Path Following Control of Rhombic Like Vehicles

Performance Assessment with Dynamic Vehicle Model

Nuno Silva

, Alberto Vale

and Luca Baglivo

Instituto de Plasmas e Fus˜ao Nuclear, Instituto Superior T´ecnico, Universidade T´ecnica de Lisboa,

Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Laboratory of Robotics and Systems in Engineering and Science, Instituto Superior T´ecnico,

Universidade T´ecnica de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Keywords:

Rhombic Like Vehicles, Path Following, Dynamic Vehicle Modelling.

Abstract:

This paper addresses the path following problem of a wheeled mobile robot with rhombic like kinematics

(drivable and steerable wheels both at front and rear) operating in cluttered environments. Four path following

controllers are developed to steer the kinematic model of a rhombic like vehicle (RLV) along a desired path:

three are based on feedback laws derived at a kinematic level with geometrical inspiration; the fourth is a

nonlinear controller built upon a kinematic model of a RLV using Lyapunov functions. All the developed

controllers are capable of performing under two situations: when both wheels follow the same path, or when

each wheel follows a different path. The dynamic modelling of a RLV is lastly introduced. Simulated results

present a comparative performance assessment of the controllers while dealing, or not, with vehicle dynamics.

The main conclusions of these controllers are summarized, leading to a possible application in the actual

vehicles that will operate in the remote handling missions of the International Thermonuclear Experimental

Reactor (ITER).

1 INTRODUCTION

Rhombic like vehicles (RLVs) as described in

(Ribeiro and Lima, 1997) and depicted in Fig. 1,

distinguish themselves from the typical unicycle and

car like vehicles, by allowing to control the linear

speed, v

, and orientation, θ

, of each wheel i ∈

{Rear,Front}. In fact, vehicles with rhombic like

kinematics are a set of nonholonomic vehicles char-

acterized for enabling a decoupling of the vehicle’s

orientation from the vehicle’s velocity vector, creat-

ing a controllable sideslip angle, β (Fig. 1). In a

RLV both wheels can follow the same path, referred

as Line Guidance (LG), or each wheel can follow a

different path, referred as Free Roaming (FR) (Vale

et al., 2012). A RLV is particularly suitable for op-

erations in cluttered scenarios where a high degree of

vehicle mobility is essential. This issue is detailed in

(Ribeiro and Lima, 1997) and strengthened in (Vale

et al., 2012), where the applicability of a RLV in the

cluttered environments of ITER (International Ther-

monuclear Experimental Reactor) was studied. Sev-

eral RLVs will operate in Tokamak Building (TB) and

Hot Cell Building (HCB) of ITER, where Fig. 2 il-

lustrates one of the levels of TB. Remote Handling





{I}





(

))



())









{M}

Figure 1: Rhombic like vehicle (RLV) kinematic (left) and

dynamic (right) variables and parameters.

(RH) systems will play an important role in ITER

and one of such systems is the Cask and Plug Re-

mote Handling System (CPRHS) depicted in Fig. 2:

a RLV responsible for RH operations of transporta-

tion of activated components and equipment inside

the TB. The CPRHS can reach dimensions of 8.50m

x 2.62m x 3.70m (length, width, height), a mass of

100T when fully loaded and has to operate within a

narrow safety margin of 10 to 30 cm. It has to be

noticed that the kinematic capabilities of RLVs can

further be extended to load transport in a wide set of

scenarios, such as storage warehouses, industrial sites

and cargo harbours. In (Vale et al., 2012), a motion

planning of the CPRHS in ITER is detailed. The tra-

Silva N., Vale A. and Baglivo L..

Path Following Control of Rhombic Like Vehicles - Performance Assessment with Dynamic Vehicle Model.

DOI: 10.5220/0004439500480057

In Proceedings of the 10th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2013), pages 48-57

ISBN: 978-989-8565-71-6

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

Cask

Pallet

CPRHS

CTS

Palle

CTS

Figure 2: The CRPHS composition (left) and its typical op-

erating scenario in a 3D and 2D view (middle and right,

respectively).

jectories (path and speed proﬁle) outputted from the

aforementioned work are considered to be the opti-

mal motion solution in the scenarios of ITER. It is the

purpose of the current paper to study different path

following solutions for a RLV with the characteris-

tics of the CPRHS, along previously optimized tra-

jectories. Moreover, this paper focuses on path fol-

lowing and not on trajectory tracking, since the ob-

jective is to steer a RLV along the optimized trajecto-

ries, while the vehicle’s speed tracks the desired speed

proﬁle embedded in the trajectory. In contrast, tra-

jectory tracking requires the vehicle to track a time-

parameterized reference (Luca and Oriolo, 1998).

