SHARED VEHICLE CONTROL USING SAFE DRIVING ENVELOPES … · safety like Anti-Lock Braking Systems...

SHARED VEHICLE CONTROL USING SAFE DRIVING

ENVELOPES FOR OBSTACLE AVOIDANCE AND STABILITY

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF MECHANICAL

ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Stephen M. Erlien

March 2015

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/pr371mm0121

© 2015 by Stephen Michael Erlien. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/pr371mm0121

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

J Gerdes, Primary Adviser


Stephen Boyd


Marco Pavone

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

This dissertation is dedicated to my family.

iv

Abstract

Recent trends in automotive crash statistics suggest a dual role of technology in both

saving and threatening the lives of American drivers. Advancements in automotive

safety like Anti-Lock Braking Systems and Electronic Stability Control have led to

a significant reduction in automotive fatalities over the last decade. However, the

ubiquity of technology, mainly the cellular phone, has led to a dramatic increase in

fatalities attributed to distracted driving.

To address this challenge, auto manufacturers are empowering modern vehicles

with even more technology. Advanced sensors provide real-time information about

the surrounding environment. By-wire actuators, which allow drivers indirect com-

mand of the vehicle through an electronic pathway, enable vehicle safety systems to

share control with a driver through augmentation of the driver’s commands. This

technology combination gives safety systems an unprecedented amount of authority

to react to the vehicle’s newly perceived world.

Leveraging these advancements in vehicle actuation and sensing, this dissertation

presents a shared control framework for obstacle avoidance and stability control using

safe driving envelopes. One of these envelopes is defined by the vehicle handling

limits while the other is defined by spatial limitations imposed by lane boundaries

and obstacles. A Model Predictive Control (MPC) scheme determines at each time

step if the current driver command allows for a safe vehicle trajectory within these

two envelopes, intervening only when such a trajectory does not exist. A sparsity

seeking objective in the MPC formulation serves as a simple and effective approach

to shared control between a human driver and an automated machine. In this way,

the controller seeks to identically match the driver’s commands whenever possible

v

while avoiding obstacles and preventing loss of control.

Computationally efficient models of the environment, the vehicle, and the handling

limits allow for real-time prediction of dangerous scenarios over a 4 (s) horizon. This

advanced warning enables the use of brake actuation to ensure safe vehicle trajecto-

ries that adhere to both safe envelopes, providing an envelope of protection through

augmentation of a driver’s steering, braking, and throttle commands. The optimal

control problem underlying the controller is inherently non-convex but is solved as

a set of convex problems allowing for reliable, real-time implementation that is ex-

ecuted at 100 (Hz). This approach is validated on an experimental vehicle working

with human drivers to negotiate obstacles in low friction environments.

vi

Acknowledgment

Writing a PhD dissertation is a humbling experience that provides a unique perspec-

tive on the realization that one’s accomplishments are exceedingly dependent on the

help and support of others. I would like to take a moment to thank some of the people

that helped make this research possible and greatly influenced my time at Stanford.

First of all, I would like to thank my advisor, Chris Gerdes. I never had the

pleasure of having Chris as a professor in class; however, over the last five years,

he taught me many valuable insights into control theory, vehicle dynamics, and life

in general. Chris does an outstanding job of providing incredible resources for his

students, giving me the opportunity to work on exciting projects and exceptional

test beds. His enthusiasm to jump into those test beds and experience the research

results first-hand provided me great encouragement and motivation throughout my

PhD. The healthy work-life balance Chris works hard to maintain provides both a

great lab atmosphere and an inspiring goal for his students. It’s hard to imagine a

better PhD advisor than Chris.

I would also like to thank my defense committee members. Professor Per Enge

enthusiastically agreed to serve as the chair of my committee. His classes on GPS

were some of the most enjoyable I had at Stanford. It was a great pleasure to have

Professor Kochenderfer on my committee, after having read his work on airplane

collision avoidance in the early years of my PhD, which greatly influenced my own re-

search. Serving also as my reading committee, Professor Stephen Boyd and Professor

Marco Pavone significantly expanded my understanding of the technical aspects of

this dissertation. The classes I had with these two professors were some of the most

mathematical courses I have ever taken, enjoyed, and applied to real-world problems.

vii

Next I would like to thank NISSAN MOTOR Co., Ltd. and the project team

members Yoshitaka Deguchi, Hikaru Nishira, and Susumu Fujita for sponsoring this

research. I would like to specifically thank Hikaru Nishira for his previous work

on predictive collision avoidance and his technical guidance throughout this project,

both of which greatly influenced this research. I had the distinct pleasure of working

closely with Susumu Fujita while he was a visiting researcher at Stanford. He was a

great resource, providing invaluable insights from industry and a sounding board for

technical brainstorming.

I would also like to thank the people that had the largest impact on my day-to-day

graduate school experience: the members of the Dynamic Design Lab. Having been

a part of the lab for the better part of a decade, I’ve had the pleasure of watching the

faces in the lab change, but the core values remain the same. The DDL has always

been an exceptionally supportive and welcoming lab, making the otherwise daunting

PhD experience actually a lot of fun. The DDL is full of exceptionally bright people

who are always willing to help out, and there is no way I could have done the cool

things I did in my PhD without their help and support. I would also like to thank

the amazing administrative team behind the scenes keeping the lab running smoothly,

including Adele, Erina, Jo, Elizabeth, and Jennifer.

I would also like to thank my friends at Stanford. I think a school is defined by

the amazing people who attend it, and the friends I’ve met here have far exceeded

even the high expectations I had for Stanford. You are all exceptional people who

played a huge role in my growth and development over the last few years. I would

especially like to thank the Chipotle Crew for your support, encouragement, and

frequent burrito dinners.

I would like to also thank my family, to whom this dissertation is dedicated.

I know it took much effort to continue to be such a large part of my life while I

was many states away from home. The support and guidance you provided was

invaluable. My brothers, Alex and Jake, came out to visit many times throughout

my time at Stanford, bringing a little of home with them every time. My parents,

Karen and Mike, have always been extremely supportive and patient with me. They

are always willing and available to listen to my concerns and struggles, even when

viii

those conversations get exceedingly technical, and provide much needed advice and

encouragement. And lastly, I would like to thank the newest member of my family,

Elisha, who, after building robots with me in my early years at Stanford, eventually

agreed to be my wife. You’ve been such an important pillar of my life in the last six

years. Your overflowing optimism and enthusiasm for learning are infectious, helping

me stay positive and motivated throughout my graduate school career.

Without the support and help of all these exceptional people, this research would

not have been possible.

ix

Contents

Abstract v

Acknowledgment vii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Shared Control of Vehicles . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Envelope Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Dissertation Contributions . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.1 Combined Obstacle Avoidance and Vehicle Stability . . . . . . 12

1.4.2 Convex Approach to Shared Human-Machine Control . . . . . 12

1.4.3 Convex Approximation of Tire Nonlinearity Over a Long Horizon 13

1.4.4 Braking in Response to Steering Infeasibility . . . . . . . . . . 13

1.5 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Safe Envelopes for Shared Steering Control 16

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Velocity States . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2 Position States . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Envelope Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.1 Stable Handling Envelope . . . . . . . . . . . . . . . . . . . . 24

2.3.2 Environmental Envelope . . . . . . . . . . . . . . . . . . . . . 26

2.4 MPC Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

x

2.4.1 Two Time Scale Prediction Horizon . . . . . . . . . . . . . . . 30

2.4.2 Convex Optimization Problem(s) . . . . . . . . . . . . . . . . 31

2.4.3 Varying Time Steps in Prediction Horizon . . . . . . . . . . . 35

2.4.4 Matching the Driver’s Command . . . . . . . . . . . . . . . . 36

2.4.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 Design Decisions, Justifications, and Extensions 47

3.1 Considering Vehicle Shape . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 Terminal Cost on Vehicle Heading . . . . . . . . . . . . . . . . . . . . 56

3.3 Curved Roads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4 Quadratic Environmental Slack Cost . . . . . . . . . . . . . . . . . . 66

3.5 Biasing to Match the Driver . . . . . . . . . . . . . . . . . . . . . . . 69

3.6 Cooperation Between Controller and Driver . . . . . . . . . . . . . . 74

3.6.1 Steering Feel Design for Active Steering . . . . . . . . . . . . 75

3.6.2 Predictive Haptic Feedback . . . . . . . . . . . . . . . . . . . 80

3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4 Predicting Rear Tire Saturation 89

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.2 Modified MPC Plant Model . . . . . . . . . . . . . . . . . . . . . . . 91

4.3 Comparison to Other Approaches . . . . . . . . . . . . . . . . . . . . 94

4.4 Safe Driving Envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.4.1 Stable Handling Envelope . . . . . . . . . . . . . . . . . . . . 94

4.4.2 Environmental Envelope . . . . . . . . . . . . . . . . . . . . . 95

4.5 MPC Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.7 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

xi

5 Envelope Control using Braking and Steering 112

5.1 Challenge of Longitudinal and Lateral Control . . . . . . . . . . . . . 112

5.2 Braking as Steering Feasibility Problem . . . . . . . . . . . . . . . . . 114

5.3 Braking to Prevent Envelope Violations . . . . . . . . . . . . . . . . . 118

5.3.1 Braking Policies . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.3.2 Constant Speed Assumption . . . . . . . . . . . . . . . . . . . 121

5.3.3 Ensuring the Horizon Always Recedes . . . . . . . . . . . . . . 122

5.3.4 Tire Force Coupling . . . . . . . . . . . . . . . . . . . . . . . 125

5.4 MPC Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.4.1 Friction Circle Constraint . . . . . . . . . . . . . . . . . . . . 128

5.4.2 Considering Front Longitudinal Forces Only . . . . . . . . . . 129

5.4.3 Optimal Control Problem . . . . . . . . . . . . . . . . . . . . 131

5.4.4 Feasibility of the Constant Speed Trajectory . . . . . . . . . . 134

5.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.5.1 Proportional Braking Policy . . . . . . . . . . . . . . . . . . . 135

5.5.2 Fixed Braking Policy . . . . . . . . . . . . . . . . . . . . . . . 140

5.6 Extension to Brake- and Throttle-By-Wire . . . . . . . . . . . . . . . 143

5.7 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6 Conclusion 150

6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.1.1 Fully Autonomous Vehicles . . . . . . . . . . . . . . . . . . . . 151

6.1.2 Haptic Feedback User Studies . . . . . . . . . . . . . . . . . . 152

6.1.3 Application to Racing . . . . . . . . . . . . . . . . . . . . . . 152

6.1.4 Implementable Ethics . . . . . . . . . . . . . . . . . . . . . . . 153

6.1.5 Leveraging Advances in Parallel Computing . . . . . . . . . . 153

6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Bibliography 155

xii

List of Tables

2.1 Vehicle Model Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Steering Controller Parameters . . . . . . . . . . . . . . . . . . . . . 38

2.3 X1 Vehicle Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1 Prediction errors for recommended minimum radius curves for a lateral

offset of one lane, e = 3.6 (m) . . . . . . . . . . . . . . . . . . . . . . 65

4.1 Prediction Horizon Parameters . . . . . . . . . . . . . . . . . . . . . . 99

4.2 P1 Vehicle Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.3 Maximum Speed without Collision . . . . . . . . . . . . . . . . . . . 106

5.1 Braking and Steering Controller Parameters . . . . . . . . . . . . . . 133

5.2 Proportional Braking Parameters . . . . . . . . . . . . . . . . . . . . 139

5.3 Fixed Braking Parameters . . . . . . . . . . . . . . . . . . . . . . . . 140

xiii

List of Figures

1.1 Vehicle states and axes . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Bike model schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Brush tire model with affine approximation at α . . . . . . . . . . . . 23

2.3 Stable handling envelope . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Generating the environmental envelope. Start with a) a collection of

obstacles along the reference line, b) discretize in the s direction, c)

extend objects in s direction to align with discretization and identify

feasible gaps between objects, and then d) connect adjacent gaps into

tubes (two of them in this example) which define a maximum (e(k)max)

and minimum (e(k)min) lateral deviation from the reference line at each

time step, k. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Representation of the environment re-evaluated a short time later (a)

without a correction time step and (b) with a correction time step . . 36

2.6 Comparison of the l1 and l2 norms illustrating the different behavior

of the norms at small values of(Fyf,driver − F (0)

yf,opt

)versus larger values 37

2.7 An environment with 3 obstacles arranged in a manner that results in

the worst-case number of possible tubes, 2n = 8. A single trajectory

from each tube is illustrated. Many of the tubes unnecessarily weave

around obstacles and can be ignored without affecting the optimal

steering command of the controller. . . . . . . . . . . . . . . . . . . . 40

2.8 Stanford’s X1, an all electric, throttle- and steer-by-wire research testbed

with automatic brakes and haptic force feedback steering system . . . 40

xiv

2.9 Experiment on a low friction surface with a single obstacle in the middle

of the road with a larger obstacle further down and on the left side of

the road . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.10 Double lane change (ISO 3888-1) on a low friction surface . . . . . . . 44

3.1 Simulation: Conservative behavior at road boundary as a result of the

crude vehicle width model . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Schematic of required distance from vehicle’s CG to environmental

envelope for various vehicle orientations . . . . . . . . . . . . . . . . . 50

3.3 Plot of required distance from CG to environmental envelope for vari-

ous vehicle orientations . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Plot of approximation error, (dL − dL), vs vehicle orientation relative

to the reference line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.5 Simulation: Less intrusive behavior at road boundary when using en-

vironmental constraint (3.6) with ∆ψ0 = 8 [deg]. . . . . . . . . . . . . 54

3.6 Simulation: Controller properly orients the vehicle when confronted

with narrowly spaced obstacles . . . . . . . . . . . . . . . . . . . . . 55

3.7 Simulation: Without a terminal cost, the planned trajectories arc into

the road boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.8 Simulation: With the terminal cost, the planned trajectories take the

form of lane changes. In this example, qT = 50 . . . . . . . . . . . . . 59

3.9 Curvature at point p that is located at a distance s along the reference

path is K(s) = 1R(s)

, where K > 0 for left hand turns . . . . . . . . . 61

3.10 Simulation: Controller augments driver’s command to navigate around

an obstacle while negotiating a turn . . . . . . . . . . . . . . . . . . . 63

3.11 Experiment: Driver bouncing off environmental boundary; l1-norm

slack penalty function leads to harsh interventions by controller . . . 67

3.12 Experiment: Driver bouncing off environmental boundary; quadratic

slack penalty function leads to smooth interventions by controller . . 68

xv

3.13 Simulation: Avoidance scenario with the driver turning into the ob-

stacle and road boundary, with ρ(0) = 1.2 and ρ(i) = 0 for i > 0.

The steering command and predicted trajectory are smooth but the

steering command rarely tracks the driver. . . . . . . . . . . . . . . . 71

3.14 Simulation: Avoidance scenario with the driver turning into the obsta-

cle and road boundary, with ρ(0) = 4.8 and ρ(i) = 0 for i > 0. Steering

matches driver but at the expense of aggressive steering commands. . 72

3.15 Simulation: Avoidance scenario with the driver turning into the obsta-

cle and road boundary, with ρ(0) = 1.2, ρ(1:3) = 0.5, and ρ(i) = 0 for

i > 3. This tuning provides a good balance between driver autonomy

and smooth steering. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.16 Conventional Steering System . . . . . . . . . . . . . . . . . . . . . . 75

3.17 Force Feedback (FFB) Steering System . . . . . . . . . . . . . . . . . 76

3.18 Steering feel during augmentation . . . . . . . . . . . . . . . . . . . . 77

3.19 Experiment: Steering feel emulator using δ, the actual road wheel

angle, to generate the artificial steering feel . . . . . . . . . . . . . . . 78

3.20 Experiment: Steering feel emulator using δdriver, the driver’s com-

manded road wheel angle, to generate the artificial steering feel . . . 79

3.21 Simulation: At t = 1 [s], controller matches driver’s steer command

but plans to augment the command thereafter with almost 2 [deg] of

augmentation planned for the future t = 2 [s] . . . . . . . . . . . . . 81

3.22 Simulation: At t = 2 [s], controller still matches driver’s steer command

despite previously planning an augmentation. The planned trajectory

has increased in severity relative to the planned trajectory at t = 1 [s] 82

3.23 Simulation: Haptic signal generated from (3.34) with Khaptic = 150

[Nm/rad] and khaptic = 4 for single obstacle avoidance scenario using

an open-loop driver model . . . . . . . . . . . . . . . . . . . . . . . . 84

3.24 Simulation: Haptic signal generated from (3.34) with Khaptic = 150

[Nm/rad] and khaptic = 6 for single obstacle avoidance scenario using

an open-loop driver model. Note the start delay in the haptic signal

relative to Figure 3.23 . . . . . . . . . . . . . . . . . . . . . . . . . . 85

xvi

3.25 Experimentation: With predictive haptic feedback enabled, augmen-

tation of the driver’s steer command by the controller is reduced as

the driver and controller cooperate to navigate the environment. Hap-

tic signal was generated from (3.34) with Khaptic = 150 [Nm/rad] and

khaptic = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.1 Bicycle model schematic . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.2 Brush tire model with affine approximation at α . . . . . . . . . . . . 92

4.3 Stable handling envelope . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.4 The environmental envelope is a representation of a) a collection of

obstacles along the nominal path using b) tubes (two of them in this

example) which define a maximum (e(k)max) and minimum (e

(k)min) lateral

deviation from the nominal path at each time step, k. . . . . . . . . . 97

4.5 Stanford’s P1, an all electric, throttle- and steer-by-wire research testbed100

4.6 Arrangement of obstacles and road boundaries used in the following

simulations. Vehicle travels in the direction indicated by the arrow. . 101

4.7 Double lane change maneuver on low friction surface at 12 [m/s] . . . 102

4.8 Double lane change maneuver on low friction surface at 16 (m/s) . . . 103

4.9 Double lane change maneuver on low friction surface at 18 (m/s) . . . 104

4.10 Comparison of the planned safe trajectory midway through the double

lane change manuever on low friction surface at 18 [m/s] . . . . . . . 105

4.11 Stanford’s X1, an all electric, throttle- and steer-by-wire research testbed

with automatic brakes and haptic force feedback steering system . . . 107

4.12 Experiment using X1 in double lane change on low friction µ = 0.55

with no environmental envelope violation . . . . . . . . . . . . . . . . 109

4.13 Experiment using X1 in double lane change on low friction µ = 0.55

with slight violation of the environmental envelope at instance 3 . . . 110

5.1 Controller plans a single lane change in response to blocked lane . . . 115

5.2 Predicted envelope violations for same maneuver at three different ve-

hicle speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

xvii

5.3 Maximum predicted envelope violation for the same maneuver as a

function of vehicle speed . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.4 Block diagram of steering only envelope controller presented in previ-

ous chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.5 Block diagram of braking and steering envelope controller . . . . . . . 120

5.6 Obstacle blocking the lane enters the prediction horizon as the vehicle

travels at 8 (m/s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.7 Braking in excess of (5.4) moves the end of the prediction horizon to

toward the vehicle causing the obstacle to leave the horizon . . . . . . 123

5.8 Brake acceleration required to fix the end of the prediction horizon in

space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.9 Friction circle concept illustrated using a g-g diagram . . . . . . . . . 126

5.10 Coupled tire force model . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.11 Friction circle approximated as the intersection of n = 12 half-spaces 130

5.12 An experimental ISO 3888-2 double lane change on a low friction sur-

face (µ ≈ 0.5) without braking . . . . . . . . . . . . . . . . . . . . . . 136


face (µ ≈ 0.5) with braking in proportion to predicted envelope violation137

5.14 Predicted envelope violation and corresponding predicted rear tire lat-

eral forces at instance 1 from the experiment illustrated in Figure 5.13 138






face (µ ≈ 0.5) with a fixed brake amount in response to predicted

envelope violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.18 A simulated panic brake scenario where a pop-up obstacle appears at

instance 1 causing the driver to immediately steer the vehicle and, at

instance 2, apply aggressive braking in a panic attempt to avoid collision145

xviii

5.19 Friction circle for the front axle during the simulated panic brake ma-

neuver on a low friction surface (µ ≈ 0.5) with Tbrake = 9 . . . . . . . 146

5.20 Friction circle for the front axle during the simulated panic brake ma-

neuver on a low friction surface (µ ≈ 0.5) with Tbrake = 0 . . . . . . . 147

xix

Chapter 1

Introduction

1.1 Motivation

The automobile is the primary means of transportation in the United States, providing

an unprecedented degree of mobility that millions of Americans rely on daily [60]. This

mobility comes at a cost: in 2012, an estimated 5.6 million traffic accidents resulted

in 33,561 traffic fatalities and over 2.3 million injuries in the United States alone [57],

making motor vehicles the leading cause of death for young Americans aged 11 to

27 [30] [57]. This marks an increase in the number of traffic fatalities over the prior

year, which is fortunately only a recent trend. For six years in a row, the overall

number of motor vehicle fatalities in the United States steadily declined, from 43,510

in 2005 to 32,367 in 2011, a historic low. Although many factors contributed to this

decline including safety improvements to roadways, reduction in drunk driving, and

increased use of safety belts, part of the decline is attributed to advances in vehicle

safety systems like Antilock Brakes (ABS) and Electronic Statibly Control (ESC)

[15]. In particular, the National Highway Traffic Safety Administration (NHTSA)

estimates that ESC systems reduce single vehicle car crashes by 36 percent and fatal

rollover crashes by 70 percent [18], and these systems are estimated to have saved

1,144 lives in 2012 alone [58].

Despite these significant gains in vehicle safety, the number of motor vehicle fa-

talities attributed to distracted drivers has risen dramatically in the United States

1

CHAPTER 1. INTRODUCTION 2

over the last decade. This increase is largely attributed to cell phone use, specifically

texting, according to a recent study by Wilson and Stimpson, which estimated that

from 2001 to 2007 texting alone led to 16,000 additional driving fatalities. From 2005

to 2008, when the overall number of fatalities was on the decline, the number of fa-

talities attributed to distracted driving rose 28 percent [75]. Safety systems like ABS

and ESC, which seek to prevent loss of control of the vehicle, can do little to pre-

vent collisions resulting from a distracted driver. In response to this growing trend,

recent Advanced Driver Assistance System (ADAS) developments aim at taking the

surrounding vehicle environment into account to avoid traffic accidents. The first

of these systems utilize sensors like RADAR, LIDAR, and/or cameras to perceive

the environment and employ emergency braking in response to pending collisions

[41]. Manufacturers like Volvo already provide these types of systems in production

vehicles with a focus on collision mitigation [3].

Another new technology further empowering modern vehicles is steer-by-wire.

Steer-by-wire removes the mechanical connection between the hand wheel and the

road wheels, decoupling the driver’s steering command from the actual steer angle of

the vehicle’s front wheels. In addition, the forces felt at the hand wheel are decoupled

from those present at the road wheels, enabling more control over the design of

the steering feel of the vehicle [9]. In 2013, Nissan Motor Co. introduced the first

production vehicle, the Inifiniti Q50, equipped with steer-by-wire with the primary

features of side wind correction and rejection of road noise in the hand wheel [16].

Beyond these initial features, steer-by-wire also provides vehicle control systems the

possibility of unprecedented control over the lateral dynamics of the vehicle. In

addition, tire road friction estimates and vehicle states can be made available in real-

time in vehicles equipped with steer-by-wire as demonstrated by Hsu et al. [39].

Steer-by-wire greatly enhances the actuation and sensing capabilities of a vehicle.

With these capabilities appearing in more modern vehicles, active safety systems

have the potential to make cars much safer.

The best way to utilize these advancements in technology to improve passenger

safety however remains an open-ended question. Fully automated driving is a tempt-

ing solution, but it requires automating the many social and contextually dependent


aspects of driving. Urmson et al. [72] of the winning team of the DARPA Urban

Grand Challenge describe the difficulty in replacing the human driver in a fully au-

tonomous system. They suggest that autonomous systems would benefit significantly

from at least some form of human assistance. Considering the statistics presented at

the beginning of this chapter relative to the miles driven illustrates the exceptional

capabilities of human drivers on the whole: in the U.S. in 2012, the average number

of vehicle miles traveled per fatality was just shy of 90 million [57]. This indicates

the exceptional ability of human drivers in the aggregate. Shared control, in which

a human driver and an automated system work together, presents an opportunity

to retain these critical abilities of human drivers while still leveraging the improved

capabilities of automated vehicle systems.

Humans and automation are uniquely suited for different tasks, and when teamed

up, a synergistic effect is possible [67]. Automated systems can respond precisely and

quickly to well-defined tasks where humans tend to respond less consistently and make

more mistakes as the complexity and the frequency of these tasks increase. On the

other hand, humans have a unique ability to detect and contextualize new patterns

and to reason inductively, whereas automated systems struggle with these tasks [6].

In addition, shared control enables human drivers who enjoy driving to continue to

do so with improved safety. For these reasons, there has been much interest in how

best to share control between a human driver and a highly automated vehicle.

1.2 Shared Control of Vehicles

Prior work in shared vehicle control explores various levels of human involvement,

employing a number of approaches to address the balance between human versus

computer control. In addition, many approaches differ on the assumptions made

about the future behavior or intentions of the driver.

Some of the earliest work in this area focuses on passive guidance. Kawabe et al.

[43] present a Model Predictive Control (MPC) framework that leverages information

about the surrounding environment to generate optimal paths to help guide a human

driver. This optimal path, which involves both longitudinal and lateral maneuvers,


serves as a summary of environmental information that is intuitive to a human driver.

Simulation results validate the approach, and specifics on how the generated paths

could be presented to the driver is left open-ended. This approach provides guidance

to the driver without actively controlling the vehicle and represents an extreme end

of the spectrum of human involvement that leaves the human driver always in full

control of the vehicle.

Another approach is to explicitly switch control between a human driver and an

automated controller. Minoiu-Enache et al. [21] present a lane departure avoidance

system for steer-by-wire vehicles that switches control between a human and a number

of control laws depending on the measured attentiveness of the driver. The steering

torque applied to the hand wheel by the driver indicates the driver’s attentiveness,

although the authors suggest a number of other methods for gauging driver’s atten-

tiveness could be used as well. The system is implemented on a prototype vehicle

equipped with an electric power steering system with the controller influencing the

steer angle of the vehicle through an assistance torque on the steering column. A fo-

cus of the work is on the stability analysis of the switching system with an unknown

human driver in the loop.

