A Fully Integrated Sensor Fusion Method Combining a Single Antenna GPS
Unit with Electronic Stability Control Sensors
by
Jonathan Ryan
A thesis submitted to the Graduate Faculty of
Auburn University
in partial ful llment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
August 6, 2011
Keywords: sensor fusion, vehicle dynamics, sideslip estimation, roll estimation, indirect tire
pressure monitoring, steering misalignment detection
Copyright 2011 by Jonathan Ryan
Approved by
David Bevly, Chair, Associate Professor of Mechanical Engineering
John Y. Hung, Professor of Electrical Engineering
Subhash Sinha, Professor of Mechanical Engineering
Abstract
This work presents a method for incorporating GPS (Global Positioning System) and
standard roll stability control (RSC) sensors into the electronic stability control (ESC) and
RSC systems. It is an adaptation of the very well known loosely-coupled GPS/INS (Inertial
Navigation System) integration strategy which has been modi ed for the purposes of ESC
systems. The rst modi cation is the removal of the pitch rate gyroscope, a sensor which
is unavailable on commercial vehicles. The second modi cation deals with the observability
problems of the standard loosely coupled lter by adding heading constraints when the ve-
hicle is not turning. The structure and algorithm of this method is presented. Observability
conditions are evaluated, and the convergence of the estimates are analyzed via simulations.
The conclusions from these simulations are compared with the expectations from the lit-
erature and observability condition checks. An experiment which illustrates the long term
performance of the bias estimation was performed, followed by an experiment showing the
roll and sideslip estimation performance during dynamic events. It is shown that over the
long term the inertial bias estimates will converge if the vehicle experiences adequate dy-
namics, and that the system is able to accurately estimate sideslip and roll during dynamic
maneuvers. The system is also able to estimate slow sideslip buildup, an important capabil-
ity for ESC systems. The uni ed system is compared with a less integrated or \modular"
approach for both experiments.
Furthermore, a method for using GPS to detect tire pressure changes is presented based
on the hypothesis that the tire e ective radius varies according to tire pressure. A technique
using GPS and wheel speed signals to estimate the e ective radius of the tires is discussed
ii
and validated in simulation and experiment. Experiments are given to show how the ra-
dius estimate varies according to tire pressure, and a simple pressure loss detection law is
discussed. A method to detect steering misalignment is also presented.
iii
Acknowledgments
This material is based upon work supported by the National Science Foundation Grad-
uate Research Fellowship under Grant No. DGE-0809382. I would rst like to acknowledge
my advisor, Dr. Bevly, for his obvious role in my graduate education. In many ways he is
the reason I started down this path of grad school, urging me to consider graduate studies in
the rst place and also challenging me to go after fellowships I never would have otherwise
dreamed of. In many ways this exempli es his main contribution to my education. That
is, he believes in his students and expects them to achieve excellence. For this, and for his
investment in me as a student, I am grateful. I also acknowledge his contribution to the
GAVLAB intramural basketball team as one of our primary o ensive producers. I also wish
to acknowledge my committee, Dr. John Hung and Dr. Subhash Sinha, for their oversight
on this thesis.
I would now like to acknowledge my parents, who have been encouraging and supportive
throughout my entire life, and no less in graduate school. There is no doubt that I would
not be where I am today without them. Likewise I want to thank my grandparents for their
constant love and support, for always being there, and for always encouraging me to do my
best, something which they have each modeled with great integrity.
I also owe much to my fellow students in the GAVLAB. Rob, Broderick, and Will were
especially helpful when I rst began and each frequently o ered their own time to help me
with this or that. Likewise Hodo, Ben, Lashley, and Wei were always there when I had
questions. All of these guys set an example both in excellence in research and also in the
importance of working with and investing in coworkers around you. I owe a great deal also
to the men of L2: Lowell, Ryan, Scott and Wei. Those were good times, and I dare say we
took L2 to new heights and ushered in its golden era (mostly because Ryan and Lowell took
iv
the e ort to deep clean the place). I also want to thank Jeremy, Jordan, Chris, and John for
their input and help over the last months. Jordan especially deserves special gratitude for
taking charge of the new G35 and making it the sleek, 007 -esque data collection machine
that it is.
I wish to thank Ford Motor Company for sponsoring this work and giving me the
opportunity to come to Dearborn for the summer. It was an invaluable experience, and I
enjoyed meeting all of the good people up there. Special thanks goes to Jianbo Lu for his
involvement in this work and all of the assistance he provided over the years.
Finally, I wish to acknowledge my wife Ginger Ryan with deepest gratitude for her
unwavering love and support during this time. She has always completely supported me and
believed in me throughout my studies, even when it meant delaying some of her own dreams.
She has truly exempli ed to me the love of Christ in the way that she has sacri ced for our
family and cared for my interests above her own. Her companionship is sweet, her wisdom is
deep, and her baking is delicious. This work would not be what it is without her, for I would
not be who I am without her. For all of these people and this opportunity I give thanks to
the Lord Jesus Christ.
v
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1 Introduction and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 GPS/INS Integration Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Kalman Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.2 Sensor Rotations and Coordinate Frames . . . . . . . . . . . . . . . . 15
2.1.3 Inertial Sensor Models . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.4 GPS Sensor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 GPS/INS: The Modi ed Modular Estimator . . . . . . . . . . . . . . . . . . 20
2.2.1 Estimation Strategy Overview . . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 Road Grade Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.3 Heading Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.4 Lateral Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 GPS/INS: The Automotive Navigation (AUNAV) Estimator . . . . . . . . . 29
2.3.1 The Loosely Coupled Algorithm . . . . . . . . . . . . . . . . . . . . . 29
2.3.2 Modi cations to the Loosely Coupled Algorithm . . . . . . . . . . . . 34
2.3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Observability of the AUNAV Estimator . . . . . . . . . . . . . . . . . . . . . . . 40
3.1 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
vi
3.2 Applications in Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Observability Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Experimental Validation of the AUNAV Estimator . . . . . . . . . . . . . . . . 59
4.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Slowly Growing Sideslip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5 Experimental Comparison with the MME Estimator . . . . . . . . . . . . . . . 79
5.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6 Other GPS Applications: Tire Radius Estimation, Tire Pressure Monitoring, and
Steering Misalignment Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.2 Tire Rolling Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.3 Tire Pressure Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.4 Steering Misalignment Detection . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
vii
List of Figures
1.1 Sideslip De nitions in the Navigation and Body Coordinate Frames. . . . . . . . 2
1.2 SAE Coordinate System, Body Frame [32]. . . . . . . . . . . . . . . . . . . . . . 4
2.1 Diagram of the Modi ed Modular Estimator. . . . . . . . . . . . . . . . . . . . 20
2.2 Diagram of the Standard Loosely Coupled Integration Strategy. . . . . . . . . . 30
3.1 Observability of the Loosely Coupled and AUNAV Filters During Longitudinal
Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Observability of the Loosely Coupled and AUNAV Filters During Lateral Dynamics. 48
3.3 Observability of the LC Filter Compared with the AUNAV Filter During Longi-
tudinal Dynamics when the Course Measurement is Conditionally Added to the
AUNAV Filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4 Observability of the LC Filter Compared with the AUNAV Filter During Lateral
Dynamics when the Course Measurement is Conditionally Added to the AUNAV
Filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Speed and Yaw Rate Pro le of Convergence Test Simulation. . . . . . . . . . . . 51
3.6 Convergence of the AUNAV Accelerometer Bias Estimates During Simulation. . 51
3.7 Convergence of the AUNAV Gyroscope Bias Estimates During Simulation. . . . 52
viii
3.8 Convergence of the AUNAV Gyroscope Bias Estimates During Experimental
Testing with Limited Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.9 Velocity and Yaw Rate from Experimental Testing. . . . . . . . . . . . . . . . . 54
3.10 AUNAV Roll Angle Estimate Convergence During Simulated Test. . . . . . . . 55
3.11 AUNAV Pitch Angle Estimate Convergence During Simulated Test. . . . . . . . 55
3.12 AUNAV Yaw Angle Estimate Convergence During Simulated Test. . . . . . . . 56
4.1 Vehicle Trajectory During Initialization Experiment. . . . . . . . . . . . . . . . 61
4.2 Pro le of Dynamic Conditions During Initialization Experiment. . . . . . . . . . 61
4.3 AUNAV Accelerometer Bias Estimation and Convergence During Initialization
Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4 AUNAV Roll Angle Estimation and Convergence During Initialization Experiment. 64
4.5 AUNAV Gyroscope Bias Estimation and Convergence During Initialization Ex-
periment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.6 Vehicle Trajectory for Dynamic Test on the NCAT Skid Pad. . . . . . . . . . . 66
4.7 Pro le of Dynamic Conditions During Dynamic Test on the NCAT Skid Pad. . 67
4.8 AUNAV Velocity Estimates for Dynamic Test. . . . . . . . . . . . . . . . . . . . 68
4.9 Velocity Estimation Residuals for Dynamic Test. . . . . . . . . . . . . . . . . . 68
4.10 AUNAV Sideslip Estimation During Dynamic Test. . . . . . . . . . . . . . . . . 69
4.11 Velocity Innovations During Dynamic Test. . . . . . . . . . . . . . . . . . . . . 69
ix
4.12 AUNAV Roll Estimate for Dynamic Test. . . . . . . . . . . . . . . . . . . . . . 71
4.13 AUNAV Pitch Estimate for Dynamic Test. . . . . . . . . . . . . . . . . . . . . . 71
4.14 Pitch Estimate when Using Septentrio Pitch Information as an Extra Measurement. 73
4.15 Comparison of the AUNAV Pitch Estimate with the Septentrio Measurement and
the Road Grade Estimate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.16 Rate of Sideslip Growth During Simulated Test. . . . . . . . . . . . . . . . . . . 75
4.17 AUNAV Sideslip Estimate During Simulated Test. . . . . . . . . . . . . . . . . 77
4.18 AUNAV Sideslip Estimate During Experimental Test with Slowly Growing Sideslip. 77
5.1 Yaw Rate and Yaw Constraint Signals of Initialization Simulated Test. . . . . . 82
5.2 MME Estimate of the Lumped State Compared with True Simulation Values. . 82
5.3 Convergence of the MME Roll Angle Estimate During Simulated Initialization
Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.4 Convergence of the MME Accelerometer Bias Estimates During Simulated Ini-
tialization Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.5 Convergence of the MME Gyroscope Bias Estimates During Simulated Initial-
ization Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.6 Convergence of the Pitch Angle Estimate During Simulated Initialization Test. . 84
5.7 Yaw Rate and Yaw Constraint Signals During Initialization Experiment. . . . . 86
5.8 Comparison of Velocity Estimates During Initialization Experiment. . . . . . . . 86
x
5.9 Comparison of Roll Angle Estimation Convergence During Initialization Experi-
ment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.10 Comparison of Accelerometer Bias Estimation Convergence During Initialization
Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.11 Comparison of Pitch Angle Estimation Convergence During Initialization Exper-
iment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.12 Comparison of Gyroscope Bias Estimation Convergence During Initialization Ex-
periment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.13 Vehicle Trajectory During Dynamic Test on NCAT Skid Pad. . . . . . . . . . . 91
5.14 Pro le of Dynamic Conditions During Dynamic Test on NCAT Skid Pad. . . . . 91
5.15 Comparison of Forward and Vertical Velocity Estimates During Dynamic Test. . 92
5.16 Comparison of Roll Angle Estimates During Dynamic Test. . . . . . . . . . . . 93
5.17 Comparison of Pitch Angle Estimates During Dynamic Test. . . . . . . . . . . . 94
5.18 Comparison of Sideslip Angle Estimates During Dynamic Test. . . . . . . . . . 94
6.1 Undriven Wheel E ective Radius Estimate Convergence in Simulation. . . . . . 101
6.2 Undriven Wheel E ective Radius Estimate Convergence with Faster Tuning in
Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.3 Driven Wheel E ective Radius Estimate Convergence in Simulation. . . . . . . 102
6.4 Undriven Wheel E ective Radius Estimate Convergence During Figure-8 Turning
Maneuvers in Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
xi
6.5 Front Left Tire Radius Estimates for Di erent Pressures. . . . . . . . . . . . . . 104
6.6 Rear Right Tire Radius Estimates for Di erent Pressures. . . . . . . . . . . . . 105
6.7 Estimates for All Four Tires for All Four Experiments. . . . . . . . . . . . . . . 105
6.8 Steer Angle Signals for All Four Misalignment Experiments. . . . . . . . . . . . 109
6.9 Averaged Steer Angle Signals and Yaw Rate Signals For All Four Misalignment
Experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
xii
Chapter 1
Introduction and Background
Intelligent safety systems are an increasing focal point in today?s automotive industry.
The motivation for these systems stems from the tragic reality that tens of thousands of
people are killed in motor vehicle accidents every year. For example, over 37,000 people
were killed in vehicle accidents in 2008. Rollover accounted for 33% of these deaths. In fact,
motor vehicle crashes are the leading cause of death among Americans between the ages
of 1 and 34 [17]. The problem has received so much attention that Congress has enacted
legislation requiring all new vehicles after 2012 to include certain intelligent safety systems
as standard features such as electronic stability control (ESC).
Electronic stability control systems and roll stability control (RSC) systems are key
elements of the modern e ort to improve and increase the safety capabilities of passenger
vehicles. ESC systems seek to control unsafe yaw and lateral motions of the vehicle. These
motions can occur when the vehicle begins to lose traction and a dangerous over steer
situation arises. One extreme example would be the back end of the vehicle \sliding out" or
\ sh-tailing" during a turn. ESC systems apply control in these situations via di erential
braking of the individual tires to create the desired control moment. This is also true of RSC
systems, which as their name suggests seek to minimize unsafe levels of vehicle roll. Both of
these systems utilize feedback control systems, requiring information about particular states
of the vehicle. Two of the most critical states for these systems are the sideslip and roll
angles, which are also two of the most expensive to directly measure. In order to obtain
accurate information about these states without greatly increasing the production cost of
the vehicle, state estimation theory must be applied with the sensors which are already on
board. This is the goal of this thesis.
1
Figure 1.1: Sideslip De nitions in the Navigation and Body Coordinate Frames.
Now consider the two de nitions of sideslip followed by a discussion of the vehicle
coordinate frame. Figure 1.1 shows a simple diagram of a single track vehicle situated in
the North-East plane of the North-East-Down (NED) coordinate frame. Not shown is the z
axis of the NED frame, which goes into the page to complete a right handed system. The
blue dashed lines show the body coordinate frame, with the x axis aligned with the forward
direction of the vehicle, the y axis perpendicular to right side, and the z axis pointing into
the page to complete the right handed system. For the purposes of this diagram, the two z
axes are collinear, but this is not true generally. In Figure 1.1, the bold vector V represents
the velocity vector of the vehicle?s center of gravity, while the blue vectors Vx;Vy represent
the components resolved into the body frame. The angle is de ned as the \course" angle,
and represents the angle of V from North. The angle is de ned as the \heading" angle
2
or yaw angle and represents the angular direction that the vehicle is facing from North.