To solve the path following problem for unicycles,

in (Micaelli and Samson, 1993; Lapierre et al., 2006)

are employed nonlinear control techniques, either in

the form of Lyapunovbased methods or throughfeed-

back linearization. Noticeable performance results

and proof of stability in relation to a feasible path are

detailed. Concerning car like vehicles, (Luca and Ori-

olo, 1998) details a path following controller based

on feedback linearization methods, which exploit the

chained form representation of the kinematic model

of a car like vehicle. Despite achieving good per-

formances, in order to guarantee stability, some con-

straints must be made on the initial pose (position and

orientation) of the vehicle and in the path itself. For

All-Wheel-Steering vehicles, class to which RLVs be-

long to, (Hiraoka et al., 2009) exploresthe application

of sliding mode controllersto the path following prob-

lem, improving robustness to uncertainties at a cost of

compromising steering smoothness.

The only references to path following for RLVs

are made in (Micaelli and Samson, 1993; Morin and

Samson, 2011) at a kinematic level only. In (Mi-

caelli and Samson, 1993) nonlinear control methods

are used to derive feedback control laws that enable

one wheel of the RLV to converge to a reference path,

whereas the other wheel determines the vehicle’s ori-

entation while following the path. In (Morin and

Samson, 2011) the application of transverse functions

to the control of RLVs is studied, placing a comple-

mentary constraint when the steering angles of both

wheels are equal to ±π/2, which is contoured using

practical stabilization (Morin and Samson, 2011) via

speciﬁc maneuvering embedded in the control design.

In this paper, and motivated by the abovemen-

tioned considerations, the authors propose four path

following controllers for RLVs, which can be divided

in two groups: three are categorized as geometrical

controllers and are based on feedback laws derived at

a kinematic level with geometrical inspiration; while

the fourth is a nonlinear controller, built upon a kine-

matic model of the vehicle using Lyapunovfunctions.

The geometrical controllers represent a good compro-

mise between complexity vs. performance and enable

a ﬁrst line of awareness to the control problem. In re-

lation to the nonlinear controller, two nonlinear feed-

back control laws are derived, acting independently

on the vehicle’s orientation and sideslip angle, and

thereby extracting the full capabilities of a RLV. In

striking contrast with the work done in (Micaelli and

Samson, 1993), the nonlinear controller proposed in

this paper allows for a RLV to follow a path formed

by poses in a simple manner. Moreover, in relation

to (Morin and Samson, 2011) the RLV kinematic con-

straint θ

,θ

6= ±π/2 is lifted by considering as con-

trol inputs v

,θ

and θ

, as opposed to (Morinand

Samson, 2011) where only

and the longitudi-

nal speed of the vehicle are used.

While operating in the scenarios of ITER, a RLV

like the CPRHS is estimated to have a considerable

speed for a vehicle with a mass up to 100T. Such

kinetic energy in a vehicle with the dimensions and

weight of the CPRHS, while operating under a 10 to

30 cm safety margin, represents a major challenge.

Hence, a dynamic model of an RLV is detailed to en-

able a more realistic CPRHS simulation. The impact

of simulated vehicle dynamics, mainly in the from of

wheel slippage, in the developed four path following

controllers, is compared against the strictly kinematic

situation, where no slippage occurs.

The paper is organized as follows: Section 2 de-

tails the kinematic model of a RLV and formulates the

path following control problem; Section 3 presents

the four path following controllers; Section 4 presents

the dynamic model of a RLV; Section 5 lays out the

results; and Section 6 outlines the conclusions and di-

rections for future work.

2 PROBLEM FORMULATION

This section comprises a mathematical formulation of

the kinematic model of a RLV, essential for the de-

PathFollowingControlofRhombicLikeVehicles-PerformanceAssessmentwithDynamicVehicleModel

sign of the path following controllers of Section 3,

followed by a precise problem statement.

2.1 Vehicle Kinematic Modelling

The formulation of a kinematic model for a RLV es-

tablishes the mathematical equations which relate the

temporal variations of the vehicle pose, with the linear

velocities on the wheels (i.e. steering plus the wheel

linear speed). It consists on a pure geometrical study

that is carried out without considering vehicle dy-

namic propertiessuch as mass, inertia, slip or friction.

On this line, and referring to Fig. 1, consider the

state vector q = [x

] as a representation of the

vehicle pose in the frame {I}, with (x

) the coor-

dinates of the center of the vehicle and θ

the orien-

tation of the vehicle. Also, consider v

as the vehi-

cle’s total speed and β the controllable sideslip angle,

both deﬁned in {I} (Fig. 1). A kinematic model for

a RLV in {I}, that allows the simulation of the vehi-

cle motion directly through the desired total speed v

of the vehicle, instead of imposing an individual lin-

ear speed for each wheel, was introduced in (Wang,

2001) as follows:







˙x

˙y













cos(θ

+ β)

sin(θ

+ β)

cosβ[tanθ

−tanθ

]







. (1)

This modelling entails that the wheels of the ve-

hicle roll without slipping, a constraint inherent to

the nonholonomy of RLVs, and also considers a rigid

body constraint, common to this type of vehicles, as

follows:

cosθ

= v

cosθ

. (2)

The possibility of directly control β, allows the ve-

hicle to move either with both wheels following the

same path (LG), or with each wheel following a dif-

ferent path (FR) (Vale et al., 2012).

