Speed-Adaptive Model-Free Lateral Control for Automated Cars Marcos Moreno-GonzalezAntonio Artu nedo

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Speed-Adaptive Model-Free Lateral

Control for Automated Cars ?

Marcos Moreno-Gonzalez ∗Antonio Artu˜nedo ∗

Jorge Villagra ∗C´edric Join ∗∗,∗∗∗∗ Michel Fliess ∗∗∗,∗∗∗∗

∗Centre for Automation and Robotics (CSIC-UPM), ctra. de Campo

Real, km 0,200, 28500 Arganda del Rey, Spain, (e-mail: {marcos.moreno,

antonio.artunedo, jorge.villagra}@csic.es).

∗∗ CRAN (CNRS, UMR 7039), Universit´ee de Lorraine, BP 239,

54506 Vandoeuvre-l`es-Nancy, France

∗∗∗ LIX (CNRS, UMR 7161), ´

Ecole Polytechnique, 91128 Palaiseau,

France, (e-mail: Michel.Fliess@polytechnique.edu)

∗∗∗∗ A.L.I.E.N., 7 rue Maurice Barr`es, 54330 V´ezelise, France,

(e-mail: {cedric.join, michel.fliess}@alien-sas.com)

Abstract: In order to increase the number of situations in which an intelligent vehicle can

operate without human intervention, lateral control is required to accurately guide it in a

reference trajectory regardless of the shape of the road or the longitudinal speed. Some studies

address this problem by tuning a controller for low and high speeds and including an output

adaptation law. In this paper, a strategy framed in the Model-Free Control paradigm is presented

to laterally control the vehicle over a wide speed range. Tracking quality, system stability and

passenger comfort are thoroughly analyzed and compared to similar control structures. The

results obtained both in simulation and with a real vehicle show that the developed strategy

tracks a large number of trajectories with high degree of accuracy, safety and comfort.

Keywords: Autonomous vehicles, Model-free control, Adaptive control applications

1. INTRODUCTION

Over the last years, autonomous driving capabilities have

been developed, but in order to increase the situations in

which the vehicle operates without driver intervention, ac-

curately controlling the vehicle in any scenario is essential.

In a decoupled control architecture, lateral control keeps

the vehicle on the path without losing stability or impair-

ing passenger comfort, regardless of the road or speed. This

problem has been addressed with diﬀerent approaches, the

ﬁrst being to improve the available model of the vehicle in

order to better tune a model-based regulator. Another is

the design of controllers for diﬀerent longitudinal speeds,

and then provide a ﬁtting law between their diﬀerent

parameters or outputs. To avoid the problems induced by

a complex system identiﬁcation, some approaches opted to

rely on model-independent strategies.

In this paper, a new lateral control strategy for au-

tonomous vehicles is presented within the Model-Free

Control (MFC) paradigm (Fliess and Join, 2013). The

proposed control law is based on the adaptation of one

of the key parameters of MFC as a function of driving

speed, allowing thus the vehicle to cope with a variety

of situations without having to re-tune the controller. To

thoroughly evaluate the potential of this strategy, metrics

of tracking quality, stability of the feedback system and

?This work is partially supported by the joint CNRS (France)-CSIC

(Spain) PICS Program through the project UbiMFC (CoopIntEer

203 073)

passenger comfort are deﬁned. A multi-objective optimiza-

tion has been applied in simulation to determine which

control structure provides the best trade-oﬀ among the

three criteria in a wide spectrum of situations. The main

contribution of the work relies on the introduction of an

easy-to-implement variation of MFC, which has proven to

be very eﬀective for lateral control of automated vehicles.

The rest of the paper is structured as follows. Section II

presents a brief review of the lateral control strategies in

the literature. A theoretical introduction to Model-Free

Control is presented in Section III. Section IV motivates

the proposed Speed-Adaptive Model-Free Control strat-

egy. The results from simulation and real-world tests are

presented in Section V. Finally, the last section draws some

concluding remarks and the references.

2. STATE OF THE ART

Lateral control of autonomous vehicles is studied from

diﬀerent approaches, most of them based on a somewhat

realistic model of the vehicle, being the single-track model

the most popular (Ariﬁn et al., 2019), which is linear

and assumes a constant longitudinal velocity. Some real

applications show that this simpliﬁcation may be inappro-

priate to control the vehicle in any scenario. To overcome

these limitations, diﬀerent strategies have been proposed

in the last years. Zainal et al. (2017) applied the single-

track model to ﬁt two PID regulators, one for low and one

for high speeds; Liu et al. (2018) used it to synthesize a

robust LQR. Alternatively, Zanon et al. (2014) rely on

arXiv:2210.01414v1 [eess.SY] 4 Oct 2022

a more complex model to develop a Model Predictive

Control (MPC) strategy, and Laghmara et al. (2019) focus

on jointly solving the path planning and control problems;

but this strategies are mainly tested on speciﬁc situations.

