CALIFORNIA PATH PROGRAM INSTITUTE OF TRANSPORTATION STUDIES UNIVERSITY OF CALIFORNIA, BERKELEY

On Fundamental Issues of Vehicle Steering Control for Highway Automation Jürgen Guldner Han-Shue Tan Satyajit Patwardhan

California PATH Working Paper UCB-ITS-PWP-97-11

This work was performed as part of the California PATH Program of the University of California, in cooperation with the State of California Business, Transportation, and Housing Agency, Department of Trans- portation; and the United States Department Transportation, Federal Highway Administration.

The contents of this report reflect the views of the authors who are responsible for the facts and the accuracy of the data presented herein. The contents do not necessarily reflect the official views or policies of the State of California. This report does not constitute a standard, specification, or regulation.

March 1997

ISSN 1055-1417

On Fundamental Issues of Vehicle Steering

Control for Highway Automation



Jurgen Guldner, Han-Shue Tan and Satyajit Patwardhan

California PATH

University of California at Berkeley

Institute of Transportation Studies

1357 S. 46th Street, Richmond, CA 94804-4698

Abstract

Automatic steering control for passenger cars, a vital control subsys-

tem of Automated Highway systems AHS, has b een studied for several

decades. Di erent reference systems have b een examined for detecting

the lateral vehicle displacement from the lane center, employing a vari-

ety of mo dern control metho ds to design automatic steering controllers.

Implementations of the AHS relevant sub class of `lo ok-down' reference

systems, however, have encountered practical constraints and limitations,

esp ecically for highway sp eed driving. By analyzing the lateral vehicle

dynamics in the light of these practical constraints and limitations, we

establish a general framework for automatic steering control in an AHS

environment and contrast various directions for control design.



List of authors was chosenrandomly

Keywords: Automated highway systems, automatic steering control, lane keeping control,

lateral vehicle control 1

1 Intro duction

Automatic steering control is a vital comp onent of highway automation, currently inves-

tigated worldwide in several programs, e.g. ITS in the US see e.g. [1] and ASV, SSVS

and ARTS under ITS Japan [2, 3]. Highway automation is a main sub ject of research and

development of Advanced Vehicle Control Systems AVCS. AVCS refers to the sub class

of Intelligent Transp ortation Systems ITS aimed at increasing safety and throughput of

road trac while decreasing environmental impacts. Overviews of highway automation were

given, for example, by Bender [4], a review of AVCS was presented by Shladover [5], and a

p ossible AHS scenario was outlined by Varaiya [6].

For individual vehicles within an Automated Highway System AHS, two control tasks

arise. The rst task, longitudinal control, involves controlling the vehicle sp eed to maintain a

prop er spacing b etween vehicles. This pap er concentrates on the second task, lateral control,

which is concerned with automatic steering of vehicles for lane keeping to follow a reference

along the lane center.

Automatic steering control approaches can b e group ed into look-ahead and look-down sys-

tems, according to the p oint of measurement of the vehicle lateral displacement from the

lane reference. Lo ok-ahead systems replicate human driving b ehavior by measuring the lat-

eral displacement ahead of the vehicle. The lo ok-ahead distance usually is increased with

increasing velo city, similar to human b ehavior. A number of research groups have sucess-

fully conducted highway sp eed exp eriments with lo ok-ahead systems like machine vision.

Examples are VaMoRs-P [7], VITA-I and I I [8, 9], and related pro jects e.g. [10] within the

Europ ean Prometheus Program, Carnegie Mellon University's PANS [11], and California

Path's stereo-vision based system [12]. In an e ort to remedy the susceptibility of machine

vision to variation of light and weather conditions, radar re ective strip es with lo ok-ahead

capability have b een develop ed and tested at The Ohio State University OSU [13]. Al-

ternatively, lo ok-ahead laser radar systems [14] and \energy emitting or re ecting" road

reference systems [15] have b een studied.

Lo ok-down reference systems, on the other hand, measure the lateral displacement at a

lo cation within or in the close vicinity of the vehicle b oundaries, typically straight down

from the front bump er. Examples are electric wire guidelines rst tested at OSU [16] in

the US for passenger cars and later by Daimler-Benz and MAN in Germany for city buses

[17], radar re ecting guard rails studied at OSU [18], and magnetic markers used in the

California Path pro ject [19, 20] and, most recently, in Japan [2]. Comprehensive overviews

were given e.g. by Shladover [5] and Fenton [21]. However, despite an impressive amount

of literature on theoretical control designs not listed here, no highway sp eed exp eriments

have b een rep orted to the b est of the author's knowledge. Furthermore, most of the practical

requirements like accuracy, comfort, and robustness could not b e met simultanously in most

exp erimentally veri ed approaches. Notable exceptions are the exp eriments p erformed at

OSU in the 70's [16], achieving up to 35 m/s 126 km/h, approx. 80 mph under idealized 2

conditions and with a lo ok-down sensor lo cated at more than 1 m in front of an already very

long vehicle, e ectively resulting in a lo ok-ahead system.

Despite the low sp eed limitation in exp eriments with true lo ok-down reference systems to

low sp eed, it was generally assumed p ossible to transfer the low sp eed results to highway

sp eeds by appropriate tuning of the controllers. Unfortunately, this may not b e an easy task

due to signi cantchanges in the vehicle dynamics for increasing sp eed. Wehave shown in [22]

that the extension of lo ok-down systems to practical conditions of an AHS environment with

sp eeds ab ove 30 m/s 108 km/h, approx. 67:5 mph is nontrivial and requires complete re-

thinking of the approach. The goal of this pap er is to study the fundamentals of automatic

steering control for highway driving in a general framework based on physical insight. Rather

than directly p ersuing a numeric control design, an in-depth analysis of the system and of

the design requirements is rst presented. This analysis is shown to lead to a distinct basis

for control design. Various design directions are examined, stressing their interrelations and

discussing resp ective practical implementation issues.

Section 2 describ es the system mo del and control design requirements. The plant and the

controller to be designed are analyzed in Section 3 using various to ols of dynamic system

analysis. Section 4 is devoted to a study of main design directions based on the insight

gained from the system analysis.

2 Problem Description

In this section, we present a description of the automatic steering control problem. Emphazis-

ing on physical interpretation rather than on complexity and accuracy of details, linearized

mo dels are used throughout this study. The design requirements concentrate on practical

and implementational issues based on the AHS exp erience gained at California Path for

almost a decade [20, 23].

2.1 Mo del for automatic steering control

A linearized mo del has proved sucient for studying car steering under `normal' conditions,

i.e. non-emergency situations [24]. Assuming small angles, this allows to use the classical

single-track mo del of Riekert-Schunk [25] shown in Fig. 1, also referred to as the \bicycle

mo del". The two front wheels and the two rear wheels are resp ectively lump ed together into

a single wheel at the front and rear axles, and the center of gravity CG is lo cated in the

plane of the road surface. A linear tire mo del is used under the assumption of small angles

during normal highway driving conditions within the physical limits of tires. The mo del

variables denote: 3

v vehicle velo city vector with v = jvj > 0; sp eed v is assumed measurable,

side slip angle between vehicle longitudinal axis and velo city vector v at CG,

vehicle yaw angle with resp ect to a xed inertial co ordinate system,

y lateral acceleration at CG,

CG

y lateral acceleration at sensor S, and

S

 front wheel steering angle.

f

_

The linearized 2D mo del with two interal states, side slip angle and yaw rate , and

front wheel steering angle  as an input can be found e.g. in [24, 26] as

f

3 2

2 3

c ` c `  c + c 

c

r r f f f r

f

2 3 2 3

1+

7 6

2

6 7

7 6

Mv Mv

Mv

d

6 7

7 6

4 5 4 5

+ 6 7  ; 1 =

f

7 6

2 2

c `

4 5

_ _

f f

dt

c ` + c ` 

5 4

c ` c ` 

f r

r r f f

f r

I

I I v

with parameters de ned as:

M total vehicle mass,

I total vehicle inertia ab out vertical axis at CG,



` `  distance of frontrear axle from CG with ` = ` + ` ,

f r f r

d \lo ok-ahead" distance between sensor lo cation S and CG,

S

c c  frontrear tire cornering sti ness,

f r

 

 road adhesion as a factor of e ective tire cornering sti ness c = c c = c .

f r

f r

v CG δ . β f . . S Ψ

l r l f .. d .. y S y

CG S

Fig. 1: Single-track mo del for car steering

The analysis in later sections is based on parameters of one of the cars used in the California

Path pro ject, a 1986 Pontiac 6000 STE sedan, summarized in Table 1. 4

M I ` ` d c = c

f r S f r

2

1573 kg 2873 kg m 1.1 m 1.58 m 1.96 m 80000 N/rad

Table 1: Car parameters of a 1986 Pontiac 6000 STE sedan

The transfer function from the steering angle  to side slip angle is given by

f

2

c I v s + c ` c ` Mv 

f f f r

 s; 2 s =

f

Ds

where D s is the second order denominator

2 2 2 2

D s = I Mv s + v I c + c +Mc ` +c `  s+

f r f r

f r

3

2 2 2

M v c ` c ` + c c ` :

r r f f f r

_

The Laplace-variable is denoted by s . The dynamics of yaw rate are

2 2

c ` Mv s +  c c `v

f f f r

_

s  =  s ;

f

D s

4

1

= W s  s ;

f

s

where W s denotes the yaw acceleration transfer function



s  = W s s : 5

f

De ning lateral acceleration at CG as an output of the system 1, a transfer function can

be derived as

2 2 2 2 2

c v I s +  c c `` v s +  c c `v

f f r r f r

y s =  s 

CG f

D s 

6

= V s s:

CG f

At displacement sensor S, lateral acceleration y comprises lateral acceleration at CG and

S

yaw acceleration scaled by lo ok-ahead distance d :

S



y s  = y s+d s

S CG S

2 2 2 2 2

c v M` d + I  s +  c c `v d + `  s +  c c `v

f f S f r S r f r

=  s 

f

7

D s

= V s+d Ws s 

CG S f

= V s  s :

S f 5

For lo ok-down systems, the lateral displacement sensor is usually lo cated at the front bump er

and lo ok-ahead distance d is approximately half the car length. The front wheel steering

S

angle  is the output of a p osition controlled actuator with limited bandwidth, denoted by

f

8

 s  = As   s;

f

where  is the commanded steering angle.