Both angles are positive in the clockwise direction. The angle is the sideslip angle, which
can now be de ned in two ways. First, it can be thought of as the di erence between the
direction that the vehicle is moving and the direction that it is facing as described in (1.1).
= (1.1)
Sideslip can also be thought of as the ratio of the lateral and forward velocities, as in (1.2).
= atan(VyV
x
) (1.2)
In both de nitions, the sideslip angle is positive in the clockwise direction. Equation (1.2)
is more suitable to intuition, because it basically shows that sideslip is simply an angular
representation of the vehicle?s lateral velocity, scaled according to forward speed. If the
sideslip angle is large, it means that the vehicle is sliding no matter what the forward speed
is. While there is always a small amount of sideslip during any turn, it holds true that cars
are not intended to go sideways. Therefore the goal for safety systems is to keep sideslip
angles low.
Now consider the de nitions of the attitude angles and the body xed coordinate axis.
Figure 1.2 shows the SAE coordinate axis for the body frame (original image courtesy of [32]).
It is a right handed system, with the X axis pointing in the forward direction, the Y axis to
the right of the vehicle, and the Z axis pointing down. The angle marks the positive roll
angle direction, the positive pitch angle direction, and the positive yaw angle direction.
An interesting side note involving the pitch angle is that if the vehicle is driving on a road
with a positive grade (pitch) angle, the Z velocity will actually be negative and vice versa.
This is a point of potential confusion which should be remembered when considering road
grades.
3
Figure 1.2: SAE Coordinate System, Body Frame [32].
1.1 Literature Review
State estimation approaches for vehicle dynamics can be broadly categorized into two
groups: model-based approaches and kinematic approaches. Both approaches use some sort
of \model", but the di erence lies in what parameterizes the equations which make up the
model. In the model-based case, the model parameters correspond to vehicle parameters such
as mass, inertia, wheel base, tire sti ness, etc. Strictly speaking, model based estimators
might even employ some kinematic equations in the model. What matters, however, is that
these equations are inevitably parameterized by vehicle parameters. By contrast \kinematic
estimators" do not use any vehicle parameters. Instead all terms in their \model" equations
are constituted by sensor signals such as acceleration, rotation rate, velocity, or position.
Many of these accelerations and rotation rates are directly measurable with common ESC
sensors, while GPS provides information about the velocities and positions. Kinematic
estimators do rely on knowledge of sensor location, although this has been shown possible to
estimate accurately when it is di cult to measure. This distance known as the \lever arm"
does not change, making it a relatively easy problem to overcome [27]. Kinematic estimators
also rely on parameters of the sensors, such as bias time constants and noise variances, yet
these are possible to identify o line. Furthermore, these parameters do not change. While it
4
is true that some of these sensor parameters are a function of temperature, this relationship
can be accounted for as well. The primary advantage, then, of \kinematic estimators" is that
they do not rely on changing vehicle parameters. The system is robust to di erent loadings,
after market suspension modi cations, tire wear, or even a completely di erent set of tires.
The signi cance of the roll and sideslip angles has led to many publications documenting
various methods of their estimation. This includes a large body of research involving the use
of GPS to estimate sideslip kinematicly, as opposed to using vehicle models. An important
distinction among these is the number of GPS antennas used, as many current vehicles
are already instrumented with a single antenna system as opposed to double or even triple
antenna systems. Several researchers have proposed using single antenna GPS systems to
improve sideslip estimation methods. A simple approach is presented in [2] and contrasted
with results from a model based estimator. A more cascaded approach is presented in [8],
where a sideslip estimate is rst obtained with a yaw rate gyroscope. This estimate is
used to aid sideslip estimation with a lateral accelerometer. This approach is expanded
even further in [6]. Even more authors approach kinematic sideslip estimation using dual
antenna GPS systems. Ryu uses a dual antenna GPS receiver with INS to estimate vehicle
velocities, sideslip, roll, and road grade [40]. In [9], Bevly and Ryu present Kalman ltering
methods for vehicle state estimation using both single and dual antenna GPS/INS systems.
A planar model is no longer assumed, as the inertial measurements are compensated for the
roll e ects [9]. In [7] and [15], the authors use a dual antenna system to obtain estimates
of the lateral states, including sideslip. These estimates are then used to determine tire
parameters.
The kinematic approach to state estimation has also been a popular way to approach
roll angle estimation. Tseng [44] presents a novel method of estimating both roll and pitch
based on the inertial mechanization equation without using GPS at all. Yet far more authors
elect to take advantage of GPS information. The previously referenced works [9], [6], and [40]
focus on roll estimation as well sideslip. In [6] the lumped e ects of roll and the accelerometer
5
bias are estimated using a single antenna GPS and then separated using a low pass lter.
The approach in [9] rst uses a single antenna system. The estimate of the roll is then
extrapolated from the estimate of the lateral accelerometer bias. Since the accelerometer
is not compensated for roll, the roll e ects dominate the bias estimate and therefore this
provides a good method for approximating roll. The authors then compare this method
with results from integration with a dual antenna GPS system. Dual antenna systems can
measure roll directly, and as a result GPS/INS integrations which use such systems can
produce more accurate roll estimates. The dual antenna approach is also adopted in [40],
which was one of the rst papers to demonstrate this potential. The roll information can
also be used with a vehicle model to separate suspension roll from the road bank, as done
in [38]. Those authors go on to show in [39] that roll and pitch have signi cant e ects on
sideslip estimation. It follows then that it is important to include pitch information into the
estimation algorithm. In urban areas the design limit for road grade can be up to 9% at
speeds of 60 mph, and up to 12% for lower speeds around 30 mph. Rural roads can go up to
10% at design speeds of 40 mph and 8% for 60 mph design speeds [33]. Therefore knowledge
of the pitch will be necessary to fully exploit these GPS/INS estimation schemes on steeper
roadways.
Jansson estimates the road grade by combining GPS information with barometer and
torque measurements into a Kalman lter [29]. Sahlholm and Johansson take a similar
model based approach using drive line sensors and GPS; however they additionally present
a method for recursively improving the grade estimate with new passes over the same road
[41]. Lingman and Schmidtbauer also use a longitudinal vehicle model and Kalman ltering
techniques to estimate both vehicle mass and road grade. This is done without any GPS
information [30]. All of these are done under the context of longitudinal vehicle control, as
opposed to lateral dynamic control applications. Bae and Ryu describe two methods for road
grade estimation using GPS that are much more suitable for lateral estimation and control
purposes [3]. These methods involve measuring total pitch directly with a dual antenna
6
GPS receiver or taking the arctangent of the ratio of the vehicle up and forward velocities
obtained from a single antenna [3]. This method is expanded in [4], where the up and forward
velocities are estimated using a simple Kalman lter. Since pitch rate sensors are uncommon
on most commercial vehicles, the road grade estimate a ords the opportunity of replacing
the pitch rate gyro with the increasingly common single antenna GPS unit.
All of the above authors use \modular" sensor fusion approaches as opposed to those
that are \uni ed" to estimate the roll, pitch, and sideslip angles. Speaking very speci cally,
the term \uni ed" is meant to signify a lter which incorporates all position, velocity, and
attitude information of the three dimensional, six degree-of-freedom (DOF) model into one
single lter. The term \modular" is meant to signify two things. First, it means that all of
the states are not coupled together into one single lter, rather there are separate or cascaded
lters for various sets of states. These separate lters may or may not be coupled together
indirectly, sharing information about certain variables, but the distinction here is that this
sharing is done outside of the lter. An example of a modular approach is [39], where the
authors use one lter for the heading state and gyro bias and another for the forward and
lateral velocities and accelerometer biases. Second, the term modular conveys that although
there might be one single lter in the overall estimator, this lter does not include states for
all six degrees of freedom. An example of this would be [15], where the lter is based on a
planar model (the authors compensate for roll e ects outside of the lter).
The following works consider approaches which are based on a uni ed integration
scheme. The authors of [10] present a navigator based on a single antenna GPS integrated
with a low cost INS in a loosely coupled integration. They evaluate its performance for
positioning and present some experimental results. The authors of [16] use vehicle con-
straints, such as assuming no lateral or vertical velocity, to improve a standard loosely cou-
pled GPS/INS implementation. These assumptions are used for performance improvement
by many authors. In [18] the authors apply the constraints to a tightly coupled GPS/INS
architecture. The authors of [20] also apply to velocity constraints to the loosely coupled
7
lter and evaluate its performance. The di erence between these works and this thesis is
three-fold. First, all of these works use a 6 DOF IMU/INS, whereas this work only uses a
5 DOF INS. Second, all of these works assume that the lateral velocity is zero at all times,
whereas that assumption is made in this thesis only on the basis of certain conditions. Fi-
nally, all of these works evaluate the performance primarily with regard to position, velocity,
or attitude as opposed to sideslip. Indeed they make the assumption of zero sideslip at
all times, whereas in this thesis the sideslip estimation performance is the chief objective.
In [26] the authors perform an observability analysis on the loosely coupled GPS/INS lter
and nd that under certain maneuvers, all states can be observed. It is also possible to use
magnetometers and magnetic sensors embedded into the roadway to aid the lter. Yang and
Farrell demonstrate this in [45] by creating a vehicle state estimation system having three
layers of redundancy which uses magnetometers, GPS, and INS to determine the vehicle
states. The accuracies and observabilities of the di erent estimates are discussed regarding
the availability of each of the sensors. They show that adding the magnetometers increases
the observability and eliminates the acceleration requirements.
The performance of the loosely coupled lter for position and velocity determination
is well studied. However, the loosely coupled lter uses a pitch rate gyroscope which is
not available on commercial vehicles. Therefore the lter?s performance must be evaluated
in light of this sensor reduction. Furthermore, the sideslip estimation performance of the
loosely coupled lter is not well documented, because the loosely coupled lter is usually
employed for general navigation purposes. The potential for sideslip estimation using this
lter is brie y discussed in [19], but the performance is not analyzed. Some example plots
of sideslip estimates are indeed shown, although these were not generated using a loosely
coupled lter, rather they are estimates from the method presented in [15]. There is even less,
if any, documented studies on the sideslip estimation performance of the loosely coupled lter
when the pitch rate gyroscope is removed. Evaluating this performance is a primary goal
of this thesis. The yaw information which is added to the lter based on certain conditions
8
(as described later) is also a new element. This constraint is practically the same as the
one added to the estimators in [8], [2], but there is no documentation regarding how this
additional yaw information a ects the performance of the loosely coupled lter. Such analysis
is an important part of understanding the sideslip estimation performance, and it is included
in this thesis. Finally, there is a fair amount of literature documenting the observability
characteristics of the system described by the loosely coupled lter, speci cally noting that
the observability of the system depends on the dynamics. Yet these results cannot be taken
for granted regarding the lter used in this thesis, because of the aforementioned changes
to the system. This thesis presents a summary of the observability results of the loosely
coupled system, followed by an observability analysis of the presented system. Therefore
this thesis is distinguished from the previous works by the lack of the pitch rate gyroscope,
the sideslip estimation performance analysis, the conditionally included yaw constraint, and
the observability analysis of this new system.
1.2 Contributions
The goal of this work is to achieve good sideslip, attitude, and velocity estimation
in addition to inertial sensor bias identi cation by combining GPS with sensors which are
already present in current RSC system sensor clusters. Furthermore, the operating window
of the system needs to be expanded by making the system robust to all road geometries.
The sensors used in this work are the GPS (single-antenna, 1Hz), accelerometers in x;y;z,
a roll rate gyroscope, and a yaw rate gyroscope. There is no pitch rate gyroscope, because
these are not present on commercial vehicles. Additionally, this work shows the capability
of using GPS to identify changes in tire pressure without using pressure sensors. A method
of using the yaw rate signal, improved with the bias estimates from the state estimator, and
the steer angle sensor to estimate steering misalignment is also presented. All of these goals
seek to use GPS to increase the intelligent safety capabilities of commercial vehicles without
any additional sensor costs. Following are the contributions contained in this thesis:
9
Development of an algorithm (initial results presented by the author in [37]) to provide
sideslip and attitude estimates using ESC sensors and single antenna GPS.
Analysis of sideslip and roll estimation performance of the algorithm.
Analysis of performance improvements provided by conditionally added course mea-
surements.
Observability analysis of the algorithm.
Performance analysis of modi ed \modular" method (initial results presented by the
author in [36]).
Development of a method to detect tire pressure changes using GPS and wheel speed
sensors.
Development of a method to detect steering misalignment using bias estimates from
the state estimator, a yaw rate sensor, and the steer angle sensor.
The rst contribution of this work is the development of a fully integrated state esti-
mation algorithm, for the speci c purpose of estimating sideslip, roll, and inertial biases,
which uses only GPS and sensors present in RSC systems. The algorithm will be referred
to throughout the thesis as the (au)tomotive (nav)igation (AUNAV) estimator. The AU-
NAV estimator is originally based on the well known \loosely coupled" GPS/INS integration
scheme [19], [23]. The di erences between the AUNAV estimator and the basic loosely cou-
pled lter are that the AUNAV estimator does not use pitch rate gyroscope information and
that it incorporates course information from the GPS, when it detects that the vehicle is not
turning, in order to improve the sideslip estimation performance. The \navigation" term
in the acronym points back to the original loosely coupled lter?s purpose, and to the fact
that the AUNAV estimator still possesses the same navigation functionality even though
navigation is not its primary purpose. The AUNAV algorithm was rst presented in [37],
10
showing initial sideslip and roll estimation results. Experimental data from two tests on low
friction surfaces was used to compare the sideslip and roll estimates from the AUNAV lter
with those produced by the commercially available Oxford RT3000 GPS/INS unit. It was
shown that for both maneuvers the AUNAV lter produced roll angle estimates within one
degree of the RT estimate. The sideslip estimate was within one degree of the RT estimate
for the rst run and within 2 degrees on the second run, which was much more challenging
from an observability standpoint.
In this thesis the estimation performance of the AUNAV lter for sideslip, roll, and
inertial bias estimation is analyzed with experimental data. The observability of the AU-
NAV lter is also analyzed, and the convergence expected from this analysis is studied by
simulation and experiment. A comparison of the estimation performance of the modi ed
\modular" estimator (MME) with the AUNAV lter is also given. The modular lter was
developed by Bevly in [6] and represents a di erent approach to GPS/INS sensor fusion.