2.2 Error Coordinates

A path following controller should reduce to zero i)

the distance from the vehicle to the path and ii) the

angle between the current orientation of the vehicle

and the desired orientation at the path. This motivates

the development of a kinematic model for a RLV in

a Serret-Frenet {F} frame that moves along the path,

as depicted in Fig. 3. This type of approach can be

found in (Micaelli and Samson, 1993) for wheeled

mobile robots, where {F} is attached to the path point

closest to the vehicle. This implies that the initial

position error of the vehicle in relation to the path,

would have to be smaller than the smallest radius of

Path described

by the center

of the vehicle





{I}

{F}

Desired

Pose

Figure 3: RLV variables and {F} frame deﬁnitions.

curvature present in the path. The current work fol-

lows the approach taken in (Lapierre et al., 2006)

for wheeled mobile robots and in (Lapierre and Jou-

vencel, 2008) for underwater vehicles, which starting

from the grounds established in (Micaelli and Sam-

son, 1993), lifts the initial condition constraint en-

tirely, by attaching {F} to a virtual moving target

along the path. This introduces an extra degree of

freedom at the controller design stage.

Consider Fig. 3, where P is an arbitrary path point

to be followed and C is the CoG of the vehicle. As-

sociated with P, consider the corresponding Serret-

Frenet frame{F}, with the signed curvilinear abscissa

of P along the path denoted as s. One can either ex-

press C as q = (X,Y,0) in a selected inertial reference

frame {I} or as (s

,0) in {F}. Stated equivalently,

C can be given in (X,Y) or (s

) coordinates. Fur-

thermore, deﬁne θ

as the angle of the tangent to the

path at P,

= c

(s)˙s with c

(s) the path curvature as

a function of s, and θ

as the desired orientation of the

vehicle at the path. Referring to the aforementioned

literature and using (1), one can obtain ( ˙s

, ˙y

) as











˙s

= − ˙s(1− c

(s)y

) + v

cosθ

˙y

= −c

(s)˙ss

+ v

sinθ

θ = ω

β− c

(s)˙s

, (3)

where θ = θ

+ β− θ

and ω

This kinematic model of a RLV in {F}, with s

not

necessarily equal to zero, plays an important role in

the development of the nonlinear controller presented

in Section 3.4.

2.3 Path Following

With the background given in the current section, and

with reference to Fig. 3, the path following problem

under study can be formulated as:

Consider the kinematic model for RLVs given by

(1) and (3). Given a LG or FR path to be fol-

lowed parametrized in terms of its length and a de-

sired speed proﬁle v

(t) > v

min

> 0 for the vehicle

ICINCO2013-10thInternationalConferenceonInformaticsinControl,AutomationandRobotics

total speed v

, derive the feedback control laws for

the linear speeds v

and orientations θ

,θ

of the

wheels, or alternatively β and ω

, so that s

, y

and

θ tend to zero.

In the current paper, the desired speed proﬁle

mentioned above is embedded in the trajectory pro-

vided by the motion planning described in (Vale et al.,

2012). Moreover, the formulation of the control laws

for β and ω

requires a transformation [ω

,β,v

] →

,θ

], detailed in Section 3, since, ulti-

mately, one must determine the linear speeds and ori-

entations of the wheels that drive the vehicle to the

path.

3 PATH FOLLOWING CONTROL

DESIGN

In what follows are unveiled the four path following

controllers developed for RLVs, which represents the

major novelty of this paper.

3.1 Alonzo Kelly Modiﬁed Controller

Two feedback control laws are here introduced by

the authors that solve the problem enunciated in Sec-

tion 2.3. The Alonzo Kelly Modiﬁed(AKM) path fol-

lowing controller draws its inspiration on the Inverse

Kinematic Model (IKM) for RLVs proposed by A.

Kelly in (Kelly, 2010). The formulation of the AKM

controller herein presented is capable of both LG and

FR, requiring only as input from the motion planning

stage the path poses to be followed.