R ref

v

CG S β ∆y CG ∆y S Reference path ∆Ψ

d S

Fig. 2: Vehicle following a reference path

Fig. 2 shows a vehicle following a circular reference path with a radius R . Lateral

ref

displacement of the vehicle from the reference path is denoted
y at CG and
y at

CG S

sensor lo cation S. California highways consist of circular arcs matched together without

transitions. The vehicle longitudinal axis spans the sideslip angle with the vehicle velo city

vector v and the angular displacement error
with the tangent to the reference path. For

1

convenience, we de ne the reference curvature as  := .

ref

R

ref

From ab ove, we extract the plant subsystems: a kinematic subsystem I, a geometric road

reference subsystem I I, and the lateral vehicle force generation III in 7 with actuator

dynamics IV in 8, all viewed at the sensor lo cation S . The reference subsystem prescrib es

the desired lateral acceleration at S ,

2

y s  = v  s ;

ref ref

where  is the road reference curvature at S. The kinematic subsystem reduces to a double

ref

integrator for the lateral displacement measurement
y :

S

1

y s y s: 9
y s  =

S S

ref

2

s 6

A blo ck diagram of the closed lo op with controller C s  using output feedback of
y is

S

shown in Fig. 3.

ρ ref II

v 2 IV III

.. y I Actuatorδ Vehicle ref _ δ f + 1 _ A(s) V (s) _ ∆y S .. 2 S y s S

C(s)

Controller

Fig. 3: Blo ck diagram of an output feedback steering control system

2.2 Performance requirements

Human driving p erformance readily adapts to trac situations. In heavy trac and on

narrow lanes, e.g. near construction sites, drivers concentrate on lane keeping with small

errrors. On wide, op en highways, however, concentration decreases and so do es the tracking

p erformance. The total width of passenger cars including side-mirrors may vary within 1.8-

2.4 m approx. 6-8 ft from sub compacts to full-size vans and pick-ups. US highway lanes

have a width of 3.6 m approx. 12 ft, leaving a worst-case margin of 0.6 m approx. 2 ft

on each side for lane keeping errors.

An automatic steering system is exp ected to achieve tight tracking p erformance [27] to

allow for sucient safety margins in case of emergencies. Assuming a one-to-one prop ortion-

ing b etween margins for normal conditions and for emergency situations like tire bursts [28],

a maximum error of 0.3 m approx. 1 ft for normal conditions and for emergencies, re-

sp ectively, app ears reasonable. For normal conditions, this error is further sub divided into

nominal op eration and extreme situations. Hence a nominal maximum error of 0.15 m ap-

1

prox. ft and an extreme situation maximum error of 0.3 m approx. 1 ft are used as

2

p erformance requirement in this study.

The nominal reference input of an automatic steering system is the road curvature  .

ref

Disturbances include wind gusts, p ot holes etc. The vehicle sp eed v = jvj should b e chosen

to limit the lateral acceleration a during steady state curve riding to a  = 0:2g,

max

ref ref 7

2

where g = 9:81 m/s is the gravity constant. We assume that vehicle sp eed v is adjusted

according to the resp ective curvature of a sp eci c stretch of road with

a 

max

2

ref

: 10 v =

 

max

ref

Under normal highway driving conditions, the lateral error should not exceed j
y j =

S

0:15 m, e.g. for a step input 10. Extreme situations, whichwould allow for maximum lateral

errors of 0:3 m include signi cant simultaneous changes of road adhesion , concurrent strong

braking/accelerating, and steps from left maximum curvature to right maximum curvature

or vice-versa without intermediate straight segments.

Ride comfort is an imp ortant concern for automatic steering control design. Human pas-

sengers are likely to reject automatically steered vehicles, and AHS as a whole, if the ride

comfort is inferior to manual driving. Research in comfort for automobiles so far has fo cused

on susp ensions. A few concepts, however, carry over to steering control and can b e translated

into p erformance requirements.

Comfort is mainly constituted by the acceleration acting on the passenger. More than the

magnitude of acceleration, its rate, also called jerk, determines the comfort level of riders.

A continuous, steady state acceleration up to 0.3-0.4 g can be comfortably counteracted by

humans. Sudden changes in lateral acceleration, however, may cause a feeling of discomfort.

For one, acceleration comp onents in the 5-10 Hz frequency range can excite internal body

resonances, without the e ected p erson b eing able to allo cate the cause of discomfort.

For deriving consequences for lateral control design from ab ove, consider Fig. 3. When

closing the control lo op from measurement
y to  via a controller C s , reference accel-

S

eration y is to be matched by y without excessive oversho ot. To ful ll the discussed

S

ref

frequency constraints, the bandwidth of the closed lo op transfer function from y to y

S

ref

should be b elow 5 Hz, but may exceed the approx. 1Hz human reaction time since signi -

cant steps in road reference curvature are usually rare. Furthermore, the e ects of steps in

y can be decreased by previewing road curvature for a feedforward term in the steering

ref

controller.

2.3 Practical constraints

An additional practical constraint relating to comfort is system noise, the main source b eing

lateral displacement sensor S . Assuming white sensor noise, no frequency ab ove approx.

0.1-0.5 Hz should be ampli ed extraordinarily in the path to lateral acceleration y . Good

S

damping in the closed lo op is required for suppressing uncomfortable lateral acceleration and

jerk. A reasonable compromise is to require an automatic steering feedback lo op to have at

least a similar amount of damping as the conventional, manually steered car. Furthermore, 8

high frequency noise in lateral displacement measurement
y should not be allowed to

S

propagate through the controller C s  to the input  of steering actuator As toavoid me-

chanical wear, requiring roll-o in the controller b eyond the designated cross-over frequency.

The contradicting p erformance requirements of maximum lateral error and comfort have

to be achieved under a wide range of op erating conditions. Most plant parameters like car

mass and inertia, and lo cation of CG are slowly time varying and can b e estimated on-line by

appropriate algorithms. These parameters as listed in Table 1 for a sample car are assumed

to be known for the remainder of this study. Only the road adhesion factor  is assumed

unknown within its range of uncertainty. The physical upp er b ound is  = 1 for dry road

with a good surface. Empirical data of various studies suggests   0:5 for wet slipp ery

road and a lower b ound of  =0:1 for pure ice. Since changes may be abrupt, stability has

to b e guaranteed robustly without adaptation, at least for sets of the ab ove range.

A set of `normal' road conditions is de ned for this study as 0:5    1 for wet or

b etter road. This range is chosen conservatively to include di erent road surface and tire

qualities with varying adhesion factors. `Adverse' road conditions, e.g. snow and ice, are

not considered explicitly here. On one hand, it is assumed that such extreme conditions

would either lead to discontinued op eration of automated highways or to signi cant sp eed

reductions. On the other hand, the analysis presented in the sequel can be easily extended

to more extreme conditions.

Two ma jor implementational design constraints have to b e considered. The rst constraint

is related to the steering actuator dynamics. The necessity of b eing able to implement an

automatic steering system on an average passenger car without ma jor constructional mo di -

cation of currently used steering mechanisms limits the available actuator bandwidth. Cost

and power considerations imp ose bandwidth limitations of maximum 10 Hz on a p osition-

controlled, third or higher order steering actuator, regardless whether it is electric or hy-

draulic.

The actuator bandwidth is sub ject to uncertainty due to op erating conditions like tem-

p erature and command amplitude. A worst-case bandwidth of 5 Hz proved realistic in the

exp eriments p erformed at California Path. Compared with a human driver b elow 1 Hz

and the car dynamics ab out 1 Hz, the bandwidth limitations constitue a serious factor,

considering the discussed p erformance, comfort and robustness requirements. Furthermore,

the inherent uncertainty of the actuator dynamics imply that the steering controller has to

provide roll-o itself, taking into account the worst-case actuator dynamics.

3 Problem Analysis

A thorough system analysis is based on an appropriate degree of abstraction for highlighting

the key characteristics and on utilization of multiple angles of view for describing the problem. 9

For this study, we chose various to ols of linear systems theory to examine the plant and

the controller design requirements. The discussion in this section is founded entirely on

linear system theory. Using the linearized mo del of Section 2.1, a linear control design is

viewed as the basis for a p ossibly more involved design using nonlinear techniques. Any

nonlinear controller, however, can be linearized around an op erating p oint, e.g.
y = 0

S

for any  and v . The linearized version of a nonlinear controller must of course satisfy all

stability requirements and should at least partially ful ll the p erformance requirements given

in Section 2.2, under the practical constraints of Section 2.3. Note that this holds b oth for

analytical nonlinear techniques like sliding mo de control and for more heuristic metho ds like

fuzzy and neural-network control.

Any controller, linear or nonlinear, will rely on the available information content of the

feedback signals. We argue that basic systems theory considerations, without using a

sp eci c control design metho dology, are sucient, if not required, to understand the fun-

damental problems of automatic steering control. Later, re nement of the controller will

be aided by mo dern to ols of control design. To some extend, the fundamental problem of

automatic steering control under practical conditions has b een underestimated in the past,

partly due to the lack of a detailed system analysis. In particular, automatic steering of

vehicles driving at highway sp eed with lo ok-down systems needs concurrent consideration of

all p erformance requirements and practical constraints already at the system analysis and

control design stage.