In [36], the author of this thesis modi ed the lter by removing the pitch gyroscope and
substituting the road grade estimate for the Euler pitch angle. The sideslip estimation per-
formance of the MME when in the presence of larger road grades was analyzed. In this
thesis, the MME estimation performance is analyzed for the same conditions as the AUNAV
system for comparison.
Another contribution is a method to indirectly detect changes in tire pressure using only
the sensors stated above in addition to wheel speed measurements, which are ubiquitous on
commercial vehicles today. The approach is based on the idea that the \e ective" or \rolling"
radius of the tire will vary as a function of tire pressure. First, a simple method of estimating
the rolling radius using GPS and wheel speed sensors is discussed. This is a recursive version
of the batch least squares used in [13]. The e ects of tuning and slip on the estimate are
investigated, and simulations are presented to validate the estimate. Further validation is
shown with experimental data. Next, experimental data is shown which illustrates how the
radius estimate varies according to tire pressure. The repeatability and potential problems
11
of this method for inferring tire pressure are discussed. A simple method of detecting tire
pressure loss well within the TREAD Act requirements using only the estimate of the rolling
radius is put forward. The TREAD Act is a law requiring new vehicles to have Tire Pressure
Monitoring Systems (TPMS) as standard features, and it contains some required performance
speci cations [1]. Future work to statistically improve the pressure loss detection algorithm
is also discussed.
The nal contribution is a method of detecting steering misalignments using the yaw
rate sensor and the steering wheel angle sensor. The yaw rate signal is improved using the
yaw rate bias estimate produced by the AUNAV estimator. The premise of the detection
logic is that if the vehicle is driving straight, then the steer angle ought to be very close to
zero. Steering e ects from road crown can cause an o set, but these are ignored in this work
as these e ects remain the same throughout all experiments. It is shown that adjustments to
the front right tire toe angle, which cause a misalignment, are detectable using the corrected
yaw rate signal and the steering wheel angle signal.
12
Chapter 2
GPS/INS Integration Algorithms
2.1 Background
2.1.1 Kalman Filtering
A linear Kalman lter is simply a classical estimator in state variable format that
incorporates the statistical knowledge of the system and sensors into the calculation of an
\optimal" estimator gain L. In doing so the Kalman lter also calculates an estimate of
the variances of the state estimation errors at each time step. This covariance matrix P
can provide reliable con dence bounds on the estimates under certain conditions. Kalman
ltering consists of two steps, referred to here as the measurement update and the time
update. For GPS/INS integration applications the measurement update usually runs at a
lower frequency than the time update. A new estimator gain L is calculated every time a
new measurement arrives. This is contrary to basic pole-placement estimation, where the
gain is constant. The innovations (the di erences between the measurements and the state
estimates) are taken and multiplied by the estimator gain, and this new quantity is added
to the state estimate, just like a basic estimator. The covariance is updated at this time
interval as well. The following equations describe the measurement update step [14]:
L = Pk CT C Pk CT +R 1 (2.1)
Pk = (I LC)Pk (2.2)
13
^Xk = ^Xk +L Y C ^Xk (2.3)
In Equations (2.1-2.3), C represents the measurement matrix, ^Xk represents the current
state estimate, and Y represents the measurement vector. It should be noted that neither
the gain nor the covariance matrix depends on the measurement innovations or any inputs
to the system. This follows from the strict list of assumptions that must be true to satisfy
both the optimality of the Kalman lter and the accuracy of the covariance estimates. The
time update consists of taking a model of the system, in either continuous or discrete repre-
sentation, and propagating it forward in time just like a traditional estimator. Continuous
equations are used during the time update in this thesis, therefore the overall lter is a
continuous-discrete Kalman lter. At each time step the system input is measured, the rate
of change of the state estimate is calculated, and the result is integrated. The rate of change
of the variances must also be calculated and integrated. Trapezoidal integration is found to
be su cient for this work. The following are the equations for the time update [14]:
_^X
k = A ^Xk +Buuk (2.4)
^Xk+1 = ^Xk + 1
2 t
_^
Xk + _^Xk 1
(2.5)
_Pk = APk +PkAT +BwQBTw (2.6)
Pk+1 = Pk + 12 t
_
Pk + _Pk 1
(2.7)
The matrices Bu and Bw are the system input and noise input matrices, respectively.
It can be seen here that other factors a ect the quality of the estimates, the optimality
of the Kalman lter, and the accuracy of the error variance estimate P. First, the lter
14
assumes a perfect model in A;Bu; and Bw. Absolute perfection in this regard is highly
unlikely. However, given that the models used for this work are kinematic sensor models,
modeling uncertainty is not a major problem. Second, it is assumed that the statistics
of the noise are perfectly known as well. This is not that bad of an assumption either,
since the process noise is actually sensor noise which can be approximated from sampled
data. Although such an approximation is not perfect, it is possible to achieve su cient
accuracy by analyzing sampled data. Third, it is assumed that the measurement errors are
uncorrelated. The measurements in this work are the position and velocity solutions from
a stand alone GPS receiver. These are likely the outputs of Kalman lters themselves, and
would therefore have time correlated error, violating this assumption. Yet it has been shown
that the impact from this is minimal, and may be overcome by simply increasing the tuning
values for the measurements [23]. Therefore despite the stringent assumptions required to
satisfy the lter?s optimality and error variance estimation, the Kalman lter does perform
very well for this application. Furthermore the structure of the Kalman lter o ers many
advantages for GPS/INS integration. GPS receivers o er highly accurate, unbiased estimates
of a vehicle?s velocity, but these are output at slower update rates on most receivers. They
can also su er from loss of signal in certain environments such as \urban canyons" or heavily
wooded regions. INS systems boast much higher update rates but su er from biases and
errors that grow over time. Combining the two truly o ers the best of both: an accurate,
unbiased estimate of the vehicle states at a high update rate which can handle a loss of GPS
satellite coverage for short periods of time.
2.1.2 Sensor Rotations and Coordinate Frames
The accelerometers and the gyroscopes of the IMU provide measurements which are
resolved in the coordinate frame of the IMU (the sensor frame), and not the reference coor-
dinate frame (the navigation frame). The two methods of GPS/INS integration presented
in this work each use a di erent navigation frame. The MME lter uses the \local tangent"
15
frame. This X axis of this system always points in the forward direction of the vehicle (see
Figure 1.1), but it is not aligned in the pitch direction with the X axis of the body frame.
The Z axis of the vehicle points to the center of the reference ellipsoid. The origin of this
frame is at projection of the center of gravity onto the local tangent plane. This frame can
be thought of as traveling with the vehicle, but it does not pitch or roll. It?s XY plane
is always aligned with the local tangent plane, and it yaws with the vehicle. The AUNAV
lter uses the North-East-Down frame. The X axis of this frame points due North, and the
Z axis points to the center of the reference ellipsoid. The only di erence between the two
coordinate frames is a yaw angle rotation, speci cally the yaw angle of the vehicle. In both
navigation frames the Y axis is chosen to complete the right-handed coordinate systems.
The body frame is in general not aligned with the navigation frame, potentially due to hills,
banked roads, or suspension de ections caused by dynamic maneuvers or vehicle loading. It
is important, then, to resolve both the IMU accelerometer and gyroscope signals into the
common navigation frame through a series of steps known as IMU mechanization.
The following equations show how to resolve the IMU into the navigation frame. The
accelerometers are rotated using the standard series of three body xed Euler rotations. This
not only brings them into the common navigation frame but also removes the gravitational
e ects present in the X and Y accelerometers. For rotations into the local tangent plane,
the yaw angle in (2.10) is 0. For rotations into the NED frame, represents the vehicle
heading. Equation (2.8) shows how the rotation matrix Rnb is used to rotate the measured
acceleration signals ab from the body frame b into the navigation frame n.
an = Rnbab (2.8)
Technically, the inertial sensors only provide measurements in the body frame if the IMU
is perfectly aligned with the vehicle body. This is rarely the case, however misalignments
are usually very small and are not a focus of this thesis. The rotation matrix Rnb is formed
by a series of three body- xed rotations which are yaw ( ), pitch ( ), and roll ( ). Equation
16
(2.9) de nes the rotation matrix in terms of the three individual rotations, while equations
(2.10 - 2.12) de ne the matrices for each of the three rotations.
Rnb = Rbn 1 = (R R R ) 1 (2.9)
R =
2
66
66
4
cos ( ) sin ( ) 0
sin ( ) cos ( ) 0
0 0 1
3
77
77
5
(2.10)
R =
2
66
66
4
cos ( ) 0 sin ( )
0 1 0
sin ( ) 0 cos ( )
3
77
77
5
(2.11)
R =
2
66
66
4
1 0 0
0 cos ( ) sin ( )
0 sin ( ) cos ( )
3
77
77
5
(2.12)
The angular signals are resolved into navigation frame using the mechanization equa-
tions, expressed in the mechanization matrix F . Equation (2.13) shows how the measured
gyroscope signals !b are resolved from the body frame b into the navigation frame n.
!n = F !b (2.13)
The mechanization matrix is a function of the level angles (roll and pitch). It is de ned
in equation (2.14).
F = 1cos ( )
2
66
66
4
1 sin ( ) sin ( ) cos ( ) sin ( )
0 cos ( ) cos ( ) sin ( ) sin ( )
0 sin ( ) cos ( )
3
77
77
5
(2.14)
17
2.1.3 Inertial Sensor Models
Inertial sensors are advantageous in that they do not su er from any sort of loss of signal,
as long as they don?t fail, and that they output signals at a very high update rate, which
is necessary for control. However they do have several disadvantages. Both accelerometers
and rate gyroscopes have static or \turn on" biases in addition to a moving bias which
is known as \drift". The accelerometer (and gyroscope) drifts are modeled as rst order
Markov processes [6], being driven by white noise and having a time constant . They
additionally su er from noise which can be modeled as additive Gaussian white noise. The
sensors also potentially have a scale factor which scales the true value being measured [6].
It has been shown that this scale factor can be estimated, and the model assumed in this
work assumes that no scale factor error is present or that it has already been estimated.
Equations (2.15-2.18) show the inertial sensor models:
abmeas = abtrue +Rbng + f + accel (2.15)
!bmeas = !btrue + ! + gyro (2.16)
fi; !i = b\turn on00 +bwalk (2.17)
_bwalk t 1
( bwalk) (2.18)
Note that Equations (2.17-2.18) apply for the biases of both the accelerometers and gy-
roscopes. The vectors a and ! represent the acceleration and rotation rate vectors. The
vectors f and ! are the accelerometer and gyroscope biases. The \b" terms represent
speci c components of the biases, and the vectors represent the noise.
2.1.4 GPS Sensor Models
When only inertial sensors are available, their biases are usually compensated through
certain assumptions such as a at road. If these do not hold, for example driving on a hill,
18
other information such as GPS is needed to compensate for the biases. Even after all of
the biases are accounted for, the white noise remains and will still corrupt the solutions.
Integrating this noise will potentially result in unbounded error growth. An additional
sensor such as GPS is therefore necessary to bound the error growth. The GPS sensor model
highlights the well documented advantages of GPS [6]. In the GPS/INS integration strategy
used in this thesis, the GPS receiver outputs position solutions as latitude, longitude, and
altitude while the velocity solutions are output in the North-East-Down (NED) navigation
frame. The velocity signals are unbiased and contain no scale factor error. The errors present
on the GPS positions and velocities can be modeled as uncorrelated Gaussian random noise,
as seen in Equation (2.19).
2
66
66
4
?
h
3
77
77
5
meas
=
2
66
66
4
?
h
3
77
77
5
+ lla
Vnmeas = Vn + v (2.19)
In this equation, represents the latitude, ? represents the longitude, and h represents
the altitude. Vnmeas is the measured velocity, Vn is the true velocity, and i is the noise vector
for each solution. There do exist more detailed models of GPS errors, but this simpli ed
model is representative of the error behavior at the position and velocity level and is su cient
for this work.
19
Figure 2.1: Diagram of the Modi ed Modular Estimator.
2.2 GPS/INS: The Modi ed Modular Estimator
2.2.1 Estimation Strategy Overview
The modi ed modular estimator is an extension of the estimator presented in [6]. The
estimation strategy is largely directed by the sensors and corresponding measurements avail-
able. This thesis assumes the following sensors to be available: a ve degree of freedom
(DOF) IMU consisting of accelerometers mounted in the body x;y; and z axes along with
roll rate and yaw rate gyroscopes, and a single antenna GPS receiver. While 5 DOF IMU
clusters are not ubiquitous in the automotive world, they can be found in certain production
vehicles with a roll stability control system or with a rollover curtain system. The following
paragraph gives a broad, big picture outline of the estimation strategy.
Figure 2.1 is a block diagram showing an overview of the process. First the IMU signals
are resolved into the navigation frame using the current attitude angle estimates. This
removes the biases resulting from gravity which are present in the x and y accelerometer
measurements. Second, Kalman lters are used to estimate the vertical and longitudinal
velocities as described in [4]. These velocities are then used to calculate an accurate estimate
20
of the road grade, which is used as a substitute for the true pitch angle. Next, a Kalman
lter which estimates the vehicle heading and the yaw gyroscope bias is used, together
with the GPS course measurement, to obtain an initial estimate of the sideslip. Finally,
the initial sideslip estimate is combined with the GPS velocity measurement to produce a
derived \measurement" of the lateral velocity for a lateral state Kalman lter. This Kalman
lter estimates the vehicle?s lateral velocity, lateral accelerometer bias, roll angle, and roll
rate gyroscope bias. The nal sideslip estimate is calculated in a straightforward manner
from the lateral velocity estimate [8], [9]. It is available at the frequency of the IMU, while
the initial sideslip estimate is only available at the frequency of the GPS.
2.2.2 Road Grade Estimation
The slope of the road in the forward direction is generally referred to as the road grade.
It can be presented in two forms: either as the actual slope of road (% grade) or as the
angle that the road makes with the horizon; where conversion between the two is a matter
of simple trigonometry. Determination of this angle can be accomplished using the ratio
of the vertical and horizontal speeds, assuming that the vehicle is moving (so as not to
have a zero in the denominator). GPS receivers output a vertical speed and a speed-over-
ground velocity vector, where the speed over ground is the vehicle velocity vector in the local
navigation frame. The magnitude of this vector can be taken to be the longitudinal speed of
the vehicle in the navigation frame, assuming little to no sideslip. Therefore the arctangent
of the two speeds can be taken to nd the grade angle. This method has been proven to
produce high quality, unbiased estimates of the road grade, yet it should be noted that any
bounce motions that the vehicle experiences will a ect the grade estimate. However, this
has been shown not to signi cantly diminish the estimator?s performance [4].