Referring to Fig. 1, the IKM described in (Kelly,

2010) and presented here in (4), enables the trans-

formation [ω

,β,v

] → [v

,θ

]. To obtain

the values of v

,θ

, while respecting the

rigid body constraint (2) and, therefore, guarantee-

ing that the wheels roll without slipping, one can

use the trigonometric relations θ

= arctan





, for

Path described

by the center

of the vehicle

Desired Pose

on the path





{I}

Figure 4: Alonzo Kelly Modiﬁed controller representation

and variables deﬁnition.

the wheels’ orientation, and v

+ v

, for the

wheels’ speed, with i ∈ {R,F}.



















cosβ 0

sinβ M

cosβ 0

sinβ −M







(4)

Hence, this IKM enables the formulation of a path

following controller that only requires as input the de-

sired pose of the vehicle at the path, [P

,θ

], and

upon the control of β and ω

, outputs the values of

,θ

that enable the vehicle to converge and

follow the desired path (Fig. 4). Moreover, the kine-

matic singularity θ

,θ

6= ±π/2 present in (Morin

and Samson, 2011) is lifted entirely with the addi-

tional v

control inputs introduced here. As men-

tioned previously, v

is assumed to follow the desired

speed proﬁle embedded in the trajectory as described

in (Vale et al., 2012).

A control law for β is used to direct the vehicle’s

velocity towards the desired path point, as depicted in

Fig. 4. Referring to the IKM (4) and excluding, for

the moment, the effect of ω

in the velocity of each

wheel, the β control law orientates the wheels in a

manner that they point towards the desired path point.

Such an objective can be embodied in the geometric

law that follows:

β = arctan



−C



− θ

. (5)

where (P

) are the coordinates of the center of the

path pose being followed and (C

) the coordinates

of the center of the vehicle (Fig. 4). By replacing (5)

in (4), the wheels’ velocity and, therefore, the vehi-

cle’s velocity is, at all times, directed towards the de-

sired path pose.

A control law for ω

is responsible for orientating

the vehicle with the desired orientation at the path. A

function that embodies this objective, can be formu-

lated as follows:

(θ

− θ

, (6)

with t

the simulation step and k

a positive gain that

tunes the vehicle’s response rate to orientation varia-

tions.

Another degree of freedom inherent to the AKM

controller is the choice of the path point to follow. In

the formulation of (5) and (6) is chosen the closest

path point P, this being the orthogonal projection of

the C on the path (Fig. 4). Though, by introducing

an offset on the path point to be followed, referred in

the literature as lookahead distance, the performance

PathFollowingControlofRhombicLikeVehicles-PerformanceAssessmentwithDynamicVehicleModel

of the controller can be signiﬁcantly improved. In

fact, due to the impossibility of the vehicle’s wheels

to instantaneously change orientation or speed, the in-

troduction of a lookahead distance might guarantee

that when the vehicle actually reaches the path point

”looked at”, the pose becomes the desired one. More-

over, the lookahead distance can be ﬁxed or speed

dependant, depending on whether there is a varying

speed proﬁle, as it happens in the present case.

In brief, as tunable parameters the AKM con-

troller has the choice of the lookahead point P and

the gain k

3.2 Arc Path Following Controller

A geometric control method has been developed by

the authors, named Arc Path Following (APF) con-

troller, capable of both LG and FR. The key idea of

APF is to exploit the high degree of mobility of RLVs

by computing a simple but effective feedback con-

trol law based on geometric and kinematic reasoning.

Since the vehicle is free to deﬁne its Instantaneous

Center of Rotation (ICR) in R

, it is in any case possi-

ble (unless speciﬁc physical constraints exist) to com-

pute the generalized circular path that steers the vehi-

cle from the current pose to the desired one. Thereby

this strategy solves the problem enunciated in Sec-

tion 2.3. The concept is captured in Fig. 5. A sim-

ilar approach using circular path manifolds in a more

general framework has been found in (Kim and Mi-

nor, 2007), whereas the use of clothoids is proposed

in (De Cecco et al., 2007).

The general solution to the desired ICR (ICR

) ex-

pressed in the mobile reference frame {M}, arises as

(ICR

)









tan

(

)





tan

(

)









, (7)

where θ

= θ

−θ

and T = [T

] is the vector P−C

expressed in {M} (see Fig. 5).

The desired radius of curvature for the front (Rd

)

and rear (Rd

) wheels, expressed in {M}, are

(Rd

)





= (F)

− (ICR

)

, (8)

(Rd

)





= (R)

− (ICR

)

, (9)

with (F)

= [

] and (R)

= [

−

Thus the feedback control laws, one for each

wheel, can be written, using (7)-(9), as

= arctan





−

sign





, (10)

= arctan





−

sign





. (11)

Desired Instantaneous Center of Rotation (



)

{I}

Path described

by the center

of the vehicle

Desired Pose on

the Path

(

)

{M}

(

Figure 5: Arc Path Following controller representation and

variables deﬁnition.

Expanding (10) and (11), the feedback control

laws become:

= arctan



−T

tan(θ

/2) − T

(M − T

)tan(θ

/2) + T



sign



tan(θ

/2)



, (12)

= arctan



−T

tan(θ

/2) − T

−(M + T

)tan(θ

/2) + T



sign



tan(θ

/2)



. (13)

Equations (12) and (13) present singularities

which can be easily solved. They come from three

singular cases:

1. both numerator and denomitator in the arctan

functions are zero;

2. θ

= 0 in the sign function, that is when the cur-

rent vehicle attitude is equal to the desired one.