Gray et al. [36] use a hierarchical nonlinear MPC (NMPC) approach for path

planning and path tracking that switches control to and from the driver as a function

of driver attentiveness and/or aggressiveness of the planned maneuver. Aggressive

maneuvers are defined by a max front tire slip angle, and the driver’s attentiveness

is monitored using auxiliary vehicle sensors such as in-vehicle cameras. When the

planned optimal trajectory exceeds this aggressiveness threshold or if the driver is

deemed to be inattentive, control switches from the driver to the MPC controller.

The NMPC path planner uses motion primitives, which are generated offline, to build

a dynamically feasible trajectory in real-time to avoid obstacles in the environment.

This path is then tracked by a low level NMPC path tracking controller. Experiments

in a test vehicle on icy and slippery test tracks validate the approach. The motion

primitives in this work address challenges in previous work that rely on a simplified

point-mass model to generate the trajectories. This simplified representation of a

vehicle led to planned trajectories that were not dynamically feasible and led to poor


path tracking performance. However, even with the motion primitive approach, Gray

et al. acknowledge that separating the path tracking from the path following in the

presence of model mismatch and external disturbances allows for tracking errors that

may become large enough to render the planned maneuver infeasible to track.

Instead of switching control, other approaches to shared control involve the inter-

pretation of the driver’s intention and a controller that seeks to track this interpreted

intent. Gao et al. [35] present both a single and a hierarchical NMPC approach

to vehicle control that assume a given trajectory represents the driver’s intent. The

single NMPC approach deviates from the driver’s intended path to avoid obstacles

and is shown to have poor performance or even be unstable without the use of an

invariant set terminal cost or a sufficiently long prediction horizon. The hierarchical

approach includes a second NMPC controller that recomputes a new trajectory in

the presence of obstacles using a simplified point-mass vehicle model. The feasibility

issues resulting from the use of the point-mass model are addressed in a follow up

work that uses a motion primitive based NMPC path planner instead [34]. Although

successful experiments on icy roads demonstrate the effectiveness of this approach

to vehicle stability and obstacle avoidance, the validity of the assumption that a

predefined path represents the driver’s intentions is not addressed.

Saleh et al. [65] present a lane keeping driver assistance system formulated as an

H2-preview horizon optimal control problem using a driver-vehicle-road (DVR) model

to model the driver’s behavior and intentions. The presented DVR model is based

on the hypotheses that drivers use visual information to navigate in their lane. This

model attempts to fully capture many aspects of the human driver including, but not

limited to, processing delays of the human visual system, the neuromuscular system

of the human arm, and human reflex gains and time constants. The parameters of

this driver model are estimated using experiments from multiple drivers on a driver

simulator. A significant portion of the work focuses on robustness guarantees in the

presence of uncertainty in the driver’s behavior and/or mismatch between the actual

driver and the driver model. Experiments on a driving simulator reveal improved

lane keeping performance, and future work will address expanding the driver model

to possibly include driver adaptation to the steering assistance system itself.


Another approach to shared control uses a final steer command which is a blend of

a human driver and an optimal controller. Anderson et al. [6] use a constraint-based,

pathless MPC approach to shared control of a teleoperated ground vehicle where the

controller’s influence on the final command increases with the severity of the predicted

maneuver. The front wheel slip angle defines the severity of the predicted maneuver,

and the final steer command applied to the vehicle is a linear blend of the MPC

optimal command the driver’s command based on this severity metric. When the

MPC optimal peak front slip angle is low, a majority of the final steering command

comes from the driver’s command, and when it is high, a larger portion of the final

steer command comes from the MPC optimal solution. The driver is not restricted

to a predefined path; however, in the presence of obstacles, the controller restricts

the driver to a heuristically defined safe corridor through the obstacles. Experimental

results with an off-road, teleoperated vehicle with simulated losses of communication,

which disrupted the visual feed to the teleoperator, demonstrate the effectiveness of

this approach to shared control of teleoperated vehicles.

In contrast to all of these approaches, the approach proposed in this dissertation

ensures vehicle safety by defining and enforcing safe driving envelopes. The predictive

nature and constraint handling capabilities of MPC make it an attractive framework

for implementing this approach to shared control. The driver’s commands are directly

incorporated into the MPC problem formulation, and matching the driver’s present

command becomes a control objective that is evaluated against the additional ob-

jectives of collision avoidance and vehicle stability. By considering only the driver’s

present command, no model or interpretation of the driver’s intentions is required.

Additionally, no logic or heuristics are required to determine when to switch control

between human and controller; the controller is always in control of the vehicle but is

biased to identically match the driver’s command whenever it is safe to do so. In this

way, the proposed controller implements a form of envelope control, which is charac-

terized by safe regions, or envelopes, of the state space in which a human operator

is free to operate with the controller intervening only to ensure operation remains

within these safe regions.


1.3 Envelope Control

Envelope control is widely used in the aircraft industry. These systems allow pilots

to freely operate the aircraft within a safe operating regime defined by aircraft load,

pitch, bank, and speed limitations [73]. The control system intervenes to prevent

aircraft instability near and beyond the edges of this safe envelope. Both Airbus and

Boeing implement envelope control in their aircraft, but their implementations differ

on the extent of human versus computer control. Boeing’s system makes use of haptic

feedback via the yoke to inform pilots of the safe envelope bounds, but allows the

pilot to override these bounds by applying more force on the yoke. This leaves the

pilot with ultimate control over the aircraft. In comparison, the approach taken by

Airbus implements the safe envelope as hard constraints that the pilot is unable to

override. This leaves the automated system always in control of the aircraft [59].

Stability envelopes have also been applied to the automotive field, and a number of

envelopes have been proposed for use with vehicle stabilization schemes. Inagaki et al.

[40] present one of the earliest analyses of vehicle stability that defines a stable region

with respect to vehicle states in the phase plane. Inagaki chooses vehicle sideslip and

sideslip rate as the phase plane variables for his analysis because of their relatively low

variability with vehicle speed and because vehicle stability is intrinsically related to

the side slip motion of the vehicle. The sideslip of a vehicle, β, is defined as the angle

between the vehicle’s heading and the vehicle’s velocity vector as shown in Figure 1.1.

He proposes an open region between the saddle points of this phase plane as a safe

envelope for vehicle stability, and validates this choice with experimental results using

a direct yaw moment control system to enforce this safe envelope.

Focusing on limiting the saturation of the tires to prevent loss of control, Hsu

and Gerdes [38] propose a vehicle stabilization envelope that limits the peak forces

of the front and rear wheels. A real-time approach to friction estimation in addition

to the proposed envelope proves effective in stabilizing a test vehicle on a limited

friction surface. Building upon this work, Beal and Gerdes [10] present a stable

handling envelope that combines the phase plane design of Inagaki’s approach and

Hsu’s explicit consideration of tire saturation. Beal’s envelope is defined in the phase


Z

XY

r Yaw Rate

Z

XY

!Sideslip V

Figure 1.1: Vehicle states and axes

plane and limits the yaw rate of the vehicle as well as bounds the rear tire forces below

their peak to prevent rear tire saturation. Vehicle sideslip and yaw rate serve as the

phase plane variables that capture this stability envelope. Yaw rate, r, is defined

as the rotational velocity of the vehicle as illustrated in Figure 1.1 and is easier to

measure than sideslip rate. A model predictive controller enforces the envelope using

front steering on vehicles equipped with steer-by-wire.

Bobier and Gerdes [13] also present a closed envelope for safe driving defined in

the sideslip and yaw rate phase plane. Bobier’s envelope is defined using isolines in

the phase plane. Although derived separately, the envelopes of Beal and Bobier agree

quite closely. Bobier’s envelope is slightly larger to incorporate more naturally stable

regions of the phase plane in vehicles with strong under-steering characteristics. Bo-

bier demonstrates this envelope on a steer-by-wire vehicle using a sliding-mode control

framework. Recent applications of MPC for vehicle control enforce constraints on the

maximum allowable rear tire force for stability [42]. In addition to these rear tire

constraints, Turri et al. [71] explores a model predictive controller for path tracking

and stabilization using different tire models throughout the prediction horizon. In


the latter portion of the horizon, the tire models are much more conservative to en-

courage trajectories with a level of robustness with respect to modeling this critically

important property for vehicle stability. In all of these approaches, the nonlinear

tire dynamics present at high slip angles are a dominant consideration for vehicle

stability. The envelopes proposed by Beal and Bobier include additional constraints

beyond tire force limitations that result in closed envelopes which define safe sets.

Safe sets ensure system stability [61] and are also used in aviation envelope design

[5].

The concept of envelope protection can be extended to collision avoidance in ad-

dition to stability. Again, the aviation industry provides an example of this already

in production. Now mandated world-wide on larger aircraft, the Traffic Alert and

Collision Avoidance System (TCAS) provides warnings and guidance to pilots if they

approach too close to other aircraft [47]. Although this system does not actively con-

trol the aircraft, its goal is to ensure safety without interfering with normal, otherwise

safe operations, and can therefore be thought of as a form of envelope protection. The

next generation system slated to become the national standard is called the Airborne

Collision Avoidance System X (ACAS X) and seeks to greatly improve upon the ca-

pabilities of TCAS. In particular, an emphasis of the new system is to reduce the

occurrence of false positive alerts, emphasizing the envelope control paradigm of min-

imum interference while still ensuring safety [44].

With regards to safe envelopes for collision avoidance for ground vehicles, a num-

ber of approaches have been proposed to generate in real-time collision-free trajec-

tories that lie in the obstacle-free regions in space. In the trajectory generation for

Stanley, the winner of the DARPA Grand Challenge, Thrun et al. [69] perform a

2D search over a number of base maneuvers which consist of swerves and nudges.

The chosen trajectory minimizes interference with obstacles, avoids leaving the lane,

and minimizes deviation from the base trajectory while adhering to kinematic and

dynamic constraints of a vehicle model. The base maneuvers and final trajectory are

defined as lateral offsets from a fixed base trajectory that may not be obstacle-free.

Hundelshausen et al. [24] use virtual, tentacle-like structures as constructs for the per-

ception and identification of the safe regions in the environment with these tentacles


also serving as motion primitives in the generation of safe trajectories. Generating

collision-free trajectories by piecing together motion primitives is also the approach

used by Gao et al. [34] and Gray et al. [36] as described previously in Section 1.2.

Attia et al. [7] present an approach to generating collision-free trajectories with-

out the use of pre-computed motion primitives. In their approach, a collision-free

trajectory is generated using parametric cubic splines that ensure a smooth final tra-

jectory. The splines are constrained to a validity area, or reachable driving area within

the vehicle’s lane, to ensure the vehicle does not collide with the environment. The

smoothness of the computed trajectory implicitly adheres to kinematic and dynamic

constraints related to the vehicle.

All of these approaches to collision avoidance use a path planning/path tracking

paradigm. There are fewer examples in the literature of vehicle navigation frame-

works that do not rely on path tracking. The shared control framework presented by

Anderson et al. [6], which was previously discussed in Section 1.2, uses a pathless,

MPC approach that represents safe regions of the environment by subdividing it into

homotropies. These homotropies are defined by the set of trajectories that lies within

them, and they greatly facilitate trajectory generation. Although safe trajectories

are generated, they are not tracked. Rather, the controller continuously responds

to the driver’s inputs by computing a new trajectory at every time step. In this

way, the driver is not restricted to a predetermined path; however, the driver cannot

move between homotropies and is restricted to operate within a single, heuristically

determined homotropy.

In comparing the envelope approaches to stability with the envelope approaches to

collision avoidance, it is observed that the collision avoidance applications commonly

leverage predictive, model-based planning and control techniques. Although these

predictive approaches appear in some stability applications, most notably the work

by Beal and Gerdes [10], almost all the collision avoidance applications require some

model-based planning in order to ensure vehicle safety. The design of the proposed

envelope control framework follows this lead in using model predictive control and

draws much inspiration from the envelope control examples described above.


This dissertation describes an approach to shared vehicle control using two en-

velopes to represent safe driving: one envelope is defined by environmental obstacles

and lane boundaries and the other is defined by the vehicle handling limits. With

regards to the production aircraft envelope controllers, the proposed framework fol-

lows the lead of Airbus, with ultimate control coming from the control system and

not the human operator. This ensures vehicle stability and safety regardless of the

human driver’s input. However, the stability envelope used in the proposed envelope

controller is enforced as a softened constraint to allow for precedence of other objec-

tives, which is similar to the approach of Boeing. In the case of Boeing’s design, the

overriding objective is the autonomy of the human pilots; whereas in the proposed

envelope controller, the overriding objective is collision avoidance. The proposed en-

velope controller builds directly on the work of Beal and Bobier with regards to the

design of the stability envelope. The extension to their work comes in considering

this stability envelope over a much longer horizon. This allows for simultaneous con-

sideration of both vehicle stability and obstacle avoidance. In addition to collision

avoidance and vehicle stability objectives, the proposed envelope control framework

prioritizes minimizing intrusiveness to the driver; the controller intervenes only when

the driver’s steering command does not allow for a safe trajectory within the two safe

envelopes. This follows a key motivation of the next generation aircraft avoidance

system, ACAS X, which seeks to minimize unnecessary interventions. To this end,

the proposed controller allows the driver much freedom before making any change to

the driver’s commands even if this results in subsequent aggressive maneuvers. With

regards to environment representation, the proposed framework was in part inspired

by the approach taken by Anderson et al. in subdividing the world into homotopies

to facilitate efficient trajectory generation. The coordinate system used to model the

motion of the vehicle in space is described as lateral offsets from a fixed base trajec-

tory, or reference line, similarly to the approach taken by Thrun et al. for Stanley’s

path planner. In addition to these aspects that were inspired by previous research,

many facets of the proposed envelope control framework are unique in design.


1.4 Dissertation Contributions

The contributions of this dissertation focus on a number of modeling and control

formulations which enable real-time trajectory optimization with consideration of

driver autonomy, obstacle avoidance, and vehicle stability.

1.4.1 Combined Obstacle Avoidance and Vehicle Stability

The proposed envelope controller implements Model Predictive Control (MPC) and

uses two envelopes to explicitly consider the sometimes competing objectives of vehicle

stability and collision avoidance. The challenge of the different time scales over which

these objectives should be evaluated is addressed using variable length time steps in

the prediction horizon of the MPC implementation. This enables look ahead times

long enough for obstacle avoidance while still capturing the fast dynamics of the

vehicle in the near term without excessive computational burden overall. Applying

an explicit stability envelope over this long horizon enables prediction of and early

adjustments to vehicle stability challenges that arise from environmental conditions.

In addition, the simple representations of the environment and the vehicle’s handling

limits enables fast, reliable real-time implementation as demonstrated on a vehicle

test bed in limited friction environments.

1.4.2 Convex Approach to Shared Human-Machine Control

In the proposed approach, the envelope controller determines if the current driver

command allows for a safe vehicle trajectory within two safe drive envelopes, inter-

vening only when such a trajectory does not exist. Use of a sparsity seeking objective

in the MPC formulation enables a computationally efficient implementation of this

approach to shared control with a human operator. Not only does this provide a min-

imally invasive implementation of envelope control that identically matches the driver

when it is safe to do so, it results in MPC solutions which can be directly used to

produce directional haptic feedback signals. This haptic guidance intuitively commu-

nicates the controller’s intentions to the driver and encourages cooperation between


them as demonstrated in experimentation on real vehicles. This simple and effective

approach to shared control could be more broadly applied to general human-machine

applications.

1.4.3 Convex Approximation of Tire Nonlinearity Over a

Long Horizon

The saturation of the tires is a key consideration in the stability of a vehicle, but

this nonlinearity poses a significant challenge to real-time optimization and control.

Previous work by Beal and Gerdes [10] presented a method for capturing the effects of

tire saturation in a model suitable for real-time optimization; however, with regards

to the rear tires, their approach only works for short time horizons of a few hundred

milliseconds. In this dissertation, the successive linearization technique provides an

affine approximation of the nonlinear tire model throughout a prediction horizon

of multiple seconds. This enables consideration of rear tire saturation in an MPC

implementation suitable for obstacle avoidance in addition to vehicle stability.

1.4.4 Braking in Response to Steering Infeasibility

To ensure vehicle safety in situations where the vehicle’s speed is unsafe for condi-

tions, the proposed control framework incorporates brake actuation ensure vehicle

safety in a wide range of driving scenarios. The challenging task of combined lateral

and longitudinal control is cast as a much simpler feasibility problem whose solution

is the maximum safe vehicle speed for the given combination of road conditions and

environmental hazards. In this way, the controller determines the amount of longi-

tudinal force that can be safely commanded at the current time step, ensuring that

braking to prevent long-term envelope violations does not produce new violations in

the near-term. In addition, this desired longitudinal force can come directly from the

driver in vehicles equipped with brake- and throttle-by-wire, creating a comprehen-

sive envelope controller capable of ensuring vehicle safety through augmentation of

the driver’s steering, throttle, and brake commands.


1.5 Dissertation Outline

This dissertation presents the development, analysis, and validation of the model

predictive envelope control framework introduced in this chapter. The remaining

chapters are organized as follows:

Chapter 2: Safe Envelopes for Shared Steering Control

Chapter 2 provides an overview of the envelope control framework used throughout

this dissertation. This chapter focuses on the definitions of the safe envelopes and the

formulation of the control problem as a set of convex optimization problems that can

be solved reliably in real-time. Experiments using vehicle test beds provide validation

of the proposed approach to shared steering control.

Chapter 3: Design Decisions, Justifications, and Extensions

Chapter 3 introduces a number of additional complexities to the basic steering

only framework presented in Chapter 2 and addresses both theoretical and practical

implementation details. Motivated by simulation and experimental results, these

additions do not change the underlying formulation introduced in Chapter 2, but

provide improved performance at the expense of additional complexity.

Chapter 4: Predicting Rear Tire Saturation

The nonlinear nature of tire dynamics poses a challenge in predicting and modify-

ing vehicle behavior in real-time. Chapter 4 describes how the successive linearization

technique can approximate nonlinear tire dynamics along the entire prediction hori-

zon without additional computational burden. Enabled by this modeling complexity,

the envelope controller can identify situations in which violation of the safe driving

envelopes is unavoidable using only steering actuation. Simulation and experimen-

tal results demonstrate interesting interactions between the occasionally competing

objectives of vehicle stability and collision avoidance.

Chapter 5: Envelope Control using Braking and Steering

Previous chapters focus on control using steering only; however, situations may

arise in which steering alone cannot ensure safe vehicle operation within both safe


driving envelopes. In these situations, brake actuation is required. In this chapter,

the envelope control problem is cast as a convex feasibility problem with regards to

vehicle speed. Using predicted envelope violations as a feedback signal for longitudinal

actuators, a comprehensive steering and braking envelope controller is presented along

with simulation and experimentation results.

Chapter 6: Conclusion

The dissertation concludes with an evaluation of the envelope control framework

presented in the previous chapters along with a discussion of future developments and

research directions.

Chapter 2

Safe Envelopes for Shared Steering

Control

This chapter introduces the envelope control framework which will serve as the foun-

dation of this manuscript. Although subsequent chapters present additional exten-

sions, the ideas and analysis of Chapter 2 are central and applicable to the remainder

of this dissertation. In this chapter, the envelope controller focuses on a single actua-

tor: front steering. In addition to the safety benefits provided by this single actuator

controller, the steering-only controller presented in this chapter enables the devel-

opment of a more comprehensive envelope controller capable of influencing both the

lateral and the longitudinal dynamics of the vehicle. The basic envelope framework

presented in this chapter will serve as the foundation for this comprehensive envelope

controller. The majority of this chapter has been submitted and is under review for

publication in the IEEE Transactions on Intelligent Transportation Systems in 2014

[22].

2.1 Introduction

The envelope control framework presented in this chapter uses two envelopes to repre-

sent safe driving. One envelope incorporates environmental obstacles and lane bound-

aries and the other is defined by the vehicle handling limits. The driver’s present

16

CHAPTER 2. SAFE ENVELOPES FOR SHARED STEERING CONTROL 17

steering command is directly incorporated into the problem formulation, resulting in

a controller which determines at each time step if the driver’s command allows for a

safe trajectory within these two safe envelopes, intervening only when such a trajec-

tory does not exist. In this way, the driver is allowed to control the vehicle’s steering

so long as his actions will not lead to collision or loss of control. The predictive nature

and constraint handling capabilities of MPC make it an attractive framework for im-

plementing this approach to shared control. The MPC implementation used in this

work makes use of variable length time steps in the prediction horizon to enable look

ahead times long enough for obstacle avoidance while still capturing the fast dynam-

ics of the vehicle in the near term without excessive computational burden overall.

This results in a control scheme that is simple enough for fast, real-time consideration

of vehicle stability and obstacle avoidance which is validated on a vehicle testbed in

limited friction environments.

The remainder of this chapter is structured as follows. Section 2.2 outlines the

vehicle model used by the real-time controller. Section 2.3 derives the safe driving

envelopes and describes the methodologies for generating these envelopes in real-

time. Section 2.4 presents the MPC formulation along with the underlying convex

optimization problem(s) to be solved at each time step. Lastly, experimental results

demonstrate smooth integration of the driver’s and controller’s commands as well as

the combined stabilizing and obstacle avoidance capabilities of the control framework.

2.2 Vehicle Model

The vehicle model used in the MPC controller is a bicycle model with five states: two

velocity states and three position states. In this chapter, front steering is the only

actuator considered, and the vehicle is assumed to be equipped with steer-by-wire

technology which enables the steer angle of the front road wheels (δ) to differ from

the driver’s commanded front steer angle (δdriver) which is inputted to the hand wheel.

Also, the controller shares control with the driver in steering only, leaving the driver in

full control of the vehicle’s longitudinal dynamics. Without a direct influence on the

vehicle’s speed, the controller simply reacts to changes in speed dictated by the driver


ab

r

UUx

y

Fyr

δαf

αrFyf

β

Reference Lines

e∆

Figure 2.1: Bike model schematic

and, for simplicity, assumes the present vehicle speed will be maintained throughout

the prediction horizon. Therefore, the vehicle model used by the controller assumes

a constant longitudinal speed.

2.2.1 Velocity States

The velocity states are sideslip (β) and yaw rate (r) as defined in Figure 2.1. The

vehicle’s sideslip can be expressed as:

β = arctan

(Uy

Ux

)(2.1)

≈ Uy

Ux

(2.2)

where Uy and Ux are the lateral and longitudinal velocities in the body fixed frame,

respectively, and the assumption that Ux � Uy gives the simplified expression.

Assuming Ux is constant, the vehicle’s velocity states have the following equations

of motion:


β =Fyf + Fyr

mUx

− r (2.3)

r =aFyf − bFyr

Izz

(2.4)

where Fy[f,r] is the lateral tire force of the [front, rear] axle, m is the vehicle mass,

Izz is the yaw inertia, and a and b are the distances from the center of gravity to the

front and rear axles, respectively.

The tire slip angle in the front (αf) and rear (αr) can be expressed as:

αf = arctan

(β +

ar

Ux

)− δ

≈ β +ar

Ux

− δ (2.5)

αr = arctan

(β − br

Ux

)≈ β − br

Ux

(2.6)

where small angle approximations give linear expressions. This approximation is

validated by the stability constraints used by the controller as discussed in Section

2.3.1.

The brush tire model proposed by Fiala [29] and presented in the following form

by Pacejka [62] gives a useful model of the relationship between Fy[f,r] and α[f,r]:

Fy =

−Cα tanα + C2

α

3µFz| tanα| tanα

− C3α

27µ2F 2z

tan3 α, |α| < arctan(

3µFz

Cα

)−µFzsgn α, otherwise

= ftire (α) (2.7)

where µ is the surface coefficient of friction, Fz[f,r] is the normal load, and Cα is the

tire cornering stiffness. This tire force model is illustrated in Figure 2.2.

The nonlinearity of the tire forces poses a significant challenge to real-time opti-

mization. To address this challenge, the vehicle model used by the MPC controller


describes the vehicle’s behavior by using tire forces and not steer angles. Front tire

force (Fyf ) serves as the input to the model and is mapped to δ using (2.5) and (2.7):

δ = β +ar

Ux

− f−1tire (Fyf) (2.8)

where real-time estimates of β and r are assumed to be available and f−1tire is computed

numerically and implemented as a 2D lookup table with inputs of rear slip angle and

surface friction estimate. Use of Fyf as the model input allows for a linear vehicle

model which considers front tire saturation.

For the rear tires, a linearization of the brush tire model at a given rear tire slip

angle (αr) models rear tire force (Fyr) as an affine function of αr:

Fyr = Fyr − Cαr(αr − αr) (2.9)

where Fyr = ftire (αr) and Cαr is the equivalent cornering stiffness at αr. This approx-

imation is also illustrated in Figure 2.2. Choosing the current rear slip angle, αr, to

be αr in the initial time steps of the prediction horizon allows the MPC controller

to explicitly consider rear tire saturation in the near-term prediction [10]. This will

be discussed further in Section 2.4. For simplicity, tire model (2.7) is only a function

of slip angle; however, force coupling due to the driver controlled longitudinal force

could be directly included in the tire model as demonstrated in a real-time MPC

scheme by Beal and Gerdes [10].

The equations of motion of the velocity states can now be expressed as affine

functions of the states and input, Fyf :

β =Fyf + Fyr − Cαr

(β − br

Ux− αr

)mUx

− r (2.10)

r =aFyf − b

[Fyr − Cαr

(β − br

Ux− αr

)]Izz

(2.11)


2.2.2 Position States

The position states of the vehicle are all in reference to a reference line that need not

be obstacle-free. These states are heading deviation (∆ψ), lateral deviation (e), and

distance along the path (s) as defined in Figure 2.1.

The equations of motion of the position states can be written as:

∆ψ = r (2.12)

e = Ux sin (∆ψ) + Uy cos (∆ψ) (2.13)

s = Ux cos (∆ψ)− Uy sin (∆ψ) (2.14)

Using small angle assumptions for ∆ψ and β, the above nonlinear equations can be

approximated as linear functions of the vehicle states:

e ≈ Ux∆ψ + Uxβ (2.15)

s ≈ Ux − Uxβ∆ψ

≈ Ux (2.16)

where, for small values of β and ∆ψ, the product β∆ψ ≈ 0. The small angle assump-

tion for β is a weaker assumption because, as described in Section 2.3.1, the controller

bounds sideslip explicitly. The small angle assumption for ∆ψ is a stronger assump-

tion because the controller does not directly bound this state; however, to avoid

collision with the road boundaries, the controller indirectly maintains ∆ψ around

zero as demonstrated in the experimental results.

