JIRS Suppression of Off-tracking Multi-Articulated Vehicles Through Movable Junction Technique

8/16/2019 JIRS Suppression of Off-tracking Multi-Articulated Vehicles Through Movable Junction Technique

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Journal of Intelligent and Robotic Systems 37: 399–414, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

399

On the Suppression of Off-tracking inMulti-articulated Vehicles through a MovableJunction Technique

S. MANESIS, N. T. KOUSSOULAS and G. N. DAVRAZOS Division of Systems and Automatic Control, Department of Electrical & Computer Engineering,University of Patras, Rion 26500, Greece; e-mail: [email protected]

(Received: 18 November 2002; in nal form: 27 May 2003)

Abstract. The motion of a multi-articulated robotic vehicle as well as of a train-like vehicle ischaracterized by the deviation of the path of each intermediate vehicle from that of the leading one(off-tracking phenomenon). In this paper, we propose the use of an innovative movable junction,which allows the kingpin to slide along the rear axis of the pulling vehicle, a technique that proved tobe efcient in reducing or completely eliminating off-tracking. New kinematic equations are derivedand a nonlinear controller is analytically developed, based on the steady-state off-tracking behaviorof the n-trailer system. Simulations and a comparison study for various cases without/with thisinnovative “sliding kingpin” junction technique showed that its use together with an analyticallyderived controller can make possible the elimination of off-tracking.

Key words: n -trailer systems, multi-articulated vehicles, off-tracking, nonlinear controller.

1. Introduction

Multi-articulated (or train-like or multi-body) vehicles or n -trailer systems can befound in two different research elds: autonomous robotics and transportation sys-tems (referring to vehicles). In autonomous robotics, the goal is to build mobilemulti-body robots that accomplish useful tasks without human intervention whileoperating in unknown environments. On the other hand, in intelligent transporta-tion systems the goal is similarly to construct transportation vehicles intelligentenough to be driven with as less as possible human intervention. In both of theabove areas, one major common problem is the undesired excess in motion dueto off-tracking. This term refers to the deviation of the path of each articulatedvehicle from the paths of preceding vehicles, especially that of the tractor’s. The

reduction or elimination of off-tracking will result in much improved performancein terms of safety during turns, cornering, overtaking other especially small cars,and backtracking.

The motion of the n -trailer system is subject to nonholonomic constraints (roll-ing without slipping) so it has been studied as a class of nonholonomic systems bymany researchers and has both theoretical and practical interest. The research work


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400 S. MANESIS ET AL.

[4] constitutes an excellent survey of the recent advances in control of nonholo-nomic systems. The main problem that has attracted most of the attention is pathfollowing while only few works consider the off-tracking problem. A closed formexpression for the off-tracking of the rear pivot point of a simple tractor–trailervehicle can be found in [1] while off-tracking bounds for a car pulling trailers havebeen derived in [3]. In [2] the path following problem with reduced off-tracking isaddressed for the n-trailer system. This is achieved by keeping track of the errordistance of each of the middle points of the axles of the vehicle from the path usingdifferent moving frames. In [8] different passive steering mechanisms as well ascontrol laws are presented for nonholonomic trailer systems. The main focus of these mechanisms is on reducing the passive tracking error with respect to thetractor’s trajectory and little attention was paid to active motion control.

The rest of the paper is organized as follows. Section 2 presents the n-trailervehicle model, the off-tracking problem and its consequences in the motion of such vehicles. In Section 3 the description of the innovative junction technique

called “sliding kingpin mechanism” is given together with the new kinematic stateequations of the resulting n -trailer system. In Section 4, a controller is analyticallydesigned based on the compensation for the steady-state off-tracking deviation. InSection 5, simulation results are presented comparing the performance with andwithout sliding using alternately an analytically derived nonlinear controller anda linear one of heuristic origin. The last section ends the paper with concludingremarks about the results and some open research problems.

2. The Multi-articulated Vehicle

In this section, we describe the kinematic model of the multi-articulated vehicle,

which is suitable for both transportation and robotic vehicles. Furthermore, thenotion of off-tracking and its consequences in the motion of an n-trailer systemis explained analytically for each domain of application. The n-trailer system isdened as a long and complex vehicle system consisting of a suitable power tractor(leading vehicle) pulling a number of passive robot bodies or semi-trailers as shownin Figure 1.

