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566 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 3, MAY 2007

Predictive Active Steering Control for Autonomous

Vehicle Systems

Paolo Falcone, Francesco Borrelli, Jahan Asgari, Hongtei Eric Tseng, and Davor Hrovat, Fellow, IEEE

Abstract—In this paper, a model predictive control (MPC)

approach for controlling an active front steering system in an

autonomous vehicle is presented. At each time step, a trajectory is

assumed to be known over a ﬁnite horizon, and an MPC controller

computes the front steering angle in order to follow the trajectory

on slippery roads at the highest possible entry speed. We present

two approaches with different computational complexities. In

the ﬁrst approach, we formulate the MPC problem by using a

nonlinear vehicle model. The second approach is based on suc-

cessive online linearization of the vehicle model. Discussions on

computational complexity and performance of the two schemes

are presented. The effectiveness of the proposed MPC formulation

is demonstrated by simulation and experimental tests up to 21 m/s

on icy roads.

Index Terms—Active steering, autonomous vehicles, model pre-

dictive control, nonlinear optimization, vehicle dynamics control,

vehicle stability.

I. INTRODUCTION

R

ECENT trends in automotive industry point in the di-

rection of increased content of electronics, computers,

and controls with emphasis on the improved functionality

and overall system robustness. While this affects all of the

vehicle areas, there is a particular interest in active safety,

which effectively complements the passive safety counterpart.

Passive safety is primarily focused on the structural integrity

of vehicle. Active safety on the other hand is primarily used to

avoid accidents and at the same time facilitate better vehicle

controllability and stability especially in emergency situations,

such as what may occur when suddenly encountering slippery

parts of the road [10].

Early works on active safety systems date back to the 1980s

and were primarily focused on improving longitudinal dy-

namics part of motion, in particular, on more effective braking

(ABS) and traction control (TC) systems. ABS systems in-

crease the braking efﬁciency by avoiding the lock of the braking

wheels. TC systems prevent the wheel from slipping and at the

same time improves vehicle stability and steerability by maxi-

mizing the tractive and lateral forces between the vehicle’s tire

and the road. This was followed by work on different vehicle

Manuscript received November 10, 2006. Manuscript received in ﬁnal form

January 12, 2007. Recommended by Associate Editor K. Fishbach.

P. Falcone and F. Borrelli are with the Universitá del Sannio, Dipartimento di

Ingegneria, Università degli Studi del Sannio, 82100 Benevento, Italy (e-mail:

falcone@unisannio.it; francesco.borrelli@unisannio.it).

J. Asgari, H. E. Tseng, and D. Hrovat are with Research and Innovation

Center, Ford Research Laboratories, Dearborn, MI 48124 USA (e-mail: jas-

gari@ford.com; htseng@ford.com; dhrovat@ford.com).

Color versions of Figs. 1, 2, and 4–12 are available online at http://ieeexplore.

ieee.org.

Digital Object Identiﬁer 10.1109/TCST.2007.894653

stability control systems [34] (which are also known under

different acronyms such as electronic stability program (ESP),

vehicle stability control (VSC), interactive vehicle dynamics

(IVD), and dynamic stability control (DSC)). Essentially, these

systems use brakes on one side and engine torque to stabilize

the vehicle in extreme limit handling situations through con-

trolling the yaw motion.

In addition to braking and traction systems, active front

steering (AFS) systems make use of the front steering com-

mand in order to improve lateral vehicle stability [1], [2].

Moreover, the steering command can be used to reject ex-

ternal destabilizing forces arising from

-split, asymmetric

braking, or wind [21]. Four-wheel steer (4WS) systems follow

similar goals. For instance, in [3], Ackermann et al. present a

decoupling strategy between the path following and external

disturbances rejection in a four-wheel steering setup. The

automatic car steering is split into the path following and the

yaw stabilization tasks, the ﬁrst is achieved through the front

steering angle, the latter through the rear steering angle.

Research on the AFS systems has also been approached

from an autonomous vehicle perspective. In [16], an automatic

steering control for highway automation is presented, where

the vehicle is equipped with magnetic sensors placed on the

front and rear bumpers in order to detect a lane reference im-

plemented with electric wire [13] and magnetic markers [36].

A more recent example of AFS applications in autonomous

vehicles is the “Grand Challenge” race driving [5], [23], [30].