3. T

= T

= 0 in the sign function, that is when the

current reference point is equal to the desired one.

Case 1) occurs when either the current position of

the forward or the rear wheel are coincident with the

desired ones. To solve this case it is sufﬁcient to set,

respectively, ICR

= (F)

or ICR

= (R)

Case 2) is solved by substituting (12)-(13) by their

limit as θ

→ 0:

= θ

= arctan





. (14)

Case 3) occurs when the referencepoint ofthe cur-

rent and desired vehicle pose are coincident (C = P).

This is solved by setting ICR

= (C)

It has to be noticed that all the singularities de-

scribed above do not consist in discontinuities of the

ICINCO2013-10thInternationalConferenceonInformaticsinControl,AutomationandRobotics

control laws. They are degeneracy cases of the solu-

tions expressed by (7) and (12)-(13) due to special ge-

ometric conﬁgurations in the problem deﬁnition. By

the above considerations, the control law is smooth

for every possible vehicle pose.

Similarly to the AKM controller, the APF con-

troller formulation enables the introduction of a

lookahead distance, i.e. it is not mandatory that P is

the orthogonal projection of C on the path (Fig. 5).

This represents an extra degree of freedom in the con-

troller that can be exploited to achieve smoother and

better path following performances.

In brief, the APF controller has the choice of the

lookahead point P as its only tunable parameter.

3.3 Stanley Modiﬁed Controller

The Stanley Method Modiﬁed (SMM) path following

controller, is an extension to RLVs of the path fol-

lowing controller used by Stanford University’s au-

tonomous car entry in the DARPA Grand Challenge,

named Stanley (Thrun et al., 2006). This controller

is capable of both LG and FR, though, it requires

as inputs the path to be followed by both wheels, in

contrast with the other controllers here presented that

only require the path poses.

Considering the path following controller de-

scribed in (Thrun et al., 2006), the SMM employs two

nonlinear feedback functions of the cross-track errors

and e

, respectively, which are a measure of the

distance from the center of each wheel to the nearest

point on the respective wheel path, as shown in Fig. 6.

The SMM approach results in two control laws, one

for each wheel, responsible for converging e

and e

to zero and for which exponential convergence can be

shown (Thrun et al., 2006).

The feedback control laws for θ

and θ

can be

written, with reference to Fig. 6, as

= θ

+ arctan





, (15)

= θ

+ arctan





, (16)

where: θ

,θ

are the required outputs; v

are the

desired front and rear wheel speeds, respectively, em-

bedded in the trajectory as described in (Vale et al.,

2012); k is a gain parameter, chosen equally for both

wheels to maintain vehicle symmetry; e

repre-

sent the cross-track errorsfor the frontand rear wheel,

respectively; and θ

,θ

describe the orientation of

the nearest path segment, measured relative to the ve-

hicle’s own orientation (see Fig. 6). The ﬁrst term

of (15) and (16) simply keeps the wheel parallel to the

respective wheel path. When the cross-track error is

Path described by the

center of the vehicle

{I}













Path described by

the rear wheel

Path described by

the front wheel

Figure 6: Stanley Method Modiﬁed controller representa-

tion and variables deﬁnition.

non-zero, the second term adjust the steering of each

wheel in (nonlinear) proportion to the cross-track er-

ror: the larger this error, the stronger the steering re-

sponse towards the respective wheel path.

Against to what was mentioned for the AKM con-

troller, the formulation above for the SMM controller

precludes the introduction of a lookahead distance.

The referencepath point to be followed by each wheel

must inevitably be the orthogonal projections of the

contact points of the wheels in the respective wheel

path.

In brief, the SMM controller has the gain k as its

only tunable parameter.

3.4 Nonlinear Based Controller

Two nonlinear feedbackcontrol laws have beendevel-

oped by the authors to steer the kinematic model (3),

along a desired path and at constant speed (˙v

= 0).

The Nonlinear Control Based (NCB) path follow-

ing controller, has its grounds on the work done in

(Micaelli and Samson, 1993; Lapierre et al., 2006)

on path following control for kinematic models of

wheeled mobile robots and in (Lapierre and Jouven-

cel, 2008) on the same topic but for autonomous un-

derwater vehicles. In contrast to the work described in

(Micaelli and Samson, 1993), the usage of the virtual

target principle enables the formulation of the path

following problem in a non-singular manner, thus

guaranteeing global convergence to the path. More-

over, the NCB controller sets apart from the kine-

matic controller developed in (Lapierre and Jouven-

cel, 2008) by deliberately controlling the rate of evo-

lution of the vehicle’s sideslip angle,

β, separately

from the vehicle’s angular speed, ω

. This enables

the vehicle to convergeand follow a desired LG or FR

path, exploiting the full maneuverability of a RLV.