The kinematic equations of the n-trailer system with a two-axle tractor (of which the front one provides driving and steering) are:

˙x0 = U 1 cos θ 0 , ˙y0 = U 1 sin θ 0 ,˙ϕ = U 2 , θ̇ 0 =

U 1L

tan ϕ,

˙θ 1 = U 1L sin (θ 0 −θ 1),(1)

θ̇ 2 = U 1

Lcos (θ 0 −θ 1) sin (θ 1 −θ 2),

...


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OFF-TRACKING SUPPRESSION IN MULTI-ARTICULATED VEHICLES 401

Figure 1. Illustration of the multi-articulated vehicle coordinates.

θ̇ i

=

U 1

L

i−1

j =1cos (θ j

−1

−θ j ) sin (θ i

−1

−θ i ),

where x0 , y 0 are the Cartesian coordinates of the leading vehicle and U 1 , U 2 are thetwo control inputs, the linear velocity and the steering angle rate, respectively, [7].The other state variables represent the orientation angles for each vehicle i , asshown in Figure 1. The above state equations are derived from the algebraic ma-nipulation of the 2 n holonomic constraints:

xi+1 = xi −L i cos θ i+1 ,yi+1 = yi −L i sin θ i+1(2)

and the n +1 nonholonomic constraints

˙xi sin θ i − ˙yi cos θ i = 0. (3)Off-tracking relates to the question of how much road is needed for the rear

wheels of a vehicle during a turn and can be dened for cars as well as for multi-articulated vehicles, where the interest focuses on the last vehicle where the phe-nomenon achieves its strongest demonstration. There is a close connection betweenoff-tracking and the curvature of the followed path, and the sharper the curvaturethe more intensive it is. Off-tracking can be easily visualized as the deviation of thetrajectory of a semi-trailers’ axle center (where the kingpin hitch assumedly lies)from the path of the axle center of the leading vehicle during a turn. However, whenit comes to modelling and exact quantication for each time instant, the denition

is not unique. We may dene the minimal distance between the two trajectoriesor the distance from the ideal position of the semi-trailer axle center. This issue isdealt with in Section 4 below.

In the research eld of mobile robotics, the major problems are to nd anobstacle free path and a path following control law. When nding an obstacle-free path for multi-articulated robotic vehicles, we must take into consideration


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the presence of off-tracking. The reason is that the last trailer may collide withobstacles if the vehicle attempts to follow the designed path for the leading vehiclewith off-tracking neglected. One efcient way to solve this problem is to nd anobstacle-free path for the leading vehicle, add a controller for path following anduse another kingpin controller for off-tracking elimination.

In the case of truck-trains it is imperative that the last semi-trailer followsexactly the path of the (lead) tractor during a turn for lane change or a turn dueto the curvature of the highway. Otherwise, it will be possible for at least thelast semi-trailer to violate the outer boundary of the highway or to crash with anadjacent car during a lane change even when both tractor and the car keep theirrelative velocity within safe limits. It is known that the driver of any long tractor–trailer vehicle because of the off-tracking of the rear wheels turns the tractor faraway into the desired path in order to avoid the unpleasant consequences of thisphenomenon.

3. The Sliding Kingpin System

We will present here the sliding kingpin technique, the realization of which re-quires a hydraulic (or pneumatic) mechanism. The principle of its operation is asfollows. Off-tracking is eliminated by actively displacing each semi-trailer withrespect to the previous one, a technique rst described in [5]. This is realized bysliding the kingpin hitch in each semi-trailer in a direction perpendicular to thelongitudinal axle (i.e., along the rear axle) of the trailer by a distance S i and in adirection opposite to the curvature of the path. The sliding distance is expectedto be a function of some quantities related to the motion of the vehicle, mosteminently the steering angle of the leading vehicle. Furthermore, a proportional

type of relation is expected, i.e., the sliding distance will increase at sharp turnsperformed by the driver to compensate for expected higher off-tracking. Of course,the question remains how to determine the exact required magnitude and directionof the sliding distance. The investigation of this matter and the derivation of aclosed-form equation for the controller can be found in the next section.