Coupling the GPS system and the IMU together in a Kalman lter structure o ers
many advantages. The following equations, which are simply scalar versions of the vector
Equation (2.15) , show how this is possible:
21
az _Vz +gz + fz + accel (2.20)
ax _Vx +gz sin + fx + accel (2.21)
where az;ax represent the accelerometer measurements in the z and x directions, _Vz and _Vx
are the true accelerations, gz is gravity, is the total Euler pitch, fz;x is the inherent sensor
bias, and is the sensor noise. The Kalman lter now takes the following form, where the
accelerometer measurement is the input and the GPS velocity is the measurement:
2
64 _^Vz
_^fz
3
75 =
2
640 1
0 1Tm
3
75
2
64 ^Vz
^fz
3
75+
2
641
0
3
75(a
z gz) +
2
641 0
0 1Tm
3
75 (2.22)
y =
1 0
26
4
^Vz
^fz
3
75+
z (2.23)
2
64 _^Vx
_^fx
3
75 =
2
640 1
0 1Tm
3
75
2
64 ^Vx
^fx
3
75+
2
641
0
3
75(a
x gsin )
2
641 0
0 1Tm
3
75 (2.24)
y =
1 0
2
64 ^Vx
^fx
3
75+
x (2.25)
where and are the measurement noise and process noises, respectively, ^Vz;x represents
the velocity estimates, and ^fz;x represent the bias estimates. Tm represents the bias time
constants. Both of these estimator models are observable. The road grade estimate is the
arctangent of the two speed estimates.
^ = arctan ^Vz^
Vx (2.26)
22
Since the input to the Kalman lter is an accelerometer, the process noise is taken to be
noise from the accelerometer where the noise driving the bias Markov process is included as
well. Both are taken to be Gaussian white noise, as shown below.
=
2
64 accel
3
75 (2.27)
accel N(0; 2accel) (2.28)
N(0; 2m) (2.29)
Ef Tg= Q =
2
64 2accel 0
0 2m
3
75 (2.30)
The measurement noise is the noise on the GPS velocity signal, which is also assumed to
be Gaussian white noise. Due to satellite orientations the noise is generally higher on the
vertical speed measurement than on the speed over ground measurement. The measurement
noise is taken to be
x N(0; 2GPSx) (2.31)
z N(0; 2GPSz) (2.32)
Ef x Txg= Rx = 2GPSx (2.33)
Ef z Tzg= Rz = 2GPSz (2.34)
This lter structure combines the advantages of the high accuracy, unbiased GPS velocity
measurements with the high update rate signals from the IMU, thereby o ering high quality
velocity estimates (and therefore a high quality road grade estimate) at a frequency suitable
for control signals. The GPS measurement enables correction for any accelerometer biases,
while the accelerometer allows for continued velocity tracking during GPS outages over short
time intervals [4].
23
2.2.3 Heading Estimation
Vehicle sideslip is de ned in two ways. It is the di erence between the vehicle course
(the direction that the vehicle is traveling) and the vehicle heading (the direction that the
vehicle is pointed). It is also the arctangent of the ratio of lateral speed to longitudinal speed
(all references to course and velocities \of the vehicle" are with respect to the IMU location,
which is near the center of gravity.) The vehicle course and longitudinal speed are readily
available with a single antenna GPS receiver, but neither heading nor lateral velocity are
measurable with only one antenna. This renders sideslip technically unobservable, but it
does not make it impossible to produce useful sideslip estimates under certain conditions. A
yaw rate gyro could be integrated to obtain a heading estimate which could be subtracted
from the course measurement to obtain sideslip. However doing so rst requires overcoming
several problems resulting from integration. The rst obstacle is any bias that may be present
in the sensor. This can be estimated and removed using a Kalman lter under the condition
that the vehicle is driving straight. When this is true, the course angle and the heading
angle are the same. A notable exception occurs on a banked road, causing a small steady
sideslip, although this would be a rare scenario for straight sections of roadway. Common
road crowns will yield on a very slight steady sideslip angle. Since GPS measures course
and the course is equal to heading when the vehicle is driving straight, the estimation can
be \switched on" during this scenario. The Kalman lter could then estimate any biases in
the gyroscope and remove them. When a turning maneuver is initiated the estimation is
\switched o " and the yaw gyro is integrated to determine heading. The di erence in this
heading estimate and the course measurement becomes the sideslip estimate. This process
is seen the equations below, beginning with the gyroscope model, which is the scalar version
of (2.16).
!r _ + !r + gyro (2.35)
24
The previous equation, along with (2.17), now yields the Kalman lter.
2
64 _^
_^!r
3
75 =
2
640 1
0 1Tm
3
75
2
64 ^
^!r
3
75+
2
641
0
3
75(!
r) +
2
641 0
0 1Tm
3
75
r (2.36)
When driving straight:
y =
1 0
2
64 ^
^!r
3
75+
= GPS (2.37)
Otherwise, the estimation is switched o :
y =
0 0
2
64 ^
^!r
3
75+
= GPS (2.38)
When the vehicle is driving straight, the C matrix is set to [1 0] and measurement updates
are performed. The C matrix is set to [0 0] when turning. In both cases the sideslip estimate
is the di erence between the course measurement and the heading estimate:
^ 0 = GPS ^ (2.39)
As in the case of the road grade estimation lters, the input to the system is an inertial
sensor measurement. The process noise is the noise on the gyroscope signal and the noise
driving the random walk. Both are assumed to be zero mean Gaussian white noise:
r =
2
64 gyro
3
75 (2.40)
gyro N(0; 2gyro) (2.41)
N(0; 2m) (2.42)
25
Ef r Trg= Q =
2
64 2accel 0
0 2m
3
75 (2.43)
The measurement noise is the noise on the GPS course measurement, and is assumed to
be zero mean Gaussian white noise as well. However, the accuracy of this measurement
increases with speed, giving a variance that is a function of speed.
N(0; 2GPS ) (2.44)
Ef T g= R = 2GPS (2.45)
2GPS =
2
V (2.46)
The second problem lies in determining whether or not the vehicle is going straight. It
is imperative to the estimator performance that the biases are estimated accurately prior
to periods of integrating the gyroscope signal. This makes it necessary to include logic
statements to determine whether or not the vehicle is driving straight, which is done using
the the yaw gyro signal. The basic logic is that if the absolute value of the yaw rate is less
than some threshold then the vehicle is deemed to be driving straight. Complications arise,
however, from the noise on the signal. The resulting situation becomes a trade-o in the
sensitivity to turning motion verses false alarms caused by the noise. If the thresholds are
set too close to the noise oor the estimator will \believe" the vehicle is turning during many
time instances in which it is not. This means that the heading estimation is turned o , and
no gyro bias correction is done. In this work the GPS measurement only comes in at 1Hz, so
losing that measurement can be very costly. The primary limitation of this sideslip estimator
is its dependency on periods of straight driving to zero out the gyro bias, and every missed
course measurement results in a lost second of straight driving. At moderate speeds on
winding roads, for example, one second of straight driving can be rare. Therefore the tuning
of these thresholds plays an important role in the overall performance of the estimator.
26
The other obvious problem with this method comes from integrating a noisy sensor
signal. Some improvement could be found by low pass ltering the signal, but all of the
noise cannot be removed. Therefore this estimation scheme is limited as well in terms of the
amount of time that the gyro can be integrated before the error grows too large.
2.2.4 Lateral Velocity
As previously noted, sideslip can be de ned as the arctangent of the ratio of the lateral
speed of the vehicle to its longitudinal speed. The longitudinal speed can be estimated
accurately by combining wheel speed sensors, longitudinal accelerometers, and GPS, but the
lateral speed is not measurable with only a single antenna GPS system. Using the sideslip
estimate from the heading Kalman lter described above, a lateral velocity \measurement"
can be generated and therefore a lateral state estimator can be introduced in the form of a
Kalman lter. The following equations form the foundation of this estimator.
ay _Vy +Vx _ +gz sin + fy + accel (2.47)
!p _ + !p + gyro (2.48)
As was the case before, this sensor model assumes no scale factor error. The biases for both
the lateral accelerometer and the roll rate gyro are taken to be Markov random process driven
by Gaussian white noise as described previously. There are two sources of bias in the lateral
accelerometer measurement, one being the roll component of gravity and the other being the
sensor?s random walk. Both of these in uences have the exact same e ect on the velocity
error, therefore the lter will not be able to distinguish the two. If a direct measurement
of roll were available, from a double antenna GPS system for example, the two would be
independently observable. Yet since roll is not measurable with a single antenna, the two
states must be lumped together into one. The resulting simpli cation of the equation follows
from using the small angle approximation and lumping the two terms together.
27
ay _Vy +Vx _ +gz ( + fy) + accel (2.49)
This results in an estimator in the form:
2
66
66
4
_^V
y
_^ + _^f
y
_^!p
3
77
77
5
=
2
66
66
4
0 gz 0
0 0 1
0 0 1=Tm
3
77
77
5
2
66
66
4
^Vy
^
+ ^fy
^!p
3
77
77
5
+
2
66
66
4
1 0
0 1
0 0
3
77
77
5
2
64 ay ^Vx (!r ^!r)
!p
3
75
+
2
66
66
4
1 0 0
0 1 0
0 0 1=Tm
3
77
77
5
2
66
66
4
accel
gyro
3
77
77
5
(2.50)
y =
1 0 0
2
66
66
4
^Vy
^
+ ^fy
^bp
3
77
77
5
+ y = VGPS sin
^
0
(2.51)
^ = arctan
^
Vy
^Vx
!
(2.52)
The process noise for the system arises from the IMU measurements. Since the measurement
is constructed using the sideslip estimate from the heading lter, the measurement noise is
approximated as the noise on the GPS course measurement.
E !!T = Q =
2
66
66
4
2accel +V2 2gyro 0 0
0 2gyro 0
0 0 2m
3
77
77
5
(2.53)
28
Ef T g= R = 2GPS (2.54)
It was stated above that the Kalman lter cannot separate the roll and the lateral accelerom-
eter bias. The reason is because the two states look exactly the same in terms of velocity
errors. However, they do not share the same frequency characteristics. That is, they do not
change in the same way. This di erence can be exploited using complimentary low and high
pass lters as shown in [6]. The following equations show the Laplace representation of the
complimentary lters.
^ = Tms
Tms+ 1
^
+ ^fy
^fy = 1T
ms+ 1
^
+ ^fy
(2.55)
Since the roll angle changes much faster than the bias, this ltering approach yields an
accurate estimate of both states independently.
2.3 GPS/INS: The Automotive Navigation (AUNAV) Estimator
The AUNAV estimator is most easily understood as a modi ed version of the generic
loosely coupled GPS/INS blending strategy. Therefore this section is divided into two sub-
sections, the rst of which details the standard loosely coupled algorithm. After this, the
modi cations which are made to complete the AUNAV lter are discussed.
2.3.1 The Loosely Coupled Algorithm
The \loosely coupled" GPS/INS method of integration is a well documented technique
for blending the GPS and INS navigation solutions [19]. Figure 2.2 gives a high level view
of the integration. The INS and GPS position and velocity solutions are compared, and
the di erence is input into the extended Kalman lter (EKF) as a measurement. This is
29
because the states of the lter are the di erences between the true states and INS estimates,
as opposed to actual position and velocity states. The lter outputs estimated corrections
to the INS solution, which are added to the INS solution. The corrected INS solution serves
as the nal estimate. The EKF also estimates the inertial errors of the INS, which are
fed back into the INS to continuously improve the estimation process. The system functions
basically as two independent navigators (the GPS and INS) and one EKF. When using lower
grade inertial navigation systems, the loosely coupled approach treats the GPS solutions as
truth [19]. Therefore the states of the EKF practically become the di erences between the
INS and GPS solutions. It should be noted that the inertial bias estimates are not error
states, but rather they represent estimates of the actual biases themselves.
Figure 2.2: Diagram of the Standard Loosely Coupled Integration Strategy.
The details are of the loosely coupled approach are described as follows. The states
of the Kalman lter, ^X =
^r; ^V; ^ ; ^f; ^!
0
, are the estimates of the errors
in the INS position-velocity-attitude (PVA) solution and the biases of the inertial sensors.
The rst step of the overall system is to calculate the INS solution. The inertial sensors
are rotated into the navigation frame and then propagated forward in time via trapezoidal
30
integration to obtain the PVA estimates as in Equations (2.56-2.62), a process also known
as IMU mechanization.
_^
k = F (!
INS ^!) (2.56)
_^V
k = R
n
b(a
INS ^f) g (2.57)
_^rk = T ^Vk (2.58)
^
k+1 =
^
k +
1
2 t
_^
k + _^ k 1
(2.59)
^Vk+1 = ^Vk + 1
2 t
_^
Vk + _^Vk 1
(2.60)
^rk+1 = ^rk + 12 t
_
^rk + _^rk 1
(2.61)
T =
2
66
66
4
1
Rn+h 0 0
0 1(Re+h)cos( ) 0
0 0 1
3
77
77
5
(2.62)
In Equations (2.56-2.57), Rnb represents the rotation matrix from the body frame to the
NED frame, and F represents the mechanization matrix to align the gyroscopes with the
NED frame. These are outlined in Section 2.1.2 in more detail. The matrix T transforms
the velocities from the NED frame into rates of latitude, longitude, and altitude. Re and
Rn are parameters of the reference ellipsoid. During this step the EKF also propagates the
uncertainty of the error estimates forward using standard Kalman lter covariance update
equations linearized about the current state estimates. This is shown in Equation (2.63),
31
where J is the Jacobian (de ned later), and Bw is simply a 15 15 matrix with the vector
([03x3;Rnb;Rnb;Rnb;Rnb]) on the diagonal and 0?s everywhere else.
_Pk = JPk +PkJT +BwQBTw (2.63)
Equation (2.63) is then integrated according to Equation (2.7) to complete the covariance
propagation. Equations (2.56-2.63) are done at the update rate of the INS. The error esti-
mates are not propagated during this time, because they are reset to zero each time they
are added to the INS solutions. When GPS measurements arrive, the di erence between the
GPS and INS position and velocity solutions are taken in Equation (2.64).
Y =
2
64 r
V
3
75
GPS
2
64 ^r
^V
3
75 (2.64)
This di erence serves as measurement for the error state Kalman lter. At this point the
Kalman lter calculates the gain L, error residuals, and the covariance matrix as per the stan-
dard Kalman lter measurement equations outlined in Section 2.1.1; followed by updating
the error states as in Equation (2.65).