The blo ck diagram in Fig. 3 shows that the double integrator 9 is to be controlled via

the lateral vehicle acceleration y , based on displacement measurement
y , and with input

S S

y . The op en lo op characteristics Gs, obtained by combining controller C s , vehicle

ref

lateral acceleration V s in 7, and actuator As in 8, are written as

S

Gs  = C s As V s  11

S

1

and constitutes the `control' for the `plant' . The closed lo op input-output relationship

2

s

between road curvature  and lateral displacement
y is denoted by

S

ref

2

v

y =  ;

S

ref

2

s + Gs 

12

= H s :

ref

Using multiple p ersp ectives, the following subsections concurrently analyze plant dynamics

and resulting control synthesis requirements to obtain suitable Gs. First, step resp onses

are employed to give a avor of the vehicle dynamics and their variation with vehicle sp eed

and road adhesion . Second, Bo de diagrams are used for frequency domain analysis to pin-

p oint the dominantvehicle characteristics. Third, these characteristics are traced backtothe

p ole/zero lo cations in eigenvalue plane. Last, and most ambitious, aphysical interpretation

is o ered to enhance the overall understanding and to motivate the control design directions

for Gs in Section 4. 10

3.1 Time Domain Analysis

The analysis commences with a set of illustrative step resp onses with zero initial conditions

shown in Fig. 4. In the top two plots, the dep endency of lateral vehicle acceleration y t

CG

at CG 6 on sp eed v left plots and on road adhesion  right plots is depicted for steering

angle steps of  =0:01 rad, neglecting the actuator dynamics As  in 8. The two plots in

f

_

the b ottom row show the asso ciated yaw rate resp onses t.

Dependence on velocity Dependency on raod adhesion 2.5 2.5 µ v 1.0 40m/s 2 2 2 2 32.5m/s µ=1 1.5 25m/s 1.5 0.5 1 17.5m/s 1 v=40m/s 0.5 10m/s 0.5 Lat. acc. CG (m/s ) Lat. acc. CG (m/s ) 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Time (s) Time (s) 10 < v < 40 m/s 0.5 < µ < 1 0.08 0.08

v 40m/s µ 1.0 0.06 32.5m/s 0.06 25m/s 17.5m/s 0.5 0.04 0.04 10m/s

Yaw rate (rad/s) 0.02 Yaw rate (rad/s) 0.02

0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2

Time (s) Time (s)

_

Fig. 4: Step resp onses of lateral acceleration y t at CG top row and yaw rate t

CG

b ottom row for variation of sp eed v within 10 m/s v  40 m/s on good road  =1 left

column and for variation of road adhesion 0:5    1 at high sp eed v = 40 m/s right

column

Observations

 The lateral acceleration transfer function y t has a direct throughput, i.e. zero

CG

di erence degree in second order 6. 11

 The initial value y 0 is sp eed indep endent, but varies linearly in , whereas steady

CG

state y 1= lim y t is b oth sp eed and -dep endent.

CG CG

t!1

 For low sp eed, the initial value y 0 exceeds steady state y 1, featuring a tran-

CG CG

sient without oversho ot and oscillations.

 At higher sp eeds, the initial value is considerably smaller than steady state, and the

transient is sub ject to increasing oversho ot and oscillatory b ehavior.

_

 Yaw rate t follows second order transfer function 4 with rst order numerator,

i.e. di erence degree one.

_

 Steady state 1 increases signi cantly from v = 10 m/s to v = 20 m/s, but only

_

slightly from there on for higher sp eeds. However, steady state 1 dep ends almost

linearly on road adhesion .

 For increasing sp eed and degrading road quality,yaw rate step resp onses tend to over-

sho ot and oscillate, even more so than lateral acceleration.

Consequences for control

At low sp eed, the vehicle dynamics are well-b ehaved without oversho ot and oscillations.

An initial value of lateral acceleration y 0 at CG exceeding steady state y 1 for a

CG CG

step input of steering angle  indicates a gain increase of V s for increasing frequencies.

f CG

The in uence of road adhesion  on the vehicle dynamics is negligible, geometric kinematics

prevail not shown here explicitly, see [22].

Increasing sp eed leads to increasing oscillatory b ehavior and signi cant oversho ot, b oth

_

in lateral acceleration y t and in yaw rate t step resp onses. The steady state value

CG

y 1 increases with v and exceeds signi cantly the initial value y 0 for higher sp eed,

CG CG

indicating a gain drop in V s and asso ciated phase lag. The increased oversho ot and the

CG

tendency to oscillate indicates an increasing in uence of the zeros in the transfer functions

and decreased damping of the vehicle p ole-pair in 3. Furthermore, an almost linear dep en-

dency on  of b oth lateral acceleration and yaw rate is visible in the righttwo plots of Fig. 4

for v = 40 m/s.

3.2 Frequency Domain Analysis

We continue the analysis using Bo de diagrams. Frequency domain most clearly reveals the

general structure of the vehicle dynamics and resulting controller requirements. A set of Bo de

diagrams of the lateral vehicle dynamics V satCGisshown in Fig. 5. The left diagram

CG

illustrates the dep endency of V s on driving sp eed v for the range 10 m/s v  40 m/s

CG 12

on a dry road with good surface  = 1. The right diagram depicts the changes caused by

variation of road adhesion  from p o or conditions  =0:5 to go o d conditions  = 1 at sp eed

v = 40 m/s. Note that the actuator dynamics As in 8 have not b een included.

10 < v < 40 m/s 0.5 < µ < 1 50 v 60 40m/s 32.5m/s µ=1 25m/s µ 40 40 17.5m/s 1 0.5 10m/s

|V(s)| (dB) 30 |V(s)| (dB) 20

v=40m/s 20 0 −1 0 1 −1 0 1 10 10 10 10 10 10 f (Hz) f (Hz) 50 100

0 10m/s 0

17.5m/s 1.0 −50 25m/s −100 µ 0.5 ang{V(s)} (deg) v 32.5m/s ang{V(s)} (deg) 40m/s −100 −200 −1 0 1 −1 0 1 10 10 10 10 10 10

f (Hz) f (Hz)

Fig. 5: Lateral acceleration dynamics V s at CG for variation of sp eed within 10 m/s

CG

v  40 m/s on good road  = 1 left Bo de diagram and for variation of road adhesion

0:5    1at high sp eed v = 40 m/s right Bo de diagram

Observations

 Lateral acceleration dynamics V shave notchcharacteristics with a distinct natural

CG

mo de around 0.5-2 Hz.

 For low sp eed, the steady state gain

2

c c `v

f r

13 V 0 =

CG

2 2

Mv c ` c ` +c c `

r r f f f r

is smaller than the high frequency gain

c

f

V 1 = lim jV |! j = :

14

CG CG

|! !1

M 13

The overall gain increase from V 0 to V 1 for v  14 m/s provides phase lead in

CG CG

the range of the natural mo de and leads to step resp onses with initial values exceeding

steady state, recall Fig. 4.

 For increasing sp eed, steady state gain V 0 increases strongly with v while high

CG

frequency gain V 1 remains constant. The gain drop from V 0 to V 1 for

CG CG CG

v>14 m/s leads to signi cant phase lag in the range of 0:1 2 Hz. This corresp onds to

initial values y 0 b eing smaller than steady state y 1inthe top row of Fig. 4.

CG CG

 At higher sp eeds, the natural mo de is accompanied by a gain \undersho ot" around

1 2 Hz.

 Decreasing road adhesion  decreases the gain jV |! j at all frequencies, also visible

CG

in the step resp onses in Fig. 4 right column. Furthermore, the frequency of the

natural mo de decreases with deteriorating road adhesion.

 The gain variation of V s due to  increases with sp eed as shown in [22]. In

CG

particular, a step from  = 1 to  = 0:5 leads to an approximate 10 gain drop of

V 0 at v = 10 m/s, but to an approximate gain drop of 40 at v = 40 m/s. The

CG

increasing dep endency of the gain jV sj on  for higher sp eeds complicates the

CG

robustness problem with resp ect to changes of  as discussed in Section 2.3.

Viewed at a lo cation S, the lateral vehicle acceleration has an emb edded additional yaw

acceleration comp onent. According to 7, lateral acceleration y at S is a linear combination

S



of lateral acceleration y at CG and yaw acceleration , with the weight of the latter b eing

CG

d . A set of Bo de diagrams for the yaw acceleration transfer function W s in 5 is shown

S

in Fig. 6 for similar conditions as in Fig. 5.

Note that V s and W s, and hence V s  share denominator D s  in 3, but have

CG S

di erentnumerators. In particular, yaw acceleration W s  has di erentiating low frequency

characteristics with asso ciated phase lead, up to a corner frequency ! . The gain in the

c

low frequency region dep ends linearly on sp eed v , whereas the high frequency gain is sp eed

indep endent. Furthermore, the natural mo de is less markable in W s  than in V s,

CG

hinting that the zero-pair in lateral acceleration rather than the p ole-pair causes the gain

undersho ot in the 1 2 Hz region of V s.

CG

Linear combination of V s and W s  to form V s  in 7 is not straightforward in

CG S

frequency domain. We restrict the following discussion to qualitative arguments: In the low

frequency range, the magnitude of V s exceeds that of W s , leading to dominance of

CG

V s in V s. At some frequency ! , dep ending on d , the magnitudes jV |! j and

CG S e S CG

d jW |! j b ecome comparable, implying that b oth transfer functions contribute to V s 

S S

according to their exact resp ective weight. 14 Variation of speed Variation of road adhesion 40 40 40m/s 1.0

30 30 v µ 0.5

10m/s µ=1

|W(s)| (dB) 20 |W(s)| (dB) 20 v=40m/s

10 10 −1 0 1 −1 0 1 10 10 10 10 10 10 f (Hz) f (Hz)

100 10m/s 150

100 50 µ 40m/s 50

ang{W(s)} (deg) ang{W(s)} (deg) 1.0 v 0.5 0 0 −1 0 1 −1 0 1 10 10 10 10 10 10

f (Hz) f (Hz)

Fig. 6: Yaw acceleration dynamics W s  for variation of sp eed within 10 m/s v  40 m/s

on good road  =1 left Bo de diagram and for variation of road adhesion 0:5    1 at

high sp eed v = 40 m/s rightBode diagram

Observations

 The steady state gain of V s  is solely determined by lateral acceleration at CG, i.e.