In this paper, it is anticipated that the future systems will be

able to increase the effectiveness of active safety interventions

beyond what is currently available. This will be facilitated not

only by additional actuator types such as 4WS, active steering,

active suspensions, or active differentials, but also by additional

sensor information, such as onboard cameras, as well as in-

frared and other sensor alternatives. All these will be further

complemented by global positioning system (GPS) information

including prestored mapping. In this context, it is possible to

imagine that future vehicles would be able to identify obstacles

on the road such as an animal, a rock, or fallen tree/branch, and

assist the driver by following the best possible path, in terms of

avoiding the obstacle and at the same time keeping the vehicle on

the road at a safe distance from incoming trafﬁc. An additional

source of information can also come from surrounding vehicles

and environments which may convey the information from the

vehicle ahead about road condition, which can give a signiﬁcant

amount of preview to the controller. This is particular is useful

if one travels on snow or ice covered surfaces. In this case, it is

very easy to reach the limit of vehicle handling capabilities.

Anticipating sensor and infrastructure trends toward in-

creased integration of information and control actuation agents,

1063-6536/$25.00 © 2007 IEEE

FALCONE et al.: PREDICTIVE ACTIVE STEERING CONTROL FOR AUTONOMOUS VEHICLE SYSTEMS 567

it is then appropriate to ask what is the optimum way in con-

trolling the vehicle maneuver for a given obstacle avoidance

situation.

We assume that a trajectory planning system is available and

we consider a double lane change scenario on a slippery road,

with a vehicle equipped with a fully autonomous guidance

system. In this paper, we focus on the control of the yaw and

lateral vehicle dynamics via active front steering. The control

input is the front steering angle and the goal is to follow the

desired trajectory or target as close as possible while fulﬁlling

various constraints reﬂecting vehicle physical limits and design

requirements. The future desired trajectory is known only over

ﬁnite horizon at each time step. This is done in the spirit of

model predictive control (MPC) [14], [26] along the lines of

our ongoing internal research efforts dating from early 2000

(see [7] and references therein).

In this paper, two different formulations of the AFS MPC

problem will be presented and compared. The ﬁrst one follows

the work presented in [7] and uses a

nonlinear vehicle model

to predict the future evolution of the system [26]. The resulting

MPC controller requires a nonlinear optimization problem to

be solved at each time step. We will show that the computa-

tional burden is currently an obstacle for experimental valida-

tion at high vehicle speed. The second formulation tries to over-

come this problem and presents a suboptimal MPC controller

based on successive online linearization of the nonlinear vehicle

model. This is linearized around the current operating point at

each time step and a linear MPC controller is designed for the re-

sulting linear time-varying (LTV) system. The idea of using time

varying models goes back to the early 1970s in the process con-

trol ﬁeld although it has been properly formalized only recently.

Studies on linear parameter varying (LPV) MPC schemes can be

found in [9], [18], [20], [22], and [35]. Among them, the work

in [18] and [20] is the closest to our approach and it presents

an MPC scheme for scheduled LTV models which has been

successfully validated on a Boeing aircraft. In general, the per-

formance of such a scheme is highly dependant on the nonlin-

earities of the model. In fact, as the state and input trajectories

deviate from the current operating point, the model mismatch

increases. This can generate large prediction errors with a con-

sequent instability of the closed-loop system. We will show that,

in our application, a state constraint can be introduced in order

to signiﬁcantly enhance the performance of the system. Exper-

imental results show that the vehicle can be stabilized up to

21 m/s on icy roads. Finally, an LTV MPC with a one-step con-

trol horizon is presented. This can be tuned in order to provide

acceptable performance and it does not require any complex op-

timization software.

We implemented the MPC controllers in real time on a pas-

senger car, and performed tests on snow covered and icy roads.

The last part of this paper describes the experimental setup and

presents the experimental and simulation results of the proposed

MPC controllers. It should be noted that our early work in [7]

focuses on the vehicle dynamical model and on simulation re-

sults of the nonlinear MPC scheme only.

This paper is structured as follows. Section II describes

the used vehicle dynamical model with a brief discussion on

tire models. Section III introduces a simpliﬁed hierarchical

Fig. 1. Simpliﬁed vehicle “bicycle model.”

framework for autonomous vehicle guidance. The contribution

and the research topic of this paper are described in details

and put in perspective with existing work and future research.

Section IV formulates the control problem when the nonlinear

and the linear prediction models are used. The double lane

change scenario is described in Section V, while in Section VI,

the experimental and simulation results are presented. This is

then followed by concluding remarks in Section VII which

highlight future research directions.

