design and comparison of robust nonlinear controllers for the lateral dynamics of vehicles.pdf

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IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS 1

Design and Comparison of Robust Nonlinear Controllers for the Lateral Dynamics of

Intelligent VehiclesGilles Tagne, Reine Talj, and Ali Charara

Abstract—This paper focuses on the latera l contr ol of intelligentvehicles; the aim is to minimize the lateral displacement of theautonomous vehicle with respect to a given reference trajectory.The control input is the steering angle, and the output is thelateral error displacement. We present here an analysis of com-monality of thr ee lateral nonlinear ad aptive contr ollers. The rstcontroller is a higher order sliding-mode controller (SMC). Thesecond contr oller is based on the immersion and invariance ( I & I )

pr inciple. The design of this contr oller led us to pr ove a very str ongstability cr iter ion of the closed-loop system for all controller gainschosen to be positive. Thereafter, some interesting characteristicsof passivity of the systems were pr oved following this development.Hence, the third controller is a passivity-based controller (PBC),an adaptive PI controller based on the feedback of a passiveoutpu t. To validate our contr ol laws, tests have been performed onSCANeR Studio, a dr iving simulation engine, according to severalreal driving scenarios. A comparison of these different controllersis made to highlight the advantages and dr awbacks of each contr olappr oach in latera l tr acking of a r eference trajectory.

Index Terms—Lateral control, sliding mode control (SMC),immersion and invar iance (I&I) contr ol, passivity-based con-trol (PBC), reference tracking, autonomous vehicles, intelligentvehicles.

I. I NT ROD UC TI ON

A. Motivations and Problem Statement

S EVERAL contests such as the DARPA (Defense AdvancedResearch Projects Agency) Challenges in the USA (2004,2005, and 2007); the Korean Autonomous Vehicle Competi-tions (AVC) in Korea (2010, 2012, and 2013) and many othershave been organized worldwide to favor the development of autonomous intelligent vehicles. The establishment of such ve-hicles would give rise to various advantages that will diminishroad accidents. The objective is to provide an autonomoussystem more reliable and faster to react than human drivers. It isimportant to notice that the driver’s mistakes contribute entirely

Manuscript received March 11, 2015; revised July 28, 2015; acceptedSeptember 24, 2015. This work was supported by the French Government andcarried out in the framework of Labex MS2T through the program Investmentsfor The Future managed by the National Agency for Research under GrantANR-11-IDEX-0004-02. The Associate Editor for this paper was J. E. Naranjo.

The authors are with Heudiasyc Laboratory, Sorbonne universit és , Universitéde technologie de Compiègne, CNRS, Heudiasyc UMR 7253, CS 60 319, 60203 Compiègne, France (e-mail: [email protected]).

Color versions of one or more of the gures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identi er 10.1109/TITS.2015.2486815

or partially to nearly 90% of road accidents. As result, variousresearch laboratories and rms are progressively stimulated bythe development of autonomous driving applications. Someexamples can be seen in [1]–[3]. This research eld is inexpansion and one of the current major challenges is to warranta high speed autonomous driving.

An autonomous navigation can be completed in three manda-

tory steps: the perception and localization, the path planningand the control. The vehicle control can be divided into twotasks: longitudinal control and lateral control. The objective of this paper is the lateral control of intelligent vehicles, which is avery active research eld that has been studied since the 1950s.

Lateral control consists on automatically maneuvering thevehicle using the steering wheel to track the reference tra-

jectory. Considering the high non-linearity of the vehicle dy-namics on one hand, and the uncertainties and disturbancesin automotive applications on the other hand, robustness can

be considered as a key issue in control design. The controller should be able to reject disturbances and handle parameter’suncertainties and variations.

Lately, signi cant research has been carried out to providelateral guidance of autonomous vehicles. In the literature, sev-eral control strategies have been developed. In [4] and [5], asimple PID controller has been proposed. We also nd a nestedcontroller in [6]. Furthermore, other classical approaches have

been used such as: state feedback [7]; Linear Quadratic (LQ)approach [8]; H ∞ control [9]; Lyapunov stability based control[10]; adaptive control [11]; fuzzy logic control [12], [13]; fuzzyTakagi-Sugeno LQ [14]; backstepping based approach [15] andmany others. Model Predictive Control (MPC) seems to be wellsuited to the trajectory tracking [16], [17]. Nonetheless, thecomputation time of nonlinear MPC is the main drawback of

this approach. In [18]–[21], Sliding Mode Control (SMC) has been applied. This control strategy is known for its robustnessagainst uncertainties and its capacity to reject disturbances.However, its main drawback is the chattering.

Some comparisons between existing controllers can be foundin the literature. In [22], a comparison is made between pro-

portional, adaptive, H ∞ and fuzzy controllers. Snider also presented and compared several path tracking methods whichwhere principally developed during the DARPA Challenges[23]. More recently, in [24], the authors have compared twoemergency trajectory tracking controllers. In [25], continuous-time and discrete-time switched H ∞ controllers are compared.As a conclusion, we can notice that it is dif cult to make

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2 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS

an objective classi cation, but it is clear that different results pointed out the adaptive controller’s class as a very promisingapproach for such uncertain and non-linear application.