S

2

c c `v

f r

V 0 =

S

2 2

Mv c ` c ` +c c `

15

r r f f f r

= V 0

CG

in 13, with strong dep edency on sp eed v .

 The high frequency gain of V s is velo city indep endent, but varies linearly with d ,

S S

V 1 = lim jV |! j

S S

|! !1

d `

S

1 f

16

= c +

f

M I

= V 1+ d W1

CG S 15

 The frequency ! with jV |! j = jW |! j dep ends b oth on v and d . Since low

e CG e e S

2

frequency V |! varies with almost quadratic v , but low frequency W |!  dep ends

CG

less than linearly on sp eed v , comp ensation of the gain changes due to sp eed requires

an approximately linear increase of d with v in 7. Note that the phase lag in V |! 

S CG

is only comp ensated by phase lead in W |!  for ! < ! < ! , requiring suciently

e c

large lo ok-ahead d due to xed corner frequency ! .

S c

Obviously, lo ok-ahead distance d plays a decisive role in weight distribution in 7, and in

S

particular in determining phase lead or lag in the range of the natural mo de. This in uence

of lo ok-ahead distance d is illustrated in Fig. 7. Left, Bo de diagrams for a short lo ok-ahead

S

distance of d = 2 m, typical for lo ok-down reference systems, are depicted for go o d, dry road

S

 = 1 and di erent sp eeds 10  v  40 m/s. Right, a high sp eed situation v = 40 m/s,

 =1is shown for various lo ok-ahead distances 0  d  10 m. As predicted ab ove, phase

S

lag in the range 0:1 1Hz increases for increasing sp eed when using a lo ok-down reference

system with xed d =2m left Bo de diagram. Conversely, increasing d decreases phase

S S

lag and eventually provides phase lead even at high sp eeds right Bo de diagram. Not

surprisingly, this complies with human driving b ehavior, where the lo ok-ahead distance is

adjusted approximately prop ortional to sp eed.

Consequences for control

For lo ok-down reference systems with small d , the phase lag around the natural mo de has

S

signi cant consequences for control design. One of the key problems is that the natural

mo de of the vehicle is in the frequency range needed for stabilization, due to p erformance

requirements. In order to achieve a maximum error of j
y  j  0:15 m for a step

S max

2

in road curvature of j
 j = 0:2g=v , a minimum low frequency gain of the controller is

ref

required. The minimum gain requirement leads to a minimum requirement for the cross-over

frequency, colliding with the frequency of the natural mo de of the vehicle dynamics, and for

higher sp eed, also with the actuator dynamics.

The p erformance requirement for `normal' op eration of less than j
y  j 0:15 m error

S max

for a step input in road curvature of 
   in 10 translates into a gain requirement

max

ref

for the closed lo op transfer function H s in 12,

j
y  j

S max

jH sj  ; 8s : 17

j
  j

max

ref

This estimation is based on the inequality

 

1

1

H s j
y tj = jL 
  j  jH sj
j
  j; 18

S max max

ref ref

s 16 Variation of speed for d S =2m Variation of look−ahead for v=40m/s 50 60 v 40m/s 10m 32.5m/s µ=1 8m 6m 25m/s 4m

40 17.5m/s 40 2m

|V(s)| (dB) |V(s)| (dB) µ=1 0m dS 10m/s 30 20 −1 0 1 −1 0 1 10 10 10 10 10 10 f (Hz) f (Hz) 50 50 dS 10m/s 10m8m 6m 0 17.5m/s 0 4m 25m/s 32.5m/s 2m 40m/s −50 −50 v ang{V(s)} (deg) ang{V(s)} (deg) 0m −100 −100 −1 0 1 −1 0 1 10 10 10 10 10 10

f (Hz) f (Hz)

Fig. 7: Lateral acceleration dynamics V s  at sensor lo cation S, d = 2m in front of CG,

S S

for variation of sp eed within 10 m/s v  40 m/s on good road  = 1 left Bo de diagram

and for variation of lo ok-ahead 0  d  10 m for  = 1 at high sp eed v = 40 m/s right

S

Bo de diagram

1

where L f
g denotes the inverse Laplace-transform. Substituting op en lo op Gs in 11

into closed lo op H sin 12 yields

2

v

H s = ; 19

2

s + C s As V s

S

with V s denoting the lateral car dynamics at sensor lo cation S in 7, As  describing

S

the actuator dynamics 8, and C s  b eing the controller to be designed. Note that the

2

quadratic dep endency of H s  on sp eed v is comp ensated by the inverse quadratic sp eed

dep endency of 
  in 10 with sp eed indep endent a  .

max max

ref ref

An output feedback controller C s , relying on measurement of lateral displacement
y ,

S

has to comp ensate the phase lag in vehicle dynamics V s  and additionally to provide

S

sucient phase margin for stabilisation of the double integrator. The gain and phase re-

quirements for controller C s can be examined by solving 19 for C s under constraint 17

17. This p erformance constraint should be coupled with the stability constraint as shown

in Fig. 8 for v = 40 m/s,  =1, and d =2m. S

20

10

0 0 10 20 30 30 deg controller phase −10 40 60 deg controller phase 47 90 deg controller phase

Gain (db) −20 Performance curves 40

−30 30 20

−40

Stability line 10 deg min lead for stability −50 −1 0 1 10 10 10

Frequency (Hz)

Fig. 8: Requirements for controller C s  at v = 40 m/s,  =1, and with d =2m in terms

S

of gain and phase at the critical frequency range for achieving the desired p erformance

Performance curves, the solutions of the ab ove p erformance constraint for jC s j and var-

6

ious values of C s , are displayed for the parameters of the Pontiac 6000 STE sedan of

Table 1. The p erformance constraint dictates that at any frequency, jC s j should lie ab ove

6

C s . Similarly, the stability require- the p erformance curve corresp onding to the resp ective

6

ment dictates C s  to be at least the marked value at cross-over, when jC sj intersects

the stability line inverse plant. In order to guarantee go o d damping and a sucient phase

o

6

C s  should exceed the marked value by at least 50 at cross-over. This already margin,

stringent constraint on the control design is aggrevated by the requirement of controller roll-

o immediately after the cross-over frequency, but within the actuator bandwidth of approx.

5 Hz. Both the controller low-pass for roll-o and the actuator dynamics start intro ducing

phase lag from around 2 Hz.

The controller C s  should avoid excessive phase lead in the vicinity of actuator dynamics

since this could cause the actuator to saturate, with consequent harmful and p ossibly unsta- 18

ble limit cycles. In addition to actuator constraints, high controller gains at high frequencies

are extremely undesirable due to noise considerations. As is obvious from Fig. 8, muchlower

gain sucies in the low frequency range to achieve the desired p erformance. High gain at

high frequencies ampli es the noise in measurement
y , leading to poor comfort and also

S

to high wear of mechanical parts in the steering mechanism.

The controller requirements shown in Fig. 8 are already highly demanding for a single

sp eed and constant known road adhesion . Addition of the robustness requirement of

simultaneous stability and good p erformance for 0:5    1 further complicates matters.

Variation of  intro duces a 40 variation in vehicle gain jV |! j see [22] and shifts the

S

frequency of the natural mo de. In order to satisfy the p erformance requirements, the gain

variation has to b e comp ensated by increasing the low frequency gain of controller C s . The

increasing phase lag of vehicle dynamics V s  for decreasing  requires even more phase

S

lead by the controller. Most imp ortant, however, the phase lead region of the controller has

to sustain for a signi cant frequency range, since cross-over frequency varies concurrently to

variation of gain and natural mo de in V s  due to changing . An additional gain increase

S

results for higher frequencies, further increasing system bandwidth and depriving the neces-

sary ro om for designing a robust controller C s  which satis es the practical requirements

and constraints.

3.3 Eigenvalue Domain Analysis

The frequency domain analysis in the previous section indicated a notch characteristic of

the lateral acceleration transfer functions V satCGandV s at sensor lo cation S, with

CG S

a natural mo de around 0:5 2 Hz. For increasing sp eed v , the phase lag asso ciated with

the natural mo de increases due to V 0  V 1 in 13 and 14, and, for small lo ok-

CG CG

ahead d , also for V 0  V 1 in 15 and 16. Furthermore, gain \undersho ot" for

S S S

1 2Hz hints a dominant and p o orly damp ed zero-pair. The p ole-zero lo cations of V s

CG

are studied in more detail in Fig. 9.

Observations

The left graph of Fig. 9 shows the dep endency of p ole and zero lo cations of V sonspeed

CG

10 m/s  v  40 m/s on go o d road with  =1. Poles follow circular arcs for sp eed variation

while zeros follow parab olas. For increasing sp eed, b oth the p oles and the zeros approach

the imaginary axis, leading to resp ective p o or damping, with the zeros having less damping

than the asso ciated p ole-pair. Also, the p ole and zero lo cations change signi cantly from

v = 10 m/s to v  25 m/s, but vary less for increasing the sp eed further.

