can thus be obtained without an inversetransformation. The state vector ~zm serves as reference vec-tor for the disturbance control loop, while the linearized con-trol vm is passed as a feedforward signal into the linearizedpart of the disturbance control loop. Due to the feedforwardaction the stability of the control loops is independent of eachother and their controllers can be designed independently fortheir specific tasks.

As the model control loop is disturbance free, its controllerwas designed to provide good tracking performance bydefining the polynomial Nm(s) as a second order binomialpolynomial. In the absence of disturbances the feedforwardcontrol from the model loop would drive the output ofthe real plant ~xs along the trajectories of the instantaneousapproximated model ~xm, because both models have thesame steady state in ~x. Since the main tracking control isperformed in the model loop, the disturbance controller canbe designed for good disturbance rejection as its role is tosuppress disturbances acting from the environment or as theresult of approximation errors. Therefore, the polynomialNs(s) was parameterized using maximal possible dynamicsunder the given dynamics of the steering servo.

However both models, the full model and the approxi-mated model, possess the same steady state in in ~x, butdue to the approximated nature of the state transformationsystem model, the steady states in ~z are not equal. To adjustfor the incorrect state transformation, the following methodwas applied. From (8) the value for x2d is determined forz2d = 0 and the reference value of x1d. This calculationcould have been evaluated analytically with much effort, butwe have chosen to solve it numerically in closed loop inthe reference generator, see Fig. 4. Based upon the valueof x2d and the reference value of x1d the value of z1d wascalculated according to (7). The reference vector ~zd, whichwas derived this way, is used as reference for the reference-model control loop. Actually, this ensures the compliancewith the steady state compatibility condition using controlengineering techniques, similar to the geometric approachtaken by Bolzern et al. [8].

To guarantee correct steady states in spite of parametricuncertainties of the plant model, in the last step of design-ing the model-reference control scheme an overlaying PI-controller was introduced at the input of the disturbancecontrol loop, see Fig. 4.

F. Simulation results

The computer simulation was completed for the demon-strator vehicle at the Institute of Control Engineering with thefollowing parameters: l0 = 0.84m, l1 = 0.6m, l2 = 0.9m,d1 = −0.4m, φ0max = π/6 and the time constant of thesteering servo Tservo = 0.2s.

As the polynomial Nm(s) in the control law (12) ofthe reference-model control loop a second-order polynomialwas used with the characteristic frequency ω0m = (1, 35s ·m/s)−1 and b1m = 2ω0m, b0m = ω20m. The correspondingparameters for the disturbance control loop are given ω0s =(2Tservo · m/s)−1and b1s = 2ω0sD, b0s = ω20s, withnormalized damping factor D.

To demonstrate the robustness of the approach, vehicleparameters with a random variation of up to ±10% wereused in the designed nonlinear controller. Simulations of thereversing process have shown the high quality of the controlfor TTCs with negative and positive hitching offset, see thesystem states and the control efforts in Fig. 5 and Fig. 6.

(x)

State

transform.

Plant

u(t) v(t)

bT x(t)=[2(t) 1(t)]

T

z(t) zd(t)

z1d(t) +

-

- +

+

+ .. 0(t)

(x) arctan

affine in control plant

linearization loop

linear regulator

(t)

pole-placement loop

linearized plan

Fig. 3. Signal flow in the exact linearization based controller

- b sT

-

linearized real plant

disturbance control loop

xs

zs

vs

vm

-

linearized virtual plant

reference-model control loop

xm

zm

vmzd

bT

m

statetransform.

xd

xs

xmPI-

controller

servomodel

f0m

f0s

steady statecalculation

f2d

zm

Dz

� �

�

�

�

�

�

�

�

�

�

�

Fig. 4. Nonlinear reference-model controller scheme

At λ = 0 the system was subjected to a stepwise changeof the reference for angle φ2, which defines the desired pathcurvature of the combination during reversing. The referencevalue φ1d is continually calculated from the steady statecompatibility condition. As can be seen in Fig. 5 (left)and Fig. 6 (left), the desired value φ2d is met well in theapproximated and the full plant model. Due to the deviationbetween the approximated and the full plant model thedesired value φ2d is achieved with differing values for φ1mand φ1s. For this reason, the corresponding steering anglesφ0m and φ0s are also not equal, see Fig. 5 (right) andFig. 6 (right). At λ = 6m a disturbance is included in formof a stepwise reduction of the steering angle by 10% of themaximal value, which simulates a lateral slip effect. It canbe seen that this disturbance is suppressed effectively.

The intricate task of TTC reversing with a positive off-set, which is more complicated under exact linearization,has been successfully controlled using the same controlalgorithm with a comparably high quality performance, seeFig. 6. Note the same steady state steering angle for thenegative and the positive offset case, that actually justcorresponds to the Ackermann steering angle.

Although a part of the controller relies on a simplificationof the nonlinear plant model, it was shown that the designedcontroller allows to effectively steer the motion of TTC withan arbitrary offset of the first joint and in both directions ofmotion. The presented approach was also implemented andsuccessfully validated on the demonstrator vehicle, see Fig. 7and webpage [14].

III. CONCLUSION

This paper presented the design of a stabilization andtracking controller for a tractor car with two semi-trailersand off-axle trailer hitching, whose kinematic model isnot exactly input-state linearizable. The designed nonlinearcontrol law is based on input-state linearization of theinstantaneously approximated kinematic model, which canbe exactly linearized and whose behavior provides an ac-curate performance. Despite the efforts of simplification, thedesigned nonlinear stabilization and tracking law is universalregarding the hitching offset and able to prevent the loss ofstability in the closed system while reversing. The presentedapproach can also be applied to other types of truck andtrailer combinations.

Fig. 5. State trajectories (left) and control efforts (right) in plant and model loop of a system with negative hitching offset d1 = −0.4m as a consequenceof a step input in the reference and in the disturbance signal

Fig. 6. State trajectories (left) and control efforts (right) in plant and model loop of a system with positive hitching offset d1 = +0.2m as a consequenceof a step input in the reference and in the disturbance signal

Fig. 7. Vehicle used to validate the designed control algorithms

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[14] Y. Sklyarenko. Institute of control engineering. maneuvering assistant.www.ifr.ing.tu-bs.de/www/forschung/autonomesfahren/rass.html.

IntroductionStabilizing ControllerAnalysis of possible solutionsKinematic model of the vehicle with two semi-trailersApproximation and input-state linearizationControl lawControl schemeSimulation results

ConclusionReferences