Over the years, extensive control strategies based on SlidingMode Control (SMC) method have been proposed to design

better and more reliable controllers for the lateral dynamics

of intelligent vehicles. The rst-order SMC is used to controlthe vehicle with small displacement error by compensatingthe uncertainties and encountered disturbances [18]–[20]. Italso provides experimental results comparable to or better thanlinear auto-tuning controllers. This control law also possessesthe advantageof being simple, compared to other morecomplexapproaches of robust control. The major drawbacks of the SMCmethod are the following: 1) it needs a prior knowledge on themaximum bounds of the disturbances and uncertainties, 2) it isnot robust outside the sliding surface, and 3) the chattering. Thechattering phenomenon often leads to undesirable results, for example; high vibrations of moving mechanical parts and even

passengers discomfort [19]. For those reasons, it is necessaryto control and maintain the amplitude of oscillations at a lowlevel. Thus, [21] presented a higher-order SMC method toalleviate the chattering. The super-twisting algorithm is usedto minimize the lateral displacement with respect to a givenreference trajectory for an autonomous vehicle. In the present

paper, we will remind the main lines of the development of thiscontroller for a better comparison with the proposed controllers.

B. Contributions

Given the implicit resemblance between SMC and the Im-mersion and Invariance ( I &I ) principle for designing non-

linear and adaptive controllers, we develop a controller basedon the I &I approach in order to overcome the drawbacks of SMC. A preliminary work related to the I &I controller designwas presented in [26]. To enhance the performances of this I &Icontroller, in the present paper we use a different law of con-vergence of the—off-the-manifold variable— z to bring up anintegral term in the control law so as to improve the robustnesswith respect to parametric uncertainties and disturbances. Ananalysis of commonality in mathematical frameworks is made;

both controllers had similar structures. Additionally, the studyof the I &I controller shows a very strong stability criterionof the closed-loop system, for all controller gains chosen to

be positive. This strong result of stability led us to study theintrinsic properties of passivity of the vehicle lateral dynamics.

Passivity is a concept that expresses a very interesting stabil-ity property of some physical systems. Indeed, passive systemsare dynamical systems in which the energy exchanged is the

point of interest. Therefore, a passive system is not able tostore more energy than it has been supplied; this re ects astrong intrinsic stability criterion. Passivity has been studied byanalyzing the frequency behavior of several input-output maps,in order to determine the passive outputs. Then, passive outputscan be used to provide robust stability and good performances

by the development of Passivity-Based Controllers (PBC). Thisis particularly relevant in this application given the parametric

variations and uncertainties (speed, curvature, road frictioncoef cient, etc...). Very recently, we studied the passivity of the

Fig. 1. Bicycle model.

lateral dynamics in the case of the road coef cient of frictionequal to one ( μ = 1) [27]. In the present paper, we extend thisstudy by considering μ as a varying parameter.

The passivity of several input-output maps of the systemhave been proved. We have shown that the sliding surface s(SM C controller) that has been chosen equal to the—off-the-manifold variable— z ( I &I controller) is a passive output. Tomaintain the same structure with respect to the two previouslydeveloped ones, we propose as third controller, an adaptive PIcontroller with non-linear gains based on the feedback of this

passive output. Finally, a comparison of the three mentionedcontrollers is made to highlight the advantages and drawbacksof each approach in lateral tracking of a reference trajectory.

To design thecontrollers, we consider that the vehicle is ttedwith sensors and/or observers to measure sideslip angle, yawrate, lateral error and its derivative. It is well known that, tocontrol the trajectory of an autonomous vehicle tracking usingonly the lateral error is dif cult [28]. Therefore, the use of

other dynamic variables in our control laws such as, the yawrate, the sideslip angle or the equivalent term helps to quickly

bring the operating point to the desired equilibrium thus facil-itating the control. To validate the control strategies, we usedSCANeR Studio [29], a driving simulation engine.

C. Paper Structure

This paper is organized as follows. Section II presents thedynamic models of the vehicle and the control problem def-inition. Section III presents the SMC design; and the designof the I &I controller is presented in Section IV. Then, the

design of the PBC-PI controller is presented in Section V.Section VI presents results. We conclude in Section VII, withsome remarks and prospects.

II. V EHICLE D YNAMIC M ODELS AND C ONTROLPROBLEM D EFINITION

A. Vehicle Dynamic Models

To design the controllers, a simple and widely used dynamic bicycle model [7] is considered (see Fig. 1). This model is usedto represent the lateral vehicle behavior (lateral accelerationa y , yaw rate ψ̇ , sideslip angle β ) and assumes that the vehicle

is symmetrical, and tire sideslip angles on the same axle areequal. The roll and pitch dynamics are neglected and angles


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