The dep endendy of p ole-zero lo cations of V son0:5    1 for high sp eed v = 40 m/s

CG

is depicted in the right graph of Fig. 9. Note the di erent scale of the real axis compared to 19 Pole/zero locations Pole/zero locations 10 10 µ=1 v=40m/s 5 v=10m/s 5 µ=0.5

0 0 µ=0.5

Imaginary axis v=10m/s Imaginary axis −5 v=40m/s −5 µ=1 −10 −10 −15 −10 −5 0 −3 −2 −1 0

Real axis Real axis

Fig. 9: Pole `x' and Zero `o' lo cations of V s for variation of sp eed 10 m/s  v  40 m/s

CG

on go o d road with  = 1 left graph and variation of 0:5    1 for high sp eed v = 40 m/s

right graph

the left graph. For decreasing , p oles and zeros further approach the imaginary axis in a

uniform manner. This results in even lower damping of b oth the p ole-pair and the zero-pair.

The dominance of the zero-pair apparent in Fig. 9 is emphazised in Fig. 10. Damping

D of the p oles of V s right graph is contrasted with damping D of the zero-pair

p CG z

left graph. Obviously, the numerator damping is signi cantly smaller than denominator

damping. Even worse, damping D of the zeros decreases sharply with increasing sp eed to

z

values as low as D < 0:35 for v = 20 m/s and further to D < 0:2 for v = 40 m/s on

z z

any road surface . Damping of the p oles, on the other hand, drops steadily with sp eed to

D < 0:6 at v = 40 m/s. Deteriorating road adhesion  further decreases D for any sp eed,

p p

e.g. to D < 0:45 at v = 40 m/s on wet road  =0:5.

p

The yaw rate transfer function 4 shares denominator D s in 3 with lateral acceleration

transfer function 6, but has only one real zero. Yaw acceleration W s in 4 has an

additional zero in the origin. Once again, addition of V s and d W s  to form V s 

CG S S

in 7 is nontrivial. Obviously, only the lo cations of the zeros change with varying d .

S

Damping asso ciated with the zeros of the lateral acceleration transfer function V s  at

S

sensor lo cation S in 7 increases with lo ok-ahead distance d , see Fig. 11 for an example at

S

high sp eed v = 40 m/s on a go o d road. The damping increase is dramatic for initial increases

of d up to ab out twice the car length. For even larger d , the further damping increase

S S

is less signi cant. Equal damping of p oles and zeros is achieved in the ab ove example of

Fig. 11 for d  15 m.

S 20 Zero−pair damping Pole−pair damping 1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6 z p

D 0.5 D 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1 1 1 0.8 0.8 0.6 0 0.6 0 20 20 40 40

µ v (m/s) µ v (m/s)

Fig. 10: Damping of the zero-pair left and the p ole-pair right of the lateral acceleration

transfer function V s for sp eed 1 m/s  v  40 m/s and road adhesion 0:5    1

CG

Consequences for control

The lo cations of p oles and zeros in the lateral acceleration transfer function V s  in 7 have

S

signi cant consequences for control design for lo ok-down systems with small d , esp ecially

S

at higher sp eeds. The degraded damping of the p oles is a poor basis for control design to

b egin with, since V s  is a vital comp onent of the overall `control' Gs  in 11 used to

S

stabilize the double integrator plant. For closing the control lo op, C s  requires one zero,

b etter one zero-pair, in the left half plane to move the double integrator p oles away from the

origin. Simultaneously, the vehicle p oles of V s  in 7 immediately start moving towards

S

the imaginary axis when closing a feedback lo op, leading to even p o orer damping in closed

lo op H s  in 12 than in op en lo op Gs  in 11. Furthermore, the zero-pair presentinthe

lateral acceleration dynamics V s  with extremenly poor damping for small d , attracts a

S S

p ole-pair when closing the feedback lo op. This leads to a p o orly damp ed p ole-pair in closed

lo op and outrules the use of high gain.

Hence, a second zero-pair is required to temp orarily detract the vehicle p oles in 3 away

from the |! -axis to improve damping. Intro ducing two zero-pairs in C s  amounts to strong

di erentiation action in the asso ciated frequency ranges, since the controller p oles need to

be placed away from the zeros to avoid cancellation of their e ects. Strong di erentiation

action in C s  leads to high gain at high frequencies, which ampli es high frequency noise

and collides with actuator dynamics, as discussed in Section 3.2. 21 Pole/zero locations 10

dS= 0m: Dz=0.17 8 v = 40 m/s, µ = 1 6 dS= 3m: Dz=0.3 4 Dp=0.58 dS= 6m: Dz=0.38 dS= 9m: Dz=0.45 dS=12m: Dz=0.51 2 dS=15m: Dz=0.57

0

−2 dS=15m: Dz=0.57 Imaginary axis dS=12m: Dz=0.51 dS= 9m: Dz=0.45 Dp=0.58 −4 dS= 6m: Dz=0.38 dS= 3m: Dz=0.3 −6

−8 dS= 0m: Dz=0.17

−10 −3 −2.5 −2 −1.5 −1 −0.5 0

Real axis

Fig. 11: Zero lo cations and asso ciated damping for variation of lo ok-ahead distance d for

S

high sp eed v = 40 m/s on go o d road with  =1

An \almost" p ole/zero cancellation could be achieved by moving a p ole-pair close to the

dominant zero-pair of the lateral acceleration dynamics at S via feedback. Besides the

obvious concern of a p o orly damp ed p ole/zero cancellation, inacceptable comfort results.

The closed lo op p oles are present in all vehicle transfer functions, i.e. in lateral accleration

dynamics at all lo cations away from S and also in the yaw dynamics. However, only in the

lateral acceleration transfer function V s  at sensor lo cation S, the zero-pair present there

S

would comp ensate the e ect of p o orly damp ed p oles. In all other transfer functions, the

p o or damping surfaces uncomp ensated, leading to an e ect called \ sh-tailing": While sensor

lo cation S might exhibit acceptable tracking p erformance in closed lo op, the vehicle itself yaw

oscillates ab out S as if b eing hinged there. This e ect also prevents utilisation of feedback

linearization or inverse dynamics metho ds, which similarly would place controller p oles close

to p o orly damp ed vehicle zeros. Furthermore, robustness requirements with resp ect to road

adhesion  raise serious doubts ab out the reliability of using p ole-zero cancellation in the

vicinity of the imaginary axis. 22

3.4 Physical Interpretation

The describ ed vehicle dynamics and their dep endence on sp eed v and road adhesion  are

b est understo o d in a physical context. In the sequel, the dominant phenomena in the crucial

lateral acceleration transfer functions V s and V s  will b e studied: high frequency and

CG S

steady state gain, and p ole and zeros lo cations.

The high frequency gain of lateral acceleration transfer function at CG in 14 describ es

the instantaneous acceleration as a ratio of instantaneous force at the front tires for a step in

steering angle  ,c  , and the total vehicle mass M . At other lo cations away from CG,

f f f

d `

S

f

c e.g. at d as in 16, a comp onent of instantaneous yaw acceleration  is added.

f S f

I

Linear prop ortioning of lateral force at the front tires with resp ect to the instantaneous

angle between the front tire and the vehicle velo city vector v at the front axle is a result of

the linear tire mo del used in this study. Hence, instantaneous acceleration is linear in road

adhesion , but is sp eed indep endent.

Steady state lateral acceleration, on the other hand, derives from the force and torque

balance equations, see e.g. [24, 26]. For neutral steer with c ` = c ` , 13 and 15 for any

f f r r

lo cation d reduce to geometric cornering

S

V 0 = V 0

CG CG

ns c ` =c `

r r

f f

2 20

v

= ;

`

with quadratic sp eed dep endence according to centrifugal force equilibria. Understeer with

c ` < c ` is generated in all passenger cars by lo cating the CG ahead of the mid-p oint

f f r r

between the two axles. This reduces the sensitivity of the vehicle at higher sp eed by creating

a torque opp osing the geometric cornering 20. In e ect, the gain from steering angle to

2

v



lateral acceleration is decreased b elow , resulting in a v , 1 <  < 2 dep endency of

`

V 0 in 13 at mo derate sp eed and even a 0 < <1 dep endency at very high sp eeds.

CG

The b ene t for human driving is desensitization of the steering at higher sp eed. In other

words, to drive a sp eci ed radius of curvature, a larger steering angle is required at high



sp eed than at low sp eed. Due to the remaining v dep endency, however, a steady state gain

lower than the sp eed indep endent high frequency gain at low sp eed turns into a steady state

gain V 0 greater than V 1 at higher sp eeds, with asso ciated phase lag instead of

CG CG

phase lead.

Pole lo cations derive from the dynamic motion equations of the lateral and the yawing

mo des. For neutral steer, the p oles in 3,

D s = D s

ns c ` =c `

r r

f f

 !

21

2 2

c ` + c `

c + c

f r

f r

f r

s +  ; = s + 

Mv I v

23

are real and clearly separated into the lateral translational mo de and the yawing rota-

tional mo de. For example, the yaw rate transfer function 4 reduces to the rst order

yawing mo de right factor in 21, since the lateral mo de left factor in 21 app ears as a

zero for neutral steer c ` = c ` . Note that b oth mo des linearly dep end on road ahesion 

f f r r

and on the inverse of sp eed v ,in compliance with basic motion laws.

Understeer intro duces coupling b etween the lateral and the yawing mo des. For decreasing



c ` c `  < the overall steady state gain b oth in V s and V s, the understeer term

f f r r CG S

I

0 in denominator D s in 3 increases the natural frequency ! of the p oles when written

p

as

2 2 2

D ss  = I Mv s +2D ! s +! : 22

p p

p

Increasing ! for constant2D !  decreases damping D and eventually leads to a complex

p p p p

p ole pair. Since understeer at higher sp eed creates a torque opp osing the geometric cornering,

the front tires have to supp ort more lateral forces than the rear tires. Due to the shorter

lever arm of the front tires compared to the rear tires for any d > 0, understeer results in

S

decreased total lateral damping forces provided by the tires.

The zeros of a mechanical system are tightly connected to input and output lo cations.

In particular, a zero in a transfer function signi es that an energy injection with the zero

frequency at the input is absorb ed by the system and will not fully reach the output.