^Xk = ^Xk +L
Y C ^Xk
(2.65)
The term C in Equation (2.65) represents the measurement model. It comes from the fact
that the GPS outputs position and velocity measurements. The C matrix is de ned below.
C =
I6x6 06x9
6x15
(2.66)
The estimates of the INS velocity and attitude errors are then added to the current INS
solution, as per Equations (2.67-2.69), and are subsequently reset to zero.
^r = ^r ^r (2.67)
32
^V = ^V ^V (2.68)
(Rnb)k =
I +
^
(Rnb)k (2.69)
The notation ( ^ ) indicates the skew symmetric form of the vector ^ . The Euler angles
are computed from Rnb as in (2.70-2.72) [19].
k = arctan
Rn
b (3;2)k
Rnb (3;3)k
(2.70)
k = arcsin (Rnb (3;1)k) (2.71)
k = arctan
Rn
b (2;1)k
Rnb (1;1)k
(2.72)
At each time step, whether during the INS estimate propagation or during a GPS update,
the sideslip estimate is calculated in Equation (2.73) [19].
^ = arctan
^
Veast
^Vnorth
!
^ yaw (2.73)
Finally, the estimated inertial sensor errors ^f and ^! are fed back into the INS (the sub-
tracted terms in Equations (2.56-2.57). This is not necessary for high grade inertial units,
but it is absolutely imperative in this work due to the relatively large errors in the automo-
tive grade sensors. This feedback is known as a closed loop implementation as opposed to
open loop.
The extended Kalman lter is based on the inertial sensor error models, Equations
(2.15-2.18), and on the error propagation equations of the INS position, velocity, and attitude
solutions which are Equations (2.74-2.76), which are simpli ed versions of those found in [19].
These are the equations used to form the Jacobian matrix which is required to update the
covariances. signi es INS estimation error.
33
_r T V (2.74)
_V Rnb aINS +Rnb f (2.75)
_ Rnb ! (2.76)
J =
2
66
66
66
66
66
4
03 T 03 03 03
03 03 Rnb aINS Rnb 03
03 03 03 03 Rnb
03 03 03 1 I3 03
03 03 03 03 1 I3
3
77
77
77
77
77
5
(2.77)
2.3.2 Modi cations to the Loosely Coupled Algorithm
There are two primary changes that need to be made to the loosely coupled lter in order
to accurately estimate sideslip and roll using only sensors available on current vehicles. First,
the pitch rate gyroscope must be removed, as these are not currently available on commercial
vehicles. Therefore the impact of this sensor reduction on the estimation performance of
the loosely coupled estimator must be studied. It is hypothesized that the AUNAV lter
will be able to accurately estimate the sideslip and roll despite this sensor reduction. It
is also expected that even without the pitch rate gyroscope the AUNAV estimator will be
able to estimate the low frequency component of pitch. This is important since the road
grade is the low frequency component and it has much higher amplitudes than the higher
frequency suspension pitch changes. Both of these expectations are con rmed in Chapter
4. Additionally, the observability of the new system requires evaluation. This is done in
Chapter 3. It is also important to note that because the pitch rate gyroscope bias is one
of the states of the EKF in the loosely coupled lter, the number of states of the AUNAV
34
estimator is reduced by 1. The AUNAV estimator has 14 states, therefore all of the EKF
matrices must be resized accordingly.
The second change that is made to the loosely coupled estimator is that a measurement
of the vehicle?s course angle from the GPS is added when the vehicle is driving straight.
The loosely coupled estimator su ers from observability issues when there is little excitation
present [19,28]. Many authors seek to overcome the problem by adding velocity constraints
[16, 18, 20]. They make the assumptions that the vehicle?s lateral and vertical velocities in
the vehicle frame are both zero at all times. These constraints are then added in the form
of a virtual measurement update. Practically speaking, the latter assumption simply means
that the vehicle is constrained to the road. The former assumption, however, is tantamount
to assuming that there is no sideslip. This is acceptable for navigation purposes, but it will
obviously not do given the explicit goal of estimating sideslip. So in order to add constraints
that do not violate the estimation of sideslip, a virtual measurement of the course angle is
created when the vehicle is driving straight. Recall the de nition of vehicle sideslip given by
Equation (1.1). If the vehicle is driving straight, and assuming that the side slope (bank) of
the road is small, then the sideslip will be practically zero. If this is true, then from Equation
(1.1) it is clear that the course and yaw angles are equivalent. The course angle can be
calculated from the GPS north and east velocities as shown in Equation (2.78).
GPS = arctan
V
east
Vnorth
GPS
(2.78)
When the vehicle is driving straight, the course angle measurement is used as a measurement
of the yaw angle. The new measurement vector is shown below.
Y =
2
66
66
4
r
V
3
77
77
5
GPS
2
66
66
4
^r
^V
^ yaw
3
77
77
5
(2.79)
35
It is important that the lter not use the course measurement when the vehicle is
turning, because the sideslip will cause the necessary assumptions to be violated, resulting
in the course angle being unequal to the yaw angle. Using the course measurement under
these circumstances will result in the lter falsely attributing the inevitable error to other
sources, thereby corrupting the other state estimates. Therefore a means of switching is
required for the lter to toggle between using the course measurement and not using it. This
is done simply by setting the term in C corresponding to the course measurement to 1 or 0,
depending on whether or not the measurement is being used. When the vehicle is turning,
the course measurement is not used and the C matrix is de ned as shown below.
C =
2
64 I6x6 06x2 06x1 06x5
01x6 01x2 0 01x5
3
75
7x14
(2.80)
When the vehicle is driving straight, the course measurement is used and the C matrix is
de ned as shown below.
C =
2
64 I6x6 06x2 06x1 06x5
01x6 01x2 1 01x5
3
75
7x14
(2.81)
Note that the column dimension of the C matrix is now 14 instead of 15, because of the
reduction of the pitch rate gyroscope bias state. Adding the course measurement keeps the
attitude errors bounded during periods of straight driving, as there is no lateral excitation
with which to relate the north and east velocities to the accelerations.
It is not trivial to determine whether or not the vehicle is turning. At rst glance, it
may seem simple enough to use the steering angle. A simple law could perhaps be used that
says if the steer angle is above a certain threshold, then the vehicle is turning. The problem
with this is that if the wheels are misaligned, or if there is a large enough road crown (a slight
road bank that improves water drainage), then the steer angle will have some constant o set
from zero. The lateral accelerometer could also potentially be used, but it is potentially
36
subject to gravitational e ects resulting from inaccurate roll angle compensation. In this
thesis, the yaw rate gyroscope is used to determine whether or not the vehicle is driving
straight. A ag is set if the yaw rate signal has been within 3 deg=s, consecutively, for a
certain period of time. If this ag is \true", then the vehicle is assumed to be driving
straight. If the absolute value of the yaw rate signal rises above the threshold even once,
then the ag is reset to false. The detection logic requires that the absolute value yaw rate
signal be below the threshold consecutively for a certain period of time in order to toggle
the ag to true (driving straight), as opposed to simply setting the ag to true upon the
rst yaw rate signal below the threshold. This is done to avoid the problem of zero crossings
of the yaw rate during a turning maneuver. The yaw rate will be zero for a short period
of time, for example, during a sinusoidal steering maneuver. The vehicle is still turning in
this case, therefore it would be an error to assume straight driving conditions. The time
requirement on the detection logic helps mitigate this e ect. There is no corresponding time
requirement to toggle the ag back to false (turning). This is in order to be conservative, so
as not to introduce error into the states by assuming that there is no sideslip when in fact
there is. The threshold of 3 deg=s was chosen as a conservative value, because experimental
tests showed that sideslip was generally extremely close to zero for yaw rates below this.
There are several problems with the current method of turning detection. First, the
e ects of the yaw rate bias need to be studied. For now, the threshold is simply set large
enough to accommodate the bias. The bias on the gyroscope used is quite small (on the
order of hundredths of a degree per second), so this is not a major problem. For lower quality
gyroscopes, however, it could be. There are potential problems as well with using the yaw
rate gyroscope bias estimate to try to mitigate the problem. For example, it could result in
an unstable feedback type of situation, where a large initial error in the bias estimate causes
yaw rate signal to always fail the test, thereby impeding the estimation of the bias. The other
problem is the sensor noise. If the noise on the gyro is too high it becomes di cult to estimate
slowly growing sideslip, because raising the detection threshold will mean that low rates of
37
turning will fall into the zero sideslip assumption. Setting the threshold correctly is also
important. It makes sense to set it to some integer multiple of the noise standard deviation.
The problem is that drivers rarely drive perfectly straight, and setting the threshold in this
way would only trigger the yaw course measurement under perfectly straight conditions. In
e ect, the threshold becomes a trade o between the sensitivity of the estimator and how
often the course measurements are applied. Furthermore, the time requirement is a tunable
parameter. Setting the value too low means that zero crossings which occur slowly will result
in the estimator assuming zero sideslip in the middle of the turn. Setting the value too large
limits the instances where the valuable course information is used. Therefore both the signal
threshold and the time threshold are important parameters of the overall AUNAV system
which must be tuned carefully.
2.3.3 Conclusion
In this chapter some fundamentals of GPS systems, inertial measurement units, and
Kalman ltering were discussed. Following this, the algorithm for the modi ed modular
estimator was presented. In contrast to the AUNAV estimator, the MME algorithm consists
of several distinct Kalman lters, each estimating states along di erent axes. Therefore there
is qualitatively less coupling between the states of all of the di erent lters in the MME when
compared with the coupling between the states of the AUNAV estimator. The MME is a
modi ed version of the estimator presented in [6]. The modi cation consists of the removal
of the pitch rate gyroscope. The pitch angle estimate in the MME comes from the road
grade estimate, which is a function of the vertical and horizontal velocity estimates.
Finally, the development of the algorithm for the AUNAV estimator was presented.
The AUNAV estimator is developed by making two important modi cations to a loosely
coupled GPS/INS lter. The classic loosely coupled algorithm is rst discussed, followed
by a discussion of the modi cations. The rst modi cation is the removal of the pitch rate
gyroscope (and its corresponding bias state in the EKF). This is necessary because pitch
38
rate gyroscopes are not currently available on commercial vehicles. The second modi cation
is the GPS course measurement which is used when the vehicle is driving straight. This is
necessary to improve the observability when the vehicle is driving straight, because there is
no lateral acceleration with which to relate heading errors to velocity errors. Turn detection
is done using the yaw rate gyroscope.
39
Chapter 3
Observability of the AUNAV Estimator
3.1 De nitions
The notion of observability for a particular system is obviously of great importance for
estimation algorithms. The observability of a system indicates whether or not it is possible
to estimate the states of the system based on the given sensor con guration. As an overly
simplistic example, consider the problem of determining the direction that a vehicle is facing
(also known as vehicle heading). Certainly this state could not be determined if the only
sensor available is a thermometer in the cabin of the vehicle. This is obvious, because
the cabin temperature information is utterly unrelated to the vehicle?s heading. Yet what
if GPS north and east velocity measurements were available? This information is much
more related to heading, but is it enough, and under what conditions? These questions of
relating the information from the sensors to knowledge of the desired states are at the heart
of observability. For linear, time invariant, deterministic systems, observability is simply a
function of the pair of the system dynamics matrix and the measurement matrix. Changing
the measurement matrix, which implies changing the sensor con guration, or changing the
system dynamics matrix, which de nes how the states are related to one another dynamically,
will a ect the observability properties. This is intuitive. For linear time-varying or non-
linear systems things become more complicated. Let?s begin the discussion by focusing rst
on linear, deterministic systems which are not time invariant. The problems imposed by
non-linearity and by stochastic in uences will be discussed afterward. A formal de nition of
observability is given by [43].
40
De nition 1. A system is said to be observable if the initial state x(t0) can be determined
from the output y(t) over the nite time interval [t0;tf].
De nition 1 assumes that the input u(t) is known. This de nition serves as a general
de nition of what is meant by \observability" for all types of systems, yet it is far from
the only de nition in the literature. In fact the researcher cannot be too careful when
considering the terminology of observability, as more speci c de nitions and conditions also
abound. Silverman and Meadows o er three more discriminating de nitions of di erent types
of observability for linear time variant systems [42]. They are complete, total, and uniform
observability. It is worth noting, as an aside, an example of inconsistent terminology found
in the literature. In [35] the authors, while citing Silverman and speaking of the exact same
three observability de nitions, refer to them as complete, di erential, and instantaneous
observability. In this work, the author adopts the terminology of Silverman. Here, complete
observability carries the same de nition of observability as De nition 1. Total and uniform
observability are de ned as follows [42].
De nition 2. A system is said to be totally observable on an interval [t0;tf] if it is completely
observable on all subintervals of [t0;tf].
De nition 3. A system is said to be uniformly observable on an interval [t0;tf] if the matrix
Q0(t) is full rank for all t on [t0;tf].
De nition 3 is the strongest of the three conditions. The matrix Q0(t) is de ned as
follows. If a linear, continuous system is given by (3.1), then the matrix Q0(t) is de ned as
in (3.2).
_x(t) = A(t)x(t) +B(t)u(t)
y(t) = C (t)x(t) (3.1)
41
Q0 (t) =
S0 (t) S1 (t) Sn 1 (t)
Sk+1 (t) = A0(t)Sk (t) + _Sk (t)
S0 (t) = C0(t) (3.2)
The general observability condition (De nition 1) is a necessary one for estimation
purposes. For systems which are linear, deterministic, and time invariant, it is also su cient,
but this is not true of the loosely coupled lter. It is important to remember moving forward
that proofs of convergence are not in view, rather the focus is on veri cations that the system
meets the minimum condition for estimation. It will be seen that many times, the loosely
coupled lter does not. It will now be considered how these de nitions have been applied in
the literature to the problem at hand, namely, the loosely coupled GPS/INS lter.