A basic assumption in this line of thought is that the leading vehicle does notperform sharp turns so as the required sliding distance can be kept within rea-sonable levels. In mobile robotics, this assumption doesn’t seem very restrictiveaccording to our experience while in the case of truck-trains, it is generally satisedbecause of the limited radius of highways’ curvatures which is not permitted to beless than 700 m. The introduction of the sliding kingpin system completely relieves

the “driver” either human or computer from having to perform correcting maneu-vers such as forcing the leading vehicle to perform a wider turn than necessary inan effort to ensure minimal off-tracking for the last trailer.

To proceed we are going to develop a kinematic model for the n -trailer systemafter the introduction of the sliding kingpin mechanism. Consider two intermediatesemi-trailers of an n -trailer system as shown in Figure 2.


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Figure 2. The kingpin slides along the axle when the i semi-trailer turns.

The position of each semi-trailer P i , is taken to be the middle point of the ithsemi-trailer’s rear axle. Position P i is dened by the pair (x i , y i ) in the Cartesiancoordinates system while θ i is the orientation of the i th semi-trailer with respect tothe horizontal axis. It has been pointed out [3] that when the lead car of a singletrailer system is travelling along a circle of radius R l , then the trailer is travellingalong a circle of radius R t , with the same center, where R t < R l . In order to com-pensate for this path deviation of the trailer, we suppose that the kingpin hitchingpoint slides from the point P i to the point P S i , by a distance S i .

The determination of this distance is crucial for the performance of the con-guration and will be considered below. The following assumptions are made forderiving the mathematical model:

(a) All trailers have the same length L . This assumption is not restrictive at all andits main purpose here is to obtain a simpler model.

(b) Each trailer is modelled as a semi-trailer having only one rear axle.(c) Each trailer is hooked up to the midpoint of the rear axle of the preceding

trailer. In reality this is only approximately true as drivers in an effort to bal-ance vehicle stability and ease of driving, select the distance of the hook-uppoint from the rear axle center depending on the load of the trailer.

(d) By sliding the location of the kingpin, the weight of the trailer shifts towardan outer direction, which does not affect the kinematic behavior of the train.

(e) The unbalanced pulling point (when the kingpin sliding is nonzero) does notcause skidding of the whole axle. Basically, it is hypothesized that the lossof symmetry inicted by the sliding does not produce moments that are highenough to destabilize the vehicle system.


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(f) The sliding of the kingpin can be performed with the trailer fully loaded viaa hydraulic, pneumatic, or even electric mechanism. The hydraulic way isprobably the one that will most favorably balance power, efciency, weight,and cost.

(g) Sliding is always applied in the opposite direction to the steering input.

As in the case of the classical n -trailer system, we have the classical n +1 non-holonomic constraints imposed by the rolling and non-slipping condition given byEquation (3) and 2 n holonomic equations introduced by the corresponding links,which, because of the sliding kingpin mechanism are of the form:

xi+1 = xi −L cos θ i+1 +S i sin θ i ,yi+1 = yi −L sin θ i+1 −S i cos θ i . (4)

The method for deriving the state equations for the modied n-trailer systemis the same one as for the classical case. Taking the derivatives of the holonomicequations (4), combining with Equation (3) and eliminating variables ˙xi , ˙yi fori = 0 leads to a system of (n +1) equations.Due to the complex nature of the above equations the solution is computa-tionally tedious and time consuming dependent on the number of trailers. Forthe algebraic solution we have used Mathematica v4.01, a software package forsymbolic manipulation. Until now and using technologically advanced comput-ers, it has been possible to derive kinematic state equations for only ve trailers(n = 5) , the computation time lasting about 20 hours. The derivation of kinematicstate equations for larger combinations is possible in the future and depends on thecomputational power that will be available in the market or the willingness of thedesigner to wait for the necessary calculations. Table I presents the time used for

deriving the state equations for the cases of three and ve trailers with Mathematica4.01 on a Pentium III, 1.2 GHz.The kinematic equations for a multi-articulated vehicle with ve trailers and a

sliding kingpin mechanism combined with equations of the leading vehicle yields:

˙x0 = U 1 cos θ 0 , ˙y0 = U 1 sin θ 0 ,φ̇ = U 2 ,θ̇ 0 =

1L

U 1 tan φ,

Table I. Computational time for derivingkinematic state equations

Number of trailers ( n ) Time used (s)

