JIRS Flatness Conservation in the N-trailer System Equipped With a Sliding Kingpin Mechanism

J Intell Robot Syst (2006) 46: 151–162DOI 10.1007/s10846-006-9056-2

Flatness Conservation in the n-trailer System Equippedwith a Sliding Kingpin Mechanism

V. Deligiannis · G. Davrazos · S. Manesis ·

T. Arampatzis

Received: 9 June 2004 / Accepted: 3 May 2006 /Published online: 3 August 2006© Springer Science + Business Media B.V. 2006

Abstract Nonlinear systems, which are differentially flat, have several propertiesthat can be useful on designing effective controllers. In this paper we show thatthe n-trailer system equipped with a sliding kingpin mechanism is a differentiallyflat system, like its non-sliding kingpin counter part. The sliding kingpin techniqueis used to eliminate the undesired deviation of the path of each intermediate vehiclefrom that of the leading one (off-tracking phenomenon). The linearizing outputs ofthe flat system are the Cartesian coordinates of the middle of the last semi-trailer’saxle. The state space and the kinematic equations of the new modified system arederived and the conditions for flatness are examined. The flatness conservation isalso checked relatively to several kinds of dynamic sliding feedback control.

Key words flatness · multi-articulated vehicles · n-trailer system · sliding kingpin

1. Introduction

A major problem in automatic control is driving a system from one state to another.This is the motion planning problem in mobile robotics. For systems described asdifferentially flat, motion planning has simple and specific solutions [1]. This arisesfrom the fact that every system variable is described as a function of the flat outputand a finite number of its time derivatives [2]. Additionally, knowledge of the system’sflat output allows the design of open loop control and assists the design of controlloops [3, 4]. Because the behaviour of differentially flat systems is determined by thelinearizing outputs, we can design trajectories in output space and then calculate theappropriate inputs. The concept of differential flatness was originally introduced byFliess et al. [5], in the context of differential algebra and later using Lie–Bäcklundtransformations [6]. It is known that many mechanical systems are differentially flat,

V. Deligiannis · G. Davrazos · S. Manesis (B) · T. ArampatzisElectrical and Computer Engineering Department, University of Patras, Patras 26500, Greecee-mail: [email protected]

152 J Intell Robot Syst (2006) 46: 151–162

such as, the n-trailer system [3], the Bi-steerable car [4], the hopping robot, the planarrigid body chain and the towed cable system [6], pendulums in series, the heavy chain[7] or even the fed-batch bioreactor [8].

The n-trailer systems or multi-articulated vehicles can be found in two differentresearch fields: Autonomous robotics and transportation systems. In autonomousrobotics the main aim is to build mobile multi-body robots that accomplish usefultasks without human intervention while operating in unknown environments. On theother hand, in intelligent transportation systems the goal is to construct long andmulti-articulated transportation vehicles, known as a hard platooning system, whichis intelligent enough to be driven with as less human intervention as possible. Onebasic problem, in both above areas, is the undesired excess in motion due to the off-tracking phenomenon. Off-tracking is defined as the deviation of the semi-trailers’axles or the kingpin hitch from the path of the steering axle of the leading vehicle.The off-tracking deviation can be eliminated using a sliding kingpin mechanism, atechnique introduced in [9] and various types of sliding control have been examinedin related works [10, 11].

As mentioned above, the n-trailer system is a differentially flat system, whose flat(or linearizing) outputs are the Cartesian coordinates of the middle point of lasttrailer’s rear axle. In [2], Rouchon et al. show that mobile robots with trailers areflat systems as soon as the trailers are hitched to the middle point of the axle of theprevious ones. The placement of the kingpin outside the axle makes the system notflat. The kingpin sliding mechanism allows the kingpin to slide along the axle of theprevious trailer. A natural question, from a steering and control theory perspective,is whether the system remains flat or not. In this paper we prove that the n-trailersystem equipped with a sliding kingpin mechanism is a differentially flat system.

The flatness property of a system is more significant in the robotics field than thetransportation one, because it is related to various problems such as path and motionplaning, path following, steering etc. A representative example among a large numberof similar works can be found in [12]. Fliess et al. [2, 5] have shown that chained-form systems are a special case of differentially flat systems, where the bottoms ofthe chains play the role of flat outputs. The advantages of the chained form are thatmany methods are available for the open-loop steering of such systems as well as forpoint-stabilization. The conversion of the n-trailer system and of the extended one(multi-steering n-trailer) into a multi-input chained form has been done by Tilburyet al. [13].