3.2 Applications in Literature
In [21] Goshen-Meskin and Bar-Itzhack develop a theoretical method of analyzing the
observability of linear time varying systems by considering them as piece-wise continuous
systems. They rst show that often times it is valid to consider a time varying system as
a sequence of consecutive time invariant systems if the time varying system meets certain
conditions. This allows the authors to view the observability of the overall system as a simpler
function of the observability of each time segment. This idea is then applied in [22] to the in
ight alignment (IFA) problem for INS units in aircraft. Various maneuvers are reduced to
distinct and consecutive time segments of constant systems, such as a constant acceleration
or a constant radius turn, and the overall observability is studied. In this way it is possible
to consider whether certain maneuvers will result in an observable system. Said another
way, it makes it possible to theoretically decide what is an e ective sequence of motions for
IFA. They conclude that, for the 12 state loosely coupled GPS/INS, any rst segment has an
42
observability matrix of rank 9. Adding another distinct segment increases the rank to 11, and
any third increases it to full rank. The order of the segments does not matter, and repeating
segments has no e ect. All segments in that work consist of distinct, constant accelerations
in either the north, east, or down directions. This analysis is very similar in approach to
most other works in the literature. That is, researchers are not evaluating the observability
of the \system", but rather asking whether or not the system is observable along certain
trajectories. The fact is, this system and those like it are neither linear nor time invariant,
so the analysis is restricted to local observability analyses of individual trajectories. Yet this
a very useful endeavor as an analysis or design tool, allowing the designer to draw conclusive
and de ned operating boundaries outside of which the lter can be guaranteed to fail. Since
the system is actually non-linear, the results for the local trajectory represent a necessary
condition, not a su cient condition. This analysis of the observability of the nonlinear
system along a certain trajectory will be referred to hereafter as the observability \of the
trajectory" for simplicity.
Another important work is that of Rhee et al. [35]. Here the authors look at the
observability of several trajectories using the de nitions and conditions de ned in [42]. First
they look at the case of constant linear acceleration, treating it as a time invariant system as
in [22]. Using well known observability tests such as the Hautus test and the standard LTI
observability matrix, they nd that the total system is unobservable and that the attitude
angles are the unobservable modes. This brings up a very important discussion. In this case,
Rhee et al. found that the observability matrix was rank de cient by 3. This means that there
are 12 observable modes, which is not the same thing as 12 observable states. Modes may be
states or linear functions of states. In this case, the 3 unobservable modes correspond directly
to states (the 3 attitude angles). However, 6 of the 12 observable modes do not correspond
to individual states, rather they are a function of other states (speci cally the attitudes and
inertial biases). While there are 12 observable modes, there are only 7 observable states.
These are position, velocity, and the vertical accelerometer bias. The remaining 5 observable
43
modes are functions of the other 8 unobservable states. This means that the combination of
the remaining attitudes and inertial biases is observable, but that the lter is unable in this
case to separate them individually. This is consistent with [22], and is basically equivalent
to applying their test to just the one acceleration segment alone. In short, the lter cannot
distinguish between the attitudes and sensor biases (except for the vertical accelerometer
bias); however the combination of the attitude errors and the sensor biases is observable.
Yet it is extremely important for the lter to be able to distinguish sensor biases from
attitude, therefore knowing the circumstances under which the attitude and bias estimates
cannot be independently observed is crucial. Rhee also considers the uniform observability
of the case of non-constant linear acceleration or constant rotation. This is done by applying
the de nition of uniform observability (De nition 3) found in [42]. In doing so they nd
that maneuvers of either type increase the number of uniformly observable modes by two.
For the case of non-constant axial acceleration, two attitude angles are made observable.
The attitude angle around the jerk vector is the state which remains unobservable. For
the constant turn case, the authors nd that the yaw angle remains unobservable. In each
case, the attitude angles which are made observable by the maneuver might also cause the
biases to become independently observable as well. Recall that for the case of no excitation
(driving straight) there are functions of the attitudes and biases which are observable, even
though the attitudes and biases aren?t observable independently. Making the attitude angles
observable e ectively decreases the number of unknowns in the equations, making the biases
observable also. This makes sense intuitively. Consider the lateral accelerometer model,
Equation (2.47). If the only biasing e ects are from the roll angle and the sensor bias
fy, and if the roll angle is observable or known, then it follows that the bias likewise is
observable.
Uniform observability (De nition 3) is the strongest condition of the three, and the
authors do not address the general observability condition (De nition 1) for the time variant
case. This makes it di cult to quantitatively compare these results with others in the
44
literature. Hong et al. [26] also examines the single antenna loosely coupled GPS/INS lter.
They nd that the attitude and bias states are unobservable if the system can be represented
as being time-invariant (i.e. when only undergoing constant axial acceleration). Furthermore
they conclude that linear acceleration changes enhance the overall observability of these
states. These ndings are qualitatively consistent with those previously mentioned, and
they also align with the intuitive presentation of observability given in [19].
3.3 Observability Simulations
The above ndings in the literature paint a clear picture of the observability of the
loosely coupled lter concerning most of the relevant trajectories and operating conditions.
However these ndings are all for the standard algorithm, and the observability of the modi-
ed algorithm in this work still needs to be considered. The modi cations to the lter include
removing a state (the pitch rate gyro bias), an input (the pitch rate gyro), and adding a
measurement (the yaw/course measurement). Adding the measurement of yaw will certainly
improve the observability, and the degree to which it does so will be shown in this section.
When considering the observability of the modi ed algorithm, the general de nition of
observability given by De nition 1 is used. As a means of evaluating a system?s observability,
Stengel [43] gives an equation for the observability matrix of a linear time-varying system.
OLTV (tf;t0) =
tfZ
t0
T ( ;t0)HTH ( ;t0)d (3.3)
The system is observable if OLTV (tf;t0) is non-singular, where OLTV (tf;t0) represents the
observability matrix for the system on the interval [t0;tf], ( ;t0) represents the state tran-
sition matrix from t0 to tf, and H represents the measurement matrix. The state transition
matrix is obtained by multiplying in series the state transition matrices of each discrete time
step.
45
(k;0) = Jd(k)Jd(k 1) Jd(0) (3.4)
The state transition matrix of each time step is obtained by discretizing the Jacobian of
the system dynamics. This is done according to (3.5) using the matrix exponential function
provided by Matlab, expm()
Jd = e(J Ts) (3.5)
where Ts represents the sample time. The AUNAV estimator is a nonlinear system, therefore
it bears restating that the observability results are local to the trajectory about which the
system is linearized. It is also important to note that for the observability analysis the
system is linearized not about state estimates, but about the true trajectory. This is because
the problem is to determine whether or not a certain trajectory (maneuver) theoretically
results in an observable time varying system. The modi ed algorithm is investigated by
analyzing the rank of the matrix OLTV (tf;t0). If it is full rank, then it is also non-singular
and the trajectory of interest is fully observable on the time interval [t0;tf]. If the trajectory
is observable, then the lter will converge if it is tuned appropriately. The analysis from
the literature provides further insight into the case where OLTV (tf;t0) is rank de cient. If
we rst analyze the modi ed lter without including the yaw constraint (i.e. the standard
loosely coupled lter without having the pitch rate gyro or the pitch rate bias state), we
should expect to see results in accord with those discussed in Section 3.2. Several simulations
were performed in Carsim to validate this expectation. A note here regarding the following
results is necessary. The following plots show the rank of OLTV (tf;t0) over the course of
the simulations. In general, the rank test of OLTV (tf;t0) shows whether or not the system
is observable on the interval [t0;tf]. This means that if a maneuver is performed which
increases the rank to full, during some time interval, then the rank will thereafter remain
full and the system will be declared observable until the end of that interval. What is really
46
Figure 3.1: Observability of the Loosely Coupled and AUNAV Filters During Longitudinal
Dynamics.
of interest here is to show the impact of various maneuvers regarding observability. Because
of this, small intervals of one second were chosen to evaluate OLTV .
Two simulations were done in Carsim in order to test the observability of each trajectory.
All values from the simulations are true values, no noise or other errors were added, because
the analysis is done concerning the linearization about the true trajectory. The rst test
involves the vehicle driving in a straight line with a period of forward acceleration followed
by deceleration. Figure 3.1 shows the rank of OLTV during this test, with subplots showing
the yaw rate and accelerations. Full rank is 15 for the standard loosely coupled lter and 14
for the modi ed lter. Figure 3.2 shows the results from a simulation in which the vehicle
drives straight, enters a steady state turn, and resumes straight driving. The rank of OLTV
during these tests con rms the ndings in the literature. Speci cally, the system is not fully
observable under constant acceleration, but requires changes in acceleration to reach full
observability. It can be seen in Figure 3.1 that as the vehicle begins to accelerate the system
47
Figure 3.2: Observability of the Loosely Coupled and AUNAV Filters During Lateral Dy-
namics.
becomes observable, but as the acceleration becomes steady the rank drops back to 11. The
same is seen when the vehicle decelerates. Figure 3.2 con rms this also, showing the same
behavior in regard to lateral acceleration. Under constant acceleration, the observability
matrix is rank de cient by three. Constant acceleration in another direction increases the
rank by two, con rming the results in [21]. The lter only reaches full rank when a change
in acceleration occurs, con rming the results in [35]. These results show that the modi ed
system, without the extra yaw constraint, behaves just like the standard system in regards
to observability. Therefore the conclusions of Section 3.2 apply to the modi ed lter when
the yaw constraint is not applied. This gives some insight into the anticipated performance
of the lter during constant acceleration operation (straight steady driving). Rhee found
that in this situation the attitude and bias states are unobservable [35]. This means that
the estimates will likely be biased, yet because the combined e ect of the leveling angles and
48
Figure 3.3: Observability of the LC Filter Compared with the AUNAV Filter During Lon-
gitudinal Dynamics when the Course Measurement is Conditionally Added to the AUNAV
Filter.
the accelerometer biases is observable, the errors will be bounded. This cannot be said of
the yaw angle, which is known to exhibit drift during this time.
Figures 3.3 and 3.4 show the results from the same two simulations, only this time the
yaw constraint is imposed when the straight driving condition is met. There is no di erence
in the rank of OLTV under dynamics, but the rank is closer to being full during straight
driving. In this case the observability matrix is only de cient by one, as opposed to being
de cient by three. The improvement of two more observable states is easily explained.
During this time, the lter has direct yaw information available (because sideslip is assumed
to be zero), which makes the yaw angle observable. Furthermore, since the integral of the
yaw rate gyroscope is measurable, the bias of this gyro becomes observable. In this way the
modi cation overcomes the problem of drifting yaw estimates.
While the previous simulation results show when the lter does and does not meet the
necessary conditions for estimation, the convergence of the lter still requires investigation.
49
Figure 3.4: Observability of the LC Filter Compared with the AUNAV Filter During Lateral
Dynamics when the Course Measurement is Conditionally Added to the AUNAV Filter.
If the system were in fact linear and deterministic, observability would imply convergence.
However large initial errors or unmodeled disturbances could cause convergence to an in-
correct local minimum. What?s more, it has been shown in [5] that the stochastic elements
by themselves can cause the lter to diverge if the lter is improperly tuned. The authors
there describe the distinction that results between standard observability and \stochastic"
observability. The system can meet the necessary conditions for observability and yet diverge
due to poor tuning of P;Q;R, due to too large values of P0, or due to large errors in the
initial estimates. So lter convergence of the nonlinear estimator can only be expected if the
trajectory is observable, the tuning is appropriate, and the initial error is not too large.
The following simulations demonstrate the behavior of the lter under such conditions.
These simulations consist of the combination of the two prior simulations. That is, the
vehicle enters a steady state turn, disengages from the turn, and accelerates and decelerates
while driving straight. The simulation is run rst without the yaw constraint, followed by
50
Figure 3.5: Speed and Yaw Rate Pro le of Convergence Test Simulation.
Figure 3.6: Convergence of the AUNAV Accelerometer Bias Estimates During Simulation.
51
Figure 3.7: Convergence of the AUNAV Gyroscope Bias Estimates During Simulation.
a run with the constraint. Figure 3.5 shows the speed and yaw rate for the simulations.
Figures 3.6-3.11 show the estimates of the bias states and the leveling angles, where the
blue signal represents the estimates without the yaw constraint and the cyan represents the
estimates with the constraint.
Let?s rst consider the case without the yaw constraint (blue). It can be seen in Figure
3.6 that the x and y accelerometer bias estimates do in fact converge toward the true value
during periods where the observability test is full rank (during acceleration changes). The
y accelerometer bias shows in particular that bias convergence is strongest when the accel-
eration change is along the axis collinear with that bias. It is also observed that the z axis
accelerometer bias converges regardless of the dynamics. Both of these results are in accord
with the results in [22], [26] and [35]. Figure 3.7 shows the behavior of the gyroscope biases.
The yaw rate gyro bias (!z bias) also behaves as expected, converging toward the true value
during changes in acceleration. By contrast, the roll rate gyroscope bias (!x bias) converges
rapidly, regardless of the dynamics. This result is in disagreement with the results in [35].
52
There the authors nd that when there is no excitation (straight driving at a constant speed),
an unobservable mode is given by
x3x = !p z + y (3.6)
where x3x is the particular unobservable mode in question (using the author?s notation),
!p is the roll rate gyroscope bias, and and are the pitch and yaw errors respectively
(using the notation of this thesis). The terms z and y relate to the Earth?s rotation rate,
see [35] for details. It is observed that if the second two terms in Equation (3.6) are small,
then
x3x !p (3.7)
Since x3x is an observable mode even with no excitation, then the roll rate gyroscope
bias is observable under these conditions. While in reality it is the sum of the bias and the
Earth rotation terms which is observable, the rotational terms are small enough compared
with the bias that they can be disregarded. Carsim does not include any simulated e ects
from the Earth?s rotation, which explains the convergence in simulation. In order to verify
that the bias estimate converges using the true sensors, experimental data was analyzed.
In this experiment, the vehicle is driven on a mostly straight road at close to a constant
speed. Details of the sensors and vehicle are given in Chapter 4. Figure 3.8 shows the roll
rate gyroscope bias estimates. Analysis of static data for this run reveals a bias of 0.139
deg=sec. This was found by taking the mean of the roll rate signal over 25 seconds of static
data (collected at 100Hz). Is is observed that even during periods of very little excitation,
the bias estimate converges to very near 0.14 deg=sec, con rming the simulation results and
the hypothesis that the Earth rotational terms are small enough to neglect. Figure 3.9
is included to show the forward speed and yaw rate, showing that apart from the period
500s 84s. The roll angle error grows, creating an arti cial
bias in the accelerometer, and the sawtooth shapes appear in the lateral velocity. As the roll
angle begins to converge again, the magnitude of the saw teeth diminishes. The AUNAV
estimator recovers more quickly from this problem than the MME does, and it produces in
general a smoother estimate of sideslip than the MME.
5.3 Conclusion
The same experimental tests which were performed to validate the performance of the
AUNAV estimator were used to validate modular lter. The tests were divided into two
phases as in the case of the AUNAV estimator. Simulations and experimental data were
used to validate the initialization phase of the modular lter. The experimental data comes
from the same run used to validate the AUNAV estimator initialization, and the simulation
was designed in Carsim to mimic the real world experiment. The results from the simulations
95
and real data are in agreement and show that given dynamic conditions over time the modular
lter is able to accurately separate the bias and level angles despite large initial errors in the
bias state. The MME estimator performs well during the dynamic estimation phase of the
experiment, except for the instance where the lever arm e ects come into play. The modular
lter is found to be more susceptible to these e ects than the AUNAV estimator. In short,
the modular lter provides good estimation performance for ESC applications.
