1. automotive and vehicles
2. cars

Department of Precision and Microsystems Engineering
Estimation of Vehicle Handling States
Shelav Jain
Report no
Coach
Professor
Specialisation
Type of report
Date
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AUT 2014.027
Mr. Anil Kunnappillil Madhusudhanan
prof. dr. ir. E. G. M. Holweg
Automotive
M.Sc. Thesis
28 October 2014
Estimation of Vehicle Handling States
Using Tire Force Measurements From Load Sensing Bearings
Master of Science Thesis
For the degree of Master of Science in Mechanical Engineering at Delft
University of Technology
Shelav Jain
October 28, 2014
Faculty of Mechanical, Maritime and Materials Engineering (3mE) · Delft University of
Technology
c Precision and Microsystems Engineering (PME)
Copyright All rights reserved.
Delft University of Technology
Department of
Precision and Microsystems Engineering (PME)
The undersigned hereby certify that they have read and recommend to the Faculty of
Mechanical, Maritime and Materials Engineering (3mE) for acceptance a thesis
entitled
Estimation of Vehicle Handling States
by
Shelav Jain
in partial fulfillment of the requirements for the degree of
Master of Science Mechanical Engineering
Dated: October 28, 2014
Supervisor(s):
prof.dr.ir.E.G.M.(Edward) Holweg
Mr. Anil Kunnappillil Madhusudhanan
Reader(s):
dr.ir.A.L.(Arend)Schwab
dr.Barys Shyrokau
Abstract
With the ever increasing demand of cars, safety is of prime concern to automobile manufactures across the world. Companies strive for the vision of "Zero accidents" through high
quality and innovative products that reduce the frequency of accidents as well as their consequences. The auto industry has been exerting a myriad of state-of-the-art technologies to
make automotive safety systems that reduce driver’s strain and fatigue and assist safe driving.
Advanced Driver Assistance Systems (ADAS) are technologies that capture the vehicle’s surrounding environment and assist the driver by keeping him informed about the current vehicle
state, and if necessary intervene to prevent an impending danger, while the driver is in control
of the vehicle at all times. Vehicle Dynamics Control (VDC) systems enhance the handling
and safety of the car by assisting the driver in maintaining control of the vehicle. However,
these features are limited by lack of knowledge of the vehicle states. Some of the vehicle states
like sideslip angle are not measurable due to technical or economic reasons. Therefore, these
must be estimated by using the available measurements.
Most of the existing sideslip angle estimators are based on lateral acceleration, but these
estimators have estimation errors arising from roll and pitch dynamics. The purpose of this
study is to develop an algorithm that estimates the vehicle sideslip angle and tire cornering
stiffnesses using tire force measurements. This is achieved by implementing a model based
deterministic approach. Kalman filters and their extensions used for state and parameter
estimation are investigated. Finally, the developed system is implemented using a multibody
vehicle simulator and tested for different maneuvers.
Keywords: Vehicle lateral dynamics, sideslip angle, cornering stiffness, Unscented Kalman
filter
Master of Science Thesis
Shelav Jain
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Shelav Jain
Master of Science Thesis
Table of Contents
Acknowledgements
ix
1 Introduction
1-1 Lateral Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
1-2 Automobile Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1-3 Active control of Vehicle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . .
7
1-3-1
Electronic Stability Control . . . . . . . . . . . . . . . . . . . . . . . . .
8
1-4 Application of State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1-4-1
Estimation of Sideslip Angle . . . . . . . . . . . . . . . . . . . . . . . .
16
1-4-2
Estimation of Tire Cornering Stiffness . . . . . . . . . . . . . . . . . . .
17
1-5 Thesis Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
1-6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2 Physical Modeling
21
2-1 Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-1-1 Four-wheel vehicle model . . . . . . . . . . . . . . . . . . . . . . . . . .
2-1-2 The Bicycle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
22
24
2-2 Tire Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2-2-1
Magic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2-2-2 Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-2-3 Transient tire model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
29
29
3 State Estimation and Filtering
31
3-1 Modeling of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3-2 Observability of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3-3 The Luenberger Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
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Table of Contents
3-4 Linear Kalman Filter . . . . . . .
3-5 Non-linear Kalman Filter . . . .
3-5-1 Unscented Kalman Filter
3-6 Summary . . . . . . . . . . . . .
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4 Estimation using Bicycle Model
33
35
36
40
41
4-1 Nonlinear Observer: State Space Algorithm . . . . . . . . . . . . . . . . . . . .
41
4-2 Observability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4-3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-3-1 Tuning Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
43
4-3-2 Observer Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-3-3 Limitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-4 Nonlinear observer with sideslip angle as measurement . . . . . . . . . . . . . .
44
50
50
4-5 Analysis and Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
4-6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
5 Estimation using Four Wheel Vehicle Model
53
5-1 Nonlinear Observer Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-2 Observability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
55
5-3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-3-1 Tuning Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
55
5-3-2 Observer Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-3-3 Limitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-4 Nonlinear observer with lateral velocity as measurement . . . . . . . . . . . . . .
56
61
64
5-5 Analysis and Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
5-6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
6 Conclusions and Recommendations
6-1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
67
68
Bibliography
69
Glossary
75
List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Shelav Jain
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Master of Science Thesis
List of Figures
1-1 Forces acting on a vehicle. Source: [1] . . . . . . . . . . . . . . . . . . . . . . .
2
1-2 Components of tire forces. Source: [1] . . . . . . . . . . . . . . . . . . . . . . .
3
1-3 Tire slip angle. Source: [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1-4 The lateral tire force versus slip angle. . . . . . . . . . . . . . . . . . . . . . . .
4
1-5 The friction circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-6 The lateral tire force versus slip angle at different normal loads. . . . . . . . . .
5
6
1-7 Yaw stability control system for vehicle lateral dynamics. Source: [3] . . . . . . .
8
1-8 Sideslip angle of a vehicle. Source: [2] . . . . . . . . . . . . . . . . . . . . . . .
9
1-9 ESP control loop in a vehicle. Source: [1] . . . . . . . . . . . . . . . . . . . . .
10
1-10 Active chassis control. Source: [3] . . . . . . . . . . . . . . . . . . . . . . . . .
11
1-11 Yaw moment as function of sideslip angle for different steering angles. Source: [3]
12
1-12 Objective of yaw stability control system. Source: [3] . . . . . . . . . . . . . . .
12
1-13 Block diagram for integrated control objective. Source: [4] . . . . . . . . . . . .
13
1-14 The principle layout of a control loop using a state estimator. . . . . . . . . . . .
15
2-1 Four-wheel vehicle model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-2 The bicycle model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
24
2-3 Tire dynamics variables. Source: [5] . . . . . . . . . . . . . . . . . . . . . . . .
25
2-4 Four categories of possible types of approach to develop a tire model. Source: [6]
26
2-5 The Magic Formula parameters. Source: [6] . . . . . . . . . . . . . . . . . . . .
28
3-1 The principle of the Unscented Transformation. Source: [7] . . . . . . . . . . . .
36
3-2 Flowchart of the Unscented Kalman Filter (UKF) algorithm. . . . . . . . . . . .
38
4-1 Steering profile and lateral acceleration for sine steer at Vx of 80km/h. . . . . .
45
Master of Science Thesis
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List of Figures
4-2 Sideslip angle for sine steer at Vx of 80km/h. . . . . . . . . . . . . . . . . . . .
45
4-3 Cornering stiffness for sine steer at Vx of 80km/h. . . . . . . . . . . . . . . . .
46
4-4 Steering profile and lateral acceleration for double lane change at Vx of 75km/h.
46
4-5 Sideslip angle for double lane change at Vx of 75km/h. . . . . . . . . . . . . . .
47
4-6 Cornering stiffness for double lane change at Vx of 75km/h. . . . . . . . . . . .
47
4-7 Steering profile and lateral acceleration for double lane change at Vx of 85km/h
and µ = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4-8 Sideslip angle for double lane change at Vx of 85km/h and µ = 0.5. . . . . . . .
49
4-9 Cornering stiffness for double lane change at Vx of 85km/h and µ = 0.5. . . . .
49
4-10 Lateral acceleration and sideslip angle for steady state cornering for UKF without
sideslip angle measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4-11 Steering profile and lateral acceleration for steady state cornering at Vx of 90km/h
and 100m radius circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-12 Cornering stiffness for steady state cornering at Vx of 90km/h and 100m radius
circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
52
5-1 Steering profile and lateral acceleration for sine steer at Vx of 80km/h. . . . . .
57
5-2 Sideslip angle for sine steer at Vx of 80km/h. . . . . . . . . . . . . . . . . . . .
57
5-3 Front tires cornering stiffness for sine steer at Vx of 80km/h. . . . . . . . . . . .
58
5-4 Rear tires cornering stiffness for sine steer at Vx of 80km/h. . . . . . . . . . . .
58
5-5 Steering profile and lateral acceleration for double lane change at Vx of 80km/h.
59
5-6 Sideslip angle for double lane change at Vx of 80km/h. . . . . . . . . . . . . . .
59
5-7 Front tires cornering stiffness for double lane change at Vx of 80km/h. . . . . .
60
5-8 Rear tires cornering stiffness for double lane change at Vx of 80km/h. . . . . . .
60
5-9 Steering profile and lateral acceleration for double lane change at Vx of 85km/h
and µ = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
5-10 Sideslip angle for double lane change at Vx of 85km/h and µ = 0.5. . . . . . . .
62
5-11 Front tires cornering stiffness for double lane change at Vx of 85km/h and µ = 0.5. 62
5-12 Rear tires cornering stiffness for double lane change at Vx of 85km/h and µ = 0.5. 63
5-13 Lateral acceleration and sideslip angle for steady state maneuver for UKF without
lateral velocity as measurement. . . . . . . . . . . . . . . . . . . . . . . . . . .
5-14 Steering profile and lateral acceleration for steady state cornering at Vx of 95km/h
and 100m radius circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-15 Front tires cornering stiffness for steady state cornering at Vx of 95km/h and 100m
radius circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-16 Rear tires cornering stiffness for steady state cornering at Vx of 95km/h and 100m
radius circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Shelav Jain
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64
65
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Master of Science Thesis
List of Tables
4-1 D class vehicle parameters values. . . . . . . . . . . . . . . . . . . . . . . . . .
42
4-2 UKF tuning parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
4-3 RMS error values for UKF with bicycle model. . . . . . . . . . . . . . . . . . . .
48
5-1 Root mean squared (RMS) error values for UKF with Four-Wheel vehicle model
(FWVM). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
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Shelav Jain
List of Tables
Master of Science Thesis
Acknowledgements
I would like to thank my supervisors prof.dr.ir.Edward Holweg and Mr.Anil Kunnappillil
Madhusudhanan for their helpful suggestions and comments during discussions about the
topic. I would also like to thank dr.Mustafa Ali Arat for guidance and direction in successful
completion of this project.
Furthermore, my sincere thank to dr.ir.A.L.(Arend)Schwab and dr.Barys Shyrokau for accepting to be on my M.Sc. defense committee.
Delft, University of Technology
October 28, 2014
Master of Science Thesis
Shelav Jain
Shelav Jain
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Shelav Jain
Acknowledgements
Master of Science Thesis
“The Future depends on what we do in the present” — Mahatama Gandhi
Chapter 1
Introduction
”It has often said that the primary forces by which a high-speed motor vehicle is controlled
are developed in four patches-each the size of man’s hand-where the tires contact the road.
This is indeed the case. A knowledge of forces and moments generated by pneumatic
(rubber) tires at the ground is essential to understanding highway vehicle dynamics.-Thomas
D. Gillespie”
From Newton’s laws of motion, it is very well understood that inertia is the property possessed
by all bodies and by virtue of this it will remain in its state of motion or rest unless an external
force is applied to it, to bring about a change to that status. On an automobile, various forces
act upon it regardless of its state of motion, as shown in Figure 1-1.
Vehicle dynamics is the branch of engineering which relates tire and aerodynamic forces to
overall vehicle acceleration, velocities and motions using Newton’s Laws of Motion. It encompasses the behavior of the vehicle as affected by driveline, tires, aerodynamics and chassis
characteristics. It is the study of forces which affect wheeled vehicles in motion and of the
vehicle’s responses, either natural or driver induced.
Vehicle Dynamics can be broken down into the following branches representing the degrees
of freedom of a vehicle:
• Longitudinal Dynamics - The ability of a vehicle to accelerate and decelerate comes
under this branch of vehicle dynamics. Basic governing factors are: vehicle weight, net
power available at the wheels (from engine during acceleration and from braking system
during deceleration), tractive capacity of the driving tires etc.
• Lateral Dynamics - This branch of vehicle dynamics deals with cornering behavior of a
vehicle. The way in which vehicles perform traverse to their direction of motion, particularly during cornering and swerving, describes their handling performance. According
to [8], there is a subtle difference between handling and cornering ability of a vehicle.
Master of Science Thesis
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2
Introduction
Figure 1-1: Forces acting on a vehicle. Source: [1]