Referring to Fig 3, the objective of deriving a con-

trol law for

β can be embodied in the Lyapunov func-

tion candidate (see (Lapierre et al., 2006))

PathFollowingControlofRhombicLikeVehicles-PerformanceAssessmentwithDynamicVehicleModel



+ y



2γ



θ− δ

)



, (17)

under the assumptions (A.1) y

sinδ

) ≤

0,∀y

∀v

, (A.2) δ

(0,v

) = 0 and (A.3) lim

t→∞

(t) 6=

0. Also, let the desired approach angle function δ

deﬁned by

= −θ

tanh





, (18)

where 0 < θ

< π/2 and k

is an arbitrary positive

gain. Notice how equation (18) satisﬁes the ﬁrst and

second assumptions.

In (17) the ﬁrst term captures the distance between

the vehicle and the path, which must be reduced to 0.

In turn, this term also precludes the introduction of a

lookahead distance in the NCB controller. The sec-

ond term shapes the approach angle of β to the path

as a function of the ’lateral’ distance y

and speed

, by forcing it to follow a desired orientation pro-

ﬁle embedded in (18). The parameter γ accounts only

for normalization purposes. The ﬁrst assumption pro-

vides an adequate reference sign deﬁnition to drive

the vehicle to the path, i.e. turn the vehicle left when

on the right side of the path, and turn right in the other

situation. A.2 imposes the condition that β must be

such that the vehicle’s velocity is tangent to the path

when y

= 0, regardless of the vehicle orientation. Fi-

nally, the last assumption states that the vehicle does

not tend to a state of rest, case when the controllability

cannot be guaranteed.

Consider the following kinematic control laws for

s, the virtual moving target, and β as:

˙s = v

cosθ+ k

, (19)

β =

+ c

(s)˙s−

− γy

sinθ− sinδ

θ− δ

− k

(θ− δ

(20)

with k

and k

positive gains. This yields

= −k

+ y

sinδ

− k



θ− δ



≤ 0, (21)

where the presence of the term y

sinδ

in the pre-

vious equation justiﬁes A.1. Moreover, (21) depicts

the importance of the virtual target control law (19)

on guaranteeing global convergence to the path.

Therefore, the kinematic control laws (19)-(20)

drive s

and θ asymptotically to zero, i.e. the ve-

hicle converges asymptotically to the path. The proof

builds similarly to the ones found in (Micaelli and

Samson, 1993; Lapierre et al., 2006) and (Lapierre

and Jouvencel, 2008).

The second objective of deriving a control law for

can be embodied in the Lyapunov function candi-

date

2γ

(θ

− θ

− δ

))

, (22)

under the assumption (A.4) δ

(0,v

) = 0 and with

deﬁned in a similar manner as in (18).

The Lyapunov function (22) captures the orienta-

tion error between the current orientation of the vehi-

cle and the desired orientation at the path, which must

be reduced to zero. A.4 guarantees that the vehicle’s

orientation must be equal to the desired orientation at

the path when y

= 0.

By makingthe kinematiccontrol law forω

required for Lyapunov stability, as

− k

(θ

− θ

− δ

), (23)

with k

a positive gain,

becomes

= −k

(θ

− θ

− δ

)

≤ 0. (24)

Hence, control law (23) drives θ

− θ

asymptot-

ically to zero, i.e. the vehicle’s orientation converges

asymptotically to the desired orientation at the path.

Similarly, the proof is alike to the ones found in (Mi-

caelli and Samson, 1993; Lapierre et al., 2006) and

(Lapierre and Jouvencel, 2008).

To determine the values of v

,θ

and θ

, the

IKM (4) is used. The required value of ω

is obtained

directly from the control law (23), while β is obtained

by integrating the output of the control law (20). Sim-

ilarly to the AKM controller, v

is provided by the tra-

jectory.

In brief, the NCB controller has the following tun-

able parameters: the normalization parameter γ, the

gains k

, k

and k

, and the approach an-

gles θ

and θ

4 DYNAMIC MODELLING

The RLV kinematic models (1) and (3), are able

to capture the essentials of the motion of RLVs.

However, the non-slipping assumptions in which the

derivation of the kinematic model was based, will

hardly be satisﬁed in reality if the vehicle’s kinetic

energy is signiﬁcant. Hence, the dynamic formulation

that follows aims to provide a more accurate simula-

tion scenario for the test of the controllers developed

in Section 3. In the present case, the RLV is regarded

as a rigid vehicle with a strictly planar motion. Hence,

it has three degrees of freedom: x and y translation,

and a rotation about the z-axis. The Newton-Euler

equations of motion for a rigid vehicle in the vehicle

ICINCO2013-10thInternationalConferenceonInformaticsinControl,AutomationandRobotics

frame {M}, attached to the vehicle at its mass center

C, are (Jazar, 2008):











= m˙u− mω

= m˙v+ mω

. (25)

The symbols m and I

represent, respectively, the

mass and moment of inertia, around the z axis, of the

vehicle. In the frame {M}, u and v denote the longi-

tudinal and transverse speeds of the vehicle, respec-

tively.