II. M

ODELING

This section describes the vehicle and tire model used for

simulations and control design. This section has been extracted

from [7] and it is included in this paper for the sake of complete-

ness and readability. We denote by

and the longitudinal (or

“tractive”) and lateral (or “cornering”) tire forces, respectively,

and are the forces in car body frame, is the normal tire

load,

is the car inertia, and are the absolute car position

inertial coordinates,

and are the car geometry (distance of

front and rear wheels from center of gravity),

is the gravita-

tional constant,

is the car mass, is the wheel radius, is the

slip ratio,

and are the longitudinal and lateral wheel veloc-

ities,

and are the local lateral and longitudinal coordinates

in car body frame,

is the vehicle speed, is the slip angle,

is the wheel steering angle, is the road friction coefﬁcient,

and

is the heading angle. The lower subscripts and

particularize a variable at the front wheel and the rear wheel,

respectively, e.g.,

is the front wheel longitudinal force.

A. Vehicle Model

A “bicycle model” [25] is used to model the dynamics of the

car under the assumption of a constant tire normal load, i.e.,

,

. Fig. 1 depicts a diagram of the vehicle model,

which has the following longitudinal, lateral, and turning or yaw

degrees of freedom:

(1a)

(1b)

(1c)

The vehicle’s equations of motion in an absolute inertial

frame are

(2)

568 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 3, MAY 2007

The wheel’s equations of motion describe the lateral (or cor-

nering) and longitudinal wheel velocities

(3a)

(3b)

(3c)

(3d)

where

and are front and rear wheel steering angle, respec-

tively, and

(4a)

(4b)

The following equations hold for rear and front axes by using

the corresponding subscript for all the variables. Longitudinal

and lateral tire forces lead to the following forces acting on the

center of gravity:

(5a)

Tire forces

and for each tire are given by

(6)

where

, , , and are deﬁned next. The tire slip angle

represents the angle between the wheel velocity vector and

the direction of the wheel itself, and can be compactly expressed

as

(7)

The slip ratio

is deﬁned as

if for braking

if for driving

(8)

where

and are the radius and the angular speed of the wheel,

respectively. The parameter

represents the road friction coef-

ﬁcient and is assumed equal for front and rear wheels.

is the

total vertical load of the vehicle and is distributed between the

front and rear wheels based on the geometry of the car model

(described by the parameters

and )

(9)

The nonlinear vehicle dynamics described in (1)–(9), can be

rewritten in the following compact from:

(10)

where the dependence on slip ratio

and friction coefﬁcient

value

at each time instant has been explicitly highlighted. The

state and input vectors are

and ,

respectively. In this paper,

is assumed to be zero at any time

instant.

Model (10) captures the most relevant nonlinearities associ-

ated to lateral stabilization of the vehicle. Section II-B brieﬂy

describes the models of tire forces

and .

Fig. 2. Longitudinal and lateral tire forces with different

coefﬁcient values.

B. Tire Model

With the exception of aerodynamic forces and gravity, all of

the forces which affect vehicle handling are produced by the

tires. Tire forces provide the primary external inﬂuence and, be-

cause of their highly nonlinear behavior, cause the largest vari-

ation in vehicle handling properties throughout the longitudinal

and lateral maneuvering range. Therefore, it is important to use

a realistic nonlinear tire model, especially when investigating

large control inputs that result in response near the limits of the

maneuvering capability of the vehicle. In such situations, the

lateral and longitudinal motions of the vehicle are strongly cou-

pled through the tire forces, and large values of slip ratio and

slip angle can occur simultaneously.

Most of the existing tire models are predominantly “semi-em-

pirical” in nature. That is, the tire model structure is determined

through analytical considerations, and key parameters depend

on tire data measurements. Those models range from extremely

simple (where lateral forces are computed as a function of slip

angle, based on one measured slope at

and one measured

value of the maximum lateral force) to relatively complex al-

gorithms, which use tire data measured at many different loads

and slip angles.

In this paper, we use a Pacejka tire model [4] to describe the

tire longitudinal and cornering forces in (6). This is a complex,

semi-empirical nonlinear model that takes into consideration

the interaction between the longitudinal force and the cornering

force in combined braking and steering. The longitudinal and

cornering forces are assumed to depend on the normal force, slip

angle, surface friction coefﬁcient, and longitudinal slip. Fig. 2

depicts longitudinal and lateral forces versus longitudinal slip

and slip angle, for ﬁxed values of the friction coefﬁcients. We

remark that the front tire of the “bicycle” model represents the

two front tires of the actual car.

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