Conversely, a disturbance at the output with zero frequency do es not show much e ect

at the input lo cation. This implies that the mo des of the system obtained by mechanically

constraining an input constitute the overall system zeros for that input.

For the lateral vehicle dynamics, mechanically constraining the front tires, i.e. \hinging"

the vehicle at the front axle, describ es the zeros of the lateral transfer functions for any d

S

as the p oles of the hinged system. Consequently, the zeros are comprised of a translational

mo de coupled with a rotational mo de ab out a vertical axis at the front axle and thus do

not dep end on front tire characteristics like sti ness c . Also, damping of the hinged system

f

is only provided by lateral damping of the rear tires. Hence for small d , the zeros in the

S

lateral acceleration transfer function in 7 have less damping than the vehicle p oles, which

can draw up on damping from all four tires. The smaller zero damping compared to the p ole

damping is most noticable for higher sp eed, when lateral damping from the tires b ecomes

more critical.

Understanding the zeros of V s  in 7 as the mo des of the vehicle b eing hinged at the front

S

axle also explains the in uence of lo ok-ahead distance d on damping of those zeros: The

S

larger the lever arm d `  to the front axle compared to the rear lever arm ` = ` + ` , the

S f f r

b etter the damping of the zeros. Initial increases of absolute d out from the front bump er

S

have a larger impact on the lever arm ratio and hence on the damping of the zeros than for

large lo ok-ahead d `  >`, as already observed in Section 3.3.

S f 24

4 Control Design Directions

This section is devoted to an investigation of p ossible control design directions. Since this

pap er concentrates on fundamental issues in automatic steering control, no explicit con-

trol design is discussed here. Rather, basic design directions are examined and contrasted

with resp ect to their advantages and p ossible drawbacks, emphasizing on the p erformance

requirements and practical constraints discussed in Sections 2.2 and 2.3.

The analysis in Section 3.3 indicates that the zero-pair of the lateral acceleration transfer

function V s in 7 is one of the ma jor obstacles towards automatic steering control design

S

for lo ok-down reference systems with small lo ok-ahead d . Recall that the zero-pair has

S

extremely p o or damping for higher sp eed, e.g. less than 0.2 damping for v  40 m/s on any

road surface. In a closed lo op system, a p ole-pair is inevitably attracted to the zero-pair,

leading to low p ole damping. Since these p oles dominate other vehicle transfer functions like

lateral acceleration away from sensor lo cation S and vehicle yaw rate, poor comfort results.

Feedback of additional internal states like vehicle yaw rate cannot solve this problem of

attractive, but p o orly damp ed zeros, since only p ole lo cations can b e in uenced.

The necessary mo di cation of zero lo cations requires changing the overall plant structure as

outlined in Section 3.4. In general, two alternatives exist: mo difying the inputs or mo difying

the outputs of the system.

For changing the plant inputs, various metho ds of generating a lateral force or a yaw

torque acting on the vehicle can be exploited [29]. Prominent examples include single-

wheel braking as b eing used for Anti-Skid-Control ASC by Mercedes-Benz and Bosch in

Germany [30] and the robust yaw decoupling control for four-wheel steered 4WS cars [31].

ASC generates a yaw torque to avoid skidding in emergency situations by braking of selected

wheels and thus can hardly be used in continuous op eration for automatic steering. An

appropriate equivalentwould b e single wheel driving torques on all four wheels, a technically

rather involved and exp ensive solution.

The 4WS control lawbyAckermann [31], on the other hand, is suited for continuous AHS

op eration at the price of an additional rear wheel steering mechanism. Robust decoupling of

yaw motion from lateral acceleration at the front axle is achieved by feedback of the vehicle

yaw rate via an integrator. The control law exploits structural prop erties of the vehicle

dynamics and is robust with resp ect to parameters like car mass and inertia, road condition,

and even nonlinear tire characteristics.

The approaches mentioned ab ove aim at mo difying the inputs to the lateral vehicle dy-

namics and thus require exp ensive changes to the mechanical structure of the car. In the

discussion b elow, we will fo cus on a less involved solution, the mo di cation of the system

outputs to in uence the lo cation of the zeros of the lateral acceleration transfer function for

automatic steering control. 25

4.1 Mo di cation of lo ok-ahead: The dual roles concept

The new problem description in Fig. 3, with op en lo op Gs  in 11 and closed lo op

1

H s in 12, reveals an interesting phenomenon. Plant in 9, the double inte-

2

s

grator from lateral acceleration error
y to lateral displacement
y , is controlled by

S S

Gs  = As V s C s  where C s  comprises the controller, V s  represents the vehicle

S S

lateral dynamics at S and As  denotes the actuator dynamics. Instead of isolated design of

controller C s , the vehicle dynamics V s  are now included in the design of Gs  to control

S

the double integrator. In fact, for practical implementation, also actuator dynamics As 

have to b e taken into account, although it might not b e p ossible to mo dify them signi cantly

for a given car.

A set of new design directions evolves from this novel p oint of view. First, an appropriate

Gs  is designed, taking in account all practical requirements and constraints on closed lo op

H s. Then, Gs is realized via concurrent design of vehicle dynamics V s  and controller

S

C s . In this second step, V s  and C s  play complementary roles and may be regarded

S

dual to each other.

1

Plant in 9 is known exactly. Combined \controller" Gs  should be designed for

2

s

closed-lo op H s  to ful ll the p erformance requirements discussed in Section 2.2, in particular

with resp ect to maximum lateral displacement, damping for ride comfort, and robustness

with resp ect to road adhesion . Furthermore, zero steady-state errors and controller roll-o

to prevent exciation of actuator dynamics are required. Reference acceleration y can be

ref

known from 9 via appropriate transmission of preview information of up-coming curvature

from the road to the vehicle. For example, in the magnetic marker system used at California

Path, binary co ding of magnet p olarity is used to preview road curvature b efore encountering

a road curvature change [19, 32]. In case of availability of preview, feed-forward of road

curvature can b e added to the overall control structure, but is sub ject to dynamic uncertainty.

Since curvature preview do es not signi cantly in uence stability, this option is not p ersued

in the sequel.

The double integrator 9 has to be stabilized by Gs  via a lead comp ensator with ap-

o

proximately 60 phase lead at cross-over for appropriate damping. Atlow frequencies, b elow

roll-o , a minimum gain according to 17 is necesssary to achieve the p erformance require-

ments. Also, Gs  needs to include integral action for zero steady state lateral displacement

and low-pass characteristics for the desired roll-o .

The simplest linear solution is a PID T structure for Gs , where the prop ortional gain

m n

P is determined to satisfy 17, the integral I-term is chosen not to interfer with stability,

i.e. do es not intro duce phase lag in the vicinity of cross-over, the di erential term D is an

m

m-th order lead comp ensator to provide sucient phase margin, and T is an n-th order

n

low pass for roll-o with n> m. This PID T structure is synthesized by design of lateral

m n 26

vehicle dynamics V s  and controller C s  to form Gs  in 11. Actuator dynamics As 

S

in 8 are not included in the design, however, bandwidth and damping requirements for the

p osition control of the actuator derive from the requirements on Gs .

Lateral acceleration V s  at S has second order dynamics with two p oles and two zeros.

S

Dep ending on sp eed v , road adhesion  and lo ok-ahead distance d with xed parameters

S

of the vehicle dynamics e.g. as in Table 1, V s  shows charactersistics of either a lead

S

comp ensator or a lag comp ensator, see also Fig. 7. In particular, at low sp eed, e.g. for

v = 10 m/s, small lo ok-ahead d already provides substantial phase lead for the full range of

S

considered road adhesion 0:5    1. For higher sp eeds, small d as present in lo ok-down

S

o

reference systems, results in phase lag, e.g. up to 70 on poor road surface at v = 40 m/s.

Increasing lo ok-ahead d reverses this trend. For example, for

S

2 2 2

I =M v c + c `  c c ` 

f r f f r

d = 23

S

0

2 2

` Mv c ` c ` + c c `

f r r f f f r

V 0 = V 1 and hence the overall phase balance is equal to zero. Furthermore, phase

S S

lead is present in V s  for suciently large d >d in the frequency range of the natural

S S S

0

mo de.

In addition to vehicle dynamics V s , controller C s  may also provide phase lead for

S

stabilization. The interchangeability of the roles of V s  and C s  in Gs  to stabilize

S

the double integrator plant 9 with sucient phase margin is the core of the dual roles

concept.

Case I: Lo ok-ahead distance d is chosen such that V s  contributes sucient phase

S S

2

lead to Gs  for stabilization of 1=s  for the range 0:5    1. Controller

C s  is chosen with PI-typ e characteristics to match the phase lead region of

V s  with the cross-over region for plant 9. The sp ecial case of Hurwitz

S

stability for a good road surface was solved for a true lo ok-ahead system by



Unyelioglu et al. [13].

Case II: Lo ok-ahead distance d is chosen similar to 23 such that V s  contributes no

S S

or little overall phase lag/lead to Gs  and is approximately a P-typ e transfer

function for the range 0:5    1. Controller C s is chosen with a PID T

m n

2

characteristic to provide sucient phase lead to Gs  for stabilization of 1=s ,

according to the p erformance requirements.

Intermediate solutions include sharing phase lead requirements between V s and C s ,

S

and comp ensating phase lag in V sby Cs or visa versa. Note that comp ensation of phase

S

lag in V s requires strong di erentiating action of C s  which may collide with noise and

S

actuator constraints. For studying the resp ective merits of the di erent approaches, contrast

the two cases on an abstract level. The main di erences are: 27

 Case I requires larger lo ok-ahead distances d than case II, which generally leads to a

S

higher noise level in the lateral displacement measurement
y .

S

 Case II requires more phase lead of controller C s  than case I, leading to higher

controller gain in the mid-frequency range, around and after cross-over, which generally

ampli es higher frequency noise.