Cornering refers to objective properties of the vehicle when changing direction and sustaining lateral acceleration in the process, it may be quantified by the level of lateral
acceleration that can be sustained in a stable condition. On the other hand, handling
improves the vehicle’s quality by giving feedback to the driver, affecting the ease of the
driving task or affecting driver’s ability to maintain control. Handling implies, not only
to the vehicle’s explicit capabilities but also its contributions to the performance of the
driver-vehicle combination.
• Vertical Dynamics - It is also referred as the ride quality of a vehicle. While driving
on an uneven road surface, the vehicle body moves upward and downward. According
to [8], ride is a subjective perception, normally associated with the level of comfort
experienced when traveling in a vehicle.
The focus of this study is lateral dynamics. Therefore, longitudinal and vertical dynamics
will not be treated hereafter.
1-1
Lateral Dynamics
As mentioned before, lateral dynamics involves stability and handling of vehicle during a
cornering maneuver. Lateral dynamics is governed by the lateral tire forces. Lateral tire force
(also known as side or cornering force) is the force necessary to sustain a vehicle through a
turn. It is generated by the lateral tire deformation in the contact patch as shown in Figure 12. In the Figure 1-2, Fz is the vertical load, Fy is the lateral force and Fx is the longitudinal
force.
Shelav Jain
Master of Science Thesis
1-1 Lateral Dynamics
3
Figure 1-2: Components of tire forces. Source: [1]
The steady-state force generation of the tire is highly nonlinear with respect to tire slip angle.
The slip angle of the tire α is the angle between the velocity vector Vf of the tire and the
orientation of the tire, as shown in Figure 1-3.
Figure 1-4 shows typical lateral tire force under pure lateral slip condition i.e. without any
longitudinal tire force (no braking or acceleration). For small values of slip, the force generation is approximately proportional to slip. As the slip increases the force reaches a point of
saturation after which it declines.
The slope of the curve in the linear region of force generation is denoted as the linear cornering
stiffness Cα . The Instantaneous Cornering Stiffness (ICS) Cins is defined as the slope of this
curve at current tire slip angle. For small tire slip angles it is equivalent to the linear cornering
stiffness. For higher tire slip angles it becomes smaller and becomes 0 when lateral tire force
saturation is reached and even become negative beyond that [9]. The ICS can be given as:
Cins (α) =
dFy (α)
.
dα
(1-1)
The lateral friction coefficient µy is the relationship between the lateral force Fy and vertical
load Fz . It can be given as:
µy = Fy /Fz .
(1-2)
The lateral force generation is also influenced by the longitudinal slip and vice-versa. This
effect can be seen by examining combined slip which depicts longitudinal and lateral forces for
Master of Science Thesis
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4
Introduction
Figure 1-3: Tire slip angle. Source: [2]
Figure 1-4: The lateral tire force versus slip angle.
Shelav Jain
Master of Science Thesis
1-2 Automobile Safety
5
Figure 1-5: The friction circle.
simultaneous longitudinal and lateral slip. The lateral force is maximum when no longitudinal
slip is present and with increasing longitudinal slip, either due to brake or drive slip, the lateral
force declines. This is shown with the help of Friction circle in Figure 1-5. The Friction circle
combines the longitudinal and lateral forces and it represents the limits of the tire for a given
set of operating conditions (load, temperature etc)[10].
As shown in Figure 1-5, the friction circle has lateral force Fy on the x-axis and longitudinal
force Fx on the y-axis. In general, the use of a circle is a simplification, because the force
generation in longitudinal direction is usually higher than in lateral direction. Therefore, the
enveloping curve will be elliptical.
The other important aspect is the influence of wheel load on tire forces. From the basics of
coulomb friction it is evident that the maximum friction force Ff ric is proportional to the
value of friction µ and the normal load Fz . Figure 1-6 shows lateral force Fy as a function
of tire slip angle for different normal loads Fz . For different wheel loads, the curves follow
the same pattern but for lower Fz it saturates at lower value of Fy . It can also be seen from
this figure that the value of cornering stiffness changes for different normal loads. In [8],
author represents the sensitivity of the cornering stiffness to the normal load by a second
order polynomial given as:
Cα = aFz − bFz2 ,
(1-3)
where a and b are empirically determined and depends on the tire and road conditions.
1-2
Automobile Safety
Nowadays cars have become an indispensable part of our life. We use them daily to commute
to our work and for pleasure. Since the inception of the Ford Model T in the 1900s, automotive
industry has undergone myriad change in terms of design and safety features of cars. Modern
day cars are equipped with technologically advanced safety features that can takeover the
Master of Science Thesis
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6
Introduction
Lateral Force Characteristics
12000
Lateral tire force [N]
10000
8000
6000
Fz = 1594N
Fz = 3187N
Fz = 4781N
Fz = 6374N
Fz = 7968N
Fz = 9561N
Fz = 11155N
Fz = 12749N
4000
2000
0
0
5
10
15
20
Slip angle [deg]
25
30
Figure 1-6: The lateral tire force versus slip angle at different normal loads.
control of the car from the driver in case of emergency situations. In the automotive industry,
safety features of a car are divided into two categories:
Passive Safety: It is referred to the components present in a car that helps to protect the
passengers during an accident. These primarily include airbags, seat belts and the physical
structure of the car. These safety features mitigate the effect of the crash.
Active Safety: These are the safety features present on a car that assist in preventing a
crash. These features actively control the vehicle dynamics and assist the driver in case
of an emergency situation. A variety of driver assistance systems are being developed by
automotive manufacturers to automate mundane driving operations, reduce driving burden
and reduce accidents. Examples of such driver assistance systems are:
1. Collision Avoidance System,
2. Anti-Lock Braking System (ABS),
3. Adaptive Cruise Control (ACC),
4. Traction Control System (TCS),
5. Electronic Stability Control,
6. Lane Departure Warning System,
7. Lane Keeping System,
8. Night Vision System,
Shelav Jain
Master of Science Thesis
1-3 Active control of Vehicle Dynamics
7
9. Driver Condition Monitoring System.
These technologies help in reducing driver burden and make them less likely to be involved
in accidents [11]. Vehicle stability control systems prevent a vehicle from drifting out of the
desired path [12]. Such systems are referred by different names such as yaw stability control
systems, electronic stability control systems or direct yaw moment control systems. To control
the yaw rate, which is defined as the rate of change of its heading angle, three types of systems
have been proposed widely in the literature:
1. Differential braking system that uses ABS that applies asymmetric braking between the
left and the right wheels to control yaw moment.
2. Active Steering System that modifies the driver’s steering angle input and applies a
correction to the steering angle.
3. Active torque distribution that uses active differential to control the drive torque distributed to each wheel.
The focus of this study is to understand the functioning of active safety features especially
those related to the control of lateral dynamics. Therefore, passive safety features are not
discussed hereafter.
1-3
Active control of Vehicle Dynamics
Vehicle handling behavior is not consistent in every situation mainly because of the nonlinearity in the tire characteristics. When driving in the linear range, the tire slip angle is
small and the behavior of the vehicle can be easily predicted. While driving in the non-linear
range i.e with larger slip angles, vehicle handling behavior is much harder to predict. It is
widely known from empirical studies that the vehicle becomes unstable as it is near the limit of
lateral acceleration during cornering. During cornering, acceleration increases the understeer
tendency while deceleration during cornering decreases this trend. Understeer and oversteer
are the undesired movements of a vehicle. When the vehicle goes beyond the steering input
of the driver, it is said to be oversteered. When the vehicle does not wholly follow the drivers
steering input, it is said to be understeered. This implies that acceleration or deceleration
affects vehicle behavior significantly, especially near the limit of lateral acceleration during
cornering. This is the major cause of vehicle instability [13].
Vehicle Dynamics Control (VDC) system is a closed-loop system designed to improve driveability through programmed intervention in the brake system, drivetrain, steering system,
suspension system or a combination of any of these. Longitudinal dynamics control of road
vehicles has been studied for quite some time. This led to the development of systems like
ABS and TCS. These important longitudinal dynamics control systems can be described as
given in [14]:
Anti-lock Braking system - The loss of yaw response of the vehicle to steering inputs during
full braking while the wheels are locked has lead to very early investigations to prevent wheel
lock. ABS system detects wheel lock and manipulates the brake pressure to maintain high
Master of Science Thesis
Shelav Jain
8
Introduction
Figure 1-7: Yaw stability control system for vehicle lateral dynamics. Source: [3]
level of handling performance during full braking. It prevents the wheels from locking and
the vehicle remains steerable. It was first introduced on a volume-production vehicle in 1978
[1].
Traction Control System - If the driven wheels spin with excess engine torque, handling
becomes difficult, particularly if the driven and steered wheels are identical. A TCS preserves
high level of handling performance during driving with excess engine torque. In addition to
this safety relevant task of ensuring stability and steerability of the vehicle when accelerating,
TCS also improves the traction of the vehicle by regulating the optimum slip. The upper limit
here is, of course, set by the traction requirement stipulated by the driver [1]. It regulates
the slip of driven wheels to the optimum level as soon as possible. TCS is not a braking
system but it makes use of and actively operates the braking system to prevent a wheel
from spinning. TCS is designed to prevent loss of traction of driven wheels. By extending
the TCS with additional sensors: steering angle sensor, brake pressure, yaw rate and lateral
acceleration, feedback control of the vehicle motion is possible.
The important aspect of a VDC is to control the lateral dynamics. The most prominent
approach in controlling the lateral vehicle dynamics is a yaw stability control system. These
yaw stability control systems generate the yaw moment around the vehicle’s vertical axis to
improve the lateral vehicle dynamics. As explained in [3], the elements of yaw stability control
system are as shown in Figure 1-7.
This study is focused around lateral dynamics of a road vehicle. Therefore, lateral dynamics
control systems are discussed in detail in the subsequent sections.
1-3-1
Electronic Stability Control
Skidding or lack of stability is one of the main causes of major traffic accidents and it has
motivated engineers to develop Electronic Stability Control (ESC). There is a tremendous
increase of interest in advanced safety features in automobiles. These systems have proven
to be helpful in reducing vehicle accidents. The goal of active control systems is to warn the
driver and/or induce some actions directly on the actuators.
ESC or Electronic Stability Program (ESP) is an active safety feature for motor vehicles which
aims at improving driving dynamics and at preventing crashes that result from loss of control.
Shelav Jain
Master of Science Thesis
1-3 Active control of Vehicle Dynamics
9
Figure 1-8: Sideslip angle of a vehicle. Source: [2]
Since 2006, ESC has become more common and more vehicles that are not luxury vehicles
have become equipped with ESC. ESC was first introduced as optional safety equipment in
passenger cars on the European market in 1995, and was increasingly installed in passenger
cars from 1998 [15].
ESC affects crash risk by enhancing the steerability and stability of vehicles. Crash types
that are typically associated with ESC are crashes that are caused by high speed cornering
maneuvers, collision avoidance maneuvers, low friction conditions etc. The types of crashes
that are typically affected by ESC are often more serious than other crashes. The results
indicate that ESC prevents about 40% of all crashes involving loss of control [15]. The
greatest reductions were found for rollover crashes (50%), followed by run-off-road (40%) and
single vehicle crashes (25%) [15].
Traditionally, understanding the steering response of a car in different operating conditions is
a task of the driver. The driver should be able to judge the physical limits of various handling
maneuvers. However, most drivers are not able to detect these limits until the vehicle reaches
its physical handling limits. To improve the handling performance, depending on the steering
wheel angle the yaw moment on the car needs to be controlled. ESC limits the slip angle β
of the vehicle in order to prevent vehicle spin. The sideslip angle β of a vehicle is the angle
between its velocity vector at the Centre of Gravity (COG) and with the longitudinal axis of
the vehicle, as shown in Figure 1-8.
There exists a large variety of ESC systems. They have in common that they enhance the
controllability of vehicles and can prevent skidding and loss of control in cases of oversteering
or understeering. ESC systems differ with respect to how they regulate driving parameters
(yaw rate or sideslip) and how they counteract deviations (e.g. by braking individual wheels
and reducing engine power) [16].
Master of Science Thesis
Shelav Jain
10
Introduction
Figure 1-9: ESP control loop in a vehicle. Source: [1]
Figure 1-9 shows the architecture of ESP control provided by BOSCH. It shows the sensors
required to determine the controller input parameters. The sensors used are: (1) yaw rate
sensor and lateral acceleration sensor, (2) steering wheel angle sensor, (3) brake-pressure
sensor and (4) wheel-speed sensor. The figure also shows: (5) ESP control unit, featuring a
high-level vehicle dynamics controller and the low-level slip controller. The actuators used to
fulfill the controller’s demand are also shown. The choice of actuator depends on the type of
lateral dynamics control system used. The possible actuating principles are: (7) brakes (9)
fuel injection system (Diesel Powertrain) (10) sprak plug and (11) throttle valve. Apart from
these actuators, other possible actuators are active front and rear steering, active differential