3 3225 65891


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θ̇ 1 =1

L 2U 1 sin(θ 0 −θ 1)[L +S 1(t) tan φ] −L Ṡ 1(t) cos (θ 0 −θ 1) ,

θ̇ 2

=

1

L3 sin (θ 1

−θ 2) U 1

[L cos (θ 0

−θ 1)

+S 2(t) sin (θ 0

−θ 1)

]× [L +S 1(t) tan φ] +L Ṡ 1(t) [−S 2(t) cos (θ 0 −θ 1) +L sin (θ 0 −θ 1)]−L 2 Ṡ 2(t) cos (θ 1 −θ 2) ,

θ̇ 3 = 1L 4

S 2(t) [L cos (θ 1 −θ 2) +S 3(t) sin (θ 1 −θ 2)]sin (θ 2 −θ 3)× U 1 sin(θ 0 −θ 1)[L +S 1(t) tan φ] −L Ṡ 1(t) cos (θ 0 −θ 1)+L sin (θ 2 −θ 3) [L cos (θ 1 −θ 2) +S 3(t) sin (θ 1 −θ 2)]× U 1 cos(θ 0 −θ 1)[L +S 1(t) tan φ] +L Ṡ 1(t) sin (θ 0 −θ 1)+L Ṡ 2(t) [−S 3(t) cos (θ 1 −θ 2) +L sin (θ 1 −θ 2)]

−L 3

˙S 3(t) cos (θ 2

−θ 3) ,

θ̇ 4 = 1L 5

S 2(t) [L cos (θ 1 −θ 2) +S 3(t) sin (θ 1 −θ 2)][L cos (θ 2 −θ 3)+S 4(t) sin (θ 2 −θ 3)]sin (θ 3 −θ 4) U 1 sin(θ 0 −θ 1)[L +S 1(t) tan φ]−L Ṡ 1(t) cos (θ 0 −θ 1) +L sin (θ 3 −θ 4) [L cos (θ 2 −θ 3)+S 4(t) sin (θ 2 −θ 3)] [L cos (θ 1 −θ 2) +S 3(t) sin (θ 1 −θ 2)] (5)× U 1 cos(θ 0 −θ 1)[L +S 1(t) tan φ]+L Ṡ 1 sin(θ 0 −θ 1)+L Ṡ 2(t) [−S 3(t) cos (θ 1 −θ 2) +L sin (θ 1 −θ 2)]+L 2 Ṡ 3(t) [−S 4(t) cos (θ 2 −θ 3) +L sin (θ 2 −θ 3)]

−L 4

˙S 4(t) cos (θ 3

−θ 4) ,

θ̇ 5 =1

L 6S 2(t) [L cos (θ 1 −θ 2) +S 3(t) sin (θ 1 −θ 2)]

× [L cos (θ 2 −θ 3) +S 4(t) sin (θ 2 −θ 3)][L cos (θ 3 −θ 4)+S 5(t) sin (θ 3 −θ 4)]sin (θ 3 −θ 4) sin (θ 4 −θ 5)× U 1 sin(θ 0 −θ 1)[L +S 1(t) tan φ] −L Ṡ 1(t) cos (θ 0 −θ 1)+L sin (θ 4 −θ 5) [L cos (θ 3 −θ 4) +S 5(t) sin (θ 3 −θ 4)]× [L cos (θ 2 −θ 3) +S 4(t) sin (θ 2 −θ 3)] [L cos (θ 1 −θ 2)+S 3(t) sin (θ 1 −θ 2)][U 1 cos(θ 0 −θ 1)[L +S 1(t) tan φ]+L Ṡ 1(t) sin (θ 0 −θ 1)] +L Ṡ 2(t) [−S 3(t) cos (θ 1 −θ 2)

+L sin (θ 1 −θ 2)]+L 2 Ṡ 3(t) [−S 4(t) cos (θ 2 −θ 3) +L sin (θ 2 −θ 3)]+L 3 Ṡ 4(t) [−S 5(t) cos (θ 3 −θ 4) +L sin (θ 3 −θ 4)]−L 5 Ṡ 5(t) cos (θ 4 −θ 5) .