Combining (2.10), (2.11), (2.12), (2.15), and (2.16), a continuous state-space rep-

resentation of the vehicle model can be expressed as:

x = Ac (αr)x+BcFyf + dc (αr) (2.17)


with

x =[β r ∆ψ s e

]T

Ac (αr) =

− Cαr

mUx

bCαr

mUx2 − 1 0 0 0

bCαr

Izz− b2Cαr

IzzUx0 0 0

0 1 0 0 0

0 0 0 0 0

Ux 0 Ux 0 0

Bc =

[1

mUx

aIzz

0 0 0]T

dc (αr) =[Fyr+αrCαr

mUx− b(Fyr+αrCαr)

Izz0 Ux 0

]Twhere subscript c denotes a continuous time model and Ac (αr) indicates matrix Ac

is linearized around αr.


−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−1

−0.5

0

0.5

1 x 104

Brush tire modelAffine approx

Tire slip angle α

α

Tire

Lat

eral

For

ce F

y

y

-C

F

α

Figure 2.2: Brush tire model with affine approximation at α

Table 2.1: Vehicle Model Notation

Description Symbol UnitsSteer angle δ radLongitudinal Speed Ux m/sLateral Speed Uy m/sSideslip β radYaw rate r rad/sHeading deviation from path ∆ψ radLateral deviation from path e mDistance along the path s mLateral tire force on [front,rear] axle Fy[f,r] NTire slip angle on [front,rear] axle α[f,r] rad


2.3 Envelope Definitions

To ensure safe operation of the vehicle, the controller confines the states of the vehicle

to remain within two safe driving envelopes over a finite prediction horizon. The first

of these is a stable handling envelope that ensures vehicle stability through constraints

on the velocity states. The second is an environmental envelope that constrains the

position states to ensure the vehicle trajectory is collision-free. The definitions of

these envelopes and the methodologies to generate them in real time are presented in

the following sections.

2.3.1 Stable Handling Envelope

The stable handling envelope was originally presented by Beal and Gerdes [10]. How-

ever, other vehicle stability envelopes have been proposed [13] and could be incorpo-

rated into this framework as well.

The stable handling envelope defines limits on the vehicle’s velocity states as

illustrated in Figure 2.3. This envelope is bounded by the steady-state yaw rate and

the rear slip angle at peak tire force. These bounds reflect the maximum capabilities

of the vehicle’s tires; at any point within this envelope, a steering command exists

such that the vehicle can safely remain inside.

−0.2 −0.1 0 0.1 0.2−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Yaw

Rat

e r

Sideslip β

1

2

3

4

Figure 2.3: Stable handling envelope

A steady-state analysis can be used to determine an appropriate bound on yaw


rate that does not exceed the friction capabilities of the vehicle. Considering the

steady-state condition of (2.3), the steady-state yaw rate can be expressed as:

rss =Fyf + Fyr

mUx

(2.18)

Neglecting the effects of weight transfer and assuming zero longitudinal tire forces,

the following relationship holds:

Fyf,max + Fyr,max = mgµ (2.19)

where g is the gravitational constant.

Combining (2.18) and (2.19) gives an expression for the maximum steady-state

yaw rate which defines bounds 2 and 4 in Figure 2.3:

rss,max =gµ

Ux

(2.20)

Another important consideration for vehicle stability is the saturation of the rear

tires. The final two bounds of the vehicle envelope serve to limit the rear slip angle

to the angle at which lateral force saturates (αr,sat). For brush tire model (2.7), this

is expressed as:

αr,sat = arctan

(3mgµ

Cαr

a

a+ b

)(2.21)

For the vehicle parameters and reduced friction surface used in the experiments de-

scribed in Section 2.5, αr,sat = 7.2 (deg), giving validation to the small angle ap-

proximations made in (2.5) and (2.6). Using this expression as a bound on αr, the

following bound on β can be determined from (2.6):

βmax = αr,sat +br

Ux

(2.22)

This maximum sideslip serves as the basis for bounds 1 and 3 in Figure 2.3.

Assuming real-time estimates of µ, r, and Ux are available, the vehicle envelope


described is easily calculated in real time and can be compactly represented as the

following linear inequality for each time step k into the prediction horizon:

Hshx(k) ≤ Gsh (2.23)

with

Hsh =

1 − bUx

0 0 0

0 1 0 0 0

−1 bUx

0 0 0

0 −1 0 0 0

Gsh =

αr,sat

rss,max

αr,sat

rss,max

where subscript sh denotes the stable handling envelope and x(k) indicates the vehicle

state at the kth time step into the prediction horizon.

2.3.2 Environmental Envelope

The environmental envelope is in reference to the same reference line as the position

states of the vehicle model. It is represented as time-varying constraints on e, the

lateral deviation from the reference line. At each time step, the trajectory of the

vehicle over the prediction horizon is constrained to be within this envelope to ensure

the trajectory is collision-free. As mentioned previously, this reference line need not

be obstacle-free; therefore, the environmental envelope may require the vehicle to

deviate from the reference line.

Figure 2.4 illustrates the methodology to generate the environmental envelope

from a collection of obstacles along the reference line as illustrated in Figure 2.4a.


As stated previously, the controller shares control with the driver in steering, leaving

the driver in full control of the longitudinal dynamics of the vehicle. Without a di-

rect influence on the vehicle’s speed, the controller simply reacts to changes in speed

dictated by the driver and for simplicity assumes the present vehicle speed will be

maintained throughout the prediction horizon. This allows for the environment to be

sampled at discrete points along the reference line, which correspond to the vehicle’s

future position k steps into the prediction horizon as illustrated in Figure 2.4b. In

Figure 2.4c, the objects in the environment are extended to align with this discrete

sampling, and feasible gaps between obstacles are identified producing a represen-

tation of the obstacle-free regions of the environment. Feasible gaps are defined as

distances greater than the vehicle width. In his overview of planning algorithms,

LaValle presents methods for decomposing the space around obstacles into simple

cells, which reduce motion planning problems to graph searches. The feasible gaps

described here can be thought of as a cell in a variant of the vertical cell decomposition

described by LaValle [48].

Starting at the vehicle’s current position and moving in the positive s direction,

adjacent feasible gaps are linked using a graph search algorithm to form tubes through

the environment like the two illustrated in Figure 2.4d. To avoid collision with the

environment, the vehicle’s future trajectory needs to be fully contained within one of

these tubes. This concept of feasible tubes has also been used in motion planning for

robotic arms as presented by Suh and Bishop [68].

Each tube defines a bound on e at each time step k and can be compactly written

as the linear inequality:

Henvx(k) ≤ G(k)

env (2.24)


with

Henv =

0 0 0 0 1

0 0 0 0 −1

Genv =

e(k)max − 1

2d− dbuffer

−e(k)min − 1

2d− dbuffer

where the subscript env denotes the environmental envelope, e

(k)max and e

(k)min indicate

the lateral deviation bounds for time step k, d is the vehicle width, and dbuffer speci-

fies a preferred minimum distance between obstacles and the vehicle to ensure driver

comfort. The additional distance of dbuffer also accounts for the additional minimum

gap between obstacles required as the vehicle’s orientation changes. The environmen-

tal envelope is defined as the set of tubes generated as described above, and a vehicle

trajectory is collision-free throughout the prediction horizon if and only if it satisfies

inequality (2.24) for all k for any one tube in the environmental envelope.


a)

Reference lineObstacleLane boundary

c)

d)

= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Feasible gap

Tube

b)

se

k

e (4)min

e (4)max

e (1)max

e (1)min e (4)

mine (4)

max

e (1)max

e (1)min

Figure 2.4: Generating the environmental envelope. Start with a) a collection ofobstacles along the reference line, b) discretize in the s direction, c) extend objectsin s direction to align with discretization and identify feasible gaps between objects,and then d) connect adjacent gaps into tubes (two of them in this example) which

define a maximum (e(k)max) and minimum (e

(k)min) lateral deviation from the reference

line at each time step, k.


2.4 MPC Formulation

The controller’s primary task is to ensure safe vehicle operation within the previously

defined safe driving envelopes. With these objectives met, the controller is designed

to be minimally invasive to the driver while avoiding harsh interventions. The first of

these objectives is modeled in this work as identically matching the driver’s present

steering command, and the latter is modeled as a preference for future trajectories

with smooth steering commands. These objectives can be expressed as an optimal

control problem to be evaluated over a finite prediction horizon. However, the set

of feasible, collision-free trajectories to be evaluated is a non-convex set due to the

presence of obstacles making the underlying control problem difficult to solve directly.

Instead, a simpler, convex sub-problem is solved for each tube in the environmen-

tal envelope, and the solutions to these sub-problems are compared to give the global

optimum to the non-convex control problem underlying the MPC controller. This

is possible because any convex combination of trajectories that are generated from

vehicle model (2.17) and are contained within a tube will also be contained within

that same tube. Therefore, each tube defines a convex set of collision-free trajecto-

ries which enables the use of fast convex optimization techniques to quickly identify

an optimal trajectory within a given tube. Dividing the environment into possible

tubes transforms a non-convex problem into a set of convex problems which can be

quickly solved. These sub-problems could be solved in parallel to further improve

performance.

2.4.1 Two Time Scale Prediction Horizon

The obstacle avoidance objective of the controller necessitates a long enough pre-

diction horizon to safely anticipate upcoming obstacles; however, if the controller

execution time step, ts,MPC, is used as the prediction time step throughout the full

horizon, the total number of time steps required is prohibitively large for real-time

implementation. Alternatively, selecting a large time step for the full horizon de-

grades the controller’s performance in predicting near-term vehicle behavior which is

necessary when reacting to unexpected challenges to vehicle stability. In addition,


it is important to capture the limited slew capabilities of the steering system in the

near-term when attempting to match the driver’s steering command while adhering

to the safe handling envelope.

To address this issue, the prediction horizon used in this work is split into two

portions. The initial portion is comprised of Tsplit small time steps of size ts,MPC = 0.01

(s) to accurately capture near-term vehicle behavior and steering system slew rate

limitations. The latter portion is comprised of larger time steps of size ts,long =

0.2 (s) to extend the horizon to incorporate upcoming obstacles in the long-term.

There are T steps in the complete prediction horizon, which is largely dictated by

computational limitations. These time step sizes provide a good balance between

look ahead distance and environment resolution and are comparable to those used in

other MPC implementations for vehicle control [35].

2.4.2 Convex Optimization Problem(s)

At each execution of the controller, optimization problem (2.25) is solved for each

tube in the environmental envelope, and the optimal input corresponding to the tube

with the lowest objective value is used. As is common with MPC, only the optimal

input for the first step into the prediction horizon, F(0)yf,opt, is applied to the vehicle,

and the optimization problem is re-solved for all tubes at the next time step without

any regard for which tube generated the optimal input previously. Therefore, the

driver is never restricted to a single tube, but is free to switch to whichever tube best

meets the control objectives. In this way, the tube construct is used only to facili-

tate computation of the global solution to the non-convex optimal control problem

underlying the controller, and, therefore, has no affect on the stability properties of

the MPC controller.

The control objectives outlined previously can be expressed as the following re-

ceding horizon optimal control problem:


minimize∣∣∣Fyf,driver − F (0)

yf,opt

∣∣∣ (2.25a)

+∑k

γ(k)(F

(k)yf,opt − F

(k−1)yf,opt

)2

(2.25b)

+∑k

[σsh σsh]∣∣∣S(k+1)

sh,opt

∣∣∣ (2.25c)

+∑l

[σenv σenv](S

(l)env,opt

)2

(2.25d)

subject to x(k+1) = A(k)d x(k) +B

(k)d F

(k)yf,opt + d

(k)d (2.25e)∣∣∣Fyf,opt

(k)∣∣∣ ≤ Fyf,max (2.25f)

Hshx(k+1) ≤ Gsh + S

(k+1)sh,opt (2.25g)

k = 0 . . . (T − 1)

Henvx(l) ≤ G

(l)env + S

(l)env,opt (2.25h)

l = (Tsplit + 1) . . . T∣∣∣Fyf,opt(i) − Fyf,opt

(i−1)∣∣∣ ≤ Fyf,max slew

i = 0 . . . Tsplit (2.25i)

where the variables to be optimized are the optimal input trajectory (Fyf,opt) and

the safe driving envelope slack variables (Ssh,opt, Senv,opt). Tunable parameters in this

optimization problem are γ, which establishes the trade-off between a smooth input

trajectory (2.25b) and matching the driver’s present steering command (2.25a), and

the slack variable costs (σsh, σenv). Instead of enforcing the envelopes as hard con-

straints, the slack variables penalize violations of the envelopes, ensuring optimization

problem (2.25) always has a feasible solution. As a result of the two-part prediction

horizon with different size time steps, different values of γ are used in each portion

of the horizon to give a more uniformly smooth trajectory over the entire horizon, as


given by:

γ(k) =

γnear 0 ≤ k < Tsplit

γlong otherwise(2.26)

where γnear and γlong correspond to the near and long-term portions of the horizon,

respectively, and γnear > γlong.

Cost term (2.25a) expresses the desire to match the driver’s command where

Fyf,driver is the front tire force corresponding to the driver’s commanded front steer

angle, δdriver. Brush tire model (2.7) provides the mapping from δdriver to Fyf,driver:

Fyf,driver = ftire

(β +

ar

Ux− δdriver

)(2.27)

Constraints (2.25g) and (2.25h) enforce the stable handling and environmental

envelopes, respectively. These constraints are softened with slack variables, Ssh,opt

and Senv,opt, to ensure optimization (2.25) is always feasible. With the choice of suffi-

ciently large σsh and σenv, cost terms (2.25c) and (2.25d) encourage zero-valued slack

variables resulting in optimal vehicle trajectories which adhere to both safe driving

envelopes whenever possible. A quadratic penalty function for the environmental

envelope allows for smoother interactions at the envelope boundary and, along with

the relative size of σsh and σenv, establishes a hierarchy between the two safe driving

envelopes. Choosing σenv � σsh enforces a preference for obstacle avoidance over ve-

hicle stability in situations where the controller is forced to choose between adherence

to one envelope in favor of the other as illustrated in the experimental results.

To ease the computation of optimization problem (2.25) and simplify the gener-

ation of the environmental envelope, the environmental envelope is not enforced in

the early portion of the prediction horizon. As will be illustrated in the experimental

results, this assumption does not negatively affect the MPC controller’s performance

because the steering actuation has little influence on the position states of the vehicle

over the short time duration of the initial prediction horizon time steps. Therefore,

even if the environmental envelope was explicitly considered in the near-term horizon,


the controller would have little authority to enforce the envelope over such a short

period of time. Instead, the predictive nature of the MPC controller steers the vehicle

in anticipation of approaching obstacles, guiding the vehicle to be collision-free over

the short initial time steps without directly enforcing the environmental envelope in

the near-term.

Constraint (2.25f) reflects the maximum force capabilities of the front tires and

(2.25i) reflects the slew rate capabilities of the vehicle steering system. The slew rate

constraint is not enforced during the later portion of the prediction horizon because

the larger time steps make such a constraint ill-defined.

The vehicle models used in constraint (2.25e) are zero order hold discretizations

of the continuous vehicle model (2.17) given for the kth time step by the matrix

exponential: A(k)d B

(k)d d

(k)d

? ? ?

= exp

A(k)c B

(k)c d

(k)c

0 0 0

t(k)s

(2.28)

with [A(k)

c d(k)c

]=

[Ac(α

(0)r ) dc(α

(0)r )]

0 ≤ k < Tsplit

[Ac (0) dc (0)] Tsplit ≤ k ≤ (T − 1)

where α(0)r is the current rear slip angle determined by (2.6) from real-time estimates

of β and r. In this way, the vehicle model used in the near-term prediction horizon

is a linearization of the nonlinear rear tire behavior at the current rear slip angle

(α(0)r ) allowing for consideration of rear tire saturation in the near-term horizon. As a

consequence of this changing rear tire model, discretization (2.28) is calculated on-line

by the controller for the vehicle model used in the initial portion of the prediction

horizon. In the remainder of the prediction horizon, the rear tire slip angle is not

known prior to solving optimization (2.25); therefore, a linear rear tire model is used.

The implications of this simplifying assumption are highlighted in the experimental

results.


2.4.3 Varying Time Steps in Prediction Horizon

To enable consideration of approaching obstacles in the long-term without compro-

mising the prediction of velocity states in the near-term, different length time steps are

employed in the prediction horizon of the MPC formulation as described in Sec. 2.4.1.

This results in the controller executing faster than the time step used in the long-term

prediction horizon, and requires a corrective, variable length time step to ensure a

consistent representation of the environment as illustrated in Figure 2.5. Following

the methodology for generating the environmental envelope described in Section 2.3.2,

the boundaries of an obstacle are extended to align with the discretization of the en-

vironment. Figure 2.5a shows the discretization of the same environment a short time

later without an initial correction time step resulting in a different representation of

the obstacle; however, if a correction time step of appropriate length is used initially,

the obstacle representation does not change as illustrated in Figure 2.5b.

This appropriate value of tcorr is computed on-line by the controller using:

ts,corr =

t−s,corr − ∆s

Ux

(t−s,corr − ∆s

Ux

)> ts,MPC

t−s,corr − ∆sUx

+ ts,long otherwise(2.29)

where t−s,corr is the correction time step length on the previous execution of the con-

troller and ∆s is the change in the vehicle’s distance along the reference line since

the previous execution of the controller.

Therefore, the sampling time, ts, used in (2.28) is not constant throughout the

prediction horizon, but is instead given for the kth step into the horizon by:

t(k)s =

ts,MPC 0 ≤ k < Tsplit

ts,corr k = Tsplit

ts,long Tsplit < k ≤ (T − 1)

(2.30)

where ts,MPC is the controller execution time step, ts,long is the longer time step used

in the later portion of the horizon, and tcorr is the corrective, variable length time

step.


ObstacleLane boundary

(b)

(a)

Representationof obstacle

Figure 2.5: Representation of the environment re-evaluated a short time later (a)without a correction time step and (b) with a correction time step

2.4.4 Matching the Driver’s Command

Cost term (2.25a) captures the desire to match the driver’s steering command. This

objective is best expressed using the l1 norm for two reasons. First, the desire to

identically match the driver’s command is better captured by the larger values of the l1

norm at small deviations than higher order norms as illustrated in Figure 2.6. Second,

in situations where significant deviation from the driver’s command is required to

ensure safety, it is desirable for the controller to ignore the driver as much as possible.

The l1 norm provides the best convex approximation to this objective because the

value at large deviations is as small as possible while still being convex as discussed

by Boyd and Vandenberghe [14].

2.4.5 Implementation

Optimization problem (2.25) is a convex quadratic program with a significantly sparse

structure that can be leveraged to produce an efficient solver for real-time implementa-

tion [52]. For this work, CVXGEN, developed by Mattingley and Boyd [51], generates


−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

Fyf,driver

− Fyf,opt(0)

norm

F yf

,driv

er −

Fyf

,opt

(0)

l2 norm

l1 norm

(

(

Figure 2.6: Comparison of the l1 and l2 norms illustrating the different behavior of

the norms at small values of(Fyf,driver − F (0)

yf,opt

)versus larger values


Table 2.2: Steering Controller Parameters

Parameter Symbol Value UnitsSurface friction coefficient µ 0.55 (none)Vehicle longitudinal speed Ux (measured) m

s

Driver comfort distance dbuffer 0.4 mPrediction horizon length T 30 (none)Near-term horizon length Tsplit 10 (none)Controller time step size ts,MPC 0.01 sCorrection time step size ts,corr (computed) sLater horizon time step size ts,long 0.2 sNear-term smooth input weight γnear 5 1

kN2

Long-term smooth input weight γlong 2 1kN2

Stable handing slack weight σsh [60 60][

1rad

srad

]Environmental slack weight σenv 1000 1

m2

Driver’s commanded steer angle δdriver (measured) rad

a custom, primal-dual interior point solver that is implemented on a single core of an

i7 processor utilizing MATLAB’s real-time toolbox. Table 2.2 gives the parameters

used in the controller as implemented in the following experiments. These parameters

give a look ahead time of 3.91 to 4.11 (s) depending on the present value of ts,corr.

These parameters define an optimization problem for each tube that can be solved

on a single core of an i7 processor in less than 5 (ms). The number of optimiza-

tion problems to be solved depends on the number of tubes needed to represent the

environmental envelope.

In the worst-case, the number of tubes needed to represent the environmental

envelope is 2n, with n being the number of obstacles. This occurs in scenarios where

each obstacle presents the controller with a choice of avoidance on the left or on

the right. However, as the number of tubes grows, more of them can be ignored

without affecting the performance of the controller because only the tube with the

lowest optimal solution to optimization (2.25) determines the steering input applied

to the vehicle. As illustrated in Figure 2.7, many of the tubes in these worst-case

environments unnecessarily weave between obstacles resulting in large optimal ob-

jective values to optimization (2.25) due to objective (2.25b). Therefore, heuristics


could be employed to reduce the number of tubes to a representative handful, which

could be evaluated in real time. However, to illustrate the effectiveness of this shared

control approach without this additional complexity, the environments used in the

experimental validation can all be represented using a maximum of two tubes. This

allows for time to evaluate both tubes in less than 10 (ms), which is the execution

time step length of the controller, without having to implement the controller on a

parallel processing system.


ObstacleLane boundary

Tube to evaluateTube to ignore

Figure 2.7: An environment with 3 obstacles arranged in a manner that results inthe worst-case number of possible tubes, 2n = 8. A single trajectory from each tubeis illustrated. Many of the tubes unnecessarily weave around obstacles and can beignored without affecting the optimal steering command of the controller.

Figure 2.8: Stanford’s X1, an all electric, throttle- and steer-by-wire research testbedwith automatic brakes and haptic force feedback steering system


2.5 Experimental Validation

Experiments, using an instrumented test vehicle on a low friction surface, demonstrate

the shared control scheme presented. In these experiments, the controller works with

a human driver to negotiate obstacles along the reference line. The test vehicle is

an all electric, drive-, brake-, and steer-by-wire vehicle called X1 which is shown

in Figure 2.8. The parameters for this vehicle are specified in Table 2.3 and were

obtained using similar techniques as described by Laws et al. [49].

X1 is equipped with an integrated GPS/INS system that provides real-time es-

timates of the vehicle states. In these experiments, obstacles and road boundary

locations are assumed to be known. All of the following experiments took place on

a gravel surface with variable friction. Although real-time friction estimation has

been demonstrated on vehicles equipped with steer-by-wire [39], a fixed estimate of

friction is used for simplicity. During these experiments, the driver maintains full

control of the vehicle’s acceleration and braking, and the hand wheel in the vehicle’s

steer-by-wire system does not incorporate any artificial steering feel.

Figure 2.9 illustrates a scenario in which the controller safely avoids a collision

when the human driver takes no action to do so. In this scenario, a single obstacle

lies in the middle of the road with a larger obstacle further down and on the left side

of the road. As illustrated in the bottom of Figure 2.9, the vehicle’s speed, which

is dictated by the driver, varies throughout the test. The simplifying assumption

that the vehicle is maintaining a constant speed throughout the prediction horizon,

Table 2.3: X1 Vehicle Parameters

Parameter Symbol Value UnitsMass m 1973 kgYaw moment of inertia Izz 2000 kg ·m2

Distance from front axle to CG a 1.53 mDistance from rear axle to CG b 1.23 mWidth d 1.87 mFront cornering stiffness Cαf 100 kN/radRear cornering stiffness Cαr 140 kN/rad


0 20 40 60 80 100

0

2

4

6

8

10

Distance (m)

Dis

tanc

e(m

)

0 1 2 3 4 5 6−5

0

5

10

Stee

r Ang

le, δ

(deg

)

−10 −5 0 5 10

−20

−10

0

10

20

Side Slip, β (deg)

Yaw

Rat

e, r

(deg

/s)

0 1 2 3 4 5 6

14

16

18

Time (s)

Spee

d, U

x(m

/s)

Vehicle PathVehicle WidthObstacles

Stable HandlingEnvelopeActual

1

2

3

4

1

ActualDriver

2

3

4

1 2 3

4

Figure 2.9: Experiment on a low friction surface with a single obstacle in the middleof the road with a larger obstacle further down and on the left side of the road


which was addressed in Section 2.3.2, does not negatively affect the controller’s per-

formance as the controller avoids collision with the environment while maintaining

vehicle stability.

Initially, the controller identically matches the driver’s command because doing

so will not lead to a violation of either safe driving envelope. However, at instance

1 , the driver’s steering command threatens to violate the stable handling envelope,

and the controller deviates from the driver’s command to safely stabilize the vehicle.

The controller continues to counter-steer the vehicle to avoid the approaching obsta-

cle, and, at instance 2 , the controller makes an additional steering adjustment to

enforce both safe envelopes. The controller begins the process of again matching the

driver’s command. The rate of this transition is determined by γnear. At instance 3 ,

the controller makes a quick steering correction to maintain stability of the vehicle

before identically matching the driver once again. This illustrates the importance of

considering vehicle stability when transitioning between an automated system and

a human driver. Despite the approaching parked car, the controller still identically

matches the driver’s command because his command safely avoids the obstacle. This

illustrates the minimally invasive approach of envelope control. After two minor steer-

ing augmentations around instance 4 in response to the stable handling envelope,

the vehicle safely continues down the road unscathed.

Figure 2.10 illustrates the controller’s performance in a scenario where vehicle

stability and obstacle avoidance directly compete. This is illustrated in a double lane

change maneuver which conforms to ISO standard 3888-1 [1] on a low friction surface

at speeds greater than 60 kph. In the beginning of the lane change, the controller

identically matches the driver’s steering command because doing so will not lead to a

violation of either safe driving envelope. At instance 1 , the driver prematurely ini-

tiates the first lane change and the controller makes a slight augmentation to prevent

collision with the environment. The driver’s command successfully completes the first

lane change with only a slight correction made by the controller; however, at instance

2 , the driver again prematurely initiates the lane change forcing the controller to

modify his command. Once the obstacle has safely passed, the controller attempts to

again match the driver’s command at instance 3 ; however, the driver’s command


0 20 40 60 80 100 120 140

0

2

4

6

Distance (m)

Dis

tanc

e(m

)

0 2 4 6 8−10

0

10

Stee

r Ang

le, δ

(deg

)

−10 −5 0 5 10

−20

−10

0

10

20

30

40

Side Slip, β (deg)

Yaw

Rat

e, r

(deg

/s)

0 2 4 6 816

17

18

Time (s)

Spee

d, U

x(m

/s)


ActualDriver


12

3

4

5

1

2 3

45

213

4 5

Figure 2.10: Double lane change (ISO 3888-1) on a low friction surface


is too aggressive and threatens to violate both safe driving envelopes. The controller

responds with a large augmentation of the driver’s command to avoid colliding with

the right road boundary.