and active suspension.
In ESC with asymmetric braking, braking is activated on individual wheels in a targeted
manner, such as, on the inner rear wheel to counter understeer, or on the outer front wheel
during oversteer. It helps to keep the vehicle’s course stable under all driving conditions.
ESC can also accelerate the driven wheels by specific engine-control interventions to ensure
the stability of the vehicle.
As steering and braking actuators are part of the vehicle chassis, the active control of yaw
stability control system can be achieved through active chassis control. The active chassis
control using active braking or active differential as actuator is known as Direct Yaw Moment
Control (DYC) and using steering is called as active steering control as shown in Figure 1-10.
In active steering control, the control strategy can be implemented either using Active Front
Steering (AFS) or Active Rear Steering (ARS) or Four-Wheel Active Steering (4WAS).
Selection of the stability control system depends on the control objective and on the driving
situation. In small lateral acceleration conditions, vehicle sideslip angle is small and tires
are in linear operating region. Thus, good handling performance can be obtained by the
Shelav Jain
Master of Science Thesis
1-3 Active control of Vehicle Dynamics
11
Figure 1-10: Active chassis control. Source: [3]
active steering system alone. In emergency cases however, when the lateral acceleration and
sideslip angle are significant, stability can not be guaranteed only by steering control, in this
case direct yaw moment control by differential braking can enhance the performance of the
vehicle.
Most of the stability control systems rely on the yaw stability controller to force the vehicle
to follow a desired path. These controllers give an output to the specific actuator to generate
the required yaw moment. The yaw moment can be described as the moment around the
vehicle’s vertical axis. This is caused by different longitudinal forces acting on the left and
right sides of the vehicle and different lateral forces acting at the front and rear axles. Yaw
moments are required to turn the vehicle when cornering.
However, it is well known that at large vehicle sideslip angles, changing the steering angle
produces very little change in the yaw rate of the vehicle. In designing a good stability
controller, yaw-rate control and sideslip control should be considered together. As mentioned
in [17], it is necessary to control the sideslip angle along with the yaw motion in order to
maintain the stability of a vehicle. The instability of vehicle at its limit is brought about by
decreased control of the yaw moment at a large sideslip angle.
Sensitivity of the yaw moment on the vehicle w.r.t changes in steering angle decreases rapidly
as the sideslip angle of the vehicle increases, as shown in Figure 1-11 [13]. From this curved line
it can be seen that, when the sideslip angle is small, the yaw moment required is proportional
to the slip angle. But for the larger sideslip angle, yaw moment saturates and starts to decrease
at a certain point. This means that at the larger slip angles, the required yaw moment can not
be generated using steering as an actuator for active chassis control. Therefore, to improve
the vehicle handling and stability performance, it is essential to control both yaw rate and
sideslip responses [18].
Clearly in lateral vehicle dynamics control both yaw rate and sideslip angle play an important
role. As explained in [19], control objective of yaw stability control system may be classified
into three categories; yaw rate control, sideslip control and the combination of yaw rate and
sideslip control as given in Figure 1-12.
Master of Science Thesis
Shelav Jain
12
Introduction
Figure 1-11: Yaw moment as function of sideslip angle for different steering angles. Source: [3]
Figure 1-12: Objective of yaw stability control system. Source: [3]
Shelav Jain
Master of Science Thesis
1-3 Active control of Vehicle Dynamics
13
Figure 1-13: Block diagram for integrated control objective. Source: [4]
The objective of yaw rate control system is to control the actual yaw rate close to the desired
yaw rate which is generated by the reference model. This will improve the handling or
maneuverability of the vehicle. In the steady state condition, the desired yaw rate response
γd can be obtained by [3]:
γd =
Vx
δf ,
(lf + lr ) + kus Vx 2
(1-4)
where Vx is the longitudinal velocity, lf and lr are the distance from COG to the front and
rear axle respectively, δf is the steering angle of the front wheels and kus is the stability factor
also known as the understeer gradient and is defined as:
kus =
m(lr Cr − lf Cf )
,
(lf + lr )Cf Cr
(1-5)
where m is the vehicle mass, Cf and Cr are the front and rear tire cornering stiffnesses.
From equation (1-5) the term (lr Cr − lf Cf ) plays an important part in cross coupling of
sideslip and yaw motions. This term is the yaw moment caused by unit tire slip angle. When
this value is positive, side slip motion yields a restoring yaw moment and reduces the slip
angle, but when it is negative the yaw moment results in an increase in the slip angle. The
former is termed as understeer and the later is oversteer [20]. The sideslip angle based lateral
dynamics control systems control the lateral stability of the system by keeping the desired
sideslip βd as zero in steady state condition.
To control these two control objectives effectively, two different control mechanisms are required. The block diagram for such a control system is shown in Figure 1-13. The vehicle
yaw moment controller gives the desired wheel slip λd as the output and wheel slip controller
applies the required braking torque on the wheels to generate the desired yaw rate and sideslip
angle.
Any chassis control system that aims to improve the vehicle handling performance must rely
on the tire lateral force and longitudinal force. These two forces rely upon the vertical load
and are interdependent. One of the major advantages of direct yaw moment control method
Master of Science Thesis
Shelav Jain
14
Introduction
is that the tire longitudinal force has no feedback from the vehicle lateral motion as long as
it is within the limit of the tire capacity with respect to the vertical load [20].
As explained earlier, the direct yaw moment control is one of the most effective methods of
active chassis control. It must be noted that direct yaw moment control can become highly
important and crucial for Electric Vehicle (EV) when compared with the Internal Combustion
Engine (ICE) type vehicles. In an EV with in-wheeled electric motors, it is easier to control
the yaw moment.
Most of the yaw moment based control systems applied to improve the vehicle lateral dynamics
are model-based. This means that in order to decide the control output the mathematical
model of the vehicle is used. Most of the yaw stability control system architectures use linear
vehicle models to simplify the control law. But, in the real world environment, the roadvehicle dynamics is highly nonlinear and uncertain. The main problem of yaw rate and sideslip
tracking control systems are uncertainties caused from variations of dynamic parameters such
as road surface adhesion coefficients, tire cornering stiffness, vehicle mass, vehicle speed and
moment of inertia. Therefore, the control system designed should be robust to overcome the
effect of these uncertainties, as given in [21], or these dynamic parameters should be available
for real time model update either by the use of suitable sensors or estimation algorithms.
As shown in [20], a robust control system is designed for lateral vehicle control to compensate
for change in the dynamic parameters. It is useful when the change in tire cornering stiffness is
not significant. It is conceivable that there may be a running condition in which robust control
theory cannot be sufficient to absorb conditions encountered when the tire cornering stiffness
is drastically reduced. In particular, when the tire force exceeds the maximum frictional force
and the cornering stiffness indicates a negative value, it is conceivable that the control system
will aggravate the instability. In addition the controller may be conservative.
In designing any lateral dynamics control law, determination of lateral force saturation or
nonlinear operating region of tires is very important. The lateral force saturation can be
because of heavy steering command and/or low friction road. The lateral tire force saturation
can be detected by estimation of ICS, as shown in Figure 1-4.
In [22], [23], [24], [25], authors have proposed a yaw moment control system for EV based on
adaptive DYC with cornering stiffness estimation. The adaptive DYC can be explained using
the following equations. The yaw motion of an EV can be given as:
I
dγ
= Nz − Nt − Nd ,
dt
(1-6)
where I is the vehicle inertia, γ is the yaw rate, Nz is the control yaw moment, Nt is the yaw
moment generated by tire and road contact and Nd is the disturbance yaw moment. For the
linear vehicle model and linear tire model the yaw moment Nt can be written as:
Nt = 2Cr
lf
lr
γ − β lr − 2Cf δf −
γ − β lf .
Vx
Vx
(1-7)
This yaw-moment can be estimated by the yaw moment observer. Then the cornering stiffnesses can be identified using a recursive least square algorithm. Alternatively, if real-time
cornering stiffnesses values are known from some other estimation process then the adaptive
DYC can be updated directly.
Shelav Jain
Master of Science Thesis
1-4 Application of State Estimation
15
Figure 1-14: The principle layout of a control loop using a state estimator.
1-4
Application of State Estimation
In most feedback control systems in the automotive field, the control action depends on some
important variables like vehicle lateral velocity, sideslip angle, tire-road forces etc, which are
often not measured directly because of technical and/or economic reasons. Therefore, it is
required to use estimators or virtual sensors for estimation of these important variables.
In order to implement any of the active control strategies discussed above, the vehicle’s lateral
dynamics states have to be estimated from the available sensor signals. Effective operation of
each of these systems depends on an accurate knowledge of the vehicle states, such as velocity,
lateral acceleration, yaw rate, as well as vehicle and tire side slip. To accurately estimate the
state values, mathematical model of the physical system used in the estimation algorithm
must be as precise as possible [26].
Estimation means the extraction of information of any physical variable not available directly
from sensors, by using available information. An observer is an algorithm that computes at
every instant the values of variable of interest, not directly measured. The objective of the
observer is to estimate sequentially the state of the vehicle dynamic system using a sequence
of noisy available measurements made on the system. For dynamic state estimation, the
discrete time approach is convenient for real-time application using on-board systems.
The idea of a state estimator is to implement a model of the real system in an on-board
computer in parallel with the system itself, as shown in Figure 1-14. The model of the real
system can be a full nonlinear model or a simplified linearlized model. For a linear model,
dynamics of the plant are represented by a state-space description where matrix A is the state
matrix, B and G are input vectors for the controller input u and plant noise w respectively,
and matrices C and D form the output y. The discrete time representation of A and B
are denoted by φ and Γ. In Figure 1-14, a Kalman filter is used as the stochastic state
estimator. It is a set of mathematical equations that provides an effective computational
Master of Science Thesis
Shelav Jain
16
Introduction
means to estimate the state of a process, in a way that minimizes the mean of the squared
error [27]. The prediction and correction steps of Kalman filter state estimation algorithm
are discussed in detail in Chapter 3.
The plant to be observed in Figure 1-14 is excited by noise w that is characterized by stochastic
quantities and that the sensors used are also corrupted by stochastic noise v as well. Essentially, the state estimator is driven by the same inputs as the plant with exception of the
process noise. The principle of estimating the system states x is based on the comparison of
measured outputs y and estimated outputs yˆ. With a good state estimator, the difference e
which is fed back into the Kalman filter and the estimated states x
ˆ will follow the plant states
x. A necessary condition for estimating the state variables is that the plant is observable.
This means that all major plant dynamics affect output y.
For a more accurate representation of system dynamics, a complex nonlinear model is required.
The Kalman filter explained above needs to be extended to accommodate the nonlinearities of
the model. Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) are the most
commonly used state estimators for nonlinear systems. Although, both are computationally
expensive, UKF is more accurate than EKF. Their differences are treated in detail in Chapter
3.
1-4-1
Estimation of Sideslip Angle
As discussed extensively in the previous section, tracking of sideslip angle is also required
along with tracking of yaw rate for satisfactory lateral dynamics response. But, measurement
of sideslip angle is not possible commercially. To date, no vehicle sideslip angle sensor is
available that is accurate or economical enough to be implemented on the current vehicles.
The estimation of sideslip angle has been a widely studied topic. Several strategies were
proposed, mainly based on state observers; the procedures rely on tire models and evaluation
of its parameters. These approaches can lead to good estimation if the tire parameters are
correctly identified, but if changes occur in tires’ cornering stiffness due to different friction
conditions or to the tire-wear, the estimation might be affected by errors. Most of the estimation techniques employs lateral acceleration sensor to obtain the sideslip angle. This
method is not very accurate because road bank angle and roll angle of the vehicle acts as a
bias element on the real lateral acceleration.
The conventional estimation methods of sideslip angle are based on model-based observer design and direct sensor integration [2]. The model based observer method has higher accuracy
in the linear tire region and it is robust against sensor bias. But, the estimation depends
on vehicle parameter like vehicle mass, inertia and tire parameters like cornering stiffness.
It is difficult to identify these parameters in real-time, therefore a model-based estimation
algorithm cannot provide reliable results over all driving situations. The direct sensor integration method is a kinematic based approach, a differential relation between the sideslip
angle and vehicle’s measurable dynamic parameters can be easily obtained. Since, the relation
is differential, its application leads to a progressive drift during the integration process.
In [28], [2], authors have proposed a new methodology by combining a vehicle model based