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Now if we assume that Ṡ i (t) remains equal to zero for all times (which is a ratherstrong assumption), the equations are simplied signicantly and it is possible toderive a recursive form for the general case of n trailers:

˙x0 = U 1 cos θ 0 , ˙y0 = U 1 sin θ 0 , ˙φ = U 2,θ̇ 1 =

U 1L 2

sin(θ 0 −θ 1)[L +S 1(t) tan φ],θ̇ 2 =

1L 3

U 1[L +S 1(t) tan ϕ]sin (θ 1 −θ 2)[L cos (θ 0 −θ 1)+S 2(t) sin (θ 0 −θ 1)],

θ̇ 3 =1

L 4U 1[L +S 1(t) tan ϕ]sin (θ 2 −θ 3)[L cos (θ 0 −θ 1) (6)

+S 2(t) sin (θ 0 −θ 1)][L cos (θ 1 −θ 2) +S 3(t) sin (θ 1 −θ 2)],...

˙θ n =1

L n+1 U 1[L +S 1(t) tan ϕ]sin (θ n−1 −θ n )

×n−1

i=1[L cos (θ i−1 −θ i ) +S i+1(t) sin (θ i−1 −θ i )].

4. Nonlinear Controller Design

In a multi-articulated vehicle operating with the sliding kingpin mechanism, twodifferent controllers need be derived: one for path planning and another that willregulate the sliding distance in the sliding kingpin mechanism for off-tracking

elimination. For path following issues, the linear velocity and the steering anglerate are the control inputs. In the transportation domain, the human driver reg-ulates the above control inputs in such a way as to achieve kinematic stabilityand the desirable trajectory tracking. For an autonomous multi-body robot movinginside a limited laboratory or in an industrial environment, the embedded controllerregulates the control inputs based on a control algorithm for path following and ob-stacle avoidance. The overall structure of the control system for a multi-articulatedvehicle either robotic or transportation is depicted in Figure 3.

To automatically determine the proper sliding distance, a number of heuristiccontrol policies have been tried. A good balance between simplicity and effective-ness is provided by the following control policy

S i (t) = K i ̇θ 0(t ), i = 1, . . . , n . (7)That is, the sliding distance is proportional to the rate of change of the tractor

orientation. Although the above controller performs in an acceptable manner, ascan be seen from the simulation results in the next section, its heuristic origin maymake it unsuitable for actual applications. It is desirable and from an engineering


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Figure 3. The overall control system for the n -trailer vehicle.

point of view imperative to derive analytically a controller that guarantees theo-retically the off-tracking elimination in an n-trailer vehicle. For that reason, weproceed to the derivation of a nonlinear controller: It is known that the curve radiusfor a vehicle is given by r = U /ω = U / θ̇ . So in general and for the i th trailer thisradius will be given by

r i =U iθ̇ i

. (8)

From the set of Equations (1) we have that

˙θ i =U 1L

i−1

j =1cos (θ j −1 −θ j ) sin (θ i−1 −θ i ). (9)

By combining (8), (9) and taking into consideration the relation

U n = U 1n−1

j =0cos (θ j −θ j +1) (10)

and after some algebraic manipulation this yields that

r i = L cot (θ i−1 −θ i ). (11)It stems from the last relation that the curve radii for different trailers are dif-

ferent so it is logical to introduce different sliding for each trailer. In [3] it wasproven that if the leading vehicle travels along a circular trajectory with radius r(where r > L ) then the trailer converges to a circular trajectory with radius R =√ r 2 −L 2 . The following lemma provides the condition for the leading vehicleand the semi-trailer of a simple tractor–trailer system to follow the same circulartrajectory.


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Figure 4. A simple tractor–trailer system depicting the trajectories of the trailer, tractor, andkingpin point.

LEMMA 1 . If the kingpin sliding is given by S =√ r 2 +L 2 −r then the trailer inthe steady state follows the circular trajectory with radius r of the leading vehicle.

Proof. According to [3] the trailer in steady-state will travel along a circulartrajectory of radius R ss

= r 2z

−L 2 (Figure 4), whereas r z

= r

+S . So we have

that

R ss = r 2z −L 2 = (r +S) 2 −L 2 = ·· ·.After some algebraic manipulations, we conclude that R ss = r .