To determine the forces F

, F

and moment M

must be noticed that the vehicle motion is a conse-

quence of forces acting at the vehicle’s wheels. In

fact, the forces acting on the vehicle can be cate-

gorized as: transverse forces, F

, and longitudinal

forces, F

, acting on the wheels, with i ∈ {R, F} . This

formulation entails, with reference to Fig. 1:

m˙u− mω

v =F

cosθ

− F

sinθ

+ F

cosθ

− F

sinθ

, (26)

m˙v+ mω

u =F

sinθ

+ F

cosθ

+ F

sinθ

+ F

cosθ

, (27)

sinθ

+ F

cosθ

− F

sinθ

− F

cosθ

. (28)

To calculate F

and F

, is followed the approach

taken in (Fonte, 2011), where a simpliﬁed linear tire

model is used, as follows:

= C

, α



− arctan



v±



, (29)

= C

, σ





− v

ef f

cosα

max





ef f

cosα









. (30)

The following deﬁnitions apply (Fonte, 2011): C

f(F

) andC

= f(F

) are the longitudinal and corner-

ing stiffness, respectively, which depend on the nor-

mal force applied at the wheel, F

, and considered

equal for both wheels (F

= F

); α

is the wheel

slip angle; σ

is the wheel longitudinal slip ratio;

ef f

is the effective rolling speed of wheel i given by

ef f



v±



; v

is the inputted wheel

speed (at the actuator level); and ± is positive for

i = F and negative for i = R.

The combined slip effects, which occur when both

6= 0, is taken into account by clipping (29) and

(30) as stated in (Jazar, 2008):

+ F

= (µF

)

, (31)

where µ is the friction coefﬁcient between the tire and

the contact surface.

Furthermore,a discrete ﬁrst ordermodel in the lin-

ear speed of the wheels and a discrete second order

model in the orientation of the wheels, is used to sim-

ulate the typical behavior of wheel actuators.

5 RESULTS

This section illustrates the simulated performance of

the four path following controllers described in Sec-

tion 3. The simulationswere performedusing the Tra-

jectory Evaluator and Simulator (TES) software de-

veloped under the scope of previous grants of Remote

Handling with the domestic agency Fusion for Energy

(F4E), as described in (Vale et al., 2012). The vehi-

cle simulated was a RLV with the characteristics of

the CPRHS, while following a trajectory to port 14 in

level B1 of TB of ITER (Fig. 7). The trajectory used

in the simulations is a discrete FR path, at constant

speed, formed by a set of consecutive poses, as de-

picted in Fig. 7. This trajectory was calculated using

the motion planning realized in (Vale et al., 2012).

The performance of the four controllers is as-

sessed by measuring the cross-track error of the cen-

ter of the vehicle, e

, along the path, as depicted in

Figs. 8 and 9. To determine the inﬂuence of vehicle

dynamics in the performance of the controllers, two

series of tests were realized: on the ﬁrst, the vehicle

total speed, v

, is set constant to 0.15 m/s and the mass

is 50T (Fig. 8); on the second test, v

is set to 0.4 m/s

and the mass is 100T (Fig. 9). On both series of tests,

each of the controllers is tested individuallyunder two

situations, these being using the kinematic model (1),

Complete Path

Lift Exit (Zoom)

Port 14 Entrance (Zoom)

Figure 7: Path used in the simulation: FR path from lift to

port 14 in level B1 of TB of ITER.

PathFollowingControlofRhombicLikeVehicles-PerformanceAssessmentwithDynamicVehicleModel

0.02

0.04

0.06

[m]

AKM (Kinematic Simulation)

AKM (Dynamics Simulation)

0.02

0.04

0.06

APF (Kinematic Simulation)

APF (Dynamics Simulation)

0.02

0.04

0.06

SMM (Kinematic Simulation)

SMM (Dynamics Simulation)

10 20 30 40 50 60 70 80 90

0.02

0.04

0.06

Distance along the path [m]

NCB (Kinematic Simulation)

NCB (Dynamics Simulation)

Figure 8: Cross-track error, e

, comparison for each of the

four controllers, while simulating, or not, vehicle dynamics,

in a trajectory to port 14 in level B1 of TB of ITER. The

vehicle speed is 0.15 m/s and the mass is 50T.

or using the dynamic model (25), for simulating the

motion of a RLV. Though, it is important to mention

that actuators dynamics is always present across the

tests and the controllers gains are kept the same.