At instance 4 , the controller allows the vehicle to operate briefly outside the

stable handling envelope to avoid collision with the road boundary. This illustrates

the capability of this control framework to evaluate the trade-off between vehicle

stability and obstacle avoidance in real time which is determined by the penalty

functions and relative weights on the envelope slack variables. At instance 5 , a

large steering augmentation re-stabilizes the vehicle. Only 0.2 (s) separate instances

4 and 5 , illustrating the importance of a fast controller execution rate in ensuring

vehicle stability. The demanding maneuver presented in Figure 2.10 also illustrates

a limitation of the linear rear tire model used in the long-term prediction horizon.

Without considering the future saturation of the rear tires, the controller attempts

to match the driver at instance 3 . If the driver’s command had been ignored,

violation of the safe envelopes may have been avoided. This can be addressed using

the successive linearizations technique for handling nonlinearities in real-time MPC

as explored by the authors [23]. However, even with the simplifying linear rear tire

model in the long-term horizon, the controller successfully stabilizes the vehicle while

safely navigating the environment in this demanding maneuver at high speed on low

friction.

2.6 Discussion

The constraint-based approach of envelope control enables the controller to ensure

vehicle safety while being minimally invasive to the driver. Experimental results

demonstrate smooth integration of the controller’s and driver’s commands, showing

promise of this approach as a shared control scheme without the need to model

or interpret the driver’s intentions. Dividing the environment into feasible tubes

allows the use of fast convex optimization techniques resulting in a controller that can

evaluate vehicle stability and obstacle avoidance at 100 (Hz) with a 4 (s) prediction

horizon. In addition, dynamic maneuvers in an experimental testbed on a low friction


surface illustrate the combined stabilizing and obstacle avoidance capabilities of the

controller even in the presence of unmodeled disturbances.

Chapter 3

Design Decisions, Justifications,

and Extensions

The safe driving envelope controller presented in the previous chapter achieves the

desired objectives of collision avoidance and vehicle stability as demonstrated in ex-

perimental validation. However, additional complexities can be introduced to expand

the capabilities and performance of the envelope controller. These additional com-

plexities include a more detailed model of the shape of the vehicle and a modified

penalty function to improve performance when the vehicle is operating near the en-

vironmental boundary in real-world experiments. A simple approach to modeling

road curvature will be presented that allows for the use of the envelope controller

on roads that are not straight. An additional terminal cost influences the nature of

the planned vehicle trajectories and improves the performance of the controller. The

last additions focus on improving cooperation between the controller and the human

driver through the use of haptic feedback derived directly from the MPC optimal tra-

jectory. This chapter presents these additional extensions and provides justification

for the added complexity they introduce. All of the modifications described in this

chapter can still be implemented using convex optimization techniques and therefore

do not significantly increase the computational requirements of the controller. This

preserves the fast execution rate that characterizes this control framework.

47

CHAPTER 3. DESIGN DECISIONS, JUSTIFICATIONS, AND EXTENSIONS 48

3.1 Considering Vehicle Shape

In Chapter 2, the collision avoidance criteria used in the controller design required the

distance from the vehicle’s center of gravity (CG) to the environmental envelope to be

greater than a fixed value, effectively modeling the vehicle as a circle. However, few

vehicles are this shape. Therefore, the simple approach shown in Chapter 2 requires

a conservative value for dbuffer in environmental envelope constraint (2.24) to ensure

collision avoidance regardless of the orientation of the vehicle for non-circular vehicles.

For convenience, constraint (2.24) is repeated here:

Henvx(k) ≤ G(k)

env

with

Henv =

0 0 0 0 1

0 0 0 0 −1

Genv =

e(k)max − 1

2w − dbuffer

−e(k)min − 1

2w − dbuffer

This conservative approach may result in unnecessary interventions by the con-

troller as the vehicle approaches the environmental envelope boundary. A simulation

of a driver attempting to merge close to the road boundary illustrates this effect as

shown in Figure 3.1. As a result of the conservative model of the vehicle’s width, the

controller unnecessarily augments the driver’s steer command, limiting the driver’s

autonomy even though her steering commands would not have resulted in collision or

loss of control.

Instead of using a fixed value for the required distance from the vehicle’s CG to the

environmental envelope in constraint (2.24), the dependency on vehicle orientation

can be directly modeled. Figure 3.2 illustrates the relationship between the required

distance between the vehicle’s CG and the environmental envelope as a function of


5 10 15 20 25 30 35 40 45−5

0

5

10

[m]

[m]

Environmental Envelope

0 0.5 1 1.5 2 2.5 3 3.5 4−20

−10

0

10

20

δ [

deg

]

[s]

Steering Command

Driver Cmd

Actual Cmd

Figure 3.1: Simulation: Conservative behavior at road boundary as a result of thecrude vehicle width model


dFL dRL=dFL dRL

Figure 3.2: Schematic of required distance from vehicle’s CG to environmental enve-lope for various vehicle orientations

vehicle orientation relative to the reference line. When considering collision on the

left side of the vehicle, both the front-left and rear-left corners of the vehicle need to

be considered. The distance of these points from the vehicle’s CG in the direction

perpendicular to the road are nonlinear functions of the vehicle heading deviation,

∆ψ, as given by:

dFL =

[(w2

)2

+ a2

] 12

sin[∆ψ + arctan

( w2a

)](3.1)

dRL =

[(w2

)2

+ b2

] 12

sin

[∆ψ + arctan

(w

−2b

)](3.2)

where dFL and dRL correspond to the distances from the vehicle’s CG to the front-left

and rear-left corners of the vehicle, respectively.

As illustrated in Figure 3.2, the minimum distance from the environmental enve-

lope occurs when the vehicle is oriented with the reference line. In this orientation,

dFL = dRL = 12w. However, as the orientation of the vehicle changes in either di-

rection, this required distance increases. To ensure the left side of the vehicle does

not collide with the environmental envelope for all vehicle orientations, the maximum

distance of the two corner points must be considered. This maximum distance on the

left side of the vehicle, dL, is given by:

dL = max(dFL, dRL) (3.3)


−100 −80 −60 −40 −20 0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Heading deviation, ∆ψ [deg]

[m]

dL

dFL

dRL

a

b

w/2

Figure 3.3: Plot of required distance from CG to environmental envelope for variousvehicle orientations

and is illustrated in Figure 3.3.

With a small angle assumption for ∆ψ, equations (3.1) and (3.2) can be approxi-

mated as linear over the range of expected heading deviations. Therefore, a piece-wise

linear approximation of dL can now be expressed as:

dL = max (λFL (∆ψ −∆ψ0) + d0,FL, λRL (∆ψ + ∆ψ0) + d0,RL) (3.4)

where ∆ψ0 is the linearization point for this approximation and λFL and λRL are the

slopes of this approximation for the front and rear of the vehicle, respectively. In

general, ∆ψ0 should be small since ∆ψ will be near zero with the envelope controller


−100 −50 0 50 100

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Heading deviation, ∆ψ [deg]

Appro

xim

ation e

rror

[m]

∆ψ0 = 0 [deg]

∆ψ0 = 15 [deg]

∆ψ0 = 25 [deg]

Figure 3.4: Plot of approximation error, (dL − dL), vs vehicle orientation relative tothe reference line

safely guiding the vehicle down the road; however, ∆ψ0 can serve as a tuning param-

eter to shape the approximation error as illustrated in Figure 3.4. Larger values of

∆ψ0 improve the range over which dL provides a useful approximation for dL, and

smaller values reduce the approximation error when the vehicle is oriented with the

road, ∆ψ = 0. Regardless of the choice of ∆ψ0, dL provides an upper-bound for dL

for all vehicle orientations.

Due to the symmetry of the vehicle, a similar approximation for the equivalent

distance on the right side, dR, is given by:

dR = max (λFL (−∆ψ −∆ψ0) + d0,FL, λRL (−∆ψ + ∆ψ0) + d0,RL) (3.5)


These expressions for dL and dR can be used to augment constraint (2.24) to give an

environmental envelope constraint which considers vehicle orientation:

Henvx(k) ≤ G(k)

env −max(FFLx

(k) + JFL, FRLx(k) + JRL

)(3.6)

with

FFL =

0 0 λFL 0 0

0 0 −λFL 0 0

JFL =

−λFL∆ψ0 + d0FL− 12w

−λFL∆ψ0 + d0FL− 12w

FRL =

0 0 λRL 0 0

0 0 −λRL 0 0

JRL =

λRL∆ψ0 + d0RL− 12w

λRL∆ψ0 + d0RL− 12w

and Henv and Genv are similarly defined as in (2.24). Constraint (3.6) consists of a

linear term less than a convex expression. This results in a convex constraint which

can be easily incorporated into optimal control problem (2.25), which was described

in Chapter 2.

Equipped with this new version of the environmental envelope constraint, a less

conservative value for dbuffer is required to ensure collision avoidance and the controller

behaves less conservatively near the road boundary. This is illustrated in Figure 3.5

where this improved environmental envelope constraint is applied to the identical

merge scenario presented in Figure 3.1.

In addition, environmental constraint (3.6) enables the controller to appropriately

orient the vehicle when confronted with narrowly spaced obstacles, as illustrated in

Figure 3.6.


5 10 15 20 25 30 35 40 45

0

5

10

[m]

[m]


0 0.5 1 1.5 2 2.5 3 3.5 4−20

−10

0

10

20

δ [

deg

]

[s]

Steering Command

Driver Cmd

Actual Cmd

Figure 3.5: Simulation: Less intrusive behavior at road boundary when using envi-ronmental constraint (3.6) with ∆ψ0 = 8 [deg].


5 10 15 20 25 30 35

−5

0

5

[m]

[m]


0 0.5 1 1.5 2 2.5 3 3.5 4−20

−10

0

10

20

δ [

deg

]

[s]

Steering Command

Driver Cmd

Actual Cmd

Figure 3.6: Simulation: Controller properly orients the vehicle when confronted withnarrowly spaced obstacles


3.2 Terminal Cost on Vehicle Heading

Model predictive controllers typically include a terminal cost on the system state at

the end of the horizon to drive the system to an invariant set. This provides sys-

tem stability in addition to the benefits of infinite horizon optimal control in a finite

horizon implementation as described by Mayne [53] and many others. The stable

handling envelope (2.3.1) was designed to be an invariant set with regards to the

velocity states of the vehicle to ensure stability and feasibility as described by Beal

and Gerdes [10] with a proof appearing in Beal’s PhD Thesis [11]. In theory, the

stable handling envelope would only have to be enforced as a terminal constraint to

achieve the stability guarantee. However, in practice, it is enforced throughout the

prediction horizon to ensure yaw stability even in the presence of un-modeled distur-

bances and mismatch between the vehicle plant and the vehicle model as illustrated

in the experimental results presented in Chapter 2.

The environmental envelope does not define an invariant set. If the state of the

vehicle at the final step of the prediction horizon adheres to the environmental enve-

lope, no guarantees are made that future vehicle states beyond the prediction horizon

will also do so. This is illustrated in Figure 3.7 where the solution to the optimal con-

trol problem is illustrated as predicted vehicle states along the reference line. Even

though all points along the prediction horizon safely adhere to the environmental

envelope, it is clear that if the vehicle were to follow this trajectory it would most

certainly collide with the road boundary at some time just beyond the prediction

horizon. Instead, this planned trajectory will have to be repeatedly modified during

subsequent executions of the controller as the prediction horizon recedes further down

the reference line.

Enforcing a terminal cost on the heading deviation, ∆ψ, of the vehicle over the

final steps in the prediction horizon improves upon this situation. This additional

cost term can be expressed as:

∑j

qT

(∆ψ(j)

)2(3.7)


5 10 15 20 25 30 35 40 45

−10

−5

0

5

[m]

[m]


0 1 2 3 4 5 6 7 8

−5

0

5

δ [

deg

]

[s]

Steering Command

Driver Cmd

Actual Cmd

Actual Traj

Safe Short Term Traj

Safe Long Term Traj

Figure 3.7: Simulation: Without a terminal cost, the planned trajectories arc intothe road boundaries


where j = (T − 3)...T and qT is chosen so that this terminal cost objective dominates

the smooth steering and driver autonomy objectives discussed in Section 2.4.2 of

Chapter 2.

Terminal cost (3.7) is enforced over the last three time steps. This ensures the

terminal state of the vehicle is safely traveling tangent to the path. The condition

∆ψ = 0 only in the final step of the horizon does not necessarily ensure the terminal

state of the vehicle is safely traveling tangent to the path because the vehicle could

have a large lateral velocity, resulting in a similar situation as illustrated in Figure 3.7.

A temping alternative is to impose a terminal cost on β; however, when operating on

curved roads, this is not the desired terminal condition as the vehicle is expected to

have a non-zero, steady-state side slip.

Figure 3.8 illustrates the effect of terminal cost (3.7) on the optimal trajectory

when the controller is confronted with the the same environment as Figure 3.7. If

the vehicle were to follow this new trajectory, it would remain collision free even

after reaching the end of the prediction horizon. In addition, the predicted trajectory

takes the form of a lane change. Lane changes are a much more intuitive avoidance

maneuver than the trajectory given in Figure 3.7 and appear extensively in vehicle

control literature, as presented by Thrun et al. for the base maneuvers used in

Stanley’s path planner [69] and by Funke and Gerdes for autonomous vehicle control

at the limits of handling [32].


5 10 15 20 25 30 35 40 45

−10

−5

0

5

[m]

[m]


0 1 2 3 4 5 6 7 8−20

−10

0

10

20

δ [

deg

]

[s]

Steering Command

Driver Cmd

Actual Cmd

Actual Traj


Safe Long Term Traj

Figure 3.8: Simulation: With the terminal cost, the planned trajectories take theform of lane changes. In this example, qT = 50


3.3 Curved Roads

For simplicity, the environments presented in Chapter 2 were all set on straight roads

using a reference line with no curvature. However, the optimal control problem can

be adapted to consider curved roads by incorporating reference paths with nonzero

curvature. Figure 3.9 illustrates a reference path with nonzero curvature. The cur-

vature at a point that is located at a distance s along the reference path, K(s), is

defined as the reciprocal of the radius R of curvature, which is radius of the osculating

circle at that point. This curvature profile parameterizes the reference path.

Given a starting position in global coordinates (E0, N0) and an initial heading

angle (ψ0), the position and heading angle along the path can be computed from this

parameterized curvature:

ψ(s) =

∫ s

0

K(x)dx+ ψ0 (3.8)

E(s) =

∫ s

0

cos (ψ(x)) dx+ E0 (3.9)

N(s) =

∫ s

0

sin (ψ(x)) dx+N0 (3.10)

where ψ, E, and N are also illustrated in Figure 3.9.

The equations of motion of the position states relative to a reference path with

nonzero curvature can be written as:

∆ψ = r −K(s)Ux (3.11)

e = Ux sin (∆ψ) + Uy cos (∆ψ) (3.12)

s = (Ux cos (∆ψ)− Uy sin (∆ψ))

(1

1−K(s)e

)(3.13)

Using small angle assumptions for ∆ψ and β as was done in Chapter 2, these nonlinear

equations of motion can be approximated as:

e ≈ Ux∆ψ + Uxβ (3.14)

s ≈ Ux

(1

1−K(s)e

)(3.15)


Reference Path

R

s

e

p

E

Nψ

Figure 3.9: Curvature at point p that is located at a distance s along the referencepath is K(s) = 1

R(s), where K > 0 for left hand turns

where, for small values of β and ∆ψ, the product β∆ψ ≈ 0.

When the vehicle travels along the path, i.e. e = 0, only state update equation

(3.11) differs from the formulation presented in Chapter 2. This modified equation has

an affine term that introduces road curvature as a known disturbance. This approach

to modeling road curvature appears in other model predictive approaches to vehicle

control as described by Turri et al. [71]. If the future speed profile of the vehicle

is known apriori, as was assumed in Chapter 2, and the vehicle travels along the

reference trajectory with e = 0 throughout the prediction horizon, the curvature at

each point along the prediction horizon is known prior to solving the optimal control

problem. Therefore, state update equation (3.11) is readily incorporated into the

affine time-varying vehicle model used in the optimal control problem formulation

presented in Chapter 2.

Incorporating road curvature in this way requires no modification to the environ-

mental envelope definition or generation, but does require mapping obstacle and road

boundary locations to e and s coordinates of a curved reference path. In the case of

a zero curvature reference path, this mapping requires only a simple linear transfor-

mation. In the case of a curved reference path, this mapping is more complicated,

but algorithms exist to do this in real-time as demonstrated by Rossetter et al. [64].

If the vehicle travels with nonzero lateral error, i.e. e 6= 0, state update equation

(3.15) differs from the zero curvature version presented in Chapter 2. Unfortunately,


equation (3.15) cannot be directly incorporated into the optimal control formulation

because, as described in Section 2.3.2, generation of the environmental envelope re-

quires knowledge of the vehicle’s future location along the reference path prior to

solving the optimal control problem. This is not possible because equation (3.15) is a

function of the vehicle’s future lateral offset which is not known prior to solving the

optimal control problem. Therefore, an approximation of equation (3.15) is required.

One approach is to assume e = 0 throughout the prediction horizon, reducing

equation (3.15) to the form presented in Chapter 2:

sapprox ≈ Ux (3.16)

If the vehicle remains near the reference path, i.e. e ≈ 0, this assumption works

well. Figure 3.10 illustrates the type of maneuvers the controller is capable of planning

using curved reference paths and the assumption that e = 0. In this simulation, the

controller plans an avoidance trajectory around an obstacle while negotiating a turn.

This approach is validated with experimental results as presented by Funke et al.

[33], who applied this modified framework to a fully autonomous vehicle operating

on curved paths.

If the vehicle deviates from the path, as would be expected when sharing control

with a human driver, this approximation results in an error in the predicted distance

of the vehicle along the reference path throughout the prediction horizon:

serr = sapprox − s (3.17)

≈ Ux − Ux

(1

1−K(s)e

)(3.18)

This error causes the controller to inaccurately predict the distance to upcoming

obstacles. To quantify the magnitude of this error, simplifying assumptions that the

road curvature is constant and the lateral offset is constant throughout the prediction

horizon allow error term (3.18) to be easily integrated over the full prediction horizon.


−20 −10 0 10 20 30 40 50

60

70

80

90

[m]

[m]


0 0.5 1 1.5 2 2.5 3−20

−10

0

10

20

δ [

deg

]

Steering Command

Driver Cmd

Actual Cmd

Actual Traj


Safe Long Term Traj

Figure 3.10: Simulation: Controller augments driver’s command to navigate aroundan obstacle while negotiating a turn


This gives the longitudinal position error at the end of the horizon:

serr =

∫ tPH

0

(sapprox − s) dt (3.19)

≈∫ tPH

0

Ux − Ux

(1

1−Ke

)dt (3.20)

≈∫ tPH

0

Ux

(−Ke

1−Ke

)dt (3.21)

≈ Ux

(−Ke

1−Ke

)tPH (3.22)

where tPH is the time length of the prediction horizon and serr is the error in predicted

vehicle position along the reference path at the end of the horizon. As seen in equation

(3.22), this prediction error is a function of vehicle speed Ux, reference path curvature

K, lateral offset from the reference path e, and prediction horizon length tPH.

Using the actual distance along the prediction horizon:

sactual = Ux

(1

1−Ke

)tPH (3.23)

(3.24)

a prediction error percentage can be computed:

serr,% = |100(serr/sactual)| (3.25)

≈ |100(Ke)| (3.26)

To get a sense of the magnitude of this error in typical driving scenarios, Table 3.1

provides minimum radii of curvature recommended by the American Association of

State Highway and Transportation Officials for the design of U.S. roadways for a

range of vehicle speeds. Table 3.1 additionally provides the corresponding prediction

error and error percentage at each speed/curvature combination assuming the same 4

(s) prediction horizon length used in Chapter 2 and a lateral offset of 3.6 (m), which

is the typical width of a highway lane in the U.S. [4].

As seen in Table 3.1, at speeds greater than 14 (m/s) the recommended minimum


Table 3.1: Prediction errors for recommended minimum radius curves for a lateraloffset of one lane, e = 3.6 (m)

Vehicle Minimum Radius Prediction Prediction ErrorSpeed Ux of Curvature R Error serr Percentage serr,%

(m/s) (m) (m) (%)6 9 11.4 51.48 26 6.7 20.011 56 4.4 10.014 106 3.1 5.617 167 2.4 3.719 257 2.0 2.522 366 1.6 1.925 510 1.4 1.428 700 1.2 1.131 953 1.1 0.933 1296 0.9 0.736 1774 0.8 0.5

radius of curvature of U.S. roadways is large enough to limit the error in predicted

distance along the reference path to less than 6% for lateral offsets of a single lane.

However, at slower speeds, this error grows significantly as the recommended mini-

mum radius shrinks. This reiterates the trend that the models used in this dissertation

are more accurate the faster the vehicle travels.

Future work will focus on two methods to reduce this approximation error. The

first approach is to assume the vehicle maintains a constant lateral offset from the

path throughout the horizon. In this case, equation (3.15) is approximated as:

sapprox ≈ Ux

(1

1−K(s)e(0)

)(3.27)

where e(0) is the vehicle’s measured lateral offset from the reference path at the

moment the optimal control problem is solved. This approximation ensures accurate

near-term predictions of the distance s to upcoming obstacles even when the vehicle

is not following the reference path. However, approximation errors of the order seen

in Table 3.1 will still result at the end of the horizon for large planned deviations


from the vehicle’s current path offset.

The second method uses a more complicated approximation for state update equa-

tion (3.15) that relies on the predicted future lateral offsets from the previous MPC

optimization to inform the current optimization:

sapprox ≈ Ux

(1

1−K(s)e(k)

)(3.28)

where e(k) is the predicted lateral offset at time step k from the optimal trajectory

computed in the previous controller execution. This approach is similar to the suc-

cessive linearizations technique that will be explored in Chapter 4 to approximate

the nonlinear behavior of tire saturation. When the optimal control problem solu-

tion does not vary significantly between executions of the controller, this provides

an accurate approximation of state update equation (3.15) throughout the prediction

horizon regardless of the vehicle’s lateral offset from the reference path. The difficulty

comes when the solution to the optimal control problem changes drastically between

executions of the controller, as might occur when a new obstacle enters the prediction

horizon. Future work will address these potential challenges.

3.4 Quadratic Environmental Slack Cost

To ensure feasibility of the optimal control problem, state constraints in MPC im-

plementations are often softened using slack variables as described by Maciejowski

[50] and many others. As presented in Chapter 2, the safe driving envelope con-

straints are softened with slack variables to ensure a feasible solution to optimization

problem (2.25) always exists. The l1 norm provides an exact penalty function for

the slack variables which preserves the optimal MPC behavior whenever the state

constraints can be enforced as observed by Oliverira and Biegler [19] and further

examined by Scokaert and Rawlings [66]; therefore, the l1 norm was initially chosen

for the penalty functions for the safe envelope slack variables. However, interactions

with the environmental envelope boundary using the l1 norm results in aggressive

steering interventions as illustrated in Figure 3.11. Human drivers reported these


−320 −300 −280 −260 −240 −220

−125

−120

−115

−110

−105

Distance (m)

Dis

tan

ce

(m

)

0 2 4 6 8 10 12 14−20

0

20

Time (s)

Ste

er

cm

d (

de

g)

0 2 4 6 8 10 12 144

6

8

10

12

Time (s)

Ve

hic

le S

pe

ed

(m

/s)

Start

Vehicle Path

Obstacles

Actual

Driver

Figure 3.11: Experiment: Driver bouncing off environmental boundary; l1-norm slackpenalty function leads to harsh interventions by controller

harsh interventions felt uncomfortable.

To address this issue, a quadratic penalty function is used instead for the environ-

mental slack variables. Quadratic penalty functions are also commonly used in the

literature as described by Scokaert and Rawlings [66] and more recently by Zeilinger

et al. in the context of robust stability [76]. The controller’s performance using the

quadratic penalty functions is illustrated in Figure 3.12. The steering interventions

by the controller are much smoother when the vehicle is up against the environmental

envelope. With this change, human drivers report a more pleasant experience when

the controller intervenes to enforce the environmental envelope.


−300 −290 −280 −270 −260 −250 −240 −230 −220 −210

−135

−130

−125

−120

Distance (m)

Dis

tance (

m)

0 1 2 3 4 5 6 7−10

−5

0

5

Time (s)

Ste

er

cm

d (

deg)

0 1 2 3 4 5 6 74

6

8

10

12

Time (s)

Vehic

le S

peed (

m/s

)

Start

Vehicle Path

Obstacles

Actual

Driver

Figure 3.12: Experiment: Driver bouncing off environmental boundary; quadraticslack penalty function leads to smooth interventions by controller


3.5 Biasing to Match the Driver

A key design consideration for the envelope controller presented in Chapter 2 is the

lack of assumptions about the future behavior of the driver. This avoided the need for

driver models in the problem formulation. To this end, the controller only considers

the driver’s present (and previous) steering commands in the optimal control problem.

However, due to inherent inertia in the steering system and the driver’s self, driver

commands in the near term horizon can be predicted and assumed without imposing

significant limitation on the driver’s autonomy. This section explores the use of these

near term predicted driver commands in the problem formulation.

Assuming a constant speed hand wheel motion is a simple approach to predicting

near term driver commands. Given the driver’s current and previous commands,

future driver commands can be approximated as:

δ(k)driver = δ

(0)driver + k

(δ

(0)driver − δ

(−1)driver

)(3.29)

where δ(k)driver is the predicted driver command at the kth step into the prediction

horizon and δ(0)driver and δ

(−1)driver are the driver’s current and previous steering commands,

respectively. As was presented in Chapter 2, these driver steering commands can be

converted to front tire force commands using (2.27). With these predicted driver

front tire force commands, objective term (2.25a) can be adapted to bias the optimal

trajectory to identically match the driver over several time steps into the prediction

horizon:

∑i

ρ(i)∣∣∣F (i)

yf,driver − F(i)yf,opt

∣∣∣ (3.30)

where ρ(i) is a tunable gain and F(i)yf,driver and F

(i)yf,opt are the driver’s commanded and

the MPC optimal front tire forces at the ith step into the horizon. In (2.25a) presented


in Chapter 2, ρ(i) was implicitly defined as:

ρ(i) =

1 i = 0

0 otherwise(3.31)

The following figures illustrate the effect of ρ(i) on the predicted trajectories and

closed-loop controller behavior. In Figures 3.13 and 3.14, the magnitude of ρ(0) is

varied while ρ(i) = 0 for i > 0. As expected, increasing ρ(0) results in more time spent

matching the driver at the expense of harsher steering commands and aggressive

predicted trajectories up against the environmental envelope. Figure 3.15 illustrates

the use of biasing to the driver further along the prediction horizon with ρ(i) 6= 0

for i > 0. As shown, this approach and tuning provide a good balance between

smooth steering and driver autonomy. In addition, as is explored in the Section 3.6,

this biasing further down the prediction horizon enables the development of haptic

signals to improve cooperation between the driver and controller.