The rest of the paper is organised as follows. Section 2 presents the mathematicalmodel of the n-trailer system equipped with a sliding kingpin mechanism and thenecessary assumptions used in the derivation. In Section 3 the flatness property ofthe defined n-trailer system with kingpin sliding is examined and proved. The lastsection presents some aspects about controller’s selection for the sliding mechanismand its influence in flatness conservation.

2. The n-trailer System Equipped with a Sliding Kingpin Mechanism

The n-trailer system is defined as a long and complex vehicle system consisting of asuitable power tractor pulling a number of passive robot bodies or semi-trailers asshown in Figure 1. Such systems are modeled as having one constraint on each axle:That the velocity vector of the axle midpoint is constrained to move perpendicular

J Intell Robot Syst (2006) 46: 151–162 153

Figure 1 Illustration of themulti-articulated vehiclecoordinates.

to the axle direction or in other words that the wheels are allowed to roll but notslip. These non-slipping constraints are nonholonomic or nonintegrable and do notreduce the configuration space of the system.

Hence, the n-trailer system is a nonholonomic system subject to n+1 nonholonomicconstraints

yi · cos (θi) − xi · sin (θi) = 0 i = 0, 1, ..., n

which lead to the relations

tan (θi) =yi

xii = 0, 1, ..., n (1)

As the term semi-trailer expresses, each trailer is hooked up to the mid-point of therear wheels of the preceding trailer or body. This means that the system is submittedto 2n holonomic equations

xi = xi−1 − ì · cos (θi)

yi = yi−1 − ì · sin (θi)i = 1, 2, ..., n (2)

Also, we have the inputs’ equations

x0 = U1 · cos (θ0)

y0 = U1 · sin (θ0)

ϕ = U2 (3)

where x0,y0 are the Cartesian coordinates of the leading vehicle (tractor), xi,yi thecoordinates of the ith trailer and U1,U2 are the two control inputs, the linear velocityand the steering angle rate, respectively. The other state variables represent theorientation angles θi for each trailer i, as shown in Figure 1, and ϕ is the tractor’ssteering angle. The only difference between a multi-body robotic system and a similarmulti-trailer vehicle (truck or road train) is the magnitude of the different physicalquantities (length, velocity, steering angle limits, weight, etc.).

Off-tracking is defined as the deviation of the semi-trailers’ axles or the kingpinhitch from the path of the steering axle of the leading vehicle. Figures 2 and3 illustrate off-tracking phenomenon showing two examples of multi-articulated


Figure 2 Trajectories for amulti-articulated vehicle withthree trailers.

Figure 3 Trajectories for amulti-articulated vehicle withfive trailers.


vehicles with three and five trailers, respectively. In the case of truck-trains, it ismore imperative than in any other case that the last semi-trailer follows exactly thepath of the lead tractor during a turn for lane change or a turn due to the curvatureof the highway. Otherwise, it will be possible for the last semi-trailer to violate theouter boundary of the highway or to crash with an adjacent car during a lane changealthough both keep invariant their relative velocity. It is known that the driver of anylong truck-train, because of the off-tracking of the rear trailers, turns the tractor fartowards the desired path in order to preserve rear trailers in acceptable boundaries.When we deal with mobile robots the major problems are to find an obstacle-freepath and path following control. However, in the case of multi-articulated roboticvehicles we must take into consideration the off-tracking phenomenon when findingan obstacle-free path. The reason is that the last trailer may collide with obstaclesif the vehicle attempts to follow the designed path for the leading vehicle with off-tracking neglected. One efficient way to solve this problem is to find an obstacle-free path for the leading vehicle, add a controller for path following and use anotherkingpin controller for off-tracking elimination.

As mentioned above, the off-tracking can be eliminated by sliding the kingpin ofeach trailer with respect to the previous one. Consider two intermediate semi-trailersof a truck train as shown in Figure 4. According to this technique the kingpin hitch ofthe i + 1th semi-trailer slides in a direction perpendicular to the longitudinal axle (i.e.along the rear axle) of the ith trailer. The position of the ith semi-trailer Pi is takento be the middle point of the ith semi-trailer’s rear axle. The original configurationspace must be reobtained since the set of holonomic constraints has been changed.

Position Pi is defined by the pair (xi,yi) in the Cartesian coordinates system whileϑi is the orientation of the ith semi-trailer with respect to the horizontal axis. It hasbeen pointed out [14] that when the lead car of a single trailer system is travelingalong a circle of radius Rl, then the trailer is traveling along a circle of radius Rt withthe same center, where Rt < Rl, as shown in Figure 5.

Figure 4 The kingpin slidesalong the axle when thesemi-trailer turns.


Figure 5 Illustration ofoff-tracking phenomenon.