 For case I, cross-over is determined by the phase lead region of V s . Since cross-

S

over and hence closed-lo op bandwidth is con ned to a range prescrib ed by the vehicle

dynamics, the P-p ortion of controller C s  cannot be chosen to satisfy arbitrary p er-

formance requirements. However, it turns out that the p erformance requirements are

almost satis ed for most passenger cars. Variation of road adhesion  simultaneously

in uences the low frequency gain of V s  and frequency range of phase lead such that

S

2

simultaneous stabilization of 1=s  in 9 is p ossible, see the left Bo de diagram in

Fig. 12 for an example for v = 40 m/s, d =16:9 m resulting in a minimum phase lead

S

o

of 50 ,  = 1 and  = 0:5, with P-control C s  = 0:03. Inherent \almost" satis-

1 2

faction of p erformance requirements and robustness with resp ect to road adhesion are

attributed to the sp eci cations of vehicle steering design to accomo date human driving

b ehavior. In fact, it can b e argued that case I resembles the steering characteristics of

a concentrated human driver.

 For case II, cross-over is determined entirely by control C s  and can be chosen to

satisfy any p erformance requirements. Since C s is invariant with resp ect to road

adhesion adaptation is not p ossible since  is not slowly time-variant, robustness is

to b e achieved by providing phase lead over a range of p ossible cross-over frequencies,

see the right Bo de diagram of Fig. 12. The example shows a similar situation as

ab ove, with d = 5:8m and C s = 0:03L50, where L50 denotes a higher order

S

o

lead comp ensator, providing 50 phase lead consistently over the necessary frequency

range.

The ab ove list p oints out the crucial trade-o s in automatic steering control design. Noise

in lateral displacement measurement
y increases at least linearly in d for sensors \lo oking

S S

ahead" of the front bump er, and even quadratically in d e.g. for maschine vision. On the

S

other hand, p oviding phase lead in C s leads to higher controller gain at higher frequencies

around and after cross-over, which may require excessive actuator bandwidth. These noise

considerations have to be traded o with the ability to satisfy p erformance requirements.

Case I, using \natural" phase lead in V s  by appropriate seletion of d o ers straight

S S

forward control design at the price of a pre-determined cross-over frequency and hence pre-

determined system bandwidth. Only two control parameters, d and the P-gain, have to

S

be selected for robustness with resp ect to the range 0:5    1, p ossibly gain-scheduled

with velo city v . Case II, on the other hand, provides exibility in cho osing cross-over, but

o

requires more involved, higher order control design. For example, the plain 50 -phase lead

controller L50 shown in Fig. 12 is not directly employable due to corruption by the vehicle

dynamics V s as apparent from the rightBode diagram.

S 28 d S = 16.9, C(s) = 0.03 d S = 5.8, C(s) = 0.03*L(50) 40 40 µ1 =1.0 µ1 =1.0 µ2 =0.5 µ2 =0.5

20 20 G1 (s) G1 (s)

G 2 (s) G 2 (s) 2 G (s)/s2 G1 (s)/s 1 0 0 2 2

Magnitude (dB) G 2 (s)/s Magnitude (dB) G 2 (s)/s

−20 −20 −1 0 −1 0 10 10 10 10 f (Hz) f (Hz)

60 80 G 2 (s) G1 (s) G 2 (s) 60 40 G1 (s) 40

20 Phase (deg) Phase (deg) 20

0 0 −1 0 −1 0 10 10 10 10

f (Hz) f (Hz)

Fig. 12: D ual r ol es concept : Robust stabilization by vehicle phase lead with long lo ok-

ahead d Case I, left Bo de diagram is dual to robust stabilization by a lead comp ensator

S

with small lo ok-ahead d Case II, right Bo de diagram

S

4.2 Virtual lo ok-ahead

In addition to the obvious solution of employing true lo ok-ahead reference systems like

machine vision [7{12] or radar [13], vitual lo ok-ahead can b e created for lo ok-down reference

systems to virtually extend d beyond the front bump er. Lo ok-down reference systems

S

are attractive with resp ect to reliability and accuracy, e.g. magnetic markers [19] are not

susceptible to weather and light conditions like machine vision and the sensor eld of view

may not be obstructed by preceeding vehicles, which is imp ortant for plato oning [6]. To

preserve the advantages of lo ok-down reference systems via virtual lo ok-ahead, consider

equation 7, which can b e re-written in error co ordinates for virtual lo ok-ahead d as

v

2



y = y v  +d  v_ 

v CG v

ref ref

24



=
y + d
:

CG v 29

Note that _ is a Dirac impulse for step changes of road curvature. However, 24 is

ref

written in terms of acceleration, which is the control variable, but not the measured variable.

_

Measurements are on p osition scale e.g.
y  or, at b est, on velo city scale e.g. .

CG

It is clear from 24 that a virtual lo ok-ahead d > d requires measurement of
or

v S

_

, e.g. by

_

Metho d I: Measurement of yaw rate , sp eed v and preview of road curvature  to

ref

obtain the rate of angular displacement,

_ _

= v : 25

ref

Metho d II: Measurement of lateral displacement
y at a lo cation near the tail of the

T

vehicle see Fig. 13 to obtain angular displacement
as

y
y

S T

= : 26

d + d

S T

Metho d III: Combination of metho ds I and I I in a Kalman lter structure.

T CG δ . . f .S ∆y T ∆y d S T dS ∆y ∆Ψ CG

Reference Path

Fig. 13: Using an additional lateral displacement sensor T at the tail of the vehicle to measure

the angular displacement

_

Metho d I has the advantage of providing a velo city scale measurement
 as compared

to a p osition scale measurement 
 of metho d II. Since, at least for controller case II,

di erentiating action is required of controller C s , measurement of an already di erentiated

_ _

variable
is attractive. However, the di erence of two absolute variables and v is

ref

_

used to calculate relative variable
, with obvious noise concerns. On the other hand,

metho d I I calculates relativevariable
via two relative measurements
y and
y . The

S T

e ect of measurement noise determines the resp ective practical merits of the two metho ds,

but is to o implementation sp eci c to b e studied in a general setting. Obviously, metho d I I I 30

combines the advantages of metho ds I and I I, and has a p otential to trade o their disadvan-

tages, i.e. supp ort di erentiating action to get phase lead in the presence of measurement

noise.

In contrast to reference systems with true lo ok-ahead, the ab ove metho ds enable utilization

of frequency functions for lo ok-ahead distance, e.g. a virtual lo ok-ahead d s  can b e de ned

v

for a virtual displacement measurement

y s =
y + d s
; 27

v CG v

to obtain a steering controller of the form

 = C s
y s 

v

28

= C s
y + C s
:

CG CG

C s 

Virtual lo ok-ahead d s  = allows to select a frequency range of measurement
,

v

C s

CG

e.g. to accomo date the Kalman lter structure of metho d III and for bandpass ltering to

obtain the necessary phase lead in the 0:1 2 Hz region to comp ensate phase lag intro duced

by V s.

CG

For implementation with a tail displacement sensor T as depicted in Fig. 13, 28 can be

re-written as

 = C s 
y + C s
y ; 29

S S T T

d C s+C s d C s C s

T CG S CG

with C s = and C s = .

S T

d +d d + d

S T S T

4.3 Mo di cation of system zeros

Virtual lo ok-ahead, i.e. d s  in 27, allows to mo dify the system zeros. We have argued

v

that the poor damping of the zero pair in the lateral acceleration transfer function 7 at

high sp eeds substantially hinders automatic steering control design for lo ok-down reference

systems under practical conditions, see also [22]. Furthermore, cancellation of the zeros by

moving the system p oles into their vicinity via feedback is inadequate b ecause the p oles in

3 are also present in all other vehicle transfer functions like yaw rate 4 and poor p ole

damping would result.

The capability to in uence the system zeros via virtual lo ok-ahead, however, provides

means to achieve p ole-zero comp ensation by moving the zeros close to the p oles instead,

preserving the original p ole lo cations. Consider Fig. 14 with an example of metho d II at

v = 40 m/s, with  =0:5 in the left graph and  =1 in the right graph. The mo di cation

of zero `o' lo cations is illustrated for variation of gain K in virtual lateral acceleration

T

K 1

T

V s V s; 30 V s  =

S T v

T s +1 T s +1

S T 31 µ=0.5 µ=1 5 10 K T = 0

K T = 0.8 K T = 0 5

K T = 0.8 0 0

Imaginary axis Imaginary axis −5

−5 −10 −2 −1.5 −1 −0.5 −5 −4 −3 −2 −1

Real axis Real axis

Fig. 14: Mo di cation of zeros `o' in virtual lateral acceleration transfer function V s  in

v

30 with gain K to comp ensate the p oles `x' in D s  3

T

with T >T and V s denoting the lateral acceleration transfer function for y at the tail

S T T T

sensor T .

For appropriate K v , gain scheduled with sp eed v , the zeros `o' are within the close

T

vicinity of the p oles `x' for the full range of 0:5    1 and V s  turns into a PT transfer

v 1

function, already providing the roll-o necessary for noise attenuation. Henceforth, controller

2

C s  can b e designed indep endently to stabilize the double integrator plant1=s  according

to the p erformance requirements of Section 2.2, considering only the gain variation in V 0

v

o

with resp ect to road adhesion and velo city. The 50 phase lead controller L50 intro duced

in the right Bo de diagram of Fig. 12 now is adequate, since Gs  is not corrupted by the

vehicle dynamics as previously with plain increase of lo ok-ahead to d =5:8m. Also, since

S

the vehicle p ole-pair is not altered in 30 and the vehicle zero-pair is placed in its vicinity,

vehicle damping and hence ride comfort is guaranteed to be similar as in the conventional,

manually steered car.