method and a kinematics-based method. The kinematic based model provides reliable results
in transient condition while vehicle model based approach is used to obtain sideslip angle in
Shelav Jain
Master of Science Thesis
1-4 Application of State Estimation
17
steady state condition. The experimental results indicate that the algorithm produces robust
estimation of sideslip angle.
In [29], [30], authors have proposed a method to estimate lateral velocity using lateral tire
force sensor for EV. But, authors have assumed that the tire cornering stiffness for left and
right tire is equal, this assumption is applicable only for EV because of lower position of
COG, hence less weight transfer. So, this method cannot be extended to ICE vehicles. In
[31], authors have presented a vehicle nonlinear model based estimation of sideslip angle using
EKF. They have used the Dugoff tire model, which incorporates tire-road friction and tire
cornering stiffness. But, in estimating sideslip angle they have used constant tire cornering
stiffness. Therefore, applicability of the results cannot be verified.
In [32], authors have proposed an adaptive approach (considering tire-road friction adaptation) for real-time estimation of sideslip angle. They have estimated the tire cornering
stiffnesses and further used them in estimating the sideslip angle. But, the proposed algorithm does not give accurate results for the vehicle running on slippery roads e.g. snowy or
icy roads. Authors in [33], have proposed a kinematic model based approach of estimation
lateral velocity. The model relates longitudinal velocity, lateral velocity, longitudinal acceleration and yaw rate. The observer based on this method is not sensitive to change in vehicle
parameters, but it produces noisy estimates. In addition, the estimates drift when yaw rate
is zero because the observer becomes unobservable.
In [34], authors have proposed an UKF based sideslip angle estimation algorithm. Since,
tire-road friction plays an important role in estimation of vehicle states. This algorithm also
estimates tire-road friction parameter as an augmented state. This estimation method using
nonlinear models shows the practical potential for calculating lateral tire forces and sideslip
angle. In [35], two block strategy to estimate tire-road forces, sideslip angle and cornering
stiffness is given. Sliding mode observer is used to estimate tire-road forces and then these
forces are used in EKF algorithm to estimate the sideslip angle and cornering stiffness. The
proposed observer gives reliable results for lateral acceleration less than 0.6g. However, for
lateral acceleration greater than 0.6g, results are not sufficiently accurate. This method is
complicated to apply in the field vehicle.
1-4-2
Estimation of Tire Cornering Stiffness
As discussed in earlier sections, a tire cornering stiffness is an important dynamic parameter. It plays an important in designing an ESC system, estimation of vehicle states and
determination of lateral tire force saturation.
In the determination of control law to enhance the handling of road vehicles, most of the
VDC systems use constant cornering stiffness as input to the system. But, in real working
situations, cornering stiffness vary due to change in tire-road friction and tire wear. Therefore,
it is important to obtain these dynamic parameters for robust working of ESC systems.
In order to minimize the effect of change in tire cornering stiffnesses, several researchers have
proposed estimation for cornering stiffness. But to estimate the stiffnesses, present values of
either tire-road friction or vehicle sideslip angle are required. Authors in [36] have proposed a
recursive parameter estimation to estimate the cornering stiffness, but the proposed method
depends on the measurement of sideslip angle and yaw rate. However, measurement of sideslip
angle using sensors is difficult and expensive.
Master of Science Thesis
Shelav Jain
18
Introduction
In [37], authors have presented a comprehensive summary of cornering stiffness estimation
based on one-track vehicle model. Estimation methods can be divided into to categories: timedomain methods and transfer-function methods. The time-domain methods use the dynamic
equations of the vehicle motion, and the underlying equations are correct even when the
vehicle state is time-varying. On the other hand, the transfer-function approach is accurate
only when the vehicle is time-invariant, and the effects of transient dynamics are ignored.
The time-domain approach presented, requires real-time information of vehicle sideslip angle
to estimate the cornering stiffness. They have also proposed a ‘beta-less’ method to estimate
the front and rear tire cornering stiffness, but the system is under-determinate. To solve the
under-determinate issue, information from multiple sensing points needs to be used and the
information has to persistently exciting for the estimation to converge to the true values.
Another way to solve the under-determinate issue is to assume the ratio of front and rear
cornering stiffness as constant.
The assumption of constant ratio of stiffnesses seems promising for real-time implementation,
as it resolves the under-determinate issue. But, after various simulations it has been confirmed
that in actual application of the vehicle, this ratio is not constant. Hence, the estimation of
parameters diverge from its true values.
According to [38], assumption of fixed front and rear cornering stiffness ratio (as described
in [37]) has no strong foundation, especially when road condition changes. So, they proposed
a new Beta-less estimation of cornering stiffness. But, this approach does not give accurate
results for higher lateral acceleration because of the assumption that the left and right tire
cornering stiffness are almost the same, as explained earlier the value of cornering stiffness
depends on vertical load Fz on the tire. This assumption is valid only for lateral acceleration
below 0.2g and for EV because of their low COG, hence less weight transfer.
In [39], a transfer function approach to estimate the cornering stiffness has been given. From
the bicycle model, the transfer function from steering angle to yaw rate was obtained and
using parameter estimation approach unknown process parameters were calculated. The
equation was numerically solved using Newton-Raphson method, which is cumbersome and
time consuming and therefore not suitable for real time application.
Recently, with the availability of cheap Global Positioning System (GPS) sensors, many
researchers have shifted their focus to estimation techniques using them. In one such approach
by authors in [40], they have used vehicle heading information from GPS along with yaw rate
to obtain reliable estimate of the sideslip angle. Then they have used the estimated sideslip
angle to obtain the tire cornering stiffnesses. The estimated tire cornering stiffness can then
be used in the model based estimator in order to provide a more accurate model of the vehicle.
In one of the research works at DLR, German Aerospace Research Center, researchers have
demonstrated the application of tire cornering stiffness to active car steering. In [9], researchers have first estimated the tire cornering stiffness and then incorporated it in designing
the control law for active steering. Basic function of cornering stiffness is to detect the saturation of lateral force, so that the controller can stabilize it. But, the estimation of cornering
stiffness requires derivatives of steering angle and lateral acceleration. These measured signals
are noisy and requires good filtering techniques. The cornering stiffness estimation is given
as:
Shelav Jain
Master of Science Thesis
1-5 Thesis Objective
19
Cf = m
a˙ y,f
lr
,
lf + lr δ˙f − ay + γf
Vx
(1-8)
where m is the mass of the front axle (according to the vehicle weight distribution), lf and lr
are the distance of front and rear axle from COG respectively, Vx is the longitudinal velocity,
ay,f is the lateral acceleration of front axle and γf is the yaw rate of front axle.
In [41], authors have proposed a betaless estimation of ICS using lateral tire force measurements. This estimation is only available if the tire slip angle is profound enough and it is
not available during constant cornering. Moreover, the proposed scheme requires derivative
of measured forces to calculate the ICS. Special signal processing techniques are required to
overcome the inaccuracy due to the sensor noise.
1-5
Thesis Objective
Significant amount of research has been conducted in the field of state estimation but future
active safety systems need more accurate information about the state of a vehicle. The main
disadvantage of most of the vehicle state estimation approaches is the requirement of prior
knowledge of tire parameters such as cornering stiffness and friction coefficient. However, the
tire models can be used to estimate these parameters but it makes the system too complex
for real-time application [42], [43].
Neglecting the aerodynamic effects, the motion of a vehicle is governed by the tire-road forces
in each direction [5]. The load sensing bearing technology from SKF [44] and NSK [30], [45]
presents an alternative approach towards vehicle state and parameter estimation. Control of
vehicle dynamics using these sensors is a new topic of research. The application of tire force
sensors has been proven successful for lateral tire force control of EV in [29], [45]. Another
group of researchers [44], [46], [47], have worked on the application of load sensing in the
area of hybrid ABS control. They have proven that the use of load sensing information along
with wheel acceleration measurement makes the algorithm simpler and robust. Along with
longitudinal dynamics control, lateral vehicle dynamics control using tire forces have also
been studied in [48], [49].
The measured tire forces give an operating state of each tire and they are the most important
variables as they are the only point of interaction of a vehicle with the road. However, most
of the approaches discussed in the literature to estimate vehicle lateral dynamics states, use
lateral acceleration measurement as one of the important input signals to the estimator. But
the roll angle φ and road bank angle λ introduce an offset in the lateral acceleration measurement due to the gravity component. In [42], the author gives an expression to compensate
the measured lateral acceleration for the road bank angle and roll angle:
ay,global =
ay,measured − g sin(φ − λ)
.
cos(φ − λ)
(1-9)
where ay,global is the actual lateral acceleration, ay,measured is the measured lateral acceleration
and g is the acceleration due to gravity.
Master of Science Thesis
Shelav Jain
20
Introduction
The use of measured tire forces for estimation eliminates the computationally expensive algorithms, which are otherwise necessary to estimate the road bank angle and roll angle. As
mentioned earlier, sideslip control along with yaw rate control is required for satisfactory
steerability and stability of a vehicle. But, measuring vehicle sideslip requires expensive sensors. Therefore, the objective of this study is to estimate the vehicle lateral dynamics states
based on measured tire forces.
The UKF algorithm is used to estimate the vehicle sideslip angle and tire cornering stiffnesses.
The vehicle model used to evaluate the performance of the estimator is a multibody mechanical
system for a four-wheeled vehicle with 15 mechanical Degree of Freedom (DOF) from CarSim
[50]. CarSim is a vehicle dynamics simulation software developed by Mechanical Simulation
Corporation in Ann Arbor, USA. It is a parametric modeling software widely used in research
and industry to simulate and analyze vehicle dynamic behavior.
1-6
Thesis Outline
This section is intended to give a brief overview of the contents of this thesis report. Chapter
1 has provided a detailed description of the VDC system. The main focus of this chapter was
to understand the concept of Active Chassis Control. It further discussed the importance of
sideslip angle tracking, which is the basis of this work. A brief description of previous work
related to sideslip angle and cornering stiffness estimation was also given.
In Chapter 2, physical modeling of the vehicle considered for the project is explained. In
addition to that it gives an overview of the important tire models. Chapter 3 is devoted to
the theory of estimation algorithms. It gives a detailed description starting from most basic
observers to the nonlinear estimation algorithms. The two extensions namely, EKF and UKF
are explained.
Chapter 4 and Chapter 5 discuss the results of this study for different vehicle maneuvers.
These chapters describe the estimation process of sideslip angle and cornering stiffness for the
bicycle model and Four-Wheel vehicle model (FWVM). The estimation algorithm, vehicle
model, observability analysis and observer error analysis constitute this chapter. The tuning
parameters for the UKF algorithm are also given.
Finally, Chapter 6 contains the conclusions and future recommendations.
Shelav Jain
Master of Science Thesis
Chapter 2
Physical Modeling
The estimator use physical model of the system to estimate the system’s states. With respect
to the automotive systems, accuracy of the estimation process mainly depends on the modeling
accuracy of the two primary subsystems i.e. vehicle model and tire model. Based on the level
of complexity, various vehicle and tire models are available in the literature. The degree of
complexity of the model also depends on the desired objective. For example, in simulator
design, it is necessary to reproduce the behavior of each individual component. But, in the
real time application, a compromise has to made due to the limited calculation capacity.
2-1
Vehicle Model
The vehicle model should represent all dynamics of interest as simply as possible. In this
analysis, the vehicle is modeled as a rigid body and is studied in a body-fixed coordinate
system with the origin located at the Centre of Gravity (COG). Complete dynamic behavior
of the vehicle can be described by three individual model or combination of them, namely,
1. Vertical dynamics model- This model is used for analyzing ride of the vehicle and it
incorporates the suspension dynamics.
2. Planar dynamics model- This model is also known as yaw plane model. It is used to
understand the lateral and longitudinal dynamics of the vehicle, which is the topic of
interest for this work. There are two commonly used vehicle models:
(a) Four-Wheel vehicle model (FWVM) or 2 track model.
(b) Single track model or Bicycle model.
3. Roll Plane model- It is used to describe the roll dynamics.
The focus of this work is to understand the lateral dynamics, which can be accurately modeled
by yaw plane model. Both the planar dynamics model i.e. the 2-track model and bicycle model
are treated in detail in the following sections.
Master of Science Thesis
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22
Physical Modeling
Figure 2-1: Four-wheel vehicle model.
2-1-1
Four-wheel vehicle model
It is commonly called as two track model and is widely used to analyze and control the
longitudinal and transverse vehicle dynamics. Figure 2-1 shows the schematic of a 3 Degree