In order to compensate for this path deviation of the trailer, we suppose that thekingpin hitching point slides from the point Pi to the point Psi by a distance Si. Thefollowing assumptions are necessary for deriving the mathematical model:

a) All trailers have the same length L.b) Each trailer is modeled as having only one axle.c) Each trailer is assumed to be hooked to the midpoint of the rear axle of the

preceding trailer.d) By sliding the location of the kingpin, the weight of the trailer shifts toward an

outer direction, which doesn’t affect the kinematic behavior of the train. Thisassumption is more significant for heavy transportation multi trailer vehicles.

e) The unbalanced pulling point (when the kingpin sliding is nonzero) does notcause skidding of the whole axle.

f) The sliding of the kingpin can be performed with the trailer fully loaded via ahydraulic mechanism.

While the nonholonomic constraints of the new system remain the same (Equation(1)), the 2n holonomic equations introduced by the corresponding links take amodified form because of the sliding distance Si = Pi Psi (see Figure 2),

xi = xi−1 − ì · cos (θi) + Si−1 · sin (θi−1)

yi = yi−1 − ì · sin (θi) − Si−1 · cos (θi−1)i = 1, 2, ..., n (4)

The inputs’ equations for the new n-trailer system with a sliding kingpin mech-anism remain the same as without it (Equation (3)). After successive algebraic


manipulations of Equations (1), (3) and (4) we can derive the mathematical model ofthe n-trailer system with sliding kingpin,

x0 = U1 cos θ0

y0 = U1 sin θ0

ϕ = U2

θ0 =U1

Ltan ϕ

θ1 =U1

L2sin (θ0 − θ1)

[L + S0 tan ϕ

]θ2 =

U1

L3

[L + S0 tan ϕ

]sin (θ1 − θ2)

[L cos (θ0 − θ1) + S1 sin (θ0 − θ1)

].

...

θn =U1

Ln+1

[L + S0 tan ϕ

]sin (θn−1 − θn)

n−1∏i=1

[L cos (θi−1 − θi) + Si sin (θi−1 − θi)]

Details can be found in [10]. The (x0,y0) position of the first vehicle along with allthe orientation angles {θ0,θ1,...,θn} and steering angle ϕ define the entire state of thesystem, while (U1,U2) is the control input.

3. Flatness of the n-trailer System with Sliding Kingpin

The notion of differential flatness was first introduced in [5] for nonlinear finitedimensional systems. A system is said to be differentially flat if there exists a finite setof outputs y = {y1,..., ym} that are differentially independent, and satisfy the followingconditions:

• The outputs yi can be expressed as functions of the system variables and of afinite number of their derivatives.

• Any system variable can be expressed as a function of the outputs yi and of afinite number of their derivatives.

The set y = {y1,..., ym} is called the flat (or linearizing) output and its number ofcomponents equals the number of independent input channels. Given that there isno general method to compute the linearizing outputs when the system is flat, someoptions for the flat outputs were tried before deriving the following proposition.

PROPOSITION. The n-trailer system with a sliding kingpin mechanism is a differ-entially flat system with the Cartesian coordinates of the middle point of last trailer’srear axle y = [xn, yn]T as flat output.

Proof. From Equation (4) after some algebraic manipulations we have

[xn

yn

]=

x0 −

n∑i=1

ì · cos (θi) +

n−1∑i=0

Si · sin (θi)

y0 −

n∑i=1

ì · sin (θi) −

n−1∑i=0

Si · cos (θi)

(5)


The first condition for proving differential flatness property is satisfied by Equation(5). Also from Equation (4) after simple calculations we have

xi = xi+1 + ì+1 · cos (θi+1) − Si · sin (θi)

yi = yi+1 + ì+1 · sin (θi+1) + Si · cos (θi)i = 0, 1, ..., n − 1 (6)

From the nonholonomic constraints (1) we have for the n-trailer system with slidingkingpin,

θn = tan−1

(yn

xn

)(7)

and equivalently for the n − 1 trailer

tan (θn−1) =yn−1

xn−1

(6)⇒ tan (θn−1) =

(yn + `n · sin (θn) + Sn−1 · cos (θn−1))′

(xn + `n · cos (θn) − Sn−1 · sin (θn−1))′⇒

tan (θn−1) =yn + `n · cos (θn) · θn + Sn−1 · cos (θn−1) − Sn−1 · sin (θn−1) · θn−1

xn − `n · sin (θn) · θn − Sn−1 · sin (θn−1) − Sn−1 · cos (θn−1) · θn−1(8)

By replacing θn in Equation (8) from (7) we have

θn−1 = f(xn, yn, xn, yn, xn, yn, Sn−1, Sn−1

)(9)

Hence, the satisfaction of the second condition depends on the existence of the slidingdistance Si and its derivative in Equation (9). If the sliding distance in each trailer isdependent on the trailer’s orientation angle,

Si = Ki · θi (10)

then the orientation angle θn − 1 can be expressed as a differential function of (xn,yn) and their derivatives up to order two. Similarly, we can prove that each θi can beexpressed as a differential function of (xn, yn) and their derivatives up to order n + 1for θ0.