5 Conclusions

This pap er discussed fundamental issues in lateral control design for automatic steering of

passenger cars within an Automated Highway System AHS. Previous research over more

than two decades employed a variety of reference systems to determine the lateral vehicle

displacement from the road center. Reference systems can be categorized into lo ok-down

and lo ok-ahead systems, dep ending on the p oint of displacement measurement with resp ect

to the vehicle's center of gravity. Among the successful exp erimental implementations of

b oth typ es of reference systems, lo ok-down systems seem to have faced limitations in driving

sp eed, indep endent of the control design. In an e ort to understand the impacts of higher 32

driving sp eed under practical constraints and limitations of an AHS, a detailed analysis

was presented using time and frequency domain to ols together with physical insightinto the

vehicle dynamics. Performance requirements and system constraints such as maximum errors

in the resp onse to a step in road curvature, robustness with resp ect to abruptly changing

road adhesion, ride comfort, measurement noise, and steering actuation considerations were

directly included in the analysis.

The analysis revealed a dramatic change of the relevant lateral vehicle dynamics of lo ok-

down system for increasing sp eed. In particular, p ole and zero lo cations vary such that

damping decreases, for zeros even more than for the p oles. Furthermore, instead of providing

phase lead as for low sp eed, the lateral vehicle dynamics of lo ok-down systems intro duce

signi cant phase lag for higher sp eed, preventing control design entirely based on feedback

of the lateral displacement measurement. Lo ok-ahead systems, on the other hand, provide

sucient phase lead even for higher driving sp eeds by adequately increasing the lo ok-ahead

distance. The resp ective in uence of phase lead from lateral vehicle dynamics and lo ok-

ahead distance led to the formulation of a dual roles concept between the feedback controller

and the vehicle dynamics in general automatic steering control design.

Several design directions were investigated, b oth for lo ok-down and lo ok-ahead systems.

In order to preserve the qualities of lo ok-down systems like reliability and accuracy, the

concept of `virtual lo ok-ahead' via additional yaw error and yaw rate error measurements

was intro duced. E ectively, virtual lo ok-ahead mo di es the system zeros and remedies the

sp eed limitations of lo ok-down systems for lateral control design. A general framework was

established for automatic steering control design based on the detailed analysis in the rst

part of the pap er.

We conclude that prop ortional-typ e feedback design sucies if the lateral vehicle dynamics

itself provide phase lead for stabilization, which is the case for lo ok-ahead systems. On

the other hand, a phase lead controller is required for shorter lo ok-ahead distances. The

metho dology of virtual lo ok-ahead for lo ok-down systems and its b ene ts for control design

were illustrated using examples at v = 40 m/s. Subsequent work is under waytoinvestigate

explicit control designs and their resp ective b ene ts, concentrating on the utilization of

physical system insight for feedback control. A number of control design approaches along

the design directions presented in this pap er are currently b eing validated in exp eriments at

California Path.

Acknowledgements

This work is p erformed as part of the Partners for Advanced Transit and Highways Path

program, prepared under the sp onsorship of the State of California; Business, Transp ortation

and Housing Agency; Department of Transp ortation CalTrans. 33

Wewould like to express our sincere thanks to S. Shladover and W.-B. Zhang of California

Path,M.Tomizuka and K. Hedrick of the University of California at Berkeley,J.Ackermann,



W. Sienel and T. Bunte of DLR Germany, and K. Astrom of Lund University Sweden

for their insightful comments and suggestions.

References

[1] W. Stevens, \The automated highway system program: A progress rep ort", in Preprints

of the 13th IFAC World Congress plenary volume, San Francisco, CA, USA, 1996, pp.

25{34.

[2] Lateral and Longitudinal Control Systems Working Group, \Overview of the R&D and

test on automated highway systems AHS", in Intel ligent Transport Systems, Tsukuba,

Japan, November 1995.

[3] S. Tsugawa, M. Aoki, A. Hosaka, and K. Seki, \A survey of present IVHS activities

in Japan", in Preprints of the 13th IFAC World Congress, Vol. Q, San Francisco, CA,

USA, 1996, pp. 147{152.

[4] J. G. Bender, \An overview of systems studies of automated highway systems", IEEE

Trans. on Vehicular Technology,vol. 40, no. 1, pp. 82{99, 1991.

[5] S.E. Shladover, \Review of the state of development of advanced vehicle control systems

AVCS", Vehicle System Dynamics,vol. 24, pp. 551{595, 1995.

[6] P. Varaiya, \Smart cars on smart roads: Problems of control", IEEE Trans. on Auto-

matic Control,vol. 38, pp. 195{207, 1993.

[7] M. Maurer et al.,\VaMoRs-P: An advanced platform for visual autonomous road vehicle

guidance", in Proc. SPIE Conf. `Mobile Robots IX', Boston, 1994.

[8] B. Ulmer, \VITA - An autonomous road vehicle ARV for collision avoidance in

trac", in Proc. of Intel ligent vehicles '92, Detroit, 1992, pp. 36{41.

[9] B. Ulmer, \VITA II - Active collision avoidance in real trac", in Proc. of Intel ligent

vehicles '94,Paris, 1994, pp. 1{6.

[10] H. Fritz, \Autonomous lateral road vehicle guidance using neural network controllers",

in ECC, Rome, 1995, pp. 285{290 Vol. 3.

[11] T. Jo chem, D. Pomerleau, B. Kumar, and J. Armstrong, \PANS: A p ortable navigation

platform", in Proc. Intel ligent vehicles Symposium, Detroit, MI, USA, 1995, pp. 107{

112. 34

[12] Q.-T. Luong, J. Web er, D. Koller, and J. Malik, \An integrated stereo-based approach

to automatic vehicle guidance", in Proc. of the 5th ICCV, June 1995.

  

[13] K. A. Unyeliog lu, C. Hatip oglu, and U. Ozguner,  \Design and stability analysis of a

lane following controller", IEEE Trans. on Control Systems Technology, to app ear in

1996.

[14] F. Andrew and M. Nakamura, \Lasar radar for a vehicle lateral guidance system", US

Patent no. 592235, pp. 1{32, Oct. 3 1990.

[15] D. Margolis and Y. Yasui, \Automatic lateral guidance control system", US Patent no.

5390119, pp. 1{18, Feb. 14, 1995.

[16] R. E. Fenton, G. C. Melo cik, and K. W. Olson, \On the steering of automated vehicles:

Theory and exp eriment", IEEE Trans. on Automatic Control, vol. 21, no. 3, pp. 306{

315, 1976.

[17] W. Darenb erg, \Automatische Spurfuhrung  von Kraftfahrzeugen in German",

Automobil-Industrie, pp. 155{159, 1987.

[18] R. J. Mayhan and R. A. Bishel, \A two-frequency radar for vehicle automatic control",

IEEE Trans. on Vehicular Technology,vol. 31, no. 1, 1982.

[19] W. Zhang and R. E. Parsons, \An intelligent roadway reference system for vehicle lateral

guidance/control", in Proc. American Control Conf., San Diego, CA, USA, 1990, pp.

281{286.

[20] S.E. Shladover, \The California PATH program and its approach to vehicle-highway

automation", in Proc. Intel ligent Vehicles Symp., Detroit, 1992, pp. 347{352.

[21] R. E. Fenton and R. J. Mayhan, \Automated highway studies at the Ohio State Univer-

sity-Anoverview", IEEE Trans. on Vehicular Technology,vol. 40, no. 1, pp. 100{113,

1991.

[22] J. Guldner, H.-S. Tan, and S. Patwardhan, \Analysis of automatic steering control for

highway vehicles with lo ok-down lateral reference systems", Vehicle System Dynamics,

vol. 26, no. 4, pp. 243{269, 1996.

[23] S. E. Shladover et al., \Automatic vehicle control developments in the PATH Program",

IEEE Trans. on Vehicular Technology,vol. 40, no. 1, pp. 114{130, 1991.

[24] H. Peng and M. Tomizuka, \Vehicle lateral control for highway automation", in Proc.

American Control Conf., San Diego, CA, USA, 1990, pp. 788{794.

[25] P. Riekert and T.E. Schunck, \Zur Fahrmechanik des gummib ereiften Kraftfahrzeugs

in German", Ingenieur Archiv,vol. 11, pp. 210{224, 1940. 35

[26] J. Ackermann, A. Bartlett, D. Kaesbauer, W. Sienel, and R. Steinhauser, Robust control:

Systems with uncertain physical parameters, Springer, London, 1993.

[27] H. Peng, W.-B. Zhang, S.E. Shladover, M. Tomizuka, and A. Arai, \Magnetic-marker-

based lane keeping: A robustness exp erimental study", in Proc. SAE Congress, SAE

paper no. 930556, Detroit, MI, USA, 1993, pp. 127{132.

[28] S. Patwardhan, M. Tomizuka, W.-B. Zhang, and P. Devlin, \Theory and exp eriments of

tire blow-out e ects and hazard reduction control for automated vehicle lateral control

system", in Proc. Int. Symp. on Advanced Vehicle Control, Japan, 1994, pp. 426{431.

  

[29] K. A. Unyeliog lu, U. Ozguner,  J. Winkelman, and T. Hissong, \Nonstandard control

inputs in the integrated design of vehicles", in Proc. American Control Conf., Baltimore,

1994, pp. 816{820.

[30] A. v. Zanten, R. Erhardt, and G. Pfa , \FDR - Die Fahrdynamikregelung von Bosch

in German", Automobiltechnische Zeitschrift,vol. 96, pp. 674{689, 1994.

[31] J. Ackermann, \Robust decoupling, ideal steering dynamics, and yaw stabilization of

4WS cars", Automatica,vol. 30, pp. 1761{1768, 1994.

[32] J. Guldner, S. Patwardhan, H.-S. Tan, and W.-B. Zhang, \Co ding of road information

for automated highways", ITS Journal,vol. submitted, 1996. 36