Real vehicles have complex dynamics that vary with speed

and steering angle, with strong non-linearities, couplings

between lateral and longitudinal dynamics and variability

of parameters that are already diﬃcult to characterize;

consequently, it is extremely hard to ﬁnd a realistic model

for a large spectrum of driving situations. As a result, the

potential of control strategies that do not rely on a vehicle

dynamic model has catched attention.

Fuzzy control is a good example of these model-free

techniques, as it absorbs some of the variability of the

system parameters and its formulation is intuitive, but

diﬃcult to tune optimally over a wide working range. Two

fuzzy regulators were integrated and validated in traﬃc-

based driving environments in Godoy et al. (2015); other

works (Jin et al., 2017) conﬁrmed the capabilities of fuzzy

logic for lateral control. Another approach is pure pursuit

control (Park et al., 2014), which is based on a kinematic

model of the vehicle, but its performance degrades when

high velocities or accelerations are requested.

The MFC framework evoked in the introduction was suc-

cessfully applied in vehicle longitudinal control (Villagra

et al., 2009) or in lateral control for low-speed AGVs

(Villagra and Herrero-Perez, 2012). Alternatively, in Men-

hour et al. (2013) the ﬂatness theory (Fliess et al., 1995),

which allows ﬁnding diﬀerentially ﬂat outputs for non-

linear systems, is applied to implement the lateral control

of a vehicle together with a model-free feedback controller.

This approach exhibited very good performance in simula-

tion, but its deployment in real vehicles requires measure-

ments that cannot be obtained with commercial sensors.

Alternatively, (Wang et al., 2022) proposes an adaptation

mechanism for MFC and apply it on a scale car, but the

resulting adaptation dynamics is too slow for automated

vehicles driving on real roads.

3. MODEL-FREE CONTROL PRINCIPLES

Fliess and Join (2013) state that the system dynamics can

be approximated by an ultra-local model

y(n) =F+α·u(1)

in which the linear relationship between the input uand

the nth derivative of the output yis ﬁtted by a variable F

that absorbs model errors and system disturbances, and

where the ratio constant αis a design parameter.

The control loop is closed by an intelligent PID controller,

iPID controller (usually iP or iPD):

u=1

α·−F+y(n)

r+Kpe+KiZe+Kd˙e(2)

where uis the control action, suﬃx rmeans reference, e

is the tracking error and Kp,Kiand Kdare the control

parameters, emulating those of a PID controller. The term

Fmust be estimated in real time, for this purpose, it can

be assumed to be the same between consecutive instants

and can be estimated from (1) as follows:

F(tk) = ˆy(n)(tk)−α·u(tk−1) (3)

where ˆ

Fis the estimator of F,tkis the current instant

and ˆy(n)is the ﬁltered nth derivative of y.

Remark 1. Note that the error dynamics derived from (1)

and (2) can be expressed as f(e, Kp, Ki, Kd) = ˆ

F−F.

If the estimation of Fis good enough ( ˆ

F≈F), then

the system dynamics could be made asymptotically stable

through an appropriate choice of the control parameters.

4. SPEED-ADAPTIVE LATERAL CONTROL

The parameter αdeﬁnes in a certain way the aggressive-

ness of the iP(D) controller, since the higher is α, the

smaller the increase in the control action between sampling

instants. Therefore, varying αmight adapt the controller

aggressiveness to diﬀerent driving situations.

(1)

(2)

Start/end point

(a) Trajectory reference and tracking

Time (s)

Speed (km/h)

10 20 30 40

(1) (2)

Vehicle speed

Speed reference

(b) Reference and vehicle speed (c) Steering wheel angle in (2)

Fig. 1. Comparison between high and low alpha in low

speed curves and higher speed straight stretches

Fig. 1 shows the same MFC regulator with two diﬀerent

α: the one with a low value performs better at low speed

curves but becomes highly oscillating at a stretch where

higher speed is allowed; the conﬁguration with high αis

stable for the straight path (with little oscillation) but

does worse tracking at curves. This ﬁnding motivated the

introduction of a model-free controller whose αvaries as

a function of speed v. This controller has a base α0which

is kept up to a given speed v0, after which it is increased

proportionally to speed variation with a constant Kα:

α= max {α0, Kα·(v−v0) + α0}(4)

This αvariation law allows to obtain a (i) more aggressive

behaviour in urban environments and (ii) smoother actions

on the highway, where the oscillations can impair comfort

and lead to system instability due to the high speed.

5. RESULTS

In this section, the control parameter space is explored

in simulation (section 5.3) using a high-ﬁdelity vehicle

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Speed-AdaptiveModel-FreeLateralControlforAutomatedCars?MarcosMoreno-GonzalezAntonioArtu~nedoJorgeVillagraCedricJoin;MichelFliess;CentreforAutomationandRobotics(CSIC-UPM),ctra.deCampoReal,km0,200,28500ArgandadelRey,Spain,(e-mail:fmarcos.moreno,antonio.artunedo,jorge.villagrag@csic.e...

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Speed-Adaptive Model-Free Lateral Control for Automated Cars Marcos Moreno-GonzalezAntonio Artu nedo

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