The linear controller of Equation (10) provides a good balance between simplicityand effectiveness. Trajectories for two multi-articulated vehicles with three and fivetrailers are shown in Figures 6 and 7, respectively.

From Equation (6) after successive substitutions we have

xn−1 = xn + `n · cos (θn) − Sn−1 sin (θn−1)

xn−2 =

xn−1︷︸︸︷(xn + `n · cos (θn) − Sn−1 sin (θn−1)) + `n−1 · cos (θn−1) − Sn−2 sin (θn−2)

...

xi = xn +

n∑k=i+1

(`k · cos (θk) − Sk−1 · sin (θk−1)) (11)

and similarly

yi = yn +

n∑k=i+1

(`k · sin (θk) + Sk−1 · cos (θk−1)) (12)

By replacing θi in Equations (11) and (12) we prove that xi,yi can be expressed asdifferential functions of (xn, yn) and their derivatives up to order n + 1 for x0,y0.


Figure 6 Trajectories for amulti-articulated vehicle withthree trailers and linearcontroller.

Figure 7 Trajectories for amulti-articulated vehicle withfive trailers and linearcontroller.


For system’s inputs and steering angle we will use Equation (3) from which we get

U1 =x0

cos (θ0)

and hence input U1 can be expressed as a differential function of (xn, yn) and theirderivatives up to order n + 2. For the steering angle we have

ϕ = tan−1

(`0 · θ0

U1

)thus steering angle ϕ can be expressed as a differential function of (xn,yn) and theirderivatives up to order n + 2. For the second input we have

U2 = ϕ

and consequently input U2 can be expressed also as a differential function of (xn, yn)and their derivatives up to order n + 3. Finally for the sliding distances Si, because wesupposed that Si = Ki · θi, the Si can be expressed as differential functions of (xn, yn)and their derivatives up to order n + 2. It must be noted that the n-trailer system withsliding kingpin has been proved to be a differentially flat system under the assumptionthat a linear controller was used for the sliding distance. However, as soon as othersliding controllers and especially nonlinear are introduced the system may be not flatanymore.

4. On Flatness and Sliding Kingpin Controller Selection

The sliding kingpin technique was proposed in order to eliminate the off-trackingphenomenon. The main issue is to design the controller with the best results. Variouscontrollers were tested through simulations for the system with three or five trailers.The nonlinear controller of Equation (13) gave us the best results. Figures 8 and9 show simulations results, for two multi-articulated vehicles with three and fivetrailers, where off-tracking is practically zero in all phases of motion. Details can befound in [11].

Si = ì1 − cos (θi − θi+1)

sin (θi − θi+1)(13)

If we use this type of controller we must again prove the satisfaction of the secondcondition of differential flatness property. By substituting Si from Equations (13) to(9) we have

θn−1 = f(xn, yn, xn, yn, xn, yn, θn, θn

)and by using (7) we can prove that the orientation angle θn − 1 can be expressed asa differential function of (xn, yn) and their derivatives up to order two. Similarly,we can prove that θi can be expressed as a differential function of (xn, yn) and theirderivatives up to order n + 1 for θ0. For the other system’s variables we have the sameresults as for the case of the linear controller Si = Ki · θi.


Figure 8 Trajectories for amulti-articulated vehicle withthree trailers and nonlinearcontroller.

Figure 9 Trajectories for amulti-articulated vehicle withfive trailers and nonlinearcontroller.


5. Conclusions

This area of work is part of an ongoing research effort in improved control ofmulti-articulated vehicles with application to problems in both mobile robotics andtransportation complex vehicles. In this paper we have shown that the n-trailersystem keeps its structural flatness property when it is equipped with a kingpin slidingmechanism, which moves dynamically the hitching point of each trailer. The goal is toexploit the advantages of flatness in order to generate effective control strategies ofthe mechanical system. In this case, problems related with the n-trailer system such asmotion planning, path following, steering etc. can be reduced to simple algebra andcomputationally efficient algorithms.

Acknowledgment This research work is partial supported by Karatheodori Program of the Re-search Commission of the University of Patras.

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JIRS Flatness Conservation in the N-trailer System Equipped With a Sliding Kingpin Mechanism