0 10 20 30 40 50 60 70 80

−10

0

10

[m]

[m]


0 1 2 3 4 5 6 7 8−20

−10

0

10

20

δ [

deg

]

FF

B [

Nm

]

[s]

Steering Command

Driver Cmd

Actual Cmd

Actual Traj


Safe Long Term Traj

Figure 3.13: Simulation: Avoidance scenario with the driver turning into the obstacleand road boundary, with ρ(0) = 1.2 and ρ(i) = 0 for i > 0. The steering command andpredicted trajectory are smooth but the steering command rarely tracks the driver.


0 10 20 30 40 50 60 70 80

−10

0

10

[m]

[m]


0 1 2 3 4 5 6 7 8−20

−10

0

10

20

δ [

deg

]

FF

B [

Nm

]

[s]

Steering Command

Driver Cmd

Actual Cmd

Actual Traj


Safe Long Term Traj

Figure 3.14: Simulation: Avoidance scenario with the driver turning into the obstacleand road boundary, with ρ(0) = 4.8 and ρ(i) = 0 for i > 0. Steering matches driverbut at the expense of aggressive steering commands.


0 10 20 30 40 50 60 70 80

−10

0

10

[m]

[m]


0 1 2 3 4 5 6 7 8−20

−10

0

10

20

δ [

deg

]

FF

B [

Nm

]

[s]

Steering Command

Driver Cmd

Actual Cmd

Actual Traj


Safe Long Term Traj

Figure 3.15: Simulation: Avoidance scenario with the driver turning into the obstacleand road boundary, with ρ(0) = 1.2, ρ(1:3) = 0.5, and ρ(i) = 0 for i > 3. This tuningprovides a good balance between driver autonomy and smooth steering.


3.6 Cooperation Between Controller and Driver

The envelope controller proposed may be required to significantly augment the driver’s

steering command to ensure the safe operation of the vehicle. During these interven-

tions, the driver may feel disconnected from the vehicle especially if the driver is

unaware of the need for the intervention. This may occur if the driver is distracted

or unfamiliar with nonlinear vehicle dynamics in the case of low friction surfaces or

excessive speeds. However, it is during these exact situations for which the proposed

envelope controller is most valuable; therefore, improving the cooperation between

controller and driver in these situations is important.

The following presents a method of communicating the planned actions of the con-

troller to the driver which is simple and intuitive. This approach leverages hardware

commonly found on vehicles equipped with steer-by-wire: a force feedback (FFB)

steering system. Due to the mechanical decoupling of the hand wheel and the road

wheels in a steer-by-wire vehicle, the steering feel to the driver through the hand

wheel is drastically altered. It has been shown in literature that drivers rely on this

steering feel to perform vehicle maneuvers as demonstrated by Forsyth and MacLean

[31]. To restore the feel of a conventional steering system in a steer-by-wire vehicle,

an artificial steering feel can be created from torques applied to the hand wheel using

the FFB steering system.

The next section describes how a FFB steering system can be used to emulate

the basic components of this artificial steering feel, focusing specifically on design

considerations to appropriately handle situations where large steering augmentations

may result from the envelope controller described in Chapter 2. The last section

presents how little to no modification to the control problem presented in Chapter 2 is

required to generate a signal suitable for haptic feedback to the driver to enable better

cooperation between the driver and the controller. This cooperation is demonstrated

in experimentation at end of this section.


3.6.1 Steering Feel Design for Active Steering

The work presented in this section was done in collaboration with Avinash Bal-

achandran and was originally presented at the ASME Dynamic Systems and Control

Conference in September, 2014 [8].

In a conventional steering vehicle, the torque at the hand wheel (τhw) is generated

from a combination of the dynamics of the steering system and forces acting at the

road wheels as illustrated in Figure 3.16. Balachandran and Gerdes [9] present the

following dynamic model of this system suitable for use in an artificial steering feel

emulator:

τhw = bsteer systemδ + Jsteer systemδ +Ktire moment (τjack + τalign) (3.32)

where bsteer system and Jsteer system are the steering system inertia and damping, respec-

tively, Ktire moment is the tire moment gain, τjack is the jacking torque, and τalign is

the aligning moment. The tire moment gain (Ktire moment) varies for different steering

systems, suspension geometries, and steering ratios.

The jacking torque (τjack) felt at the hand wheel is a function of the front normal

tire forces (Fzf) acting on the vehicle. Jacking torque creates the centering feel for a

driver at low speeds. The aligning moment (τalign) is generated by the front lateral tire

forces (Fyf) acting on the vehicle, and dominates the steering feel at higher speeds.

The steering angle of the front road wheels (δ) directly influences both the jacking

Figure 3.16: Conventional Steering System


Figure 3.17: Force Feedback (FFB) Steering System

torque and the aligning moment felt at the hand wheel.

In a steer-by-wire vehicle, the hand wheel does not have the feel of a conventional

vehicle. Instead, this feel can be emulated using a FFB steering system. Figure 3.17

illustrates the FFB system hardware whose dynamics can be modeled as:

τhw,FFB = τmotor + JFFB systemδhw + bFFB systemδhw (3.33)

where bFFB system and JFFB system are the FFB steering system inertia and damping,

respectively, τmotor is the FFB motor torque, and δhw is the hand wheel angle. It

should be noted that δhw = δdriver/Ksteer where Ksteer is the steering ratio and δdriver

is the road wheel angle commanded by the driver via the hand wheel as previously

described in Chapter 2 (2.2). Combining (3.32) and (3.33) provides an expression for

τmotor which emulates the τhw of the conventional steering system while canceling the

dynamic effects of the FFB system to give an artificial steering feel on a steer-by-wire

vehicle.

When the envelope controller identically matches the driver’s command, the hand

wheel agrees with the road wheel steer angle, i.e. δ = δdriver, and the approach

to emulating steering feel outlined above can be directly applied. However, when

the envelope controller augments the driver’s steering command as illustrated by

Figure 3.18, it is unclear if feeding back the actual road wheel forces to the hand

wheel still provides the driver with useful information.

To evaluate this affect of steering augmentation on steering feel, experiments were

conducted using X1, which is equipped with a FFB steering system that generates


Figure 3.18: Steering feel during augmentation

artificial steering feel at the hand wheel. Precise encoders obtain the hand wheel

angle and road wheel angle while the road wheel angle derivative is approximated

by differencing the encoder signal. Figure 3.19 illustrates performance of the com-

bined envelope controller and steering feel emulator using the conventional steering

feel model. As is shown, feeding back the actual road wheel forces to the hand wheel

generates a hand wheel torque that is in opposition to the action of the envelope

controller, enhancing the disparity between the driver’s and the controller’s steering

commands. Collision with the environment is still safely avoided as the envelope

controller simply commands a larger augmentation to overcome the driver’s conflict-

ing command; however, the driver is receiving mixed signals from the vehicle and

cooperation between driver and controller is not observed.

To resolve these conflicting signals to the driver, the steering feel emulator should

be designed to feedback the forces that would be generated by the driver’s commanded

steer angle (δdriver) rather than the forces generated by the actual steer angle of the

road wheels (δ). Figure 3.20 illustrates the affect of of this modification to the steering

feel emulator during an similar scenario as Figure 3.19. As is shown, generating the

artificial steering feel based on the fictional forces associated with the driver’s steering

command encourages the driver to act in the save direction as the envelope controller,

improving the agreement between driver and controller.


−310 −300 −290 −280 −270 −260 −250 −240 −230−160

−140

−120

[m]

[m]

Vehicle Path

6 8 10 12 14 16 18 20

−5

0

5

Fro

nt T

irefo

rce

[kN

]S

teer

ing

Fee

lT

orqu

e [N

m]

Time [s]

6 8 10 12 14 16 18 20

−10

0

10

Roa

dwhe

elA

ngle

[deg

]

Start

Driver CmdActual CmdSteer Feel

Driver CmdActual Cmd

Figure 3.19: Experiment: Steering feel emulator using δ, the actual road wheel angle,to generate the artificial steering feel


−310 −300 −290 −280 −270 −260 −250 −240 −230 −220−160

−140

−120

[m]

[m]

Vehicle Path

2 4 6 8 10 12 14 16 18

−5

0

5

Fro

nt T

irefo

rce

[kN

]S

teer

ing

Fee

lT

orqu

e [N

m]

Time [s]

2 4 6 8 10 12 14 16 18

−10

0

10

Roa

dwhe

elA

ngle

[deg

]

Start

Driver CmdActual CmdSteer Feel

Driver CmdActual Cmd

Figure 3.20: Experiment: Steering feel emulator using δdriver, the driver’s commandedroad wheel angle, to generate the artificial steering feel


3.6.2 Predictive Haptic Feedback

As demonstrated in the previous section, haptic feedback from a FFB steering sys-

tem can help improve the cooperation between a driver and controller during active

steering interventions. In this section, it will be demonstrated that haptic feedback

can additionally be used to improve driver and controller cooperation prior to aug-

mentation of the driver’s steering command, possibly to an extent where the steering

augmentation no longer becomes necessary.

The envelope controller’s predictive nature and objective to identically match the

driver’s command in the initial steps of the prediction horizon are key properties that

enable predictive haptic feedback. As is commonly done in MPC implementations,

a planned input trajectory over the entire prediction horizon is computed at each

execution of the controller, but only the input on the first time step is actually

applied to the plant with the remaining planned inputs discarded. For the envelope

controller presented in Chapter 2, this remaining input trajectory can be used to

generate a haptic signal to inform the driver of the controller’s planned course of

action. Figures 3.21 and 3.22 illustrate the planned vehicle states and input sequences

of the envelope controller in response to an approaching obstacle at two instances in

time, t = 1 and t = 2.

As seen in these figures, the optimal trajectory computed at t = 1 shows a planned,

non-zero steer command 1 [s] into the prediction horizon, but the optimal steer com-

mand computed 1 [s] later still matches the driver’s command of zero steer angle.

This is expected. At t = 1, the controller makes no assumptions about the future

behavior of the driver, but 1 [s] later new information about the driver’s steering

command becomes available, and this is considered in the generation of the new op-

timal trajectory. Even though at both time instances the optimal steering command

applied to the vehicle is δ = 0, the severity of the optimal input trajectory computed

at t = 2 suggests a pending steering augmentation by the controller. Therefore, it

would be desirable for a predictive haptic signal to be stronger for the case illustrated

in Figure 3.22 than Figure 3.21. A simple approach is to compute the predictive


0 10 20 30 40 50 60

−10

−5

0

5

10

[m]

[m]


0 1 2 3 4 5 6−5

0

5

δ [

deg

]

[s]

Steering Command

Driver Cmd

Actual Cmd

Actual Traj


Safe Long Term Traj

Figure 3.21: Simulation: At t = 1 [s], controller matches driver’s steer commandbut plans to augment the command thereafter with almost 2 [deg] of augmentationplanned for the future t = 2 [s]


0 10 20 30 40 50 60

−10

−5

0

5

10

[m]

[m]


0 1 2 3 4 5 6−5

0

5

δ [

deg

]

[s]

Steering Command

Driver Cmd

Actual Cmd

Actual Traj


Safe Long Term Traj

Figure 3.22: Simulation: At t = 2 [s], controller still matches driver’s steer commanddespite previously planning an augmentation. The planned trajectory has increasedin severity relative to the planned trajectory at t = 1 [s]


haptic feedback torque as:

τhaptic = Khaptic

(δ

(khaptic)opt − δ(khaptic)

driver

)(3.34)

where τhaptic is the predictive haptic feedback torque, Khaptic is a tunable gain, δ(khaptic)

driver

and δ(khaptic)opt are the anticipated driver’s commanded steer angle and optimal planned

steer angle at the khaptic time step into the prediction horizon, respectively. Equation

(3.34) provides a directional haptic signal that nudges the driver to steer in agreement

with the controller. In practice, this signal is also saturated in the interest of driver

comfort as well as to reflect the limited torque capabilities of the haptic feed back

steering system. Figure 3.23 illustrates the time series of the haptic signal generated

using (3.34) with khaptic = 4 for the same obstacle avoidance scenario used in Fig-

ures 3.21 and 3.22. As is shown, the haptic signal nudges the driver in the direction

of the steering augmentation and grows in magnitude as the controller intervention

time approaches. Saturation of the haptic signal is also illustrated.

Choice of khaptic provides another tuning knob in shaping the haptic signal. In-

creasing khaptic provides a subtle distinction in the haptic signal as illustrated in Fig-

ure 3.24. Comparing the haptic signals from Figures 3.23 and 3.24, it is seen that the

larger khaptic produces a stronger and earlier haptic signal. Earlier and strong signals

may be preferred to give advanced warning to the driver and encourage tighter co-

operation between driver and controller. Alternatively, a delayed haptic signal is less

intrusive to the driver. These are common haptic design considerations and the opti-

mal control problem underlying the envelope controller provides flexibility in making

these design decisions.

However, this simulation does not include a model of the haptic signal’s influence

on the driver; therefore, to evaluate the performance with a driver-in-the-loop, exper-

imentation on X1 is conducted with the results presented in Figure 3.25. In this test,

the driver works with the envelope controller and allows the FFB system to guide

him through the environment. Even though augmentations of his steering commands

are still required, they are small and always in the same direction as the driver’s com-

mand. As a result, the envelope controller is effectively applying a positive scaling to


0 10 20 30 40 50 60 70 80

−10

0

10

[m]

[m]


1 2 3 4 5 6 7 8

−10

−5

0

5

10

δ [

deg

]

FF

B [

Nm

]

[s]

Steering Command

Driver Cmd

Actual Cmd

τHaptic

Actual Traj


Safe Long Term Traj

Figure 3.23: Simulation: Haptic signal generated from (3.34) with Khaptic = 150[Nm/rad] and khaptic = 4 for single obstacle avoidance scenario using an open-loopdriver model


0 10 20 30 40 50 60 70 80

−10

0

10

[m]

[m]


1 2 3 4 5 6 7 8

−10

−5

0

5

10

δ [

deg

]

FF

B [

Nm

]

[s]

Steering Command

Driver Cmd

Actual Cmd

τHaptic

Actual Traj


Safe Long Term Traj

Figure 3.24: Simulation: Haptic signal generated from (3.34) with Khaptic = 150[Nm/rad] and khaptic = 6 for single obstacle avoidance scenario using an open-loopdriver model. Note the start delay in the haptic signal relative to Figure 3.23


the steering ratio to avoid the obstacles in the environment and never has to reverse

the steering direction relative to the driver.

Initial simulations and experiments show promise for this technique of deriving

haptic feedback signals directly from the MPC solution. Future work will focus on

user studies to validate the effectiveness of this haptic feedback signal in promoting

cooperation between driver and controller.


−330 −320 −310 −300 −290 −280 −270 −260 −250−130

−125

−120

−115

Distance (m)

Dis

tan

ce

(m

)

0 1 2 3 4 5 6 7 8

−5

0

5

Time (s)

Ste

er

cm

d (

de

g)

0 1 2 3 4 5 6 7 84

6

8

10

12

Time (s)

Ve

hic

le S

pe

ed

(m

/s)

Start

Vehicle Path

Obstacles

Driver

Actual

Figure 3.25: Experimentation: With predictive haptic feedback enabled, augmen-tation of the driver’s steer command by the controller is reduced as the driver andcontroller cooperate to navigate the environment. Haptic signal was generated from(3.34) with Khaptic = 150 [Nm/rad] and khaptic = 4


3.7 Discussion

This chapter presented a number of additional complexities and extensions to the safe

driving envelope controller. Many of these additions were motivated by simulation

and experimental results. Among these are improved modeling of the interaction be-

tween the vehicle and environment when the vehicle is operating near the boundaries

of the environmental envelope, as well as modified penalty functions to improve per-

formance at this boundary in the presence of model/plant mismatch. The addition

of a terminal state cost was motivated by model predictive control theory which im-

proves the stability and leads to planned vehicle trajectories that are intuitive. The

last two additions, curved roads and haptic feedback, required almost no additional

changes to the controller structure, but expanded the types of environments in which

the controller could be applied and improved the cooperation between controller and

driver. Despite the added complexity, the underlying optimal control problem is still

formulated as a set of quadratic programs which can be quickly and reliably evaluated

in real-time.

Chapter 4

Predicting Rear Tire Saturation

The proposed envelope controller has been demonstrated to successfully share control

with a human driver, augmenting the driver’s steering commands to avoid collisions

and prevent loss of control. The extent to which this can be done is limited by the

controller’s ability to anticipate dangerous scenarios in order to appropriately inter-

vene and steer the vehicle to safety. However, the nonlinear nature of tire dynamics

poses a challenge in predicting and modifying vehicle behavior in real-time. This

chapter describes another extension to the proposed envelope controller that uses

an improved method for modeling the rear tire dynamics. Enabled by this model-

ing improvement, the envelope controller can more accurately predict, and possibly

prevent, saturation of the rear tires. As referenced throughout this dissertation, the

nonlinearity of the rear tires plays a central role in vehicle stability, and an improved

model of this physical property improves the performance of the controller in highly

dynamic situations. Like the extensions presented in the previous chapter, this im-

proved modeling approach comes at the cost of complexity. However, simulation

and experimentation demonstrate improved performance that justifies this additional

complexity. In addition, the simulation and experimental results illustrate interest-

ing interactions between the occasionally competing objectives of vehicle stability and

collision avoidance.

This improved modeling technique enables the controller to identify situations

in which violation of the safe driving envelopes is unavoidable using only steering

89

CHAPTER 4. PREDICTING REAR TIRE SATURATION 90

actuation. As will be discussed in Chapter 5, this result establishes the foundation

for a comprehensive envelope controller capable of ensuring vehicle safety through

augmentation of the driver’s steering, throttle, and brake commands. The majority

of this chapter including the simulation results using P1 was originally presented at

the American Control Conference in Portland, OR in June, 2014 [23] with the balance

of the chapter focusing on subsequent experimental validation using X1.

4.1 Introduction

As described in Chapter 2, the rear tire model used by the MPC controller in the

near-term prediction horizon follows the approach presented by Beal and Gerdes [10].

This approach works well for short time scales where the state of the tire does not

change significantly. However, over the course of a few seconds, which is the time

scale required for obstacle avoidance, this assumption does not hold, and the tire state

can change dramatically while maneuvering around obstacles. Therefore, a different

approach is required if the nonlinearity of the rear tires is to be captured throughout

the prediction horizon of the proposed envelope controller.

One approach is to abandon the convexity requirement for the vehicle model and

directly consider the nonlinear tire model in the optimization. This approach comes

at the expense of run-time performance and solver reliability, limiting its use in a

real-time, safety critical application. Another approach approximates nonlinear plant

dynamics through a technique known as successive linearizations. In these schemes,

the controller optimizes the plant’s trajectory using a linear time-varying plant model

derived from the optimized trajectory from the previous time step. This enables

consideration of nonlinear dynamics while retaining the performance and reliability

of convex optimization with affine models.

Falcone et al. [25] use the technique of successive linearizations in a real-time tra-

jectory tracking controller intended for a fully autonomous vehicle. Vehicle stability

is enforced through an ad hoc constraint on maximum front tire slip angle. In their

approach, the entire vehicle model is linearized, requiring a non-trivial amount of com-

putational time to compute the linearizations at each time step. The resulting Linear


Time Varying (LTV) MPC scheme enables a 15x improvement in controller execu-

tion run-time over an equivalent implementation using the nonlinear vehicle model

directly with a nonlinear optimization solver, a Sequential Quadratic Program (SCP)

solver. Building upon this initial work, Falcone et al. [26] proposed a similar LTV

MPC scheme based on successive linearizations that replaced the application specific,

ad hoc stability criteria with a more general stability constraint. This general stabil-

ity criteria involves an additional convex constraint on the states and inputs in the

MPC formulation and is provably stabilizing for a wide range of nonlinear systems.

Applications of the successive linearizations technique also extend beyond real-time

control. For example, Timing and Cole [70] use successive linearizations as an ef-

ficient alternative to nonlinear optimization in an off-line algorithm for generating

racing lines.

In this chapter, the successive linearizations technique improves the rear-tire model

used by the MPC controller, enabling the controller to explicitly consider rear tire

saturation in predicting future vehicle behavior relative to the safe driving envelopes

presented in Chapter 2. Simulation results demonstrate more accurate prediction of

situations in which vehicle stability needs to be sacrificed in order to ensure collision

avoidance. The chapter concludes with experimental results of aggressive maneuvers

that demonstrate the usefulness of this approach in ensuring vehicle safety in highly

dynamic maneuvers in constrained environments.

4.2 Modified MPC Plant Model

The vehicle model used in the online MPC controller is the same constant speed,

planar bicycle model presented in Chapter 2 and illustrated again in Figure 4.1 for

reference. The model modification involves the treatment of the rear tires. Recall

from Chapter 2, a linearization of the brush tire model (2.2) at a given rear tire slip

angle (αr) models rear tire force (Fyr) as an affine function of rear slip angle (αr) as

illustrated in 4.2.

Following the lead of Beal and Gerdes [10], this rear tire model is linearized around

the measured rear tire slip angle, αr, in the near term of the prediction horizon. Over


ab

r

UUx

y

Fyr

δαf

αrFyf

β

Reference Lines

e∆

Figure 4.1: Bicycle model schematic

Brush tire modelAffine approx

Tire slip angle α

α

Tire

Lat

eral

For

ce F

y

y

C

F

α

Figure 4.2: Brush tire model with affine approximation at α


this short horizon, the vehicle states do not change significantly, enabling accurate

prediction of vehicle state propagation in the near term.

The appropriate choice of αr in the remainder of the prediction horizon is the

focus of this chapter. Two possibilities will be evaluated and compared. The first

is simply choosing α(k)r = 0, which results in a linear tire model for time steps k in

the latter portion of the horizon as was used in Chapter 2. The second approach

is to choose α(k)r = α

(k)r , where α

(k)r is the predicted rear slip angle at time step k

from the optimal trajectory computed in the previous controller execution. This use

of successive linearizations provides an approximation to the nonlinear tire behavior

throughout the entire prediction horizon.

For both of these approaches, the equations of motion of the velocity states can

be expressed as affine functions of the states and input:

β =Fyf + Fyr − Cαr

(β − br

Ux− αr

)mUx

− r (4.1)

r =aFyf − b

[Fyr − Cαr

(β − br

Ux− αr

)]Izz

(4.2)

The position states of the vehicle are defined as presented in Chapter 2 using small

angle approximations:

∆ψ = r (4.3)

e ≈ Ux∆ψ + Uxβ (4.4)

s ≈ Ux (4.5)

giving a discrete, time-varying vehicle model which can be expressed as:

x(k+1) = A(k)αr,tsx

(k) +B(k)αr,tsFyf

(k) + d(k)αr,ts

(4.6)

where x = [β r ∆ψ s e]T , k is a time step index, subscript αr denotes lin-

earization of the rear tire model around rear slip angle αr, and subscript ts denotes

discretization of the vehicle model using time step length ts.


4.3 Comparison to Other Approaches

Comparing this approach to that of Falcone et al. discussed previously, some interest-

ing differences emerge. In both [25] and [26], Falcone et al. linearize the full vehicle

model, which requires a non-trivial amount of computational time to compute the lin-

earizations at each time step. Simplifying assumptions were used in an approximate

method of generating these linearizations to reduce the computational complexity. In

contrast, the approach presented in this chapter linearizes only the rear tire model,

avoiding this computational complexity and the need for simplifying assumptions.

Another significant difference is the tracking of a reference trajectory, which plays

a important role in the stability proof of the general stability criteria presented in

[26]. Since the proposed envelope controller does not restrict the human driver to a

predefined path, a tracked reference trajectory is not available and the assumptions

of the more general stability criteria are not met. Therefore, an application specific

stability criteria is required and is achieved through the invariant set defined by the

stable handling envelope presented in Chapter 2.

4.4 Safe Driving Envelopes

The safe driving envelopes used in this chapter are identical to those presented in

Chapter 2 and are briefly described here for reference.

4.4.1 Stable Handling Envelope

Originally presented by Beal and Gerdes, the stable handling envelope defines limits

on the states describing the vehicle’s motion as illustrated in Figure 4.3. This envelope

reflects the maximum capabilities of the vehicle’s tires so at any point within this

envelope, there exists a steering command to safely remain inside, ensuring stability

[10].

The maximum steady-state yaw rate defines bounds 2 and 4 in Figure 4.3. The

final two bounds of the vehicle envelope serve to limit the rear slip angle to the angle

at which lateral force saturates. Given this bound on rear slip angle, a maximum


−0.2 −0.1 0 0.1 0.2−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Yaw

Rat

e r

Sideslip β

1

2

3

4

Figure 4.3: Stable handling envelope

sideslip can be defined and serves as the basis for bounds 1 and 3 in Figure 4.3.

The stable handling envelope can be compactly expressed as the inequality:

∣∣Hshx(k)∣∣ ≤ Gsh (4.7)

where xk is the vehicle state at the kth time step into the prediction horizon and

subscript sh indicates the stable handling envelope.

4.4.2 Environmental Envelope

The environmental envelope consists of a set of collision free tubes along the nominal

path like the two illustrated in Figure 4.4. To avoid collision with the environment,

the vehicle’s trajectory must be fully contained within any one of these tubes. Each

tube defines time-varying constraints on the lateral deviation of the vehicle from the

nominal path:

e(k) ≤ e(k)max −

1

2d− dbuffer (4.8)

e(k) ≥ e(k)min +

1

2d+ dbuffer (4.9)


where e(k)max and e

(k)min indicate the lateral deviation bounds for time step k, d is the

vehicle’s width, and dbuffer specifies a preferred minimum distance between obstacles

and the vehicle to ensure driver comfort.

For any two trajectories generated using vehicle model (4.6) and contained within

a tube, the linear combination of those trajectories will also be contained within that

same tube. Therefore, the set of collision free trajectories corresponding to a single

tube is a convex set, which enables the use of fast optimization techniques to quickly

identify optimal trajectories.