of Freedom (DOF) vehicle model that represents the lateral, longitudinal and yaw motions.
The front and rear track widths E are assumed to be equal. The distances from the vehicle’s
COG to the front and rear axles are lf and lr respectively. Here δij is the steering angle of
the ij tire, Fyij is the lateral tire force of ij tire and Fxij is the longitudinal tire force of ij
tire.
The side-slip at the vehicle’s COG (β) is the angle between the velocity vector (V ) and the
˙ is the angular velocity about the vertical axis
true heading of the vehicle. The yaw rate (ψ)
with Iz being the yaw moment of inertia. The longitudinal and lateral velocities are Vx and
Vy , respectively. The longitudinal and lateral forces (Fx,y,i,j ) acting during the movement,
are shown for front and rear tires of the vehicles.
Describing the inter-relationship between different vehicle dynamics parameters using Newton’s second law gives:
Shelav Jain
Master of Science Thesis
2-1 Vehicle Model
23
lf [FyF L cos δF L + FyF R cos δF R + FxF L sin δF L + FxF R sin δF R ]
ψ¨ =
Iz
lr [FyRL cos δRL + FyRR cos δRR + FxRL sin δRL + FxRR sin δRR ]
−
Iz
E
[FyF L sin δF L − FyF R sin δF R + FxF R sin δF R − FxF L cos δF L
+
2Iz
−FxRL cos δRL + FyRL sin δRL + FxRR cos δRR − FyRR sin δRR ],
ay =
ax =
(2-1)
1
[FyF L cos δF L + FyF R cos δF R + FxF L sin δF L + FxF R sin δF R + FyRL cos δRL
mv
+FyRR cos δRR + FxRL sin δRL + FxRR sin δRR ], (2-2)
1
[−FyF L sin δF L − FyF R sin δF R + FxF L cos δF L + FxF R cos δF R + FxRL cos δRL
mv
+FxRR cos δRR − FyRL sin δRL − FyRR sin δRR ], (2-3)
V˙x = Vy ψ˙ + ax ,
(2-4)
V˙y = −Vx ψ˙ + ay ,
(2-5)
where mv is mass of the vehicle, ax and ay are the longitudinal and lateral acceleration
respectively. The sideslip angle can be calculated from lateral and longitudinal velocities:
!
Vy
.
Vx
β = arctan
(2-6)
The longitudinal and lateral velocities, the steer angle of the front wheels and the yaw rate
are then used as a basis for the calculation of the tire slip angles αij :
αF L
Vy + lf ψ˙
= δF L − arctan
,
˙
Vx − E ψ/2
(2-7)
αF R
Vy + lf ψ˙
= δF R − arctan
,
˙
Vx + E ψ/2
(2-8)
αRL
Vy − lr ψ˙
= δRL − arctan
,
˙
Vx − E ψ/2
(2-9)
αRR
Vy − lr ψ˙
.
= δRR − arctan
˙
Vx + E ψ/2
(2-10)
"
"
"
"
Master of Science Thesis
#
#
#
#
Shelav Jain
24
Physical Modeling
Figure 2-2: The bicycle model.
2-1-2
The Bicycle Model
Vehicle model can be considerably simplified by using a classical bicycle model as shown in
Figure 2-2. It is also known as the single track model. It is mostly used to describe the
lateral dynamic behavior, especially for evaluation of sideslip angle [5]. It is simplified from
the nonlinear FWVM based on the following assumptions:
1. Tire forces operate in the linear region.
2. The two left and right front wheels are represented by one single wheel. Similarly, the
rear wheels are represented by one central rear wheel.
3. Longitudinal speed is constant i.e. the longitudinal acceleration is zero.
4. In this model, vertical movements are ignored, roll motion is not taken into account.
5. Both front wheels have the same steering angle and also the rear wheels.
6. No braking is applied at all wheels.
The simplified bicycle model is formulated by the following relationship:
1
ψ¨ = [lf [Fxf sin δf + Fyf cos δf ] − lr [Fxr sin δr + Fyr cos δr ]],
Iz
β˙ =
1
˙
[Fyf cos δf + Fxf sin δf + Fyr cos δr + Fxr sin δr ] − ψ,
mv Vx
(2-11)
(2-12)
where,
• ψ˙ is the yaw rate.
• β is the vehicle sideslip angle.
• Iz is the yaw moment of inertia.
• mv is the mass of the vehicle.
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Master of Science Thesis
2-2 Tire Dynamics
25
Figure 2-3: Tire dynamics variables. Source: [5]
• lf is the distance between front axle and COG.
• lr is the distance between rear axle and COG.
• Fyf and Fyr are the front and rear lateral forces respectively.
• Fxf and Fxr are the front and rear longitudinal forces respectively.
• Vx is the longitudinal velocity of the vehicle.
• δf and δr are the front and rear steering angles respectively.
Assuming small angles, front (αf ) and rear (αr ) tire slip angles are calculated using kinematic
relations with respect to the vehicle’s speed and yaw rate:
2-2
αf = δf − β − lf
ψ˙
,
Vx
(2-13)
αr = δr − β + lr
ψ˙
.
Vx
(2-14)
Tire Dynamics
Tires are the main vehicle components generating external forces that can be effectively
manipulated to affect vehicle motions. The developed forces are the function of tire properties
Master of Science Thesis
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26
Physical Modeling
Figure 2-4: Four categories of possible types of approach to develop a tire model. Source: [6]
(material, tread pattern, tread depth, profile, etc.), the normal load on the tire, road friction
and the velocities experienced by the tire.
Tires deform due to the vertical load, and makes contact with the road surface over a nonzero footprint area called the contact patch. The force that a tire receives from the road is
assumed to be at the center of the contact patch and can be decomposed along the three
wheel axes. The lateral force, Fy , is the force along the Y axis, the longitudinal force, Fx , is
the force along the X axis and the normal or vertical force, Fz , is the force along the Z axis.
Figure 2-3 shows the tire dynamics variables. In Figure 2-3, the moment along the Z axis,
Mz , is the aligning moment, moment along the X axis Mx , is the overturning moment and
the moment along the Y axis, My , is the rolling moment.
All the vehicle states depend on the accurate measurement of vehicle accelerations, which
in turn depends on the accurate knowledge of tire forces, as can be seen in equations (2-2)
and (2-3). Until very recently, the actual tire forces cannot be measured, they need to be
calculated based on the estimated and measured states.
Several types of tire models have been developed during the past half century; each type
for a specific purpose. Different levels of accuracy and complexity may be introduced in the
various categories of utilization. This often involves entirely different approaches. Figure 2-4
illustrates the effect of different modeling approaches i.e. from empirical to theoretical model.
Shelav Jain
Master of Science Thesis
2-2 Tire Dynamics
27
From left to right in Figure 2-4 the model is less experimental and more theoretical. In
the middle, the model will be simpler but possibly less accurate while at the far right the
description becomes complex and less suitable for application in the simulation of vehicle
motions and may be more appropriate for the analysis of detailed tire performance in relation
to its construction [6].
The focus of this study is to understand the lateral dynamics of a vehicle. So the lateral force
models used to understand this important characteristic are discussed further. A vehicle can
turn because of the applied lateral tire forces. While negotiating a turn, lateral force originates
at the center of the contact patch which lies in the horizontal plane and is perpendicular to
the direction in which the wheel is headed if no camber exists. To transmit these lateral
forces, the tire must turn laterally. In turn, the direction of motion of the tire deviates from
the wheel plane.
The purpose of a tire model is to obtain a structure in which the measurement data can
be fitted with the use of suitable parameters. From the point of simplicity and real-time
implementation of this model-based estimation technique, linear tire model has been used in
this study. But, for the sake of completeness most commonly used nonlinear tire model is
also briefed in this work, for detailed description interested readers may follow [6].
2-2-1
Magic Formula
According to Pacejka [6], the tire model should be:
• able to describe all steady-state tire characteristics,
• easily obtainable from measured data,
• physically meaningful; its parameters should characterize in some way the typifying
quantities of the tire,
• compact and easy to use,
• able to contribute to better understanding of tire behavior.
The basic formula for this model is [6]:
y = Dsin[Carctan(Bx − E(Bx − arctanBx))],
(2-15)
with Y (x) = y(x) + SV and x = X + SH .
In these formulas, Y is the output variable, which stands for longitudinal force Fx or lateral
force Fy or aligning moment Mz . X is the input variable, which stands for lateral slip angle
αy or longitudinal slip αx . The parameters B, C, D, E, SV and SH of these formulas as shown
in Figure 2-5 are defined as follows:
• D: the peak value.
• C: the shape factor that controls the limits of the range of the sine function and thereby
determines the slope of the resulting curve.
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Physical Modeling
Figure 2-5: The Magic Formula parameters. Source: [6]
• B: the stiffness factor. This factor determines the slope at the origin and is also called
the stiffness factor.
• E: the curvature factor; it controls the value of the slip at which the peak of the curve
occurs.
• BCD: this product corresponds to the slope at the origin (x = y = 0). For lateral
force, this factor corresponds to the cornering stiffness.
The Magic Formula typically produces a curve that passes through the origin. To allow the
curve to have an offset with respect to the origin, shifts SH and SV have been introduced.
2-2-2
Linear model
In normal driving situations, the tires are well under their saturation limit and have small
tire slip angles. For these conditions, the lateral tire force is a linear function of the slip angle
with the slope equal to the linear cornering stiffness. Lateral forces of the front and rear tires
(Bicycle model) can be written using front equation (2-13) and rear slip angle equation (2-14)
as:
ψ˙
δf − β − lf
,
Vx
(2-16)
ψ˙
δr − β + lr
.
Vx
(2-17)
Fyf = Cαf αf = Cαf
Fyr = Cαr αr = Cαr
Shelav Jain
Master of Science Thesis
2-3 Summary
2-2-3
29
Transient tire model
As tires have a complex structure, tire forces are not developed instantaneously at the maneuvering action. Owing to the flexible structure of the tire, generation of forces require some
rolling distance. In turn the force and moment response is delayed in response to the external
input namely steering angle. As explained in [11], the dynamic lateral tire force model can
be given as:
τ F˙y + Fy = F¯y ,
(2-18)
where τ is the relaxation time constant, Fy is the dynamic lateral force and F¯y is the static
lateral force calculated from one of the tire models either discussed earlier or from the literature. The relaxation length behavior of tires is not analyzed in this study but interested
readers may follow [51], [52], [53].
2-3
Summary
This chapter explained the physical models of a vehicle and tire. Different types of vehicle
models are described and classified according to the complexity of implementation and accuracy of the results. The FWVM is more accurate than the bicycle model as it captures
the load transfer from left to right wheels or vice versa during cornering maneuvers. But,
implementation of the FWVM is difficult.
Next, tire dynamics is described and different models are studied. Static tire models like the
Magic tire formula and a linear model are presented. The transient model on the basis of
relaxation tire behavior is briefly touched upon.
The dynamic modeling of the vehicle-tire system forms the backbone of the estimation problem. The estimation algorithms which use these physical models are be explained in the next
chapter.
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30
Shelav Jain
Physical Modeling
Master of Science Thesis
Chapter 3
State Estimation and Filtering
In this chapter, the state estimators used in this study are discussed. First, a short introduction is given to modeling of a general system, then its observability conditions are discussed
and then the estimator algorithm is explained.
In many dynamic mechanical systems, it is often impractical to assume that all states describing the system’s response can be measured. Sometimes, they cannot be measured by physical
means and sometimes it is not economical. Also, resources can be saved when the states may
be estimated using information from other signals and physical models. Whenever the state
of a system is to be estimated from noisy sensor information, some kind of state observer
is implemented to combine the information from different sensors to produce an accurate
estimate of the state.
State reconstruction is a two-step procedure: in the first step a physical model is identified
and then an estimator is designed using suitable observation techniques. Vehicle and tire
model used in this study are already discussed in the chapter 2.
3-1
Modeling of the system
The design of observers is based on the system dynamics. The dynamics of a system under
study can be described by a non-linear function f of the state variables x, the system inputs
u and a noise signal v known as process noise. The system outputs y can be described as
continuous nonlinear function of x, u and a noise signal w, known as measurement noise. The
system can be shown as:
x˙ = f (x, u, v),
y = h(x, u, w).
(3-1)
In general the functions f and h are nonlinear but in many cases, it is assumed that f and h
are linear, functions of x, u and the noise. The system can be described as:
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State Estimation and Filtering
x˙ = Ax + Bu + v,
y = Cx + Du + w,
(3-2)
where A is the state evolution matrix, B is the input matrix, C is the output matrix, and D
is the feed through matrix.
Assumption of linearity makes the estimation problem of easier to solve. In reality, however
linear systems are very rare and models are never free of uncertainties.
3-2
Observability of the system
As explained in [54], A system is observable if for any t > 0 it is possible to determine the state
of the system x(t) through measurements y(t) and inputs u(t) in the interval [0, t]. If a system
is observable, then there is no "hidden" dynamics inside it; we can understand everything that
is going on by observing the inputs and outputs. The observability of the system is of practical
interest because it determines if a set of sensors is sufficient for controlling a system.
The linear system given by equation (3-2) is observable if the observability matrix O given
by equation (3-3) has full rank n (i.e. n linearly independent rows) [54]:
O = [C CA CA2 ....CAn−1 ]T .
(3-3)
The question for nonlinear system is the same, whether it is possible to reconstruct the states
from measurement of outputs and inputs. The estimation problem observability of nonlinear
systems is investigated by [55], the authors have presented two sufficient conditions of global
observability. But, in this study, the observability studied for non-linear system is local, as
discussed in [56], [57]. The system is locally observable at x0 if there exists a neighborhood
of x0 such that every x in that neighborhood other than x0 is distinguishable from x0 . The
local observability analysis uses Lie derivatives of hi function. The Lie derivative at (r + 1)
order is defined as:
∂Lf r hi (x)
Lf r+1 hi (x) =
f (x, u),
(3-4)
∂x
with
∂hi (x)
Lf 1 hi (x) =
f (x, u),
(3-5)
∂x
where i = 1, ..., p and p is the number of measurements, x is the state and u is the input. The
observability function oi corresponding to the measurement function hi is defined as:


dhi (x)
 dL 1 h (x) 


i
f
oi = 
,


....
dLf n−1 hi (x)
(3-6)
where d is the operator:
dhi =
Shelav Jain
∂hi (x)
∂hi (x)
, ....,
.
∂x1
∂xn
(3-7)
Master of Science Thesis
3-3 The Luenberger Observer
33
The observability function of the system is calculated as:

o1

 
 
 
O = .... .
 
 
(3-8)
op
The system given by equation (3-1) is locally observable if the observability matrix O given
by equation (3-8) has full rank n, otherwise the system is unobservable.
3-3
The Luenberger Observer
The Luenberger observer, on which the majority of the later developments in the field of state
estimation are based, uses a mathematical model of the system to simulate it in real-time
parallel to the system, using the inputs as well as the outputs to estimate the state. It assumes
a system of the form in equation (3-2) but without considering the noise. The variables of a
state observers are commonly denoted by ‘hat’: x
ˆ and yˆ to distinguish them from the variables
of the equations describing by the physical system. The equations describing the Luenberger
observer are:
x
ˆ˙ = Aˆ
x + L[y − yˆ] + Bu,
(3-9)
In the above equation L is a feedback gain. As can be seen, the observer is simply a copy
of the system, augmented by a feedback term to correct estimation errors due to model
uncertainty and process noise v. It is not always possible to choose L such that the state
can be obtained. This depends on the so called observability of the system. The pair [A, C]
should be observable. If this condition is met and the system is noise free, the observer gain
L can be chosen such that the estimation error converges arbitrarily fast. In fact, the system
describing the error dynamics will become:
e˙ = (A − LC)e.
(3-10)
If there is noise on the system, a Luenberger observer in general cannot guarantee convergence
of the error to zero. The presence of noise on the state variables as well as on the output
measurements results in a dilemma. On one hand, it is desirable to choose L large, to quickly
compensate for errors due to the process noise. On the other hand, this will make the
estimation more sensitive to measurement noise, which makes it more attractive to choose a
smaller L.
3-4
Linear Kalman Filter
The Kalman filter is a linear observer with the observer gain L chosen such that the estimation
error is least-square optimal under the conditions that:
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State Estimation and Filtering
• The system is linear and the system matrices are known. The stochastic discrete-time
state-representation of a linear time-invariant system is given by:
Xk = AXk−1 + BUk + vk ,
(3-11)
Yk = CXk + wk .
(3-12)
• The process noise v and the measurement noise w are temporarily uncorrelated, zero
mean, Gaussian white noise signals:
vk = N (0, Qk ),
wk = N (0, Rk ),
(3-13)
where Q and R are the covariance matrices describing the second-order properties of
the state and measurement noise.
The optimal estimation problem of Xk based on input-output data and knowledge of the
model can be solved by minimizing the loss function:
ˆ k/k ) = E{(X
ˆ k/k−1 − Xk )2 }, ∀k,
J(X
(3-14)
ˆ k/k−1 and X
ˆ k/k are, respectively, the prediction and the prior estimate of Xk .
where X
A recursive estimation of Xk can be expressed in the form:
ˆ k/k = X
ˆ k/k−1 + Lk (Yk − Yˆk/k−1 ),
X
(3-15)
where Yˆk/k−1 is the prediction of Yk and Lk is the Kalman gain. The difference between Yˆk/k−1
ˆ k−1/k−1 and
and Yk is called the filter innovation at instant k. Assuming the prior estimate X
the current observation Yk to be Gaussian random variables, the optimal solution is given by
the following equations:
• Initialization: The initial state and the initial covariance are determined by:
ˆ 0 = E[X0 ], P0 = E[(X0 − X
ˆ 0 )(X0 − X
ˆ 0 )T ].
X
(3-16)
• Time update:
– The prediction of the state is given by:
ˆ k|k−1 = AX
ˆ k−1|k−1 + BUk .
X
(3-17)
– The predicted covariance is computed as:
Pk|k−1 = APk−1|k−1 AT + R.
(3-18)
• Measurement update:
– The filter gain is calculated by:
Lk = Pk|k−1 C T [HPk|k−1 C T + Q]−1 .
Shelav Jain
(3-19)
Master of Science Thesis
3-5 Non-linear Kalman Filter
35
– The state estimation is determined by:
ˆ k|k = X
ˆ k|k−1 + Lk [Yk − C X
ˆ k|k−1 ].
X
(3-20)
– The estimated covariance is:
Pk|k = [I − Lk C]Pk|k−1 .
(3-21)
The first step consists of initializing the filter by choosing a starting estimate for the state and
its variance. In general, the effect of these initial estimates diminishes with time and they do
not affect the steady-state performance of a filter. The second step introduces equations to
estimate the state vector. These equations are divided into time and measurement equations.
The time update projects the current state estimate ahead in time whereas the measurement
update adjusts the projected estimate by an actual measurement at that time [5].
3-5
Non-linear Kalman Filter
Kalman Filter is an optimal filter for estimating a linear system states. Due to its simplicity,
it has found its usage in numerous practical applications. However, most of the real-world systems are non-linear in nature. Estimation of these non-linear systems is extremely important
because all practical systems are non-linear for example target tracking, vehicle navigation,
plant control etc. In order to deal with the non-linearity, many extensions have been developed
over the Kalman filter. Two of the main extensions are:
1. Extended Kalman Filter (EKF) where non-linearities are accounted by linearising the
system about its last-known best estimate with the assumption that the error incurred
by neglecting the higher-order terms is small in comparison to the first-order terms.
2. Unscented Kalman Filter (UKF) which implements the unscented transformation. In
this work, UKF is used and its properties and advantages are discussed in detail in the
next section.
EKF is widely used in the industry for estimation of nonlinear systems. In this study, it is
not implemented but interested readers can follow [27]. Although the EKF is conceptually
simple, it has three well-known drawbacks, as given in [58]:
1. Linearisation can produce unstable filter performance if the time-step intervals are not
sufficiently small.
2. Derivation of Jacobians are non-trivial in most applications and often lead to significant
implementation difficulties.
3. Sufficiently small time-step intervals usually imply high computational overhead as the
number of calculations demanded for the generation of the Jacobian and the prediction
of state estimate and covariance is large.
Master of Science Thesis
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36
State Estimation and Filtering
Figure 3-1: The principle of the Unscented Transformation. Source: [7]
3-5-1
Unscented Kalman Filter
In Kalman filter a Gaussian Random Variable (GRV) is propagated through the system
dynamics. In the EKF, the state distribution is approximated by a GRV, which is then
propagated analytically through the first-order linearisation of the non-linear system. This
can introduce large errors in the true posterior mean and covariance of the transformed GRV,
which may lead to sub-optimal performance and sometimes divergence of the filter. The UKF
addresses this problem by using a deterministic sampling approach. The state distribution
is again approximated by a GRV, but it is now represented using a minimal set of carefully
chosen sample points known as sigma points. These sigma points completely capture the
true mean and covariance of the GRV, and when propagated through the true non-linear
system, captures the posterior mean and covariance accurately to the 3rd order (Taylor series
expansion) for any non-linearity [59]. The EKF, in contrast, only achieves first-order accuracy.
Remarkably, the computational complexity of the UKF is the same order as that of the EKF
[59].
Since UKF algorithm is based on the Unscented Transformation (UT), it is explained in the
next subsection.
The Unscented Transformation
The UT is a method of calculating the statistics of a random variable which undergoes a nonlinear transformation. It is based on the intuition that it is easier to approximate a Gaussian
distribution than it is to approximate an arbitrary non-linear function or transformation [60].
As shown in Figure 3-1, a set of sigma points are chosen so that their mean and sample
¯ and Pxx . Then the non-linear function is applied at each point to get a
covariance are x
¯ and Pyy .
cloud of transformed points, their mean and covariance are y
¯ and covariance Pxx is approximated by
The L-dimensional random variable x with mean x
2L + 1 sigma points with corresponding weights Wi , according to the following:
Shelav Jain
Master of Science Thesis
3-5 Non-linear Kalman Filter
37
χ0 = x
¯,
q
χi = x
¯ + ( (L + λ)Pxx )i ,
i = 1, ...., L,
q
χi = x
¯ − ( (L + λ)Pxx )i−L ,
i = L + 1, ...., 2L,
W0 (m) = λ\(L + λ),
W0 (c) = λ\(L + λ) + (1 + α2 + β),
Wi (m) = Wi (c) = 1\[2(L + λ)],
i = 1, ...., 2L,
(3-22)
where λ = α2 (L + κ) − L is a scaling parameter. α determines the spread of the sigma points
¯ and is usually set to a small positive value. κ is a secondary scaling parameter
around x
which is usually
set to 0 and β is used to incorporate prior knowledge of the distribution of x.
p
The term ( (L + λ)Pxx )i is the ith row of the matrix square root. In [61], it is given that the
sigma points capture the same mean and covariance irrespective of the choice of linear algebra
method used to obtain matrix square root. To obtain the matrix square root in this study,
Cholseky
decomposition is used, which is a numerically efficient and stable method. The term
p
( (L + λ)Pxx )i is obtained form lower triangular matrix of the Cholseky factorization. For
detailed explanation about matrix square root methods, [62] can be followed.
These sigma points are then propagated through the non-linear output functions. The transformation procedure is as follows as given in [61]:
1. Instantiate each point through the function to yield the set of transformed sigma points,
Yi = h(χi ),
i = 0, ..., 2L.
(3-23)
¯ is given by the weighted average of the transformed points,
2. The mean y
¯=
y
2L
X
Wi (m) Yi .
(3-24)
i=0
3. The covariance Pyy is the weighted outer product of the transformed points,
Pyy =
2L
X
¯}{Yi − y
¯ }T .
Wi (c) {Yi − y
(3-25)
i=0
The Unscented Filter
The Unscented Filter is an extension of the UT. The flowchart of the algorithm is given in
Figure 3-2.
Master of Science Thesis
Shelav Jain
38
State Estimation and Filtering
Figure 3-2: Flowchart of the UKF algorithm.
The transformation process which occur in an UKF consists of the following steps, as given
in [59]:
1. Initialize the filter with the initial value of state and its covariance:
x
ˆ0 = E[x0 ],
P0 = E[(x0 − x
ˆ0 )(x0 − x
ˆ0 )T ].
(3-26)
2. Calculate sigma points using this initial estimate of state and covariance as given in
equation (3-22):
q
q
χk−1 = [ˆ
xk−1 x
ˆk−1 + ( (L + λ)Pxx )i=1,..,L x
ˆk−1 − ( (L + λ)Pxx )i=L+1,..,2L ]. (3-27)
3. Time Update: The predicted mean and covariance are computed by instantiating
each point through the process model:
χk|k−1 = f [χk−1 , u(k)].
Shelav Jain
(3-28)
Master of Science Thesis
3-5 Non-linear Kalman Filter
39
(a) The predicted mean is computed as:
x
ˆ−
k =
2L
X
Wi (m) χi,k|k−1 .
(3-29)
i=0
(b) The predicted covariance is computed as:
Pk − =
2L
X
T
Wi (c) [(χi,k|k−1 − x
ˆ−
ˆ−
k )(χi,k|k−1 − x
k ) ] + Qk ,
(3-30)
i=0
where Qk is the process noise covariance matrix.
(c) Again obtain the sigma points using the predicted mean and covariance:
χk−1 = [ˆ
x−
ˆ−
k x
k ±
q
(L + λ)Pk − ].
(3-31)
(d) Instantiate each of the sigma points through the observation model:
Yk|k−1 = h(χk−1 , u(k)).
(3-32)
i. The predicted observation is calculated as:
yˆk− =
2L
X
Wi (m) Yi,k|k−1 .
(3-33)
i=0
ii. The innovation covariance is:
Pyk yk =
2L
X
Wi (c) [(Yk|k−1 − yˆk− )(Yk|k−1 − yˆk− )T ] + Rk ,
(3-34)
i=0
where Rk is the measurement noise covariance matrix.
(e) Finally, predict the cross correlation as:
P xk yk =
2L
X
Wi (c) [(χk|k−1 − x
ˆ−
ˆk− )T ].
k )(Yk|k−1 − y
(3-35)
i=0
4. The filter gain can be calculated as:
Kk = Pxk yk Pyk yk −1 .
(3-36)
5. Measurement Update:
(a) The estimated state is given as:
x
ˆk = x
ˆ−
ˆk− ).
k + Kk (yk − y
(3-37)
Pk = Pk − − Kk Pyk yk Kk T .
(3-38)
(b) The estimated covariance is:
Master of Science Thesis
Shelav Jain
40
3-6
State Estimation and Filtering
Summary
This chapter discussed the approaches for linear and non-linear Kalman filtering. The observability conditions for linear and non-linear systems are discussed. Then the formulation
of state observer has been given.
Next, the linear and non-linear Kalman filters are described, which constitute the backbone
of this study. The filters work under the assumption that the noise has Gaussian distribution.
For nonlinear filtering, the UKF algorithm is discussed in detail and a brief description of
EKF is also given.
In chapter 4 and chapter 5, the results obtained using the UKF algorithm described in this
chapter are discussed.
Shelav Jain
Master of Science Thesis
Chapter 4
Estimation using Bicycle Model
The focus of this chapter is to analyze the accuracy of the nonlinear filter designed to estimate
the sideslip angle and tire cornering stiffness. Two different configurations of the filter are
studied using different vehicle maneuvers. First Unscented Kalman Filter (UKF) algorithm
is without the sideslip angle as measurement signal and second UKF has sideslip angle as
measurement signal. The measurement of sideslip angle is taken from the estimation algorithm
designed in [63] using tire force measurements.
4-1
Nonlinear Observer: State Space Algorithm
To estimate the sideslip angle and tire cornering stiffness, the vehicle’s equations of motion
are considered to model these variables as function of other vehicle parameters. The nonlinear
vehicle model used is described in section 2-1-2 and linear tire force model is described in
section 2-2-2.
To build a model based UKF, the nonlinear bicycle model equations (2-11) - (2-12) and linear
tire model equations (2-16) - (2-17) are converted to discrete form by first-order Euler method
as:
xk = fk−1 (xk , uk ) + vk ,
yk = h(xk , uk ) + wk .
(4-1)
The state vector xk , at each time instant k, comprises of sideslip angle, yaw rate, front tire
cornering stiffness, rear tire cornering stiffness:
xk = [βk , ψ˙ k , Cαf,k , Cαr,k ]T .
(4-2)
The input vector uk comprises the front and rear wheels steering angles and front and rear
wheels longitudinal forces:
uk = [δf,k , δr,k , Fxf,k , Fxr,k ]T .
(4-3)
Master of Science Thesis
Shelav Jain
42
Estimation using Bicycle Model
The measurement vector comprises of yaw rate, front and rear wheels lateral forces:
yk = [ψ˙ k , Fyf,k , Fyr,k ]T .
(4-4)
The nonlinear function f (·) that relates the states at time k to the states at time k − 1 and
to the inputs uk is given as:
T
ψ˙ k−1
f1 = βk−1 +
Cαf,k−1 δf,k − βk−1 − lf
cos δf,k + Fxf,k sin δf,k
mv Vx
Vx
ψ˙ k−1
+ Cαr,k−1 δr,k − βk−1 + lr
cos δr,k + Fxr,k sin δr,k − ψ˙ k−1 ,
Vx
T
ψ˙ k−1
˙
lf Cαf,k−1 δf,k − βk−1 − lf
cos δf,k + Fxf,k sin u1,k
f2 = ψk−1 +
Izz
Vx
ψ˙ k−1
− lr Cαr,k−1 δr,k − βk−1 + lr
cos δr,k + Fxr,k sin δr,k ,
Vx
f3 = Cαf,k−1 ,
f4 = Cαr,k−1 .
(4-5)
The measurement equations are given as:
h1 = ψ˙ k ,
ψ˙ k
h2 = Cαf,k δf,k − βk − lf
cos δf,k ,
Vx
ψ˙ k
h3 = Cαr,k δr,k − βk + lr
cos δr,k .
Vx
(4-6)
The vehicle parameters are taken from the CarSim multi-body simulation software. A D-Class
sedan vehicle is chosen for the simulation. The parameters values are given in table 4-1.
S.No
1
2
3
4
5
Parameter
Mass of the vehicle
Yaw moment of inertia
Distance from Centre of Gravity (COG) to front axle
Distance from COG to rear axle
Sampling Time
Symbol
mv
Izz
lf
lr
T
Unit
kg
kgm2
m
m
s
Value
1530
2315.3
1.110
1.670
0.001
Table 4-1: D class vehicle parameters values.
4-2
Observability Analysis
As discussed in section 3-2, observability is the measure of how well the system states are
related to its inputs and measurements. The observability study for the nonlinear system
is local and uses the Lie derivatives as given in equation (3-4). For the discrete time state
Shelav Jain
Master of Science Thesis
4-3 Simulation Results
43
space system described above, observability matrix is evaluated using MATLAB symbolic
environment and it is given as:
O = [dhi (x) dL1f hi (x) dL2f hi (x) dL3f hi (x)]T .
(4-7)
For initial condition of non zero cornering stiffness the system is locally observable. The
dimension of the observability matrix is 12 × 4 and the rank of the matrix O is 4. Therefore,
the nonlinear system described by equations 4-5 and 4-6 is locally observable.
4-3
Simulation Results
In this section, the algorithm developed in the previous section is validated and simulation
results are presented for various experiments. The algorithm is simulated in CarSim and
Simulink. The measurement and input signals are taken from the CarSim environment and
then post-processed in Simulink. The vehicle model in CarSim simulation package has 15
mechanical Degree of Freedom (DOF) [50]. The CarSim package uses built-in nonlinear tire
models with dependency on slip, load and camber.
4-3-1
Tuning Parameters
For satisfactory working of the UKF, it is important to tune the process noise covariance
matrix Q and measurement noise covariance matrix R. The yaw rate and sideslip angle are
modeled using system dynamic equations, therefore low uncertainty is assigned to them. However, the cornering stiffnesses are not modeled at all, hence, they are given high uncertainties.
The uncertainties in the measurement noise covariance matrix are taken from the measurement signal. The uncertainty for yaw rate signal is taken from the available gyroscope sensor.
While the uncertainties in the forces are taken from the measurement data obtained from a
test vehicle mounted with SKF Load Sensing Bearings.
The process and measurement noises are assumed to be constant and uncorrelated, therefore,
the off-diagonal elements are assigned to 0. The measurement noise matrix R in terms of
standard deviation of the measured signal is given as:


σ ˙2
0
0
 ψ

R =  0 σFyf 2
0 .
0
0
σFyr 2
After a number of simulations and careful tuning, the following values of Q and R gave
desirable results:


0.00001
0
0
0

0
15.3
0
0 


Q = 10e − 08 
,

0
0
15e11
0 
0
0
0
11e11
Master of Science Thesis
Shelav Jain
44
Estimation using Bicycle Model