As it can be seen in Figs. 8 and 9, the controllers

performancedeterioratesas v

increases and while ne-

gotiating turns (zoomed areas in Fig. 7). More im-

portantly, Fig. 8 demonstrates how wheel slippage at

0.15 m/s for a 50T RLV is negligible, since the perfor-

mance of the controllers is the same regardless of the

utilization of a kinematic or dynamic model for ve-

hicle simulation. In contrast, Fig. 9 depicts how the

performance of the controllers worsens at 0.4 m/s for

a 100T RLV if a dynamic vehicle model, instead of a

kinematic one, is used for simulation. This result en-

hances the importance of including vehicle dynamic

considerations in the design of the controllers.

Overall, the AKM and APF controllers attain sim-

ilar performances and surpass the NCB and SMM

controllers. This demonstrates how the inclusion of a

lookahead distance contours the non consideration of

vehicle and actuators dynamics in the path following

controllers formulation. To achieve the results shown

for the AKM and APF controllers, a 0.09 m looka-

head distance was used. This allowed a superior per-

formance while using a kinematic model for vehicle

simulation. Though, it is also the reason of the strong

oscillations while using a dynamic model for simula-

tion (Fig. 9). In fact, a high lookahead distance (e.g.

1 m) increases e

, but enables the controller to better

stabilize the vehicle under high wheel slippage. Table

1 summarizes the controllers performance at port 14

entrance (Fig. 7), this being the most critical part of

0.02

0.04

0.06

[m]

AKM (Kinematic Simulation)

AKM (Dynamics Simulation)

0.02

0.04

0.06

APF (Kinematic Simulation)

APF (Dynamics Simulation)

0.02

0.04

0.06

SMM (Kinematic Simulation)

SMM (Dynamics Simulation)

10 20 30 40 50 60 70 80 90

0.02

0.04

0.06

Distance along the path [m]

NCB (Kinematic Simulation)

NCB (Dynamics Simulation)

Figure 9: Cross-track error, e

, comparison for each of the

four controllers, while simulating, or not, vehicle dynamics,

in a trajectory to port 14 in level B1 of TB of ITER. The

vehicle speed is 0.4 m/s and the mass is 100T.

Table 1: RMSE of e

for each of the controllers at port 14

entrance, while using a kinematic, or a dynamic, model for

vehicle simulation. Units are in [mm].

Controllers

Kinematic Model Dynamic Model

0.15 m/s 0.40 m/s

50T 100T

AKM 1.8 4.3 1.7 25.2

APF 2.1 3.6 1.9 22.5

SMM 12.1 23.2 12.2 36.1

NCB 25.7 16.1 25.8 26.4

the path.

In relation to the NCB controller, a comparison

between Figs. 8 and 9 reveals that, while using a kine-

matic model for simulation, the NCB performance is

better at 0.4 m/s than at 0.15 m/s. Although it seems

contradictory, this is justiﬁed by (18), which depicts

how, for the same gains, a stronger path deviation cor-

rection is achieved if the vehicle total speed, v

, is

higher. This issue can be mitigated at a kinematic

level by increasing the NCB gains. Though, while

dealing with vehicle dynamics, high gains would in-

crease vehicle instability and decrease path following

performance.

It should be noticed that all the four path following

controllers perform well within the 30 cm safety mar-

gin established for the scenario, even while dealing

with the dynamics of a 100T vehicle at 0.4 m/s. The

maximum e

registered was 11.95 cm for the SMM

controller at 0.4 m/s with dynamics.

ICINCO2013-10thInternationalConferenceonInformaticsinControl,AutomationandRobotics

6 CONCLUSIONS

Four path following controllers for RLVs were de-

veloped and implemented. Alongside, the dynamic

modelling of a RLV was realized. The tests aimed to

assess the controllers performance while using either

the kinematic or the dynamic vehicle model for sim-

ulation. All the controllers perform within the strict

safety margin constraints established for the cluttered

scenarios of ITER and attain good results, even in

the presence of some wheel slippage. The study of

four distinct controllers enables the conclusion that

both the AKM and APF controllers slightly stand out

from the other two, due to the inclusion of a looka-

head distance which contours the vehicle and actua-

tors dynamicsissue. The SMM controller still enables

a good performance, while being very simple, and is a

good choice if the path of the wheels is available. The

NCB controller also achieves a good performance and

possesses the biggest room for improvement, by en-

abling the inclusion of vehicle and actuators dynamics

through backstepping techniques.

The results were obtained only in simulation and

a future route would pass by performing experimen-

tal tests at a small scale, to begin with, and, ultimately,

at a larger scale. The results show that when the ve-

hicle’s mass and speed is signiﬁcant, the controllers

performance deteriorates. Hence, an improved ver-

sion of the controllers is being developed, that takes

into account dynamic considerationsat the controllers

design stage.

ACKNOWLEDGEMENTS

The work was supported by FCT in the frame of the

Contract of Associate Laboratories of Instituto de Plas-

mas e Fus˜ao Nuclear – Laborat´orio Associado/IST (Pest-

OE/SADG/LA0010/2011).

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