Inequalities (4.8) and (4.9) can be compactly expressed as:

Henvx(k) ≤ G(k)

env (4.10)

where subscript env indicates the environmental envelope.

A vehicle trajectory is collision free throughout the prediction horizon if and only if

it satisfies inequality (4.10) for all k for any one tube in the environmental envelope.

For simplicity, the environments considered in this chapter can all be represented

using only one tube; however, the proposed approach easily extends to the more

complicated environments discussed in Chapter 2.


a)

b)

se

e (4)min

e (4)max

e (1)max

e (1)min e (4)

mine (4)

max

e (1)max

e (1)min

Nominal pathObstacle

Lane boundaryTube boundary

Figure 4.4: The environmental envelope is a representation of a) a collection of obsta-cles along the nominal path using b) tubes (two of them in this example) which define

a maximum (e(k)max) and minimum (e

(k)min) lateral deviation from the nominal path at

each time step, k.


4.5 MPC Formulation

The controller’s objectives match those presented in Chapter 2: safe vehicle operation

within the safe driving envelopes while being minimally invasive to the driver and

avoiding harsh interventions. Therefore, these same overall objectives are expressed

as the same optimal control problem using the modified plant model:

minimize∣∣∣Fyf,driver − F (0)

yf,opt

∣∣∣ (4.11a)

+∑k

γ(k)(F

(k)yf,opt − F

(k−1)yf,opt

)2

(4.11b)

+∑k

[σsh σsh]∣∣∣S(k)

sh,opt

∣∣∣ (4.11c)

+∑k

[σenv σenv](S

(k)env,opt

)2

(4.11d)

subject to x(k+1) = A(k)αr,tsx

(k) +B(k)αr,tsF

(k)yf,opt + d

(k)αr,ts (4.11e)∣∣∣F (k)

yf,opt

∣∣∣ ≤ Fyf,max (4.11f)∣∣Hshx(k+1)

∣∣ ≤ Gsh + S(k)sh,opt (4.11g)

Henvx(k+1) ≤ G

(k+1)env + S

(k)env,opt (4.11h)∣∣∣F (k)

yf,opt − F(k−1)yf,opt

∣∣∣ ≤ F(k)yf,max slew (4.11i)

k = 0 . . . 29

where the variables to be optimized are the optimal input trajectory (Fyf,opt) and the

safe driving envelope slack variables (Ssh,opt, Senv,opt). As is commonly done in MPC,

only the optimal input for the first step into the prediction horizon, F(0)yf,opt, is applied

to the vehicle, and optimization problem (4.11) is re-solved at the next time step.

Tunable parameters in this optimization are again the slack variable costs (σsh,

σenv) and γ, which establishes the trade-off between a smooth input trajectory (4.11b)

and matching the driver’s present steering command (4.11a). Constraint (4.11f) re-

flects the maximum force capabilities of the front tires and (4.11i) reflects the slew

rate limit of the vehicle steering system. Constraints (4.11g) and (4.11h) enforce the


Table 4.1: Prediction Horizon Parameters

Near Term Long TermParameter Horizon Horizon

k 0...9 10...29

t(k)s 0.01 [s] 0.2 [s]γ(k) 30 1.5

F(k)yf,max slew 0.2 [kN] 5 [kN]

σsh 60 60σenv 1500 1500

0 (Linear Model)

α(k)r αr or

α(k)r (Approx Nonlinear Model)

stable handling and environmental envelopes, respectively. These constraints are soft-

ened with slack variables, Ssh,opt and Senv,opt, to ensure optimization (4.11) is always

feasible. With the choice of sufficiently large σsh and σenv, cost terms (4.11c) and

(4.11d) encourage zero-valued slack variables, resulting in optimal vehicle trajectories

that adhere to both safe driving envelopes whenever possible. In addition, the slack

variable costs are chosen such that σenv � σsh, which establishes a hierarchy between

the two safe driving envelopes, prioritizing collision avoidance over vehicle stability.

This trade-off will be explored in the simulations and experiments that follow.

The prediction horizon used in optimization (4.11) uses different length time steps

in the near and long terms of the horizon as described in Chapter 2. Table 4.1 gives

the values of parameters in these two portions of the prediction horizon, which are

used in the simulations that follow.

4.6 Simulation Results

Simulations of the controller using both approaches to rear tire modeling demonstrate

the value of approximating nonlinear rear tire behavior using the technique of suc-

cessive linearizations. This simulations use the vehicle parameters listed in Table 4.2

and represent Stanford’s P1 research test bed. Illustrated in Figure 4.5, this test


Figure 4.5: Stanford’s P1, an all electric, throttle- and steer-by-wire research testbed

vehicle proved to be an invaluable platform for preliminary work on the proposed

envelope controller. Although all of the experimental results obtained from P1 have

been supplanted by the more modern X1 test vehicle, the following simulations were

conducted using P1’s parameters and, for legacy and nostalgia reasons, have been

retained in this dissertation.

In the following simulations, the arrangement of obstacles, as illustrated in Fig-

ure 4.6, requires the vehicle to execute a double lane change to avoid collision with the

environment. This double-lane change maneuver conforms to ISO standard 3888-1

[1]. For simplicity, the effects of vehicle roll and weight transfer are ignored in the

following simulations.

Table 4.2: P1 Vehicle Parameters

Parameter Value Unitsm 1725 kgIz 1300 kg ·m2

a 1.35 mb 1.15 md 1.60 m

Cαf 57.8 kN · rad−1

Cαr 110 kN · rad−1


0 20 40 60 80 100

−10

0

10

20

[m]

[m]

Figure 4.6: Arrangement of obstacles and road boundaries used in the followingsimulations. Vehicle travels in the direction indicated by the arrow.

Figure 4.7 illustrates the shared control capabilities of the controller at a moderate

speed of 12 (m/s) on a low friction surface (µ = 0.55). Using either tire model, the

controller successfully augments the human driver’s steering command to negotiate

the double lane change. The driver is modeled using an open-loop steering com-

mand which, at this speed, almost navigates the obstacles without collision or loss of

control. However, as illustrated, two slight augmentations of the driver’s command

are required to avoid collision, and the controller identically matches the driver’s

command otherwise.

Figure 4.8 provides a comparison of the controller’s performance at a more aggres-

sive speed of 16 (m/s) on the same low friction surface. At this speed, the performance

improvement using the successive linearizations technique becomes apparent. At the

instance t = 6.25 (s), the controller using the improved rear tire model deviates from

the driver’s command to set the vehicle up for a smooth exit through the final lane

change. Uninformed of the saturation of the rear tires, the controller using the lin-

ear rear tire model mistakenly matches the driver at this point in time and leads

the vehicle into a state without a feasible trajectory that adheres to both envelopes.

This results in aggressive attempts to adhere to both safe envelopes, which fail on

both accounts. This simulation highlights the importance of planning with accurate

models to ensure vehicle safety. Subtle inputs at one point in time can drastically

influence the feasibility at a future time. As the vehicle moves from the linear to

the nonlinear regions, the sucessive linearization technique serves as a powerful tool


to capture these nonlinear dynamics, enabling the controller to anticipate feasibility

challenges and appropriately act to avoid them.

With only the ability to steer, situations arise when adherence to both safe driving

envelopes is impossible, as illustrated in Figure 4.9 where the speed is 18 (m/s). The

combination of low friction and high speed forces the controller to violate one of

the envelopes. According to the envelope hierarchy dictated by the relative weights

discussed in Section 4.5, the controller should prioritize collision avoidance over vehicle

0 20 40 60 80 100 120

0

10

20

[m]

[m]

0 2 4 6 8 10

−5

0

5

ste

ering a

ngle

[deg]

time [s]

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−0.5

0

0.5

side slip [rad]

yaw

rate

[ra

d/s

]

Driver

Linear

Approx Non−Linear

Figure 4.7: Double lane change maneuver on low friction surface at 12 [m/s]


0 20 40 60 80 100 120

0

10

20

[m]

[m]

0 1 2 3 4 5 6 7 8

−5

0

5

10

ste

ering a

ngle

[deg]

time [s]

−0.15 −0.1 −0.05 0 0.05 0.1 0.15−0.5

0

0.5

side slip [rad]

yaw

rate

[ra

d/s

]

Driver

Linear w/Collision

Approx Non−Linear

Figure 4.8: Double lane change maneuver on low friction surface at 16 (m/s)


stability. However, only the controller equipped with the approximate nonlinear tire

model can anticipate the need for this trade-off and avoid collision at the expense of

stability. The controller not informed of the rear tire nonlinearity ultimately fails to

satisfy either constraint and eventually stabilizes the vehicle only after allowing it to

get significantly sideways and collide with the road boundary.

0 20 40 60 80 100 120

0

10

20

[m]

[m]

0 1 2 3 4 5 6 7 8

−20

−10

0

10

ste

ering a

ngle

[deg]

time [s]

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2−0.5

0

0.5

1

side slip [rad]

yaw

rate

[ra

d/s

]

Driver

Linear w/Collision

Approx Non−Linear

Figure 4.9: Double lane change maneuver on low friction surface at 18 (m/s)

Figure 4.10 displays a comparison of the internal predictions by the controller

of the future vehicle states using each of the rear tire models. As expected, in both


−0.2 −0.1 0 0.1 0.2

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

side slip [rad]

yaw

rate

[ra

d/s

]

Linear

−0.2 −0.1 0 0.1 0.2

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

side slip [rad]

yaw

rate

[ra

d/s

]

Approx Non−Linear

Initial State

Near Term Prediction

Long Term Prediction

Actual State Trajectory

Figure 4.10: Comparison of the planned safe trajectory midway through the doublelane change manuever on low friction surface at 18 [m/s]

cases the controller accurately predicts the pending stable handling envelope violation

because the current rear slip angle is used to predict near term rear tire behavior.

However, as illustrated on the left of Figure 4.10, use of a linear model in the remainder

of the prediction horizon underestimates the extent of this envelope violation. This

leads the controller to make steering decisions that result in significant violation

of the safe handling envelope followed by a collision with the road boundary. The

approximate nonlinear rear tire model better predicts future vehicle states, and, as

seen in the right of Figure 4.10, the controller plans a slight violation of the stable

handling envelope in order to prevent significant violations in the future and avoid

collision with the environment.


Table 4.3: Maximum Speed without Collision

Linear Approx. NonlinearSurface Friction, µ Rear Tire Model Rear Tire Model

0.90 18 [m/s] 22 [m/s]0.55 14 [m/s] 19 [m/s]

Additional simulations reveal a similar trend on high friction surfaces as summa-

rized in Table 4.3. These results highlight the importance of considering future rear

tire saturation for both low and high values of surface friction.

4.7 Experimental Results

Experiments using Stanford’s X1 test bed, which is illustrated in Figure 4.11, support

the results of the simulations presented previously and enable a direct comparison

with the experimental results of Chapter 2. As in the simulations, the environment

used in these experiments conforms to ISO standard 3888-1 [1] and the surface was

gravel on asphalt with a surface coefficient of µ ≈ 0.5.

Figure 4.12 illustrates the first of two experiments with a human driver and the

proposed envelope controller sharing control of the vehicle as it negotiates the double

lane change. As shown, a number of steering augmentations by the controller are

required in order to safely navigate the double lane change without collision with the

environment. A few of those augmentations are highlighted. At instance 1 , the con-

troller reduces the magnitude of the driver’s steer command in response to a pending

stable handling envelope violation. Even with this reduction in steer command, the

vehicle safely completes the first lane change without collision with the environment.

At instance 2 , the driver prematurely begins the second lane change, forcing the

steering controller to delay the steer command by about 0.5 [s] in order to avoid

collision with the environment. At time instances 3 and 4 , two quick steering

interventions by the controller minimize violations of the stable handling envelope as

the vehicle safely completes the second lane change.

Overall, the benefit of the improved rear tire modeling that was demonstrated


Figure 4.11: Stanford’s X1, an all electric, throttle- and steer-by-wire research testbedwith automatic brakes and haptic force feedback steering system

in simulation is made evident in experimentation by comparing Figure 4.12 with

Figure 2.10 from Chapter 2. Both of these experiments used the same vehicle, same

set of gains, and same environment. However, it should be noted that with the driver

in the loop, the experimental runs are not identical, and the driver’s command differs

between the two runs. The only difference not related to the driver was the rear

tire modeling approach. Although the controller performed quite well using only

the linear tire model in the long term portion of the prediction horizon as shown in

Figure 2.10, a large violation of the stable handling envelope and a small violation

of the environmental envelope were observed. In addition, the controller using the

linear model made large steering interventions in reaction to unanticipated challenges

to vehicle stability and obstacle avoidance. Informed of future rear tire saturation,

the controller using the approximate nonlinear rear tire model better adheres to both

safe envelopes while avoiding large steering commands.

Figure 4.13 illustrates another iteration of the same test and demonstrates how

control at the limits remains challenging even with the approximate nonlinear rear

tire model. At instances 1 and 2 , the controller makes attempts to adhere to

the stable handing envelope but the vehicle slides more than the controller predicted

leading to a slight violation of the environmental envelope. The violation of the stable

handling envelope is reduced relative to Figure 2.10 illustrating better stability with


the approximate nonlinear rear tire model.


80 100 120 140 160 180 200

10

20

30

40

Distance (m)

Dis

tanc

e (m

)

0 1 2 3 4 5 6 7−5

0

5

Time (s)

Stee

r cm

d (d

eg)

−10 −5 0 5 10

−20

−10

0

10

20

Side Slip (deg)

Yaw

Rat

e (d

eg/s

)

0 1 2 3 4 5 6 714

16

18

Time (s)

Vehi

cle

Spee

d (m

/s)

ActualDriver

EnvelopeActual

StartVehicle PathObstacles

1

1

1

2

2

2

3

3

34

4

4

Figure 4.12: Experiment using X1 in double lane change on low friction µ = 0.55with no environmental envelope violation


80 100 120 140 160 180 200

10

20

30

40

Distance (m)

Dis

tanc

e (m

)

0 1 2 3 4 5 6 7−10

−5

0

5

10

Time (s)

Stee

r cm

d (d

eg)

−10 −5 0 5 10

−20

−10

0

10

20

Side Slip (deg)

Yaw

Rat

e (d

eg/s

)

0 1 2 3 4 5 6 714

16

18

Time (s)

Vehi

cle

Spee

d (m

/s)

ActualDriver

EnvelopeActual

StartVehicle PathObstacles

12

3

1

2

3

1 2

3

Figure 4.13: Experiment using X1 in double lane change on low friction µ = 0.55with slight violation of the environmental envelope at instance 3


4.8 Discussion

With only the ability to steer, situations may require violation of the stable handling

envelope to avoid a collision with the environment. In these situations, use of the

successive linearization technique enables the controller to consider future rear tire

saturation to appropriately steer the vehicle safely without additional computational

burden. In addition, the prediction of future vehicle states throughout the prediction

horizon is improved, giving advanced warning on pending challenges to the combined

objectives of vehicle stability and collision avoidance. As described in the next chap-

ter, this advanced warning allows the controller time to safely reduce the vehicle’s

speed using brake actuation, ensuring the vehicle’s future trajectory will always ad-

here to both safe envelopes. This predictive capability is the key to determining

when velocity changes are necessary and is the foundation for a comprehensive enve-

lope controller capable of ensuring vehicle safety through augmentation of the driver’s

steering, throttle, and brake commands.

Chapter 5

Envelope Control using Braking

and Steering

As illustrated in Chapter 4, situations may arise in which steering alone cannot en-

sure adherence to both safe driving envelopes. Capturing the saturation of the rear

tires using successive linearizations improves the controller’s ability to safely nego-

tiate these types of situations, enabling the controller to trade-off vehicle stability

to ensure collision avoidance. However, avoiding violation of the safe envelopes in

these situations requires additional actuation. This chapter explores the use of brake

actuation as a way of extending the steering only envelope controller presented thus

far, expanding the number of situations in which the envelope controller can ensure

vehicle safety.

5.1 Challenge of Longitudinal and Lateral Control

Adding the ability to brake opens up many new opportunities to the envelope con-

troller. Unfortunately, this expanded set of possibilities makes searching over these

possible trajectories in real-time a challenging problem. A number of approaches have

been purposed to overcome this challenge and enable the use of both steering and

braking in the control of automated vehicles.

Many approaches address this challenge of combined braking and steering control

112

CHAPTER 5. ENVELOPE CONTROL USING BRAKING AND STEERING 113

by reducing the problem to path tracking, which relies on a desired path generated

externally to the controller. Falcone et al. present a nonlinear MPC (NMPC) ap-

proach to path tracking using both steering and braking [28]. This approach achieves

good tracking performance using a higher order vehicle model that considers indi-

vidual wheel braking; however, use of this high-fidelity model is computationally

prohibitive for real-time implementation. A simplified version of their controller that

models differential braking as a single moment applied to the vehicle allows for real-

time implementation but at a reduced level of performance. Logic external to the

optimal controller converts the optimal brake moment into individual wheel brake

commands. Falcone et al. also present a linear time-varying MPC extension to the

previous NMPC approach that addresses the performance and computational limita-

tions [27]. More recently, Katriniok et al. present an optimal approach to combined

lateral and longitudinal control using steering angle and longitudinal acceleration as

control inputs to track a predefined evasive trajectory [42]. These approaches illus-

trate the ability to perform combined lateral and longitudinal control in real-time

when a predetermined desired path is available.

Moving away from path tracking, another way to reduce the search space of pos-

sible trajectories is to consider only a finite set of speed profiles. Turri et al. present

this approach using multiple pre-processed speed profiles in a series of on-line opti-

mizations for a fully autonomous vehicle [71]. In considering only a finite number

of braking or throttling profiles, the search space of possible trajectories is greatly

reduced and the lateral and longitudinal dynamics of the vehicle can be decoupled.

For each speed profile, the optimal steering command is computed by solving a simple

QP, thereby reducing the problem to a set of QPs that can be solved efficiently at each

time step. Turri et al. assume prior knowledge of which side of the obstacles it is best

to pass. The complexity would increase if multiple routes, or tubes, around obstacles

are additionally considered. Experiments in test vehicles on icy roads validate this

approach.

Approaches which do not restrict the vehicle to a pre-defined path or finite set of

speed profiles in general require the use of nonlinear optimization techniques. Gao

et al. present a NMPC path planner combined with a linear MPC path tracker for


combined steering and braking control. The nonlinear path planner uses a spatially

discretized model with a 22.5 (m) look ahead distance that executes at 5 (Hz). Al-

though this shows promise of NMPC in real-time, its limited look-ahead distance

requires heuristic augmentation of the obstacles to ensure smooth avoidance trajec-

tories and is limited to single tube environments [34]. Bevan et al. also presents a

nonlinear path planner for combined braking and steering using a multiple pass opti-

mization scheme that solves approximations to the full control problem at each pass.

Although the approach leverages fast convex optimization techniques, it requires a

simplified point-mass vehicle model, single tube environment, and the convergence

of the multi-stage solver is not guaranteed [12]. Other examples in the literature

combine path tracking and path following into a single NMPC controller which ex-

plores the full space of feasible trajectories; however, all of these have a computational

complexity that prevents real-time implementation [56] [74] [17].

These examples underline the computational challenge in optimal longitudinal

and lateral vehicle control. To address this challenge, the approach presented in

this chapter formulates braking with envelope control as a steering feasibility prob-

lem. The controller reduces the vehicle’s speed in response to anticipated envelope

violations, ensuring a feasible trajectory exists that adheres to both safe envelopes.

This approach does not restrict the driver to a predetermined path or speed profile

and therefore continues to ensure driver autonomy whenever possible. In addition,

this feasibility problem is solved using the same QP presented in Chapter 4 with an

additional convex constraint and sparsity seeking objective; therefore, nonlinear opti-

mization techniques are not required and the envelope controller maintains its speed

and reliability.

5.2 Braking as Steering Feasibility Problem

Using the approach to rear tire modeling presented in the previous chapter, the con-

troller can determine in real-time when the current vehicle speed is too fast to safely

adhere to the safe envelopes. Figure 5.1 illustrates the envelope controller’s predicted


−20 −10 0 10 20 30 40 50

−15

−10

−5

0

5

10

15

20

[m]

[m]

Figure 5.1: Controller plans a single lane change in response to blocked lane

vehicle trajectory when confronted with a blocked lane at Ux = 16 (m/s) on a sur-

face with µ = 0.5. As illustrated in the top plot of Figure 5.2, the optimal vehicle

trajectory at this speed requires violation of the stable handling envelope. This indi-

cates that the vehicle is traveling too fast for the combination of road conditions and

approaching environmental hazards. The remaining plots in Figure 5.2 illustrate the

predicted envelope violations for the optimal trajectory evaluated at different vehicle

speeds. As shown, the maximum speed at which adherence to the stable handling

envelope is possible can be framed as a convex feasibility problem. The maximum

violation of the stable handling envelope is an increasing function of vehicle speed as

illustrated in Figure 5.3.

As the vehicle speed decreases, the stable handling envelope increases in size and

the max acceleration and yaw rate required to complete the lane change decrease.

These two factors combine to give the generally monotonic trend observed in Fig-

ure 5.3: faster speeds result in larger required envelope violations. This trend is not


strictly monotonic due to the variation in discretization and sampling of the envi-

ronment which, as described in Chapter 2, is speed dependent. However, this trend

indicates that reducing the vehicle’s speed via brake actuation is a key mechanism for

reducing envelope violations. When the vehicle speed is too fast for conditions, the

MPC optimal trajectory reveals that steering alone is not sufficient to ensure vehicle

safety.


−0.2 −0.1 0 0.1 0.2

−0.4

−0.2

0

0.2

0.4

0.6

yaw

rat

e [r

ad/s

]

Ux = 16 [m/s]

−0.2 −0.1 0 0.1 0.2

−0.4

−0.2

0

0.2

0.4

0.6

yaw

rat

e [r

ad/s

]

Ux = 10.9 [m/s]

−0.2 −0.1 0 0.1 0.2

−0.4

−0.2

0

0.2

0.4

0.6

yaw

rat

e [r

ad/s

]

Ux = 10 [m/s]

side slip [rad]

Figure 5.2: Predicted envelope violations for same maneuver at three different vehiclespeeds


6 8 10 12 14 16

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

vehicle speed [m/s]

no

rma

lize

d e

nve

lop

e v

iola

tio

n [

−]

Figure 5.3: Maximum predicted envelope violation for the same maneuver as a func-tion of vehicle speed

5.3 Braking to Prevent Envelope Violations

Figure 5.4 provides a block diagram of the proposed envelope controller presented

thus far. As shown, the envelope controller has the capability to modify only the

driver’s steering command, δdriver, and the driver’s brake and acceleration commands,

represented here as a lumped longitudinal force command, Fx,driver, are unregulated.

This leaves the driver in full control of the vehicle’s speed. Measurements of this

speed, the vehicle states, x, and the driver’s steering command serve as inputs to the

MPC optimization as described in previous chapters.

Figure 5.5 illustrates the block diagram for the proposed braking and steering

envelope controller introduced in Section 5.2. As shown, the final longitudinal force

command, Fx, is no longer dictated solely by the driver. Instead, the envelope con-

troller now has the capability to override the driver’s longitudinal force command in

addition to the steering command. A new input to the MPC optimization, Fx,des, is

computed as the minimum of the driver’s command and a braking force that seeks to

reduce envelope violations, Fx,envViolation. This heuristically determined brake force,

Fx,envViolation, is a function of the maximum predicted envelope violation, dmaxEV,


Figure 5.4: Block diagram of steering only envelope controller presented in previouschapters

which is derived directly from the MPC optimal solution. When the current ve-

hicle speed results in an MPC optimal trajectory that violates the safe envelopes,

dmaxEV > 0 and the brake policy computes an Fx,envViolation that reduces the vehicle’s

speed. As described in Section 5.2, this speed reduction will reduce the anticipated

envelope violation. If instead the current vehicle speed results in an MPC optimal

trajectory that does not violate either safe envelope, dmaxEV <= 0 and a reduction

in speed is not required. In this situation, Fx,envViolation is set arbitrarily large so that

Fx,des = Fx,driver.

As will be described in Section 5.4, an MPC optimization will ultimately determine

the amount of braking and steering applied to the vehicle to ensure vehicle safety. The

desired longitudinal force, Fx,des, is determined externally from the MPC optimization,

leaving the optimal controller with the simpler decision of determining how much of

this desired longitudinal force can be safely commanded at the current time step.

5.3.1 Braking Policies

There is much freedom in determining a policy for Fx,envViolation, so long as Fx,envViolation

is chosen to adequately reduce the vehicle’s speed when envelope violations are pre-

dicted. This section describes two possible braking policies, creatively called the fixed

braking and proportional braking policies. Both of these policies command braking


Figure 5.5: Block diagram of braking and steering envelope controller

in response to predicted envelope violations represented as the scalar dmaxEV, which

is a normalized, maximum violation of the stable handling envelope from the MPC

optimal solution. This can be computed directly from the optimal slack variables:

dmaxEV = maximizek

(max

(S

(k)sh,opt./Gsh

))(5.1)

where ./ indicates element-wise division, max gives the maximum value from a given

vector, k is a time step index into the prediction horizon, Gsh is the stable handling

envelope constraint, and S(k)sh,opt is the optimal slack variables for the stable handling

envelope at time step k. Both Gsh and S(k)sh,opt were previously defined in Sections 2.3.1

and 2.4.2, respectively.

For the fixed braking policy, the controller commands a constant braking amount

whenever there is a predicted envelope violation:

Fx,envViolation =

∞ dmaxEV ≤ 0

Fx,fixed otherwise(5.2)

where Fx,fixed is a predetermined braking value. This policy is simple and, as shown

later in experimental validation, effective in reducing the vehicle’s speed to adhere to


the safe envelopes.

For the proportional braking policy, the commanded braking is in proportion to

predicted envelope violations:

Fx,envViolation =

∞ dmaxEV ≤ 0

Kbrake dmaxEV 0 < dmaxEV <Fx,maxComfort

Kbrake

Fx,maxComfort otherwise

(5.3)

where Kbrake is a proportional gain and Fx,maxComfort serves to bound the commanded

brake force to ensure driver comfort. This policy seeks to more smoothly introduce

braking in response to anticipated envelope violations.