The AUNAV estimator performs better than the MME estimator by most comparisons.
The roll angle estimates from the AUNAV estimator were found to be more accurate than
the MME estimates in both tests. There is little performance di erence when looking at the
forward and vertical accelerometer biases, but the AUNAV estimator converges faster when
estimating the lateral accelerometer bias. The velocity estimates are almost qualitatively the
same, although the AUNAV estimator is observed to be slightly more accurate. The pitch
angle estimates are also comparable, with the main di erence being the suspension pitch.
In this case the MME estimate is probably preferable over the AUNAV estimate, because it
is always clear exactly what it is outputting. The MME, in steady state, always produces
an unbiased estimate of the road grade. The AUNAV estimator, by contrast, may not have
converged to the true pitch value. It was demonstrated in this thesis that the AUNAV pitch
estimate is close to the true pitch value, and that correspondingly the forward bias estimate
is close to the true value, but some ambiguity remains. This is of course due to the lack
of the pitch rate gyroscope. There is no ambiguity in the road grade estimate from the
MME estimator, making it the preferable pitch estimate on vehicles which have negligible
suspension pitch. The AUNAV estimator strongly out performs the MME in regards to roll
rate bias estimation, as shown in Section 5.1. On the other hand, the yaw rate bias is more
accurately estimated by the MME estimator. This is because this estimate is only a function
of the yaw rate signal and the course measurement. Therefore, misalignment errors in the
yaw direction do not a ect the bias estimate. Unmodeled misalignment errors will a ect
96
the bias estimate of the AUNAV lter, as described in Section 4.1, although including these
e ects in the AUNAV model could potentially mitigate these errors.
It is no surprise that the AUNAV estimator outperforms the MME. The reason is simply
that the AUNAV estimator is based on a more complete model of the underlying kinematic
relationships. The more accurate the model is, the more accurate the estimator will be. This
obvious conclusion actually draws a parallel with model based state estimation methods.
The performance of those methods is also primarily a function of the model accuracy. The
di erence, however, is that the model parameters for the the kinematic AUNAV and MME
estimators are measurable and practically unchanging. This is the primary advantage of
the AUNAV and MME estimators. Finally, although the MME estimator is capable of
accurately estimating important vehicle states for ESC systems, it is outperformed by the
AUNAV estimator.
97
Chapter 6
Other GPS Applications: Tire Radius Estimation, Tire Pressure Monitoring, and Steering
Misalignment Detection
6.1 Introduction
GPS information can be also be used to serve vehicle functions other than state esti-
mation and navigation. In this chapter, GPS information is used to estimate the e ective
radius of the tires, to detect pressure drops in the tires, and to detect steering misalignment.
The state estimates from the AUNAV estimator can also be used to aid in these processes.
For example, road grade information is important for the pressure change detection system,
because large road grades a ect the tire loading and may impact the radius estimate. Fur-
thermore, an accurate yaw rate signal is required for the steering misalignment detection
algorithm presented here. The yaw rate gyroscope bias estimate produced from the state
estimator can be used to remove the bias from the yaw rate signal in order to provide the
needed level of accuracy. In short, this chapter will show how both direct GPS information
and the state estimates produced from the AUNAV estimator can be used to aid in other
vehicle safety functions beyond dynamic state estimation.
6.2 Tire Rolling Radius
Combining GPS with on board ESC sensors can aid in estimating the tire?s rolling
radius, which is an important parameter for various vehicle systems. An approach based
on linear estimation techniques is presented in [31], where the authors combine GPS and
wheel speed information with a vehicle model to obtain estimates of the rolling radius and the
longitudinal sti ness of the driven wheels. In [11] the authors expand on the work done in [31]
98
by considering the adverse properties of linear estimation methods. Finding that noise in the
measurement matrix (as opposed to merely having noise in the measurement vector) causes
a bias in the linear least squares estimate, the authors propose using nonlinear optimization
methods to solve the problem. Improvements to the nonlinear estimation strategy are given
in [12], [13]. The approaches in all of these are for nding the longitudinal sti ness and
rolling radius of the driven wheels. However, the rolling radius of the undriven wheels are
estimated and discussed also. Equation (6.1) is the simple, important governing relationship
which is exploited to produce the estimate of the undriven rolling radius.
Vx = Reff! (6.1)
Equation (6.1) describes the relationship between the forward velocity Vx, the rolling radius
Reff, and the wheel speed ! under the important condition that there is no slip. In [13] the
radius estimate is obtained by solving Equation (6.1) in matrix (batch) form. For reference,
Equation (6.2) expands the relationship to deal with slip by including the forward slip ratio
.
Vx = Reff!1 (6.2)
All of the above references focus on methods for estimating the driven wheel radius together
with the longitudinal sti ness. These estimates are useful in themselves, however the authors
also focus on how the sti ness in particular might be an indicator of the tire-road friction
coe cient. This follows the hypothesis raised in [24], [25] that the longitudinal sti ness
value could be an accurate predictor of the friction even at low slip values. In pursuing this
investigation, the authors nd that many things in uence the sti ness estimate including
tire pressure [13]. Therefore much discussion is given to how the sti ness might also be an
indicator of tire pressure. In this thesis the focus is on how the estimate of the rolling radius
might accurately indicate tire pressure or pressure changes, as opposed to the sti ness. While
99
minor consideration is given to this idea in [11], [12], and [13], this section of this thesis will
explore how the radius estimate alone might be a successful predictor of tire pressure.
If the rolling radius estimate of the undriven wheel can successfully predict tire pressure,
the method might be expanded to the driven wheels using relative radius estimation methods.
Therefore consider the case where there is no slip, then Equation (6.1) can be used to form a
linear Kalman lter. In this case the state estimate is held constant in between the recurring
measurements of Vx=!. For comparison, this is simply the recursive version of the approach
shown in [13] for the undriven wheel radius. Equations (6.3) and (6.4) show the state vector
and the measurement vector for the lter.
^X =
Reff fl Reff fr Reff rl Reff rr
T
(6.3)
Y =
Vx fl
!fl
Vx fr
!fr
Vx rl
!rl
Vx rr
!rr
T
+ (6.4)
In (6.4), represents the measurement noise vector. The lter operates as a normal
Kalman lter according to Equations (2.1-2.3). Since the wheel speeds are by de nition the
rotational speeds at the tire, the translational speeds also must be at the tire. These are
denoted in (6.4) by Vx fl;Vx fr;Vx rl;Vx rr. Calculating these velocities is straightforward,
and Equation (6.5) shows how this is done.
Vx fl = Vx rl = Vx + 12ltw!z
Vx fr = Vx rr = Vx 12ltw!z (6.5)
In (6.5), !z is the yaw rate, and ltw represents the track width.
Figure 6.1 shows the estimation performance in simulation. Carsim was used to simulate
a simple scenario where a front wheel drive sedan drives in a straight line for 10 minutes.
The estimate converges to the true value over time. Note that a small bias is present in the
100
Figure 6.1: Undriven Wheel E ective Radius Estimate Convergence in Simulation.
Figure 6.2: Undriven Wheel E ective Radius Estimate Convergence with Faster Tuning in
Simulation.
101
Figure 6.3: Driven Wheel E ective Radius Estimate Convergence in Simulation.
estimator. This is because even for the undriven wheel there exists a very tiny steady state
slip of around 0.04% (according to Carsim?s models). Yet the bias from this is miniscule, at
only 0.1mm. The convergence speed is a function of the tuning, although there is a trade
o between speed and smoothness. A faster tuning o ers quicker convergence. This is also
seen in Figure 6.2, where the tuning has been \sped up" for the same run. That is, the
process noise covariance matrix (Q) value is larger. The sensor noise covariance matrix (R)
value is held constant. In this case the lter converges more quickly, but it is not as smooth.
Later discussions will focus on the importance of smoothness. Figure 6.3 shows the radius
estimate of a driven wheel. The transient performance is exactly the same, (since the slip is
constant), but the bias is larger because the slip is larger for the driven wheel. It violates the
no slip assumption stated previously, and the bias is around 1.5mm. Figure 6.4 shows the
radius estimate of the undriven wheel during a simulation of the vehicle driving in a gure 8
pattern at much lower speeds, demonstrating that the lter is able to operate under turning
conditions.
102
Figure 6.4: Undriven Wheel E ective Radius Estimate Convergence During Figure-8 Turning
Maneuvers in Simulation.
6.3 Tire Pressure Monitoring
It is hypothesized that the e ective rolling radius is a function of the tire pressure [11],
[12], [13]. Since it has now been shown that GPS o ers a way to very accurately estimate
the e ective radius, it is further hypothesized in this thesis that this estimate can be used to
estimate the tire pressure and detect changes. In order to validate this hypothesis, four sets
of data were collected on the loop at NCAT. The vehicle was driven at a nearly constant
speed of 50mph for 45 minutes. Before and after each run, the pressure of all four tires was
recorded. The G35 is a rear wheel drive vehicle, so the front tires are the undriven wheels.
The front left and rear right tires were chosen as the variable tires, and they were run with a
di erent pressure each time. While the other two tires were maintained at 33psi, the front left
and rear right tires were set to 36, 32, 28, and 24 psi for each respective run. Figure 6.5 shows
the estimates for the front left tire for each run. It should be noted that only the data from
the straightaways is processed in the estimate. This is to limit any e ects of weight transfer
103
Figure 6.5: Front Left Tire Radius Estimates for Di erent Pressures.
on the tires, as the steeply banked turns might result in small tire deformations. Recall
that the radius is speci cally the e ective or rolling radius, which means it is extremely
di cult measure the true value. Yet it is known that this value lies between the loaded and
unloaded radii [34]. The radius estimates were all found to lie within the measured loaded
and unloaded radii as expected.
It can be seen that changes in tire pressure do result in changes in the rolling radius.
The rst pressure drop from 36 to 32 psi results in a radius change of 0.5mm. This might
seem too small to track, but the estimator is clearly able to do so given enough data to
average out the noise. This is because of the accuracy of the GPS velocity measurements,
which are unbiased and have noise characteristics of approximately 5 cm=s (1 ). The second
drop from 32 to 28 psi results in a change of 0.5mm. The last drop from 28psi to 24 psi
results in a change of 0.6mm. Figure 6.6 shows the estimates from the rear right tire. It
is observed from this graph that the pressure drops correspond with decreases in the radius
estimate of 0.5mm, 0.4mm, and 0.6mm. Recall that the method presented in this thesis
104
Figure 6.6: Rear Right Tire Radius Estimates for Di erent Pressures.
Figure 6.7: Estimates for All Four Tires for All Four Experiments.
105
results in a biased estimate when it is applied to the driven wheels. However, this bias does
not a ect the change in radius that results from the pressure. Therefore although the radius
estimates of the driven wheels are not as accurate as those of the undriven wheels, they are
still able to be used for pressure loss detection.
Looking at Figures 6.5 and 6.6 alone would suggest that these pressure ranges constitute
an easily dividable space for a lookup table or other classi er. Indeed the gaps between signals
is far larger than the noise, which is a good sign. However the repeatability of the estimates
must be considered. Speaking qualitatively, if there is much variation between the radius
estimates of a tire with a constant pressure, then it will be very di cult to isolate radius
changes which correspond to pressure changes. Figure 6.7 shows the estimates for all four
tires for all runs. Recall that the front right and rear left tires were maintained near 33psi
for all four runs. It appears that all of the estimates for the control tires lie within the range
331.1mm and 331.5mm. Furthermore, the estimates for the front left and rear right tires
lies within this range for the 32psi run. This indicates a degree of consistency across all four
tires. From Figure 6.7 it can be inferred that if the radius estimate is within this range, then
the tire pressure lies within 30psi to 34psi. Similar regions can be constructed for the other
pressure ranges. The space from 331.1mm to 330.7mm corresponds to a pressure between
30psi and 26psi. Below 330.7mm, the tire pressure can be considered to be less than 26psi.
Clearly these conclusions only apply to this particular data, and much more data would be
needed to construct a reliable tire pressure predictor. However, Figures 6.5 - 6.7 show that
indirect tire pressure monitoring is possible with GPS.
6.4 Steering Misalignment Detection
Steering misalignment is a problem because of the accelerated and irregular wear that
it causes on the tires. Large misalignments are obvious and easily noticed by the driver,
but smaller misalignment problems can be far more subtle. Even small misalignments have
negative e ects on tire wear and gas mileage, therefore it would be advantageous if the
106
vehicle were able to detect small misalignments and inform the driver, and GPS provides
the ability to do this.
The phrase \steering misalignment" is used here to describe any condition which causes
the steering system to produce an improper steer angle at the tires for a given steering wheel
input. This may be caused by a variety of problems in the suspension or steering systems,
but from a general standpoint the end e ect is the same. The premise of the misalignment
detection strategy presented in this thesis is that if the vehicle is driving straight, then the
steer angle ought to be zero degrees. This is not exactly true, however, because of the
steering e ects of road crown. Most roads generally have a very small side slope in order to
improve water drainage, and this slope will cause a very slight o set from zero in the steer
angle. In this thesis, all tests were done on the same section of roadway, so the e ects of
road crown will be the same for all experiments. Therefore the road crown e ects will be
ignored, although the accurate estimate of the vehicle roll angle produced by the AUNAV
estimator could be used to detect non standard road crowns.
The steering misalignment detection algorithm functions as follows. It requires the
steering wheel angle (SWA), an accurate yaw rate signal, noise characteristics for the yaw
rate sensor, and a baseline steer angle. The baseline steer angle is the steady state steer
angle for straight driving when the vehicle is perfectly aligned, which can be ascertained the
rst time a production vehicle is driven. If there is no road crown, then the baseline steer
angle will be zero. For the experiments in this thesis, the baseline steer angle was found to
be approximately 2.25 degrees. The assumption is made that the In niti G35 was perfectly
aligned before the experiments were made. Since road crown is assumed to be constant, the
baseline steer angle is adjusted to be zero degrees, meaning that a constant 2.25 degrees
is subtracted from all SWA measurements. This is because it is the relative steer angle
compared to the baseline which is important, not necessarily the absolute steer angle. When
it is determined that the vehicle is driving straight, as described in Section 2.3.2, a moving
average (100 second window) of the SWA measurement is taken. The yaw rate gyroscope
107
bias estimate from the AUNAV lter is subtracted from the raw yaw rate signal during the
process of determining straight driving conditions. If the average steer angle is o set from
zero, when the vehicle is driving straight, then it is concluded that there is a misalignment
problem.