0.001
0
0


6.3458e3
0
R= 0
.
0
0
6.3458e3
In addition to the Q and R matrices, there are some other parameters for tuning in UKF.
These parameters determine the scaling of Unscented Transformation (UT), their significance
is given in subsection 3-5-1. The values of these parameters used for this study are given in
table 4-2.
S.No
1
2
3
Parameter
α
β
κ
Value
1e-03
2
0
Table 4-2: UKF tuning parameters.
4-3-2
Observer Evaluation
The algorithm is verified by comparing the outputs of the estimator with the measurements
from CarSim for different maneuvers. The observer error is also calculated to analyze the
accuracy of the estimation results. The measurement signals, i.e. the yaw rate and lateral
tire forces are polluted with white noise having standard deviation as given in the R matrix.
Figure 4-1 to Figure 4-9 describe the comparison between the estimated and measured sideslip
angle and cornering stiffnesses.
Sine-sweep maneuver - First, the model is simulated for sine steering input with vehicle
speed of 80km/h. The applied steering profile and the lateral acceleration are shown in
Figure 4-1. Figure 4-2 and Figure 4-3 compare the estimated sideslip angle and tire cornering
stiffnesses with the output from CarSim. The estimated sideslip angle tracks the reference
well. From the Figure 4-3, it can be seen that the algorithm captures the change in cornering
stiffness. But at low steering angles the error between the estimated cornering stiffness and
the reference is significant. It is due to the low magnitude of the lateral force close to the
crossover region. Therefore, resulting in a low signal to noise ratio and thereby results in an
estimate mainly driven by sensor noise.
Double lane change maneuver- Next, the vehicle is simulated for double lane change
maneuver. The steering profile and lateral acceleration are shown in Figure 4-4. As can
be seen from the Figure 4-5 the estimated sideslip angle tracks the reference. In Figure 4-6
comparison between the estimated and reference cornering stiffness is given. The filter follows
the reference but, in case of slow dynamics, i.e. for close to zero steering angles, the estimate
is driven by noise in the lateral force signal. Hence, not resulting in an accurate estimate.
Shelav Jain
Master of Science Thesis
4-3 Simulation Results
45
Steering Angle
5
0
f
δ [deg]
Steer
−5
0
5
10
15
Lateral Acceleration
1
a
y
a [g]
y
0
−1
0
5
10
15
t [s]
Figure 4-1: Steering profile and lateral acceleration for sine steer at Vx of 80km/h.
Sideslip Angle
Reference
Estimated
3
Sideslip Angle [deg]
2
1
0
−1
−2
−3
0
5
10
15
t [s]
Figure 4-2: Sideslip angle for sine steer at Vx of 80km/h.
Master of Science Thesis
Shelav Jain
46
Estimation using Bicycle Model
Front Cornering Stiffness
Reference
Estimated
Cαf [N/deg]
2500
2000
1500
0
5
10
15
Rear Cornering Stiffness
Cαr [N/deg]
2000
1500
1000
0
5
10
15
t [s]
Figure 4-3: Cornering stiffness for sine steer at Vx of 80km/h.
Steering Angle
5
0
f
δ [deg]
Steer
−5
0
2
4
6
8
10
Lateral Acceleration
1
y
a [g]
ay
0
−1
0
2
4
6
8
10
t [s]
Figure 4-4: Steering profile and lateral acceleration for double lane change at Vx of 75km/h.
Shelav Jain
Master of Science Thesis
4-3 Simulation Results
47
Sideslip Angle
Reference
Estimated
3
Sideslip Angle [deg]
2
1
0
−1
−2
−3
0
2
4
6
8
10
t [s]
Figure 4-5: Sideslip angle for double lane change at Vx of 75km/h.
Front Cornering Stiffness
Cαf [N/deg]
4000
Reference
Estimated
3000
2000
1000
0
2
4
6
8
10
8
10
Rear Cornering Stiffness
Cαr [N/deg]
2500
2000
1500
1000
0
2
4
6
t [s]
Figure 4-6: Cornering stiffness for double lane change at Vx of 75km/h.
Master of Science Thesis
Shelav Jain
48
Estimation using Bicycle Model
Steering Angle
5
0
f
δ [deg]
Steer
−5
0
1
2
3
4
5
6
7
8
Lateral Acceleration
0.5
a
ay [g]
y
0
−0.5
0
1
2
3
4
5
6
7
8
t [s]
Figure 4-7: Steering profile and lateral acceleration for double lane change at Vx of 85km/h
and µ = 0.5.
Low friction surface- To check the applicability of the developed algorithm on low friction
surface which results in higher sideslip angles. The vehicle is simulated for double lane change
on low friction surfaces with µ = 0.5. The steering profile and lateral acceleration are shown
in Figure 4-7. The sideslip angle and cornering stiffness are given in Figure 4-8 and Figure 4-9
respectively. The cornering stiffness estimate in Figure 4-9 shows inaccurate results in the
region of slow dynamics. The cornering stiffness values are not converging because of lack of
system information. Hence, resulting in the estimate driven by sensor noise.
To measure the performance of the state estimator, Root mean squared (RMS) values of
estimation error for different maneuvers are evaluated. The formulation of RMS error is
given below:
v
u
N
u1 X
t
RM S error =
(ˆ
xn − xn )2 ,
(4-8)
N n=1
where x
ˆn is the estimated state, xn is the measured state and N is the total number of
samples. The RMS error values of sideslip angle estimation are shown in the table below.
S.No
1
2
3
Maneuver
Sine Steer at Vx of 80km/h
Double Lane Change at Vx of 75km/h
Double Lane Change at Vx of 85km/h and µ = 0.5
RMS error
0.2918 deg
0.2726 deg
0.4107 deg
Table 4-3: RMS error values for UKF with bicycle model.
Shelav Jain
Master of Science Thesis
4-3 Simulation Results
49
Sideslip Angle
8
Reference
Estimated
6
Sideslip Angle [deg]
4
2
0
−2
−4
−6
0
1
2
3
4
5
6
7
8
t [s]
Figure 4-8: Sideslip angle for double lane change at Vx of 85km/h and µ = 0.5.
Front Cornering Stiffness
Cαf [N/deg]
4000
Reference
Estimated
2000
0
0
2
4
6
8
Rear Cornering Stiffness
Cαr [N/deg]
3000
2000
1000
0
0
2
4
6
8
t [s]
Figure 4-9: Cornering stiffness for double lane change at Vx of 85km/h and µ = 0.5.
Master of Science Thesis
Shelav Jain
50
Estimation using Bicycle Model
Lateral Acceleration
1
ay
ay [g]
0.5
0
−0.5
0
2
4
6
8
10
Sideslip angle [deg]
Sideslip Angle
1
Reference
Estimated
0
−1
−2
0
2
4
6
8
10
t [s]
Figure 4-10: Lateral acceleration and sideslip angle for steady state cornering for UKF without
sideslip angle measurement.
4-3-3
Limitation
The UKF presented in the section above does not give satisfactory results in the steady
state cornering maneuver. The estimates diverge from the actual measurements, as shown in
Figure 4-10. The reason for the divergence in steady state maneuver needs to be investigated.
Therefore, to estimate the cornering stiffness of a vehicle in steady state cornering maneuver,
measurement of sideslip angle is incorporated in the algorithm. The sideslip angle is obtained
from another estimator designed in [63].
4-4
Nonlinear observer with sideslip angle as measurement
The measurement vector given in (4-4) is extended with sideslip angle as measurement signal.
The extended vector is:
yk = [β, ψ˙ k , Fyf,k , Fyr,k ]T .
(4-9)
Then the output equation (4-6) is extended as:
h1 = βk ,
h2 = ψ˙ k ,
ψ˙ k
h3 = Cαf,k δf,k − βk − lf
cos δf,k ,
Vx
ψ˙ k
h4 = Cαr,k δr,k − βk + lr
cos δr,k .
Vx
Shelav Jain
(4-10)
Master of Science Thesis
4-5 Analysis and Observations
51
Steering Angle
5
δ [deg]
Steer
f
0
−5
0
2
4
6
8
10
Lateral Acceleration
1
a
y
ay [g]
0.5
0
−0.5
0
2
4
6
8
10
t [s]
Figure 4-11: Steering profile and lateral acceleration for steady state cornering at Vx of 90km/h
and 100m radius circle.
Steady state maneuver - The vehicle is tested for steady state cornering maneuver on a
track of constant radius 100m with vehicle speed of 90km/h. The steering input and lateral
acceleration are given in Figure 4-11. With the sideslip angle information, filter tracks the
cornering stiffness accurately, as can be seen in the Figure 4-12. For other maneuvers the
UKF with sideslip angle as measurement shows the same performance as the one without the
sideslip angle as measurement signal.
4-5
Analysis and Observations
The developed algorithm tracks the reference well in transient maneuver. It is able to estimate
vehicle sideslip angle in the lateral acceleration range {−0.8, 0.8} g. The model also gives
accurate estimate of sideslip angle on low friction surface, as shown in the results for µ = 0.5.
But, for steady state maneuver it is not able to estimate the sideslip angle accurately. The
filter is more accurate for lower lateral acceleration values. This can be understood from the
fact that the linear tire model is used in this study. From the cornering stiffness curves it can
be seen that the changes in the cornering stiffness are well captured.
4-6
Summary
This chapter presented the results of the developed sideslip and cornering stiffness estimation
algorithm. First, the estimator algorithm is given for the bicycle model. The observability
analysis results show that the system is locally observable.
Next, the simulation results are given which constitutes the values of tuning parameters,
observer evaluation on different maneuvers and its RMS error. This part also gives the
Master of Science Thesis
Shelav Jain
52
Estimation using Bicycle Model
Front Cornering Stiffness
Cαf [N/deg]
2100
Reference
Estimated
2000
1900
1800
0
2
4
6
8
10
8
10
Cαr [N/deg]
Rear Cornering Stiffness
1700
1600
1500
1400
0
2
4
6
t [s]
Figure 4-12: Cornering stiffness for steady state cornering at Vx of 90km/h and 100m radius
circle.
limitation of the algorithm and presents a method to improve the developed algorithm. The
performance of the filter in steady state maneuvers can be improved by incorporating sideslip
angle as measurement signal in the filter.
The results in this chapter are based on the bicycle model. In the next chapter the UKF is
designed on the basis of Four-Wheel vehicle model (FWVM).
Shelav Jain
Master of Science Thesis
Chapter 5
Estimation using Four Wheel Vehicle
Model
The algorithm designed in the last chapter is extended to accommodate the Four-Wheel
vehicle model (FWVM). The FWVM is more accurate than the bicycle model, as it also
captures the load transfer from inside to outside wheels while cornering.
5-1
Nonlinear Observer Formulation
As can be seen from the equations (2-1) - (2-5), determination of lateral velocity requires
information of the longitudinal forces. These forces should be taken into account to accurately
estimate vehicle sideslip angle (or lateral velocity) during braking or acceleration in a turn.
The state vector xk , at each time instant k, comprises of sideslip angle, yaw rate, front and
rear tires lateral forces, front and rear tires cornering stiffness:
xk = [Vx,k , Vy,k , ψ˙ k , FyF L,k , FyF R,k , FyRL,k , FyRR,k , CαF L,k , CαF R,k , CαRL,k , CαRR,k ]T . (5-1)
Therefore, the vehicle sideslip angle can be calculated using:
Vy,k
,
Vx,k
x2,k
.
= arctan
x1,k
βk = arctan
(5-2)
The input vector uk comprises the front and rear wheels steering angles and front and rear
wheels longitudinal forces:
uk = [δF L,k , δF R,k , δRL,k , δRR,k , FxF L,k , FxF R,k , FxRL,k , FxRR,k ]T .
Master of Science Thesis
(5-3)
Shelav Jain
54
Estimation using Four Wheel Vehicle Model
The measurement vector comprises of vehicle’s longitudinal velocity, yaw rate, front and rear
wheels lateral forces:
yk = [Vx,k , ψ˙ k , FyF L,k , FyF R,k , FyRL,k , FyRR,k ]T .
(5-4)
The nonlinear function f (·) that relates the states at time k to the states at time k − 1 and
to the inputs uk is given as:
1
FxF L,k cos δF L,k − FyF L,k−1 sin δF L,k + FxF R,k cos δF R,k
mv
− FyF R,k−1 sin δF R,k + FxRL,k cos δRL,k − FyRL,k−1 sin δRL,k
f1 = Vx,k−1 + T Vy,k−1 ψ˙ k−1 +
+ FxRR,k cos δRR,k − FyRR,k−1 sin δRR,k
,
1
FyF L,k−1 cos δF L,k + FxF L,k sin δF L,k + FyF R,k−1 cos δF R,k
f2 = Vy,k−1 + T − Vx,k−1 ψ˙ k−1 +
mv
+ FxF R,k sin δF R,k + FyRL,k−1 cos δRL,k + FxRL,k sin δRL,k
+ FyRR,k−1 cos δRR,k + FxRR,k sin δRR,k
f3 = ψ˙ k−1 +
,
T
lf FyF L,k−1 cos δF L,k + FxF L,k sin δF L,k + FyF R,k−1 cos δF R,k + FxF R,k sin δF R,k
Izz
− lr FyRL,k−1 cos δRL,k + FxRL,k sin δRL,k + FyRR,k−1 cos δRR,k + FxRR,k sin δRR,k
E
+
FyF L,k−1 sin δF L,k − FxF L,k cos δF L,k + FxF R,k cos δF R,k − FyF R,k−1 sin δF R,k
2
− FxRL,k cos δRL,k + FyRL,k−1 sin δRL,k + FxRR,k cos δRR,k − FyRR,k−1 sin δRR,k
f4 = CαF L,k−1 δF L,k − arctan
Vy,k−1 + lf ψ˙ k−1
Vx,k−1 −
E ψ˙ k−1
2
,
,
Vy,k−1 + lf ψ˙ k−1
f5 = CαF R,k−1 δF R,k − arctan
,
E ψ˙
Vx,k−1 + 2k−1
Vy,k−1 − lr ψ˙ k−1
f6 = CαRL,k−1 δRL,k − arctan
,
E ψ˙
Vx,k−1 − 2k−1
Vy,k−1 − lr ψ˙ k−1
,
f7 = CαRR,k−1 δRR,k − arctan
E ψ˙
Vx,k−1 + 2k−1
f8 = CαF L,k−1 ,
f9 = CαF R,k−1 ,
f10 = CαRL,k−1 ,
f11 = CαRR,k−1 .
Shelav Jain
(5-5)
Master of Science Thesis
5-2 Observability Analysis
55
The observation system h(·) which gives the relation between the measurements at time k
with the states and inputs is:
h1 = Vx,k ,
h2 = ψ˙ k ,
h3 = FyF L,k ,
h4 = FyF R,k ,
h5 = FyRL,k ,
h6 = FyRR,k .
(5-6)
The vehicle parameters are taken from the CarSim simulation package. A D-Class sedan
vehicle is chosen for the simulation. The parameters values are given in table 4-1 on page 42
and E is the track width which is equal to 1.550m.
5-2
Observability Analysis
The observability study for the nonlinear system is local and uses the Lie derivatives as given
in equation (3-4). For the discrete time state space system described above, observability
matrix is evaluated using MATLAB symbolic environment and it is given as:
O = [dhi (x) dL1f hi (x) dL2f hi (x) dL3f hi (x) dL4f hi (x) dL5f hi (x) dL6f hi (x) dL7f hi (x)]T . (5-7)
For initial condition of non zero cornering stiffness the system is locally observable. The
dimension of the observability matrix is 66 × 8 and the rank of the matrix O is 8. Therefore,
the nonlinear system described by equations 5-5 and 5-6 is locally observable.
5-3
Simulation Results
The efficiency of the estimation process to reconstruct the sideslip angle and cornering stiffness
has been tested using CarSim simulation environment. The details of the CarSim model are
given in section 4-3. This section gives the Unscented Kalman Filter (UKF) tuning parameters
and simulation results for different maneuvers.
5-3-1
Tuning Parameters
After a number of simulations and careful tuning, the following values of Q and R gave
desirable results:
Master of Science Thesis
Shelav Jain
56
Estimation using Four Wheel Vehicle Model


55.5
0
0
0
0
0
0
0
0
0
0

1e − 7
0
0
0
0
0
0
0
0
0 

 0


 0
0
15.3 0
0
0
0
0
0
0
0 


 0
0
0
100 0
0
0
0
0
0
0 


 0
0
0
0 100 0
0
0
0
0
0 




0
0
0
0 100 0
0
0
0
0 ,
Q = 10e−07  0


 0
0
0
0
0
0 100
0
0
0
0 


 0
0
0
0
0
0
0 85.3e09
0
0
0 


 0
0
0
0
0
0
0
0
85.3e09
0
0 




0
0
0
0
0
0
0
0
85.3e09
0 
 0
0
0
0
0
0
0
0
0
0
0
85.3e09


0.001
0
0
0
0
0

 0
0.001
0
0
0
0



 0
0
6.3458e03
0
0
0


R=
.