5.3.2 Constant Speed Assumption

Throughout this dissertation, the vehicle model used in the MPC optimization as-

sumes the vehicle is traveling at a constant speed. With the introduction of braking

policies into the envelope controller, it is reasonable to question whether this assump-

tion is still valid. Given the controller’s fast execution rate of 100 (Hz), braking can,

at most, alter the vehicle’s speed by ≈ 0.1 (m/s) in between controller executions as-

suming a full 1 (g) brake event. Therefore, the vehicle’s speed remains nearly constant

between executions of the controller even during extreme braking events, validating

the constant speed assumption that is used in the MPC vehicle model.

If a variable speed trajectory is known a priori, this constant speed vehicle model

could be replaced with an linear time varying (LTV) model linearized at these known

speeds along the prediction horizon. Future work will apply this braking and steering

envelope controller to vehicle applications with known speed trajectories. One of

these applications is Shelley, an autonomous race car jointly developed by Stanford

and Audi [45]. Shelley maintains a planned speed profile generated from the racing

line, and this profile could serve as the known speed profile for the MPC vehicle

model. However, for simplicity in illustrating the basic concept of braking in response

to envelope violations, the braking and steering envelope controller proposed in this

chapter continues to utilize the constant speed vehicle model presented previously.


15 20 25 30 35 40 45

−6

−4

−2

0

2

4

6

8

10

12

[m]

[m]

Figure 5.6: Obstacle blocking the lane enters the prediction horizon as the vehicletravels at 8 (m/s)

5.3.3 Ensuring the Horizon Always Recedes

In the presence of braking, a time based, constant speed prediction horizon could

encounter a “loss of information” with the end of the prediction horizon advancing

toward the vehicle instead of receding. This situation is illustrated in Figures 5.6 and

5.7, and occurs when the brake acceleration exceeds:

−Ux

tPH

(5.4)

where Ux is the vehicle longitudinal speed and tPH is the length of time of the predic-

tion horizon.

Acceleration (5.4) can be derived by first defining the distance along the reference

line of the end of the prediction horizon:

send,PH(t) = tPHUx(t) + s(t) (5.5)


15 20 25 30 35 40 45

−6

−4

−2

0

2

4

6

8

10

12

[m]

[m]

Figure 5.7: Braking in excess of (5.4) moves the end of the prediction horizon totoward the vehicle causing the obstacle to leave the horizon

and setting its derivative to zero:

send,PH(t) ≈ tPHax(t) + Ux(t) (5.6)

0 = tPHax(t) + Ux(t) (5.7)

ax(t) =−Ux(t)

tPH

(5.8)

where it is assumed s ≈ Ux and Ux ≈ ax. If the vehicle brakes at the acceleration

specified by (5.8), the end of the prediction horizon will remain fixed in space and the

vehicle will safely come to a stop at this point. If the vehicle is braking in response to

a threat that exists at the end of the prediction horizon and the vehicle is capable of

braking at the amount specified by (5.8), then there is no reason to brake in excess of

this amount. Figure 5.8 illustrates the magnitude of this acceleration as a function of

vehicle speed. As can be seen at fast speeds and short horizon times, this acceleration

value can easily exceed the brake capabilities of typical passenger vehicles; however,

using longer horizon times, like the tPH = 4 (s) used in this work, this acceleration

is easier to achieve. Therefore, as expected, longer prediction horizons require less


0 5 10 15 20 25 30 35 40−12

−10

−8

−6

−4

−2

0

Vehicle Speed, Ux (m/s)

Bra

ke

Acce

lera

tio

n,

ax =

−U

x /

tP

H

(m/s

2)

tPH

= 1 (s)

tPH

= 2 (s)

tPH

= 4 (s)

tPH

= 8 (s)

Figure 5.8: Brake acceleration required to fix the end of the prediction horizon inspace


braking to ensure vehicle safety.

5.3.4 Tire Force Coupling

Braking to prevent future violations of the safe driving envelopes should not create

new near term envelope violations. This is a possibility because the braking and

steering of a vehicle are coupled. This coupling derives from a physical limitation of

the tires referred to as the friction circle, which dictates that the total force generated

by the tire in any direction cannot exceed the total force available from friction.

A common way to visualize this limitation with respect to the acceleration of the

vehicle is the g-g diagram as described by Rice [63] and Milliken and Milliken [54]

and illustrated in Figure 5.9. As shown, when the vehicle is engaged in full braking,

it has no ability to generate lateral force. Likewise, when it is engaged in full lateral

acceleration, it has no ability to generate longitudinal force. Kritayakirana and Gerdes

[45] further describe the vehicle capabilities conceptualized with the friction circle (g-g

diagram) as applied to path planning and control at the limits of handling.

This limitation of the tires can be expressed as the following physical constraint

on the tire forces:

F 2x + F 2

y ≤ (µFz)2 (5.9)

where Fx, Fy, and Fz are the lateral, longitudinal, and normal forces, respectively,

acting on the tire and µ is the tire-road surface friction coefficient.

The brush tire model (2.7) introduced in Chapter 2 does not capture the effect of

this force coupling. Therefore, the proposed braking and steering envelope controller

uses a modification to the brush tire model to approximate this coupling. Initially

presented by Hindiyeh and Gerdes [37] for use in the control of drifting, this modified

brush tire model serves as a simple model of tire force coupling at the limits of friction.

This simple model ignores wheel speed dynamics and therefore does not represent

extremely fast dynamics. However, the model does capture two key properties: the

derated peak lateral force in the presence of a non-zero longitudinal force and the

linear characterization of lateral tire force at low slip angles, as shown in Figure 5.10.


ay

ax

Maximumsteering

Maximum braking

Friction Limit

Figure 5.9: Friction circle concept illustrated using a g-g diagram

This modified brush model is the original brush tire model with a derated surface

friction coefficient, ξµ:

Fy =

−Cα tanα + C2

α

3ξµFz| tanα| tanα

− C3α

27ξµ2F 2z

tan3 α, |α| < arctan(

3ξµFz

Cα

)−ξµFzsgn α, otherwise

= fcoupled tire (α, Fx) (5.10)

where ξ is a derating factor (0 ≤ ξ ≤ 1) that accounts for the reduction in lateral

force capability in the presence of longitudinal forces and the remaining terms are

similarly defined as in the tire model presented in Section 2.2.1. This derating factor


Tire Slip Angle, α

Tire L

ate

ral F

orc

e, F

y

Fx = 0%

Fx = 25%

Fx = 50%

Fx = 75%

Figure 5.10: Coupled tire force model

is computed using a rearrangement of constraint (5.9):

ξ =

√(µFz)

2 − F 2x

µFz

(5.11)

The reverse lookup of the coupled tire model serves to map optimal lateral forces

to desired steer angles and is expressed as:

α = f−1coupled tire (Fy, Fx) (5.12)

Although this modified tire model has the additional dimension of Fx, it can be

implemented using the same 2D lookup table described in Chapter 2. This is possible

because ξµ appears in (5.10) as a derated surface friction; therefore, a given Fx serves

only to augment the surface friction coefficient used in the original brush tire model

(2.7).


5.4 MPC Formulation

The tire force coupling described in the previous section needs to be considered when

computing the steering and braking commands to be applied to the vehicle. Without

this consideration, attempts to prevent future violations of the safe driving envelopes

though the use of brake actuation could result in near term envelope violations, doing

more harm than good. As shown in Figure 5.5 and described previously, a desired

longitudinal force, Fx,des, is determined externally from the MPC optimization, leav-

ing the optimal controller with the simpler decision of determining how much of this

desired longitudinal force can be safely commanded at the current time step. An

additional convex constraint, which represents the friction circle, and an additional

sparsity seeking objective enable the MPC optimization to determine an optimal

steering and braking/accelerating command with explicit consideration of the cou-

pling between the two.

5.4.1 Friction Circle Constraint

Constraint (5.9) is convex and could be directly incorporated into the convex optimal

control problem defined in Chapter 4. Directly incorporating constraint (5.9) into

the previously presented QP (4.11) creates a Second Order Cone Program (SOCP),

which could be solved exactly using embedded solvers like ECOS [20].

An alternative to this exact representation is to approximate constraint (5.9) as

the intersection of a number of half-spaces. This approximation is beneficial for a

number of reasons. First, the solver used in this work, CVXGEN, supports QP-

representable problems only; therefore, the half-space approximation of (5.9) allows

for the continued use of this reliable and efficient solver. Second, the half-spaces

approximation provides modeling flexibility to capture secondary effects that alter

the shape of the friction circle. The circular shape of the friction circle is an idealized

model that ignores the effects of weight transfer. When these effects are included, the

shape of the friction “circle” is more diamond-like as described in Kritayakirana’s PhD

Thesis [46] and the intersection of half-spaces provides the flexibility to approximate

this shape better than an ellipsoid. Future work will explore these more complex


representations of the friction circle and modeling the friction circle constraint directly

in a SOCP formulation; however, for simplicity, in this work the friction circle is

assumed to be circular and modeled using the half-space approximation in a QP

formulation.

The half-space approximation of (5.9) can be expressed as:

LyFy + LxFx ≤M (5.13)

Ly =

sin((1)2π

n− π

n

)sin((2)2π

n− π

n

)...

sin((n)2π

n− π

n

)

Lx =

cos((1)2π

n− π

n

)cos((2)2π

n− π

n

)...

cos((n)2π

n− π

n

)

M =

λµFz

λµFz

...

λµFz

where n is the number of half-spaces and λ is a scale factor. This half-space repre-

sentation of the friction circle is illustrated in Figure 5.11 using n = 12. The scalar

factor, λ, ensures the approximation always lies within the friction circle, and can be

expressed as:

λ = sin

((n− 2)π

2n

)(5.14)

5.4.2 Considering Front Longitudinal Forces Only

As described in Section 2.2.1, mapping of the optimal front tire force, Fyf,opt, to

the steer command, δ, allows for direct consideration of the tire force nonlinearity

for the front tires. Coupled tire force model (5.10) captures this nonlinearity and the

derating of the lateral tire force in the presence of a longitudinal tire force. Therefore,

this coupled tire force model provides a mapping from Fyf,opt to δ in the presence

of longitudinal forces. This mapping, combined with the friction circle constraint

presented in the previous section, enables optimization of both braking and steering

of the front tires.

However, as also discussed in Section 2.2.1, the inability to steer the rear wheels


Fx

Fy

Friction Circle

Half−Space Approx.

Figure 5.11: Friction circle approximated as the intersection of n = 12 half-spaces


prevents the use of this nonlinear mapping to model the rear tire forces. Instead, an

affine approximation to the nonlinear tire curve provides a model for the rear tire

forces linearized around a given operating point, as was extensively investigated in

Chapter 4. As presented by Beal and Gerdes [10], this nominal operating point can

additionally include longitudinal forces present at the tire to capture the effects of tire

force coupling in the short term. Unfortunately, this requires a priori knowledge of

the these longitudinal forces, thereby preventing the simple optimization of combined

braking and steering that is possible with the front tires. An approximate method

for enabling consideration of rear braking is described in Section 5.7, and future

work focuses on addressing this challenge. However, to illustrate the basic concept of

braking in response to envelope violations, the proposed braking and steering envelope

controller considers longitudinal forces on the front tires only because this can be

directly incorporated into the MPC optimization.

5.4.3 Optimal Control Problem

The optimal control problem presented in Chapter 4 can now be expanded to in-

clude this combined braking and steering of the front tires along with the extensions

described in Chapter 3 to give Quadratic Program (5.15). An additional optimiza-

tion variable, Fxf,opt, represents the optimal amount of front axle longitudinal force

command that adheres to the friction limits of the front tires. This value is biased

to match the desired longitudinal force, Fx,des, which, as described previously, is a

minimum of the driver’s command and braking in response to predicted envelope vio-

lations. The resulting combined longitudinal and lateral control problem is expressed

as:


minimize ν |Fxf,opt − Fx,des| (5.15a)

+∑k

ρ(k)∣∣∣F (k)

yf,opt − F(k)yf,driver

∣∣∣ (5.15b)

+∑k

γ(k)(F

(k)yf,opt − F

(k−1)yf,opt

)2

(5.15c)

+∑k

[σsh σsh]∣∣∣S(k+1)

sh,opt

∣∣∣ (5.15d)

+∑l

[σenv σenv](S

(l)env,opt

)2

(5.15e)

+∑h

qT

(∆ψ(h)

)2(5.15f)

h = Tterm . . . T

subject to x(k+1) = A(k)d x(k) +B

(k)d F

(k)yf,opt + d

(k)d (5.15g)∣∣∣F (k)

yf,opt

∣∣∣ ≤ Fyf,max (5.15h)

Hshx(k+1) ≤ Gsh + S

(k+1)sh,opt (5.15i)∣∣∣F (k)

yf,opt − F(k−1)yf,opt

∣∣∣ ≤ F(k)yf,max slew (5.15j)

k = 0 . . . (T − 1)

Henvx(l) ≤ G

(l)env −max

(FFLx

(l) + JFL, FRLx(l) + JRL

)l = (Tsplit + 1) . . . T (5.15k)

LyF(j)yf,opt + LxFxf,opt ≤M (5.15l)

j = 0 . . . Tbrake

where the optimization variables are the input trajectory (Fyf,opt), the safe driv-

ing envelope slack variables (Ssh,opt, Senv,opt), and the front longitudinal force scalar

(Fxf,opt).

Tunable parameters in this optimization problem include ν, ρ, and γ, which estab-

lish the trade-off between matching the desired longitudinal force (Fx,des), matching


Table 5.1: Braking and Steering Controller Parameters

Parameter Value UnitsT 30 (none)Tterm 28 (none)Tsplit 10 (none)Tbrake 10 (none)ν 8 1

kN

ρ(0) 1.2 1kN

ρ(1:3) 0.5 1kN

ρ(4:T ) 0 1kN

γ(0:Tsplit) 5 1kN2

γ(Tsplit+1:T−1) 2 1kN2

F(0:Tsplit)

yf,max slew 0.79 kN

F(Tsplit+1:T−1)

yf,max slew 5 kN

qT 50 1rad2

σsh [60 60][

1rad

srad

]σenv 1200 1

m2

dbuffer 0.3 m

the driver’s present lateral force command (Fyf,driver), and a smooth steering trajec-

tory. These gains are chosen to prioritize matching the desired longitudinal force,

Fx,des, over the driver’s desired lateral force, i.e. ν > ρ, because this desired longitu-

dinal force may be important for the safety of the vehicle. Other tunable parameters

include the slack variable costs (σsh, σenv) and the terminal heading deviation cost

(qT). Table 5.1 gives the values for the controller parameters that will be used in all of

the simulations and experiments presented in this chapter unless otherwise specified.

The optimal inputs to be applied to the vehicle are Fxf,opt and F(0)yf,opt. The optimal

front brake force (Fxf,opt) is mapped to an actuator command using:

Pf,brake = Kbrake actuatorFxf,opt (5.16)

where Pf,brake is the hydraulic pressure applied to the front brakes and Kbrake actuator

is a experimentally determined brake actuator gain. The optimal front lateral force

(F(0)yf,opt) is mapped to a front steering angle, δ, using the inverse lookup of the coupled


tire model (5.12):

δ = β +ar

Ux

− f−1coupled tire

(F

(0)yf,opt, Fxf,opt

)(5.17)

Softened constraint (5.15k) enforces the environmental envelope with explicit con-

sideration of the vehicle’s shape as presented in Chapter 3. Terminal cost (5.15f)

encourages lane change trajectories as also presented in Chapter 3.

Constraint (5.15l) enforces the friction circle on the optimal front longitudinal and

lateral forces over the first Tbrake time steps into the prediction horizon. The parameter

Tbrake provides a simple method for considering delays in the braking system. Setting

Tbrake large ensures the optimal brake command will adhere to the friction circle over a

large number of planned steering commands, ensuring adherence to the friction circle

even if these brake commands are delayed in appearing at the tires. This enables

conservatism in modeling the time response of the brake actuator as illustrated in

simulations later in this chapter.

5.4.4 Feasibility of the Constant Speed Trajectory

As described in Section 5.3.2, the constant speed assumption is still valid as a result of

the fast execution rate of the controller, despite the new capability of the controller to

actively brake the vehicle. With this new capability, it is also reasonable to question

whether the planned constant speed trajectories are feasible. The constant speed

planned trajectories are always feasible because, at any point, a viable option available

to the controller is to set Fxf,opt = 0, effectively ceasing to alter the vehicle’s speed.

Therefore, at any given time step, a planned constant speed trajectory is a feasible

trajectory. In this way, even though the actions of the controller can now directly

change the vehicle’s speed, the constant speed assumption is still valid and the planned

constant speed trajectories are feasible.


5.5 Experimental Results

Experiments conducted in a limited friction environment using the X1 test bed il-

lustrate the improved capability of the controller to safely navigate the vehicle using

combined steering and braking. X1 is equipped with an automatic brake system that

operates in parallel with the manual brake system. These manual brakes are con-

trolled exclusively by the driver. As a base for comparison, Figure 5.12 illustrates the

performance of the controller navigating a double lane change without any braking.

Specifically, the maneuver is an ISO-3888-2 [2] double lane change on low friction at

14 (m/s). Due to the limited testing space available, these experiments use ISO-3888-

2 double lane change specification because it is an aggressive maneuver that reaches

the limits of the vehicle at lower speeds than ISO-3888-1.

Although the controller performs well at this demanding task without braking,

violations of both safe envelopes are evident, indicating that this speed is too fast

to safely navigate this environment. Three collisions with the environment occur at

the three instances indicated. At instance 1 , the vehicle clips the environment as

the controller attempts to minimize future collisions by prematurely initiating the

double lane change. The steering commands leading up to instance 2 correspond

to circling around the lower bound of the stable handling envelope as shown in the

second plot. The vehicle comes into contact with the environment at instance 3

before the controller safely matches the driver’s command again.

5.5.1 Proportional Braking Policy

Figure 5.13 illustrates the same maneuver with the same initial speed, but equipped

with the proportional braking policy (5.3) and coupled tire force optimal control prob-

lem (5.15). As shown, the controller safely slows the vehicle and completes the ma-

neuver without violating either safe envelope. Table 5.2 provides the braking policy

parameters used in this experiment.

At instance 1 , the desired braking saturates to the maximum brake force for

driver comfort, Fx,maxComfort, in response to the large predicted envelope violation.

This predicted violation is illustrated in Figure 5.14 along with the corresponding rear


0 10 20 30 40 50 60 70

0

2

4

6

Distance (m)

Dis

tanc

e(m

)

0 1 2 3 4 5−20

0

20

Stee

r Ang

le, δ

(deg

)

−10 −5 0 5 10

−20

0

20

40

Side Slip, β (deg)

Yaw

Rat

e, r

(deg

/s)

0 1 2 3 4 568

101214

Time (s)

Spee

d, U

x

(m/s

)


ActualDriver


2

2

1

13

3

1

2

3

Figure 5.12: An experimental ISO 3888-2 double lane change on a low friction surface(µ ≈ 0.5) without braking


0 20 40 60 80 100

0

2

4

6

Distance (m)

Dis

tanc

e(m

)

0 2 4 6 8 10−20

−10

0

10

20

Stee

r Ang

le, δ

(deg

)

0 2 4 6 8 10−4

−3

−2

−1

0

Fron

t Lon

gitu

dina

l For

ce, F

xf

(kN

)

Time (s)

Spee

d, U

x

(m/s

)


ActualDriver

ActualDesired1

1

1 2

2

2

3

3

3

0 2 4 6 8 1068

101214

Figure 5.13: An experimental ISO 3888-2 double lane change on a low friction surface(µ ≈ 0.5) with braking in proportion to predicted envelope violation


−15 −10 −5 0 5 10 15

−40

−30

−20

−10

0

10

20

30

40

Side Slip, β (deg)

Ya

w R

ate

, r

(de

g/s

)

−5 0 5

−6

−4

−2

0

2

4

6

Re

ar

La

tera

l F

orc

e,

Fyr

(kN

)Rear Slip Angle, α

r (deg)

Current State

Near Pred

Long Pred

Actual

NonlinearTire Curve

Figure 5.14: Predicted envelope violation and corresponding predicted rear tire lateralforces at instance 1 from the experiment illustrated in Figure 5.13

−15 −10 −5 0 5 10 15

−40

−30

−20

−10

0

10

20

30

40

Side Slip, β (deg)

Ya

w R

ate

, r

(de

g/s

)

−5 0 5

−6

−4

−2

0

2

4

6

Re

ar

La

tera

l F

orc

e,

Fyr

(kN

)

Rear Slip Angle, αr (deg)

Current State

Near Pred

Long Pred

Actual

NonlinearTire Curve



−15 −10 −5 0 5 10 15

−40

−30

−20

−10

0

10

20

30

40

Side Slip, β (deg)

Ya

w R

ate

, r

(de

g/s

)

−5 0 5

−6

−4

−2

0

2

4

6

Re

ar

La

tera

l F

orc

e,

Fyr

(kN

)Rear Slip Angle, α

r (deg)

Current State

Near Pred

Long Pred

Actual

NonlinearTire Curve


tire forces. At instance 1 , the optimal vehicle states do not safely return within the

stable handling envelope as a result of the prioritization of collision avoidance over

stability. Using the approximate nonlinear rear tire model, the controller correctly

identifies the current speed as too fast for upcoming conditions. At instance 2 ,

the vehicle’s speed has greatly reduced and the predicted envelope violations, which

are shown in Figure 5.15, are much smaller resulting in a smaller braking command.

At instance 3 , the vehicle emerges from the double lane change unscathed with

the vehicle states just reaching the stable handling envelope bound as illustrated in

Figure 5.16.

Throughout this experiment, the friction circle constraint largely remains inactive,

Table 5.2: Proportional Braking Parameters

Parameter Value UnitsKbrake -12 kNFxf,maxComfort -3.6 kN


Table 5.3: Fixed Braking Parameters

Parameter Value UnitsFxf,fixed -3 kN

and the desired front longitudinal force, Fxf,des, identically matches the commanded

front tire force, Fxf,opt. This will not be the case for more aggressive braking policies.

5.5.2 Fixed Braking Policy

Figure 5.17 illustrates the controller implementing the fixed braking policy (5.2) for

the same double lane change maneuver at the same initial speed. For this experiment,

the brake parameters used are specified in Table 5.3. Despite the change in braking

policy, the vehicle’s speed, as expected, is reduced to a similar level as the previous

braking policy, indicating both policies converge to the same safe maximum speed.

And, like the previous example, the controller successfully navigates the maneuver

without violating either safe envelope.

At time instance 1 , the second lane change comes into view of the prediction

horizon, which causes a slight predicted envelope violation. Following braking policy

(5.2), a fixed brake value is commanded in response to this small predicted violation.

However, the optimal brake force applied to the vehicle is much less than this desired

fixed amount because the full friction capabilities of the front tires are currently

required to steer the vehicle, and the controller is explicitly considering this tire

force coupling. The steering command at this instant is rapidly decreasing as the

controller completes the first lane change. Even though the steering angle is not

saturated, the lateral force from the front tires is saturated. This is a result of the

extreme maneuver being executed and the dependency of front lateral force on both

steer angle and vehicle states, as previously described by equation (5.17). This is

something that may not be intuitive to the average driver but is explicitly considered

by the controller. The extreme steering of the maneuver at instance 1 naturally

causes a slight drop in vehicle speed without the use of brake actuation. This drop

in vehicle speed is enough to eliminate the predicted envelope violation, and, in


accordance with the braking policy (5.2), results in no commanded braking shortly

after instance 1 .

This experiment illustrates the ability of the controller to handle the coupling

between steering and braking in real-time as it avoids collisions and stabilizes the ve-

hicle. This ability is also important during emergency evasive maneuvers as described

in the next section.


0 20 40 60 80

0

2

4

6

Distance (m)

Dis

tanc

e(m

)

0 2 4 6 8 10−20

−10

0

10

20

Stee

r Ang

le, δ

(deg

)

0 2 4 6 8 10−4

−3

−2

−1

0

Fron

t Lon

gitu

dina

l For

ce, F

xf(k

N)

0 2 4 6 8 1068

101214

Time (s)

Spee

d, U

x(m

/s)


ActualDriver

ActualDesired

1

1

1

Figure 5.17: An experimental ISO 3888-2 double lane change on a low friction surface(µ ≈ 0.5) with a fixed brake amount in response to predicted envelope violation


5.6 Extension to Brake- and Throttle-By-Wire

Up to this point, Fxf,des has been largely dictated in response to predicted envelope

violations. However, as shown in the block diagram in Figure 5.5, this desired longi-

tudinal force could additionally come from the driver. Although X1 is equipped with

automatic brakes, the driver always maintains control over a set of manual brakes;

therefore, the driver’s brake command cannot be overridden as would be possible in

a true brake-by-wire system. For this reason, simulation results illustrate the con-

troller’s application using brake-by-wire.

Figure 5.18 illustrates a scenario where a driver overreacts and panic brakes in

response to a pop-up obstacle with the controller intervening to bring the vehicle

safely to a stop. At instance 1 , a pop-up obstacle appears 30 (m) in front of the

vehicle. The driver reacts immediately with a step steer command. The controller

augments this step command to adhere to the stable handling envelope. This is

followed by an aggressive brake command by the driver a short time later at instance

2 . The controller augments this brake command to ensure the combined steering

and braking of the front wheels adheres to the friction limits as illustrated by the

friction circle plotted in Figure 5.19. Transitioning from instance 2 to 3 , the

controller commands optimal lateral and longitudinal forces near the friction limits

of the front tires providing near maximum braking while allowing for the steering

force necessary to avoid collision and loss of control. At instance 4 , the full brake

command of the driver is safely applied to the vehicle.

From Figure 5.19, it is seen that the controller commands forces near, but not

always exactly on, the friction circle during the panic brake event. This is a result

of the conservatism introduced by using Tbrake = 9. Since the optimal longitudinal

force scalar is constrained to be within the friction circle for all Tbrake time steps, the

optimal longitudinal force is decreased as the controller plans to increase the lateral

force in the near term. This hedges against delays in the brake system leading to

saturation of the front tires on subsequent time steps. If instead Tbrake = 0, the

controller would brake up to the friction limits at each time step during the panic

brake event as indicated in Figure 5.20; however, this assumes a more aggressive


model of the time response of the brake actuator.

The extension to throttle-by-wire is straight forward on front-wheel-drive vehicles.

In this case, throttle commands would result in positive values of Fxf,des with brake

commands equating to negative values of Fxf,des. This would allow the controller to

override the driver’s acceleration commands in the exact same manner as brakes to

safely avoid collisions and loss of control.