Four experiments were conducted to validate the steering misalignment detection strat-
egy. These tests were conducted using the In niti G35, which is instrumented as described at
the beginning of Chapter 4. Each test consisted of driving 10 minutes north on I-85 towards
Atlanta while maintaining a steady speed and with minimal lane changes. The rst test was
conducted without making any modi cations to the vehicle, assuming that the alignment at
that time was very good. Thus the alignment for the rst test was taken as the baseline.
For each subsequent test, misalignment error was manually introduced by adjusting the toe
angle of the front right tire. The toe was adjusted by screwing the tie road in or out of
the sleeve by some amount. In this experiment, the tie road was screwed into the sleeve,
thereby shortening the overall length. Since this linkage is located behind the center of the
tire, the change resulted in a toe out situation on the front right tire. The tie road was
adjusted by making two turns on the screw for the rst experimental run, 1/2 turn for the
second, and 1 full turn for the nal run. Figure 6.8 shows the raw SWA measurements and
the corresponding moving averages for the experiments. The baseline steer angle of 2.25
degrees is also observed in this plot. Notice that the magnitude of the steer angle increases
with the amount of alignment adjustments made. Figure 6.9 shows the moving averages
of the steering wheel angles, the yaw rate signals for each run, and the yaw rate detection
threshold for straight driving determination. It can be clearly seen that adjustments to the
toe angle of the tire results in non zero steering angles when the vehicle is driving straight.
Furthermore, the magnitude of the o set clearly corresponds with the degree to which the
toe angle is adjusted. The yaw rate threshold for straight driving detection is shown as a
reference. A threshold of 3 was used, where the assumed standard deviation of the noise on
the yaw rate signal is = 0:5deg=s. The yaw rate signals have been improved by subtracting
108
Figure 6.8: Steer Angle Signals for All Four Misalignment Experiments.
the estimates of the yaw rate bias provided by the AUNAV lter. In this case only a marginal
improvement was obtained because of the high quality of the Crossbow gyroscopes. However
less expensive sensors will have larger biases, making it necessary to use the bias estimate
in order to be able to detect straight driving. Based on Figure 6.9 it is concluded that it is
possible to detect even slight misalignments using GPS enhanced inertial sensor signals and
the steering wheel angle.
6.5 Conclusions
It has been shown that GPS information can be combined with wheel speeds to very
accurately estimate the rolling radius of the undriven wheels. The algorithm operates on the
assumption that there is no slip. This assumption is valid for the undriven wheels except in
the case of braking. This does not present an obstacle for the algorithm however, because
the braking signals are readily available on the CAN bus and can be used as a toggle on
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Figure 6.9: Averaged Steer Angle Signals and Yaw Rate Signals For All Four Misalignment
Experiments.
and o switch for the estimator. It has also been shown that the rolling radius varies with
tire pressure, presenting an opportunity for indirect tire pressure monitoring. That is, GPS
o ers the possibility of detecting pressure losses without actually needing pressure sensors in
the tire. The TREAD act requires tire pressure monitoring systems (TPMS) to notify the
driver if the tire pressure has dropped by 25% of the recommended cold pressure [1]. The
recommended pressure for the G35 is 33psi, which means that any TPMS must be able to
detect pressure drops of 8.25 psi. The results from Figure 6.7 show that this is extremely
possible with the rolling radius estimate. Such a system would require many more data
sets for analysis to provide statistically signi cant detection and alarm thresholds. However,
the operation is simple: if the radius estimate drops below a certain value, then the system
interprets that to correspond with a drop of a certain pressure. Given the data presently
available, the design would be as follows. If the radius for one of the front tires drops below
331.1mm, then a drop of 4psi (12%) can be safely inferred. This is well within the accuracy
110
requirements of the TREAD act. Again, more data needs to be collected to improve the
statistics of the detection algorithm, but the concept is shown feasible here. Future work
would also deal with studying and isolating the e ects of pressure on the radius. For instance,
the e ects of vertical loading conditions on the radius estimate needs to be studied, because
these e ects must be isolated from the pressure detection algorithm. Yet it has been shown
that GPS can be used to accurately estimate the vehicle mass online [3]. If the mass is known,
and more importantly if the change in mass is known, this information can be included in the
process of inferring the pressure from the radius. Tire wear might also present a problem.
If the change in radius due to tire wear is in the same range as the change in radius due
to pressure, it may be very di cult to separate these e ects. Yet for now it is concluded
that the GPS aided estimate of the undriven wheel rolling radius o ers great potential in
determining pressure loss without needing tire pressure sensors. It has also been shown that
the yaw rate signal, aided with the bias estimate from the AUNAV estimator, can be used
with the steer angle sensor to detect steering misalignment. Data was collected on portions
of I-85 north with various alignment adjustments to the steering linkage. The data con rmed
the hypothesis that misalignment conditions result in steady state steer angle o sets when
the vehicle is driving straight. Therefore if the vehicle is known to be driving straight via
the yaw rate information and if there is a non zero steady state steering wheel angle, then
it is accurately inferred that there is an alignment problem.
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Chapter 7
Conclusions and Future Work
This thesis has shown the advantages of combining the increasingly common single
antenna GPS systems with sensors which are already present on board vehicles equipped
with ESC and RSC systems. First, the Automotive Navigation (AUNAV) estimator was
presented. This estimator integrates GPS information with information from a 5 degree of
freedom inertial sensor cluster to produce accurate estimates of the vehicle sideslip, attitude,
velocity, and position along with accurate estimates of the inertial sensor biases. An analysis
of the observability of the AUNAV estimator was also presented, showing that this estimator
is fully observable under certain dynamic conditions. The AUNAV estimator was then
contrasted with the Modi ed Modular Estimator (MME), which is a di erent approach
to integrating the same sensors. Finally, methods of estimating the e ective tire radius,
detecting tire pressure changes, and detecting steering misalignment were presented. These
methods all take advantage of GPS information and information provided by the AUNAV
estimator.
The AUNAV estimator is a specialized form of the generic loosely coupled GPS/INS
integration lter that is shown in [19]. It consists of the GPS and IMU functioning as two
independent navigation systems, with the GPS serving as truth, and one extended Kalman
lter. The states of the EKF are the errors in the INS solution, so the measurements used by
the lter are the di erences between the GPS and INS solutions. The estimates of the INS
solution errors are added back to the original INS solutions and the corrected solution is used
as the nal estimate. The EKF also produces estimates of the inertial sensor biases, and
these are used in the INS processing stage to improve the solutions. The AUNAV estimator
di ers in its design from the loosely coupled lter in two ways. First, the AUNAV estimator
112
does not use a pitch rate gyroscope because these are not available on modern commercial
vehicles. Second, the AUNAV estimator uses the GPS course measurement as a measurement
of heading (yaw) when the vehicle is driving straight. This is made possible because when
the course and heading are equivalent when there is no sideslip. The AUNAV estimator
also di ers from the loosely coupled lter in that the AUNAV estimator is primarily used
to estimate sideslip, while the loosely coupled lter is primarily analyzed for its positioning
capabilities.
A study of the observability of the AUNAV estimator was conducted. Many authors
have analytically determined the observability of the loosely coupled lter under various
dynamic scenarios, making the important case that the observability is contingent on the
dynamics. It was hypothesized in this work that the same conclusions apply for the AUNAV
estimator, and this was tested using an observability test for linear time varying systems and
simulations. The observability tests con rmed the hypothesis that the AUNAV estimator has
the same observability characteristics as the loosely coupled lter. The AUNAV estimator
requires acceleration changes in order to be made fully observable, but combinations of the
attitude states and the inertial biases remain observable when the vehicle is not performing
any acceleration. This means that dynamics are required in order for the estimator to
be able to separate the attitude states from the bias states. Simulations were performed
which con rmed the results of the observability checks. It was also found that the roll rate
gyroscope bias converges when there are no dynamics. The yaw measurement gives the
AUNAV estimator an observability advantage over the loosely coupled lter by making the
yaw angle and the yaw rate gyroscope bias observable when the vehicle is driving straight.
The performance of the AUNAV estimator was demonstrated using experimental data
collected with an In niti G35 sedan which is instrumented with various GPS and inertial
sensors, along with an attitude determination system. It was shown that the AUNAV es-
timator is able accurately estimate the sideslip, roll, and pitch during dynamic maneuvers.
It was also shown that the estimator is able to accurately separate the attitude states and
113
the biases when dynamic maneuvers are performed over time, con rming the observability
analysis. Finally, it was shown that the AUNAV estimator is able to accurately estimate
sideslip which is increasing a rates as slow a 1.5 deg=s.
The performance of the AUNAV estimator was also compared with the performance
of the MME estimator using the same data from the previously experiments. The MME
estimator is based on the approach presented in [6]. It consists of several independent
GPS/INS lters which estimate states along the di erent axes. The roll angle must be
additionally high pass ltered from a lumped state in the lateral lter. The pitch angle is
obtained by estimating the road grade as in [4]. The sideslip angle is computed from the
lateral and forward velocity estimates. It was found that in general the AUNAV estimator
outperforms the MME estimator in almost all cases. This is because the AUNAV estimator
uses a more complete model of the kinematic relationships than the MME does. However,
the MME does a better job of estimating the yaw rate gyroscope bias. This is because
its simpli ed model is more isolated from the e ects of misalignment than the AUNAV
estimator. In fact, the MME heading lter is not subject to yaw angle misalignment e ects
at all. The AUNAV lter combines all yaw rate errors, including misalignment, into the bias
estimate, making it a less accurate estimate of the pure sensor bias.
A simple method of using GPS and wheel speed sensors to estimate the absolute e ective
rolling radius of the wheels was also shown. The method was demonstrated in simulation
to be unbiased if the no slip assumption holds. While the estimate for the driven wheels is
biased due to wheel slip, the estimate for the undriven wheels can serve as an absolute radius
for any relative calculations for the driven wheels. There is no reference method or \truth"
measurement for the e ective radius with which to compare the estimate in real experiments,
yet it is known that the e ective radius lies within the loaded and unloaded radii. Several
runs were performed on the G35 on the NCAT loop, and the radius estimates for all four tires
were within this range for all runs. This served as a sanity check, showing that while there
may not be an exact truth for comparison, the estimates from the experiment are reasonable.
114
A method for inferring the tire pressure on the basis of the rolling radius estimate is also
proposed. Tests were conducted showing that the rolling radius estimate does vary according
to tire pressure, and that these changes are distinguishable from the noise on the estimates.
This leads to the conclusion that a simple hypothesis tester can be used to detect pressure
drops of half of the percentage required by the TREAD act [1]. Future work requires many
more data runs to improve the statistics of such a detector and to investigate other sources of
change in the radius estimate. This is because the radius is only a good indicator of pressure
if the changes resulting from pressure changes can be isolated.
A method of using the yaw rate bias estimate from the AUNAV estimator, the yaw
rate signal, and the steering wheel angle to detect steering misalignment was presented.
This method assumes that an aligned vehicle will have a zero steady state steer angle when
driving straight if there is no road crown. For this thesis, the e ects of road crown are ignored,
although they could potentially be incorporating using the roll angle estimate provided by
the AUNAV estimator. Experiments were conducted by altering the toe angle of the front
right tire of the G35 and collecting data along a 10 minute section of I-85 north. It was found
that tire misalignment results in a non zero steady state steer angle when driving straight,
and that the corrected yaw rate signal and the steer angle signal can be used to accurately
detect even subtle steering misalignments. If a non zero steady state steering angle is present
when the corrected yaw angle is zero, then a ag is set indicating a misalignment condition.
There are many important avenues of future work stemming from this thesis. First, a
more detailed analysis on the sensitivity of the AUNAV estimator to sensor quality needs
to be done. The bias stability of the inertial sensors in particular plays an important role
in the overall observability and operation of the lter. It was claimed in this thesis that
once the biases are properly identi ed, the estimator can sustain accurate estimates of the
level angles and the biases for a period of time even when the vehicle is not undergoing any
dynamics. The bias stability is the main factor determining how long this period of time is.
Furthermore, the length of this period of time is an extremely important system parameter
115
of the AUNAV estimator, and it would have to be accurately known in order to safely
implement the estimator in commercial vehicles. Sensor quality also a ects the amount of
time that the estimator can sustain accurate state estimates during a GPS outage. This is
another very important system parameter because GPS outages are ultimately inevitable.
Another avenue of future work would be to investigate other methods of determining when
the vehicle is driving straight. One potential method would be to run a model switching
algorithm. The switching would be between an instance of the AUNAV lter which assumes
straight driving and one which does not. It currently remains to be seen whether there are
any advantages in using such a method or any other method beyond the current detection
strategy. The e ects of the bias stability on the straight driving detection method also
requires additional study. It might be possible for the bias to cause the lter to never detect
periods of straight driving, making it even more di cult to estimate the bias. This could
potentially cause rapid degradation in the lter performance.
More data is required to improve the tire pressure detection algorithm, in order to
make the results statistically signi cant and to more accurately quantify the accuracy and
resolution. Another current problem is that there is no way to establish a baseline radius for
the tire. The absolute radius can be accurately estimated for the undriven wheels, but there
is no way of knowing as a baseline what pressure a given radius corresponds to. A solution to
this problem would be essential to real world implementation. The e ects of tire loading on
the radius estimate also needs to be studied, as the pressure e ects have to be isolated from
these. The radius estimates have a very ne resolution, and it is likely that large loading
variations could cause true deformations in the tires within the order of magnitude of the
estimation resolution. The steering misalignment algorithm would also bene t from more
data for the same reasons as the tire radius estimation algorithm. That is, more data is
required in order to statistically set detection thresholds. The e ects of road crown also
need to be investigated. It is possible for non standard road crowns to cause the system to
116
falsely detect a steering misalignment. The roll angle estimate could be used to mitigate this
problem, but this has not yet been studied.
This thesis has shown the bene ts of combining single antenna GPS information with
the signals from standard ESC/RSC sensor clusters. These results can be applied today,
from the standpoint of sensor availability, to any vehicle equipped with a navigation pack-
age and ESC/RSC. Navigation packages are becoming increasingly ubiquitous on passenger
vehicles, making these solutions extremely cost e ective. No additional sensors are required.
Furthermore these solutions o ers the advantage of robustness to vehicle parameters, since
they does not use any such parameters. Neither changes in vehicle loading, new sets of tires,
nor after market sway bar installations will cause any problems for the systems presented
here. In conclusion, the solutions o ered here present great opportunity to improve vehicle
safety without one cent of additional materials cost, and it is all made possible with the
Global Positioning System.
117
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