The above lemma can be extended for the n trailers case following the sameprocedure, so the sliding for the i th trailer will be given by

S i = r 2i +L 2i −r i (12)and must be realized on the i th −1 trailer.By combining (11) and (12) we nd that the different sliding distances that wemust apply to the different trailers are given by the relation

S i = L1 −cos (θ i−1 −θ i )

sin (θ i−1 −θ i )(13)


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so that

Ṡ i = Lθ̇ i−1 − θ̇ i

1

+cos (θ i

−1

−θ i )

. (14)

5. Simulation Results

To test the controllers derived in the last section, the Matlab/Simulink environmentwas used. Although the main difculty in simulations is to nd the appropriate“driver” inputs (U 1 , φ̇) that correspond to real conditions, after many trials wemanaged to nd a pair of inputs that according to our opinion is close to realdriving conditions. We must note here that the controller, which was analyticallyderived, performs exceptionally well under any combination of inputs. The “driver”inputs U 1 ,

˙φ that we have used for simulation purposes are shown in Figure 5. In

Figure 6 simulation results are shown for a multi-articulated vehicle with threetrailers without/with using both controllers while the same simulation results for amulti-articulated vehicle with ve trailers are shown in Figure 7. In Figure 8 areshown the outputs of the analytically derived controller (sliding distance), for bothtrain-like vehicles with three and ve trailers.

Without loss of generality, we assume that all trailers and the leading vehiclehave a unit length. We must note that for our simulation purposes we use thekinematic equations (5), which fully describe the kinematic behavior of n-trailersystem with sliding kingpin mechanism. Simulation results using the simpliedkinematic equations (6) have been presented in [6].

Although the inputs U 1 , φ̇ shown in Figure 5 do not correspond exactly to thereal “driver” inputs that they are applied to real transportation or robotic n-trailervehicles they are used here because the maneuver that the n-trailer vehicle performsas a result of these two inputs is close to reality.

Figure 5. “Driver” inputs U 1 in [m / s] and U 2 = φ̇ in [rad / s].


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Figure 6. Trajectories for a multi-articulated vehicle with three trailers without/with linearand nonlinear controller.

Observing Figures 6 and 7, we may easily notice that the simulation results aremuch better when we use the nonlinear controller of Equation (13) instead of thelinear controller of Equation (7), in the sense that when the former (analyticallyderived) controller is in action, the off-tracking is practically zero in all phases of motion (sections of the trajectory). Observing Figure 8, we may notice that the


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Figure 6. (Continued.)

analytically derived nonlinear controller (13) has inherently the notion of delay inapplying the sliding distance and that stems from the fact that the sliding distanceis a function of both θ i −1 and θ i .

6. Conclusions

Off-tracking is one of the most signicant problems with potentially dangerous

consequences occurring in multi-articulated vehicles. This problem creates dif-culties or even renders prohibitive the wide application of multi-articulated vehiclesin both robotic and transportation domains. The sliding kingpin mechanism is amechanism whose principle of operation allows correcting such deviations. In thispaper, we derive the complete kinematic equations for multi-articulated vehicleequipped with the novel sliding kingpin mechanism. We concentrate on the threeand ve semi-trailer cases without any additional assumption other than what non-holonomy requires. The complex nature of these equations makes very difcult andtime consuming but not impossible in principle the derivation of kinematic equa-tions for more semi-trailers. Based on the analysis, we derive a nonlinear controllerfor adjusting the sliding distance of the sliding kingpin mechanism based on the

theoretical steady-state off-tracking when the leading vehicle moves along a circu-lar trajectory. Its response has been compared to another controller, which is linearand has been heuristically developed. Both controller designs have been validatedthrough simulations and they both showed satisfactory performance. However, theanalytically derived nonlinear controller has proved to be superior in all cases,eliminating off-tracking in all phases of the vehicle motion.


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Figure 7. Trajectories for a multi-articulated vehicle with ve trailers without/with linear andnonlinear controller.

Acknowledgement

This research work has been partially supported by the Caratheodory Program of the Research Commission of the University of Patras.


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Figure 7. (Continued.)

Figure 8. Sliding distances when the nonlinear controller of Equation (13) is used.


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References

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JIRS Suppression of Off-tracking Multi-Articulated Vehicles Through Movable Junction Technique

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