0 10 20 30 40 50 60 70 0

2

4

6

8

Distance (m)

Dis

tanc

e(m

)

0 2 4 6 8−20

−10

0

10

20

Stee

r Ang

le, δ

(deg

)

0 2 4 6 8−6

−4

−2

0

Fron

t Lon

gitu

dina

l For

ce, F

xf

(kN

)

0 2 4 6 80

5

10

15

Time (s)

Spee

d, U

x

(m/s

)


ActualDriver

Actual

1

1

1

2

2

2

34

Driver

3 4

3

4

Figure 5.18: A simulated panic brake scenario where a pop-up obstacle appears atinstance 1 causing the driver to immediately steer the vehicle and, at instance 2,apply aggressive braking in a panic attempt to avoid collision


−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1

0

1

2

3

4

5

Fxf [

kN]

Fyf [kN]

Friction CircleFront Tire Forces

1

2

4

3

Figure 5.19: Friction circle for the front axle during the simulated panic brake ma-neuver on a low friction surface (µ ≈ 0.5) with Tbrake = 9


−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1

0

1

2

3

4

5

Fx

f [k

N]

Fyf [kN]

Friction Circle

Front Tire Forces

Figure 5.20: Friction circle for the front axle during the simulated panic brake ma-neuver on a low friction surface (µ ≈ 0.5) with Tbrake = 0


5.7 Future Work

Considering braking as a steering feasibility problem provides a nice simplification to

the challenging control problem of combined braking and steering. The implemen-

tation of this idea presented in this chapter works well for many driving situations;

however, there are some limitations left to future work. The first limitation is the

use of front longitudinal forces only, as mentioned previously in Section 5.4.1. Since

passenger cars are predominately front-wheel-drive and brake forces come predomi-

nately from the front brakes, due to weight transfer effects during braking, the front

longitudinal force only approach presented in this chapter is well suited to typical pas-

senger vehicles as is. However, considering only front longitudinal forces does limit

the peak brake force available to the vehicle and is therefore not acceptable for a

production safety system. To enable the envelope controller access to the full braking

capability of the vehicle, future work will seek to incorporate the coupled tire model

for the rear wheels as well as the front. A starting approach could be to introduce

another optimization variable representing an optimal rear longitudinal force, Fxr,opt,

and simply impose the friction circle constraint (5.9) on this force and the rear lateral

force, which is modeled using the affine approximation of the nonlinear tire curve.

This would ensure the rear tire brake force, Fxr,opt, adheres to the friction circle. The

challenge comes with respect to the derating of the rear lateral force in the presence

of this non-zero longitudinal force. As a result of the rear tire affine model, the lat-

eral rear force computed by the MPC optimization will not reflect this derating and

therefore this approach will only be an approximation. Future work will determine

how accurate or useful this approximation can be.

Another limitation arises from the constant speed assumption. Assuming a con-

stant speed throughout the prediction horizon limits the trajectories evaluated by

the controller to only a subset of the feasible trajectories. In most cases, this is not

a problem. In more extreme maneuvers, a safe variable speed trajectory may exist

where a safe constant speed trajectory does not, resulting in unnecessary interven-

tion by the controller. Although this may be irksome to the driver, the controller still

ensures the safe operation of the vehicle. However, the constant speed assumption


does result in the controller choosing to always steer to avoid collision, and, in some

rare cases, steering alone or combined steering and braking could lead to a collision

when there may exist a collision-free trajectory using braking only. Due to the long

4 (s) prediction horizon, these situations would only arise with pop-up obstacle over

a small range of vehicle speeds [55]. Fortunately in these situations, the controller

would identify the pending collision. Therefore, external logic could be used to force

the controller to consider non-steering trajectories in these scenarios. This could be

accomplished through augmentation of the road bounds to force the controller to

remain within its present lane or the generation of a future speed profile that varies.

Therefore, if an appropriate future speed profile could be computed externally from

the optimal controller, these situations could be avoided and the approach presented

in this chapter could be more broadly applied with Fxf,des coming partially from this

planned speed profile.

5.8 Discussion

Introducing a desired longitudinal force into the optimal control formulation avoids

the computationally challenging task of generating an optimal brake and steer trajec-

tory in real-time. Instead, the controller simply determines the amount of longitudinal

force that can be safely commanded at the current time step. This desired longitudi-

nal force is generated to slow the vehicle in response to predicted envelope violations,

creating a convex feasibility problem whose solution is the maximum safe vehicle

speed for the given combination of road conditions and environmental hazards. In

addition, this desired front longitudinal force can also come directly from the driver

in vehicles equipped with brake- and throttle-by-wire creating a comprehensive enve-

lope controller capable of ensuring vehicle safety through augmentation of the driver’s

steering, throttle, and brake commands.

Chapter 6

Conclusion

Future driver assistance systems will continue to move beyond vehicle stabilization

and additionally consider environmental threats to vehicle safety. Enabled by sig-

nificant improvements in sensing and actuation capabilities, automated systems can

assume more responsibility in ensuring vehicle safety, freeing human drivers of the

burden of perfection in the safe control of their vehicles. This sharing of control per-

mits a synergy that combines the unique cognitive and reasoning abilities of humans

with the vigilance and precision of machines.

This dissertation presents a simple and effective approach to shared control be-

tween a human driver and a highly capable automated system, implementing a form of

envelope control to ensure vehicle safety. Inspired by envelope control approaches in

both aircraft and vehicle control, the design and development of the proposed frame-

work makes a number of contributions of its own. The main contributions involve

modeling and control formulations that make possible real-time trajectory optimiza-

tion with explicit consideration of driver autonomy, obstacle avoidance, and vehicle

stability. A simple representation of the environment allows for the fast generation of

safe and feasible trajectories in real-time. A sparsity seeking objective biases the MPC

optimal solution to identically match the driver’s command whenever possible. As a

result of this strong bias to match the driver, the optimal steering solution lends itself

to the generation of a directional haptic feedback signal to intuitively communicate

the controller’s intentions to the driver. As demonstrated in experimental validation,

150

CHAPTER 6. CONCLUSION 151

this feedback encourages cooperation between driver and machine.

The challenging task of combined lateral and longitudinal control is cast as a

feasibility problem whose solution is the maximum safe vehicle speed for the given

combination of road conditions and environmental hazards. Applying successive lin-

earization to a pathless MPC approach enables real-time identification of vehicle

speeds that are too fast for the safe operation of the vehicle. This allows for a com-

prehensive envelope controller capable of ensuring vehicle safety in a wide range of

driving scenarios through augmentation of the driver’s steering, throttle, and brake

commands. Extensive validation and testing with experimental vehicles provides

support to the analysis and design of this control framework. The resulting con-

trol framework is a powerful alternative to the path tracking and following paradigm

commonly employed in automated vehicle control.

6.1 Future Work

The impact and contribution of a research project is demonstrated by the new and

exciting research directions that it inspires. The envelope control framework presented

in this dissertation serves as the foundation for a number of new research projects

addressing various vehicle control problems.

6.1.1 Fully Autonomous Vehicles

The design of the proposed control framework focuses on cooperation with a human

driver. However, this framework could be used in the control of a fully automated

vehicle. Removing the objective to match the driver’s steer command results in an

autonomous steering controller capable of both obstacle avoidance and stability. Ex-

tending the heading deviation cost through the prediction horizon and adding a cost

on deviations from a desired lateral offset, a path tracking objective is easily incor-

porated, which, if prioritized much lower than the obstacle avoidance and stability

objectives, provides guidance to the controller in the absence of threats to vehicle

safety. Implementing a fully autonomous vehicle is a major undertaking, but this


modified envelope controller is well suited to be the coordinator of the steering, brak-

ing, and throttle commands to achieve higher level objectives in a fully autonomous

vehicle. This transforms the complex problem of generating safe and feasible trajecto-

ries through the environment into higher level objectives like desired path and desired

longitudinal force, simultaneously simplifying the control problem while ensuring ve-

hicle safety. Experimental results using X1 have already provided a proof-of-concept

of this approach to fully automated driving, and future work seeks to build upon this

initial success.

6.1.2 Haptic Feedback User Studies

To the author’s knowledge, using an MPC optimal trajectory as the basis for a di-

rectional haptic feedback signal has not been previously explored in the literature.

The results presented in Chapter 3 show promise that this haptic signal can provide

a useful and intuitive method of communication between the control system and the

human driver. Future research focuses on exploring the subjective user experience

and overall effectiveness of this approach to haptic feedback for shared vehicle control.

In particular, a user study is underway with the goal of evaluating drivers’ responses

to pop-up obstacles and of understanding how the proposed haptic signal influences

these responses.

6.1.3 Application to Racing

As discussed briefly in Chapter 5, situations exist where considering variable speed

trajectories would improve the performance of the controller. The challenge arose in

how to compute these speed trajectories in real-time. In certain driving situations,

like racing, these variable speed profiles could be generated off-line and used in a

modified form of the proposed envelope controller. Research efforts are underway to

apply this approach to Shelley, an autonomous race car jointly developed by Stanford

and Audi [45].


6.1.4 Implementable Ethics

The proposed envelope controller uses multiple objectives to ensure safe vehicle op-

eration. As illustrated a number of times, these objectives sometime compete and

the tiered costs establish an objective hierarchy that guides the controller in these

situations. In doing this, the envelope controller implements a type of ethical frame-

work, and current research explores how the design of these types of control systems

might change if viewed through the lens of various established, ethical frameworks.

This interdisciplinary research attempts to bridge the gap between philosophy and

engineering.

6.1.5 Leveraging Advances in Parallel Computing

Modern processing units increasingly incorporate multiple processing cores to boost

computational performance. The proposed envelope controller is well suited to lever-

age these advancements in parallel processing hardware. The method of dividing

the non-convex obstacle avoidance problem into a number of convex sub-problems

enables a parallel processing unit to evaluate all of the sub-problems very quickly.

Future research efforts are underway to port the current implementation of the enve-

lope controller to parallel processing hardware, enabling the real-time consideration

of environments with many obstacles.

6.2 Outlook

Utilizing recent innovations in vehicle actuation and sensing, the proposed envelope

control framework shows great promise for ensuring vehicle safety. This work high-

lights the capabilities and opportunities of next generation driver assistant systems in

protecting the driver against environmental hazards in addition to stabilizing the ve-

hicle. These systems will play a more active role in driving, ensuring vehicle safety in

situation where, currently, the driver shoulders much of the burden. Off-loading some

of this responsibility to an envelope controller like the one presented would leverage


the precision and vigilance of machines while still retaining the critical thinking abili-

ties of humans. These next generation driver assistance systems will radically improve

the nature and safety of driving, while still providing drivers with the mobility and

flexibility of the automobile.

Bibliography

[1] Passenger cars - test track for a severe lane-change manoeuvre - part 1: Double

lane-change, 1999.

[2] Passenger cars - test track for a severe lane-change manoeuvre - part 2: Obstacle

avoidance, 2011.

[3] Volvo collision avoidance features: initial results. Technical report, Highway Loss

Data Institute, April 2012. Vol. 29 (5).

[4] AASHTO. Aashto green book: A policy on geometric design of highway and

streets, 6th ed. Technical report, American Association of State Highway and

Transportation Officials, Washington D.C., USA, 2011.

[5] Robert C. Allen and Harry G. Kwatny. Maneuverability and envelope protection

in the prevention of aircraft loss of control. In 8th Asian Control Conference

(ASCC), pages 381–386, 2011.

[6] S.J. Anderson, S.B. Karumanchi, and K. Iagnemma. Constraint-based planning

and control for safe, semi-autonomous operation of vehicles. In IEEE Intelligent

Vehicles Symposium, pages 383–388, June 2012.

[7] Rachid Attia, Jeremie Daniel, Jean-Philippe Lauffenburger, Rodolfo Orjuela, and

Michel Basset. Reference generation and control strategy for automated vehicle

guidance. In IEEE Intelligent Vehicles Symposium (IV), pages 389–394, June

2012.

155

BIBLIOGRAPHY 156

[8] Avinash Balachandran, Stephen M. Erlien, and J. Christian Gerdes. The vir-

tual wheel concept for supportive steering feel during large active steering inter-

ventions. Proceedings of the ASME Dynamic Systems and Control Conference,

September 2014.

[9] Avinash Balachandran and J. Christian Gerdes. Designing steering feel for steer-

by-wire vehicles using objective measures. IEEE/ASME Transactions on Mecha-

tronics, PP(99):1–11, 2014.

[10] C. E. Beal and J. C. Gerdes. Model predictive control for vehicle stabilization

at the limits of handling. IEEE Transactions on Control Systems Technology,

2013.

[11] Craig E. Beal. Applications of Model Predictive Control to Vehicle Dynamics for

Active SaSafe and Stability. PhD thesis, Stanford University, 2011.

[12] G P Bevan, H Gollee, and J OReilly. Trajectory generation for road vehicle ob-

stacle avoidance using convex optimization. Journal of Automobile Engineering,

224:455–473, 2010.

[13] Carrie G. Bobier and J. Christian Gerdes. Staying within the nullcline boundary

for vehicle envelope control using a sliding surface. Vehicle System Dynamics:

International Journal of Vehicle Mechanics and Mobility, 51(2):199–217, 2013.

[14] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University

Press, Cambridge, UK, 2004.

[15] Marc Starnes & Marilouise Burgess. Passenger vehicle occupant fatalities: The

decline for six years in a row from 2005 to 2011. Technical Report DOT HS 812

034, NHTSA of the U.S. Department of Transportation, June 2014.

[16] Robert N. Charette. Nissan moves to steer-by-wire for select infiniti models.

IEEE Spectrum, October 2012.

[17] Chulho Choi, Yeonsik Kang, and Seangwock Lee. Emergency collision avoidance

maneuver based on nonlinear model predictive control. In In proceedings of the

BIBLIOGRAPHY 157

IEEE International Conference on Vehicular Electronics and Safety, pages 393–

398, July 2012.

[18] Jennifer N. Dang. Statistical analysis of the effectiveness of electronic stability

control systems - final report. Technical Report DOT HS 810 794, NHTSA of

the U.S. Department of Transportation, July 2007.

[19] Nuno M. C. de Oliveira and Lorenz T. Biegler. Constraint handling and sta-

bility properties of model-predictive control. American Institute of Chemical

Engineers, 40:1138–115, July 1994.

[20] A. Domahidi, E. Chu, and S. Boyd. ECOS: An SOCP solver for embedded

systems. In European Control Conference (ECC), pages 3071–3076, 2013.

[21] Nicoleta Minoiu Enache, Sad Mammar, Mariana Netto, and Benoit Lusetti.

Driver steering assistance for lane-departure avoidance based on hybrid automata

and composite lyapunov function. IEEE Transactions on Intelligent Transporta-

tion Systems, 11(1), March 2010.

[22] Stephen M. Erlien, Susumu Fujita, and J. Christian Gerdes. Shared steering

control using safe envelopes for obstacle avoidance and vehicle stability. IEEE

Transactions on Intelligent Transportation Systems, 2014. (submitted, under

review).

[23] Stephen M. Erlien, Joesph Funke, and J. Christian Gerdes. Incorporating non-

linear tire dynamics into a convex approach to shared steering control. In Amer-

ican Control Conference, pages 3468 – 3473, June 2014.

[24] A. Muller F. von Hundelshausen, M. Himmelsbach and H.-J.Wunsche. Diriv-

ing with tentacles - integrated structures of sensing and motion. International

Journal of Field Robotics Research, 25(9):640–673, 2008.

[25] P. Falcone, F. Borrelli, J. Asgari, H.E. Tseng, and D. Hrovat. A model predictive

control approach for combined braking and steering in autonomous vehicles. In

Mediterranean Conference on Control Automation, pages 1–6, June 2007.

BIBLIOGRAPHY 158

[26] P. Falcone, F. Borrelli, E. Tseng, J.Asgari, and D. Hrovat. Linear time-varying

model predictive control and its application to active steering systems: Stabil-

ity analysis and experimental validation. International Journal of Robust and

Nonlinear Control, 18:862–875, 2007.

[27] Paolo Falcone, Francesco Borrelli, H. Eric Tseng, Jahan Asgari, and Davor

Hrovat. Low complexity mpc schemes for integrated vehicle dynamics control

problems. In Proceedings of the 9th International Symposium on Advanced Ve-

hicle Control (AVEC), 2008.

[28] Paolo Falcone, H. Eric Tseng, Francesco Borrelli, Jahan Asgari, and Davor

Hrovat. Mpc-based yaw and lateral stabilization via active front steering and

braking. Vehicle System Dynamics: International Journal of Vehicle Mechanics

and Mobility, 46:611–628, Sept 2008.

[29] E. Fiala. Lateral forces on rolling pneumatic tires. In Zeitschrit V.D.I, volume 96,

1954.

[30] Center for Disease Control and Prevention. Injury prevention & control: Data

& statistics, September 2014.

[31] B.A.C. Forsyth and K.E. MacLean. Predictive haptic guidance: intelligent user

assistance for the control of dynamic tasks. IEEE Transactions on Visualization

and Computer Graphics, 12(1):103–113, Jan 2006.

[32] Joesph Funke and J. Christian Gerdes. Simple clothoid paths for autonomous ve-

hicle lane change at the limits of handling. In proceedings of the ASME Dynamic

Systems and Control Conference, October 2013.

[33] Joseph Funke, Mathew Brown, Stephen M. Erlien, and J. Christian Gerdes. Dis-

cretionary stability control integrated with obstacle avoidance and path tracking

for autonomous vehicles. In proceedings of the IEEE Intelligent Vehicles Sympo-

sium, 2015. (submitted, under review).

BIBLIOGRAPHY 159

[34] Yiqi Gao, Andrew Gray, J. V. Frasch, Theresa Lin, H. Eric Tseng, J. Karl

Hedrick, and Francesco Borrelli. Spatial predictive control for agile semi-

autonomous ground vehicles. In International Symposium on Advanced Vehicle

Control, 2012.

[35] Yiqi Gao, Theresa Lin, Francesco Borrelli, Eric Tseng, and Davor Hrovat. Predic-

tive control of autonomous ground vehicles with obstacle avoidance on slippery

roads. In Dynamic Systems and Control Conference, 2010.

[36] Andrew Gray, Yiqi Gao, Theresa Lin, J. Karl Hedrick, H. Eric Tseng, and

Francesco Borrelli. Predictive control for agile semi-autonomous ground vehi-

cles using motion primitives. In American Control Conference (ACC), 2012,

pages 4239 –4244, june 2012.

[37] Rami Y. Hindiyeh and J. Christian Gerdes. A controller framework for au-

tonomous drifting: Design, stability, and experimental validation. Journal of

Dynamic Systems Measurements and Control, 136(5), July 2014.

[38] Y H Judy Hsu and J. Christian Gerdes. Envelope control: Keeping the vehicle

within its handling limits using front steering. In 21st International Symposium

on Dynamics of Vehicles on Roads and Tracks, 2009.

[39] Y.-H.J. Hsu, S.M. Laws, and J.C. Gerdes. Estimation of tire slip angle and

friction limits using steering torque. IEEE Transactions on Control Systems

Technology, 18(4):896–907, July 2010.

[40] S. Inagaki, I. Kshiro, and M. Yamamoto. Analysis of vehicle stability in crit-

ical cornering using phase-plane method. In Proceedings of the International

Symposium on Advanced Vehicle Control, 1994.

[41] Rolf Isermann, Roman Mannale, and Ken Schmitt. Collision-avoidance systems

proreta: Situation analysis and intervention control. Control Engineering Prac-

tice, 20(11):1236–1246, Nov 2012.

BIBLIOGRAPHY 160

[42] Alexander Katriniok, Jan P. Maschuw, Frederic Christen, Lutz Eckstein, and

Dirk Abel. Optimal vehicle dynamics control for combined longitudinal and

lateral autonomous vehicle guidance. In In Proceedings of the European Control

Conference, number 978-3-033-03962-9, pages 974–979, July 2013.

[43] Taketoshi Kawabe, Hikaru Nishira, and Toshiyuki Ohtsuka. An optimal path

generator using a receding horizon control scheme for intelligent automobiles. In

IEEE International Conference on Control Applications, 2004.

[44] Mykel J. Kochenderfer, Jessica E. Holland, and James P. Chryssanthacopoulos.

Next generation airborne collision avoidance system. LINCOLN LABORATORY

JOURNAL, 19:55–71, 2012.

[45] Krisada Kritayakirana and J. Christian Gerdes. Autonomous vehicle control at

the limits of handling. International Journal of Vehicle Autonomous Systems,

10(4):271–296, Jan 2012.

[46] Krisada (Mick) Kritayakirana. Autonomous Vehicle Control at the Limits of

Handling. PhD thesis, Stanford University, 2012.

[47] James K. Kuchar and Ann C. Drumm. The traffic alert and collision avoidance

system. Lincoln Laboratory Journal, 16(2):277–296, 2007.

[48] Steven M LaValle. Planning Algorithms. Cambridge University Press, 2006.

[49] Shad M. Laws, Christopher D. Gadda, and J. Christian Gerdes. Frequency

characteristics of vehicle handling: Model and experimental validation of yaw,

sideslip, and roll modes to 8 hz. In Proceedings of the AVEC 8th International

Symposium on Advanced Vehicle Control, number 0211, August 2011.

[50] Jan M Maciejowski. Predictive Control with Constraints. Prentice Hall, 2000.

[51] J. Mattingley and S. Boyd. Cvxgen: a code generator for embedded convex

optimization. Optimization and Engineering, 13(1):1–27, 2012.

BIBLIOGRAPHY 161

[52] J. Mattingley, Yang Wang, and S. Boyd. Code generation for receding horizon

control. In 2010 IEEE International Symposium on Computer-Aided Control

System Design (CACSD), pages 985 –992, sept. 2010.

[53] David Q. Mayne, James B. Rawlings, Christopher V. Rao, and Pierre O. M.

Scokaert. Constrained model predictive control: Stability and optimality. Auto-

matica, 36(6):789–814, 2000.

[54] William F. Milliken and Douglas L. Milliken. Race Car Vehicle Dynamics, pages

345–359. SAE International, 1995.

[55] Nikolai Moshchuk, Shih-Ken Chen, Chad Zagorski, and Amy Chatterjee. Op-

timal braking and steering control for active safety. In Proceedings of the 15th

IEEE Conference on Intelligent Transportation Systems, pages 1741–1746, 2012.

[56] M. Nanao and T. Ohtsuka. Nonlinear model predictive control for vehicle collision

avoidance using c/gmres algorithm. In In proceedings of the IEEE International

Conference on Control Applications, pages 1630–1635, Sept 2010.

[57] NHTSA. Traffic safety facts: 2012 data. Technical Report DOT HS 812 016,

U.S. Department of Transportation, May (Revised) 2014.

[58] NHTSA. Traffic safety facts: Estimating lives saved by electronic stability con-

trol, 2008-212. Technical Report DOT HS 812 042, U.S. Department of Trans-

portation, June 2014.

[59] M. D. North. Finding common ground in envelope protection systems. Aviation

Week and Space Technology, August 2008.

[60] U.S. Department of Transportation. Transportation statistics annual report

2012. Technical report, Research and Innovative Technology Administration

Bureau of Transportation Statistics, Washington, D.C., August 2013.

[61] Meeko Oishi, Ian Mitchell, Claire Tomlin, and Patrick Saint-Pierre. Computing

viable sets and reachable sets to design feedback linearizing control laws under

BIBLIOGRAPHY 162

saturation. In 45th IEEE Conference on Decision and Control, pages 3801–3807,

Dec 2006.

[62] Hans B. Pacejka. Tire and Vehicle Dynamics. Butterworth-Heinemann, 3rd

edition, 2012.

[63] R. S. Rice. Measuring car-driver interaction with the g-g diagram. Society of

Automotive Engineers, Warrendale, PA, (730018):1–19, 1973.

[64] Eric J. Rossetter, Joshua P. Switkes, and J. Christian Gerdes. Experimental

validation of the potential field lanekeeping system. International Journal of

Automotive Technology, 5:98–108, 2004.

[65] Louay Saleh, Philippe Chevrel, Fabien Claveau, Jean-Franois Lafay, and Franck

Mars. Shared steering control between a driver and an automation: Stability in

the presence of driver behavior uncertainty. IEEE Transactions on Intelligent

Transportation Systems, 14(2):974–983, June 2013.

[66] Pierre O. M. Scokaert and James B. Rawlings. Feasibility issues in linear model

predictive control. American Institute of Chemical Engineers, 45:1649–1659,

1999.

[67] T. B. Sheridan and R. Parasuram. Human-automation interaction. Rev. Human

Factors Ergonomics, 1(1):89–129, 2005.

[68] Suk-Hwan Suh and Albert B. Bishop. Collision-avoidance trajectory planning

using tube concept: Analysis and simulation. Journal of Robotic Systems, 5:497–

525, 1988.

[69] Sabastian Thrun. Stanley: The robot that won the darpa grand challenge. Jour-

nal of Field Robotics, 23(9):661–692, 2006.

[70] J. P. Timings and D. J. Cole. Minimum maneuver time calculation using convex

optimization. Journal of Dynamic Systems Measurements and Control, 135,

March 2013.

BIBLIOGRAPHY 163

[71] Valerio Turri, Ashwin Carvalho, Hongtei Eric Tseng, Karl Henrik lohansson,

and Francesco Borrelli. Linear model predictive control for lane keeping and

obstacle avoidance on low curvature roads. In IEEE International Conference

on Intelligent Transportation Systems, Oct 2013.

[72] Chris Urmson. Autonomous driving in urban environments: Boss and the urban

challenge. Journal of Field Robotics, 25(8):425–466, 2008.

[73] Klaus H Well. Aircraft control laws for envelope protection. In AIAA Guidance,

Navigation, and Control Conference, number AIAA-2006-6055, 2006.

[74] Mortiz Werling and Darren Liccardo. Automatic collision avoidance using model-

predictive online optimization. In Proceedings of the 51st IEEE Conference on

Decision and Control, December 2012.

[75] Fernando A. Wilson and Jim P. Stimpson. Trends in fatalities from distracted

driving in the united states, 1999 to 2008. American Journal of Public Health,

100(11):2213–2219, 2012.

[76] M.N. Zeilinger, M. Morari, and C.N. Jones. Soft constrained model predictive

control with robust stability guarantees. IEEE Transactions on Automatic Con-

trol, 59(5):1190–1202, May 2014.

SHARED VEHICLE CONTROL USING SAFE DRIVING ENVELOPES … · safety like Anti-Lock Braking Systems...

Documents

Transcript of SHARED VEHICLE CONTROL USING SAFE DRIVING ENVELOPES … · safety like Anti-Lock Braking Systems...