 0
0
0
6.3458e03
0
0



 0
0
0
0
6.3458e03
0
0
0
0
0
0
6.3458e03
In addition to Q and R matrices, there are some other parameters for tuning in UKF. These
parameters are given in table 4-2.
5-3-2
Observer Evaluation
The observer is verified by comparing the outputs of the estimator with the measurements
from CarSim. The observer error is also calculated to analyze the accuracy of the estimation
results. The measurement signals i.e. the longitudinal velocity, yaw rate and tire forces are
polluted with white noise having standard deviation as given in the R matrix. The Figure 5-1
to Figure 5-12 describe the comparison between the estimated and measured sideslip angle
and tires cornering stiffnesses.
Sine sweep maneuver- First, the vehicle is tested on sine steering input with longitudinal
velocity of 80km/h. The steering profile and lateral acceleration are as shown in Figure 5-1.
The estimated sideslip angle and tires cornering stiffness are compared with the measurements
from CarSim as shown in Figure 5-2 - Figure 5-4. The estimates track the reference well. But
close to the crossover region, i.e. close to zero steering angles, cornering stiffness estimates
are driven by the sensor noise. Therefore, shows large deviation from the reference.
Double lane change maneuver - To simulate the vehicle behavior in transient conditions,
it is tested for Double Lane Change maneuver with vehicle velocity of 80km/h. The lateral
acceleration and steering input is as shown in Figure 5-5. The sideslip angle and tires cornering
stiffness are given in Figure 5-6 - Figure 5-8. The algorithm gives accurate results in region
where sufficient system information is available. But in the region where steering angle is
almost zero, the filter does not converge and estimates are driven by sensor noise.
Shelav Jain
Master of Science Thesis
5-3 Simulation Results
57
Steering Angle
5
0
f
δ [deg]
Steer
−5
0
5
10
15
Lateral Acceleration
1
a
y
a [g]
y
0
−1
0
5
10
15
t [s]
Figure 5-1: Steering profile and lateral acceleration for sine steer at Vx of 80km/h.
Sideslip Angle
Reference
Estimated
3
Sideslip Angle [deg]
2
1
0
−1
−2
−3
0
5
10
15
t [s]
Figure 5-2: Sideslip angle for sine steer at Vx of 80km/h.
Master of Science Thesis
Shelav Jain
58
Estimation using Four Wheel Vehicle Model
Front Left Tire Cornering Stiffness
CαFL [N/deg]
1500
Reference
Estimated
1000
500
0
0
5
10
15
Front Right Tire Cornering Stiffness
CαFR [N/deg]
1500
1000
500
0
0
5
10
15
t [s]
Figure 5-3: Front tires cornering stiffness for sine steer at Vx of 80km/h.
Rear Left Tire Cornering Stiffness
CαRL [N/deg]
1500
Reference
Estimated
1000
500
0
0
5
10
15
Rear Right Tire Cornering Stiffness
CαRR [N/deg]
1500
1000
500
0
0
5
10
15
t [s]
Figure 5-4: Rear tires cornering stiffness for sine steer at Vx of 80km/h.
Shelav Jain
Master of Science Thesis
5-3 Simulation Results
59
Steering Angle
Steer
0
f
δ [deg]
5
−5
0
2
4
6
8
Lateral Acceleration
1
a
y
a [g]
y
0
−1
0
2
4
6
8
t [s]
Figure 5-5: Steering profile and lateral acceleration for double lane change at Vx of 80km/h.
Sideslip Angle
4
Reference
Estimated
3
Sideslip Angle [deg]
2
1
0
−1
−2
−3
−4
−5
0
2
4
6
8
t [s]
Figure 5-6: Sideslip angle for double lane change at Vx of 80km/h.
Master of Science Thesis
Shelav Jain
60
Estimation using Four Wheel Vehicle Model
Front Left Tire Cornering Stiffness
CαFL [N/deg]
2000
Reference
Estimated
1000
0
0
2
4
6
8
Front Right Tire Cornering Stiffness
CαFR [N/deg]
2000
1000
0
0
2
4
6
8
t [s]
Figure 5-7: Front tires cornering stiffness for double lane change at Vx of 80km/h.
Rear Left Tire Cornering Stiffness
CαRL [N/deg]
1500
Reference
Estimated
1000
500
0
0
2
4
6
8
Rear Right Tire Cornering Stiffness
CαRR [N/deg]
1500
1000
500
0
0
2
4
6
8
t [s]
Figure 5-8: Rear tires cornering stiffness for double lane change at Vx of 80km/h.
Shelav Jain
Master of Science Thesis
5-3 Simulation Results
61
Steering Angle
5
0
f
δ [deg]
Steer
−5
0
1
2
3
4
5
6
7
8
Lateral Acceleration
0.5
a
ay [g]
y
0
−0.5
0
1
2
3
4
5
6
7
8
t [s]
Figure 5-9: Steering profile and lateral acceleration for double lane change at Vx of 85km/h
and µ = 0.5.
Low friction surface - Next, the vehicle is simulated on low friction surface. It is important
for an estimator to accurately estimate the sideslip angle on low friction surface because
skidding on low friction surfaces can cause accidents. To generate larger sideslip angle, the
vehicle is tested for a longitudinal velocity of 85km/h at µ = 0.5. The developed lateral
acceleration and steering input are as shown in Figure 5-9. The vehicle sideslip angle angle
cornering stiffness are given in Figure 5-10 - Figure 5-12. The filter tracks the sideslip angle
even at larger values. From the cornering stiffness curves vehicle’s handling limit can be
determined. As the value of cornering stiffness becomes very low for high sideslip angle
condition on low friction surfaces.
To measure the performance of the state estimator, Root mean squared (RMS) values of
estimation error for different maneuvers are also evaluated. The RMS error values of sideslip
angle estimation are shown in the table below.
S.No
1
2
3
Maneuver
Sine Steer at Vx of 80km/h
Double Lane Change at Vx of 80km/h
Double Lane Change at Vx of 85km/h and µ = 0.5
RMS error
0.3635 deg
0.3043 deg
0.3454 deg
Table 5-1: RMS error values for UKF with FWVM.
5-3-3
Limitation
The estimation accuracy of the algorithm is only limited to fast and transient maneuvers.
It does not give satisfactory results in the steady state cornering maneuver. The estimates
diverge from the actual measurements, as shown in Figure 5-13. Therefore, to estimate the
cornering stiffness of a vehicle in steady state cornering maneuver, measurement of lateral
Master of Science Thesis
Shelav Jain
62
Estimation using Four Wheel Vehicle Model
Sideslip Angle
8
Reference
Estimated
Sideslip Angle [deg]
6
4
2
0
−2
−4
−6
0
1
2
3
4
5
6
7
8
t [s]
Figure 5-10: Sideslip angle for double lane change at Vx of 85km/h and µ = 0.5.
Front Left Tire Cornering Stiffness
CαFL [N/deg]
2000
Reference
Estimated
1000
0
0
2
4
6
8
Front Right Tire Cornering Stiffness
CαFR [N/deg]
1500
1000
500
0
0
2
4
6
8
t [s]
Figure 5-11: Front tires cornering stiffness for double lane change at Vx of 85km/h and µ = 0.5.
Shelav Jain
Master of Science Thesis
5-3 Simulation Results
63
Rear Left Tire Cornering Stiffness
CαRL [N/deg]
1500
Reference
Estimated
1000
500
0
0
2
4
6
8
Rear Right Tire Cornering Stiffness
CαRR [N/deg]
1500
1000
500
0
0
2
4
6
8
t [s]
Figure 5-12: Rear tires cornering stiffness for double lane change at Vx of 85km/h and µ = 0.5.
velocity (related to sideslip angle by equation (5-2)) is incorporated in the algorithm. The
lateral velocity is obtained from another estimator designed in [63].
Lateral Acceleration
1
a
y
ay [g]
0.5
0
−0.5
0
2
4
6
8
10
Sideslip angle [deg]
Sideslip Angle
0
Reference
Estimated
−1
−2
−3
0
2
4
6
8
10
t [s]
Figure 5-13: Lateral acceleration and sideslip angle for steady state maneuver for UKF without
lateral velocity as measurement.
Master of Science Thesis
Shelav Jain
64
Estimation using Four Wheel Vehicle Model
Steering Angle
Steer
δf [deg]
4
2
0
0
2
4
6
8
10
Lateral Acceleration
1
a
ay [g]
y
0.5
0
−0.5
0
2
4
6
8
10
t [s]
Figure 5-14: Steering profile and lateral acceleration for steady state cornering at Vx of 95km/h
and 100m radius circle.
5-4
Nonlinear observer with lateral velocity as measurement
The vehicle lateral velocity is incorporated into the system given in equation (5-4). The
measurement vector for the updated observer is:
yk = [Vx,k , Vy,k , ψ˙ k , FyF L,k , FyF R,k , FyRL,k , FyRR,k ]T .
(5-8)
The output equation (5-6) is extended with lateral velocity as measurement.
h1 = Vx,k ,
h2 = Vy,k ,
h3 = ψ˙ k ,
h4 = FyF L,k ,
h5 = FyF R,k ,
h6 = FyRL,k ,
h7 = FyRR,k .
(5-9)
Steady state maneuver- To understand the behavior of the vehicle during steady state
cornering, it is tested on constant radius turn of 100m with vehicle velocity of 95km/h. The
steering profile and lateral acceleration are given in Figure 5-14. The estimated cornering
stiffness for front and rear tires are given in Figure 5-15 and Figure 5-16. It is evident for the
results that the lateral velocity information improves the performance of the filter especially
in steady state conditions. In all other maneuvers, filter with lateral velocity as measurement
has similar performance as compared to the one without the lateral velocity as measurement.
Shelav Jain
Master of Science Thesis
5-4 Nonlinear observer with lateral velocity as measurement
65
[N/deg]
Front Left Tire Cornering Stiffness
Reference
Estimated
600
C
αFL
500
400
0
2
4
6
8
10
8
10
CαFR [N/deg]
Front Right Tire Cornering Stiffness
1400
1200
1000
0
2
4
6
t [s]
Figure 5-15: Front tires cornering stiffness for steady state cornering at Vx of 95km/h and 100m
radius circle.
[N/deg]
Rear Left Tire Cornering Stiffness
Reference
Estimated
400
C
αRL
300
200
0
2
4
6
8
10
8
10
Rear Right Tire Cornering Stiffness
CαRR [N/deg]
1400
1200
1000
800
0
2
4
6
t [s]
Figure 5-16: Rear tires cornering stiffness for steady state cornering at Vx of 95km/h and 100m
radius circle.
Master of Science Thesis
Shelav Jain
66
5-5
Estimation using Four Wheel Vehicle Model
Analysis and Observations
The model is able to follow the reference sideslip angle in the lateral acceleration range of
{−0.8, 0.8}g. It is also able to track the reference in the case of low friction surface for example
ice, snow, water etc. The developed algorithm also accurately tracks the cornering stiffness of
all four tires. From the figures, it is evident that the cornering stiffness changes significantly
during a turn especially due to the lateral load transfer.
But the algorithm is not able to capture the sideslip angle correctly in the steady state
maneuvers. Therefore, it is incorporated with the measurement of lateral velocity to give
satisfactory results.
5-6
Summary
This chapter discussed the results of the FWVM based UKF algorithm. First the algorithm
was formulated and then the observability analysis has been performed.
Next, the observer is evaluated on the basis of simulation results from different maneuvers
along with the RMS error. The analysis of the results is given along with its limitation.
The filter without lateral velocity as measurement signal does not give satisfactory results in
steady state situations. Therefore, a different observer with lateral velocity as measurement
is proposed to overcome this limitation.
Shelav Jain
Master of Science Thesis
Chapter 6
Conclusions and Recommendations
6-1
Conclusions
The thesis report gives an overview of the Vehicle Dynamics Control (VDC) systems. The
objective of these systems can be either control of yaw rate or sideslip angle or both. The
importance of controlling the sideslip angle for satisfactory handling characteristics has been
discussed in detail. But effective measurement of sideslip angle is expensive.
This thesis work developed a model-based algorithm to estimate the vehicle sideslip angle
and tire cornering stiffness. The nonlinear model based Unscented Kalman Filter (UKF)
accurately estimates the sideslip angle and cornering stiffness for different maneuvers. The
UKF algorithm is designed for two types of vehicle models, the bicycle model and two-track
model. The results of the two-track model are more accurate than the bicycle model, as it
also captures the lateral load transfer during cornering.
The developed model is tested using multibody simulation package CarSim. From the results
of different maneuvers it has been shown that the algorithm is able to capture the dynamics of
sideslip angle in the lateral acceleration range {−0.8, 0.8}g and on different friction surfaces.
The model also accurately captures the changes in tire cornering stiffness during different
maneuvers. In the filter design, cornering stiffness is not modeled with any dynamics and
high uncertainty is assigned to it, so that the filter tracks the changes. But due to the lack of
system information close to zero steering angles, cornering stiffness estimate is not accurate.
In case of low lateral forces, the signal to noise ratio is low and the estimates are guided
by noise rather than the actual signal. The filter also does not show adequate performance
in steady state maneuvers. But the availability of lateral velocity or sideslip angle signal as
measurement improves the performance of the filter.
The accuracy of model-based estimation approach is limited by the knowledge of vehicle and
tire parameters. The cornering stiffness obtained from this study can be used to update the
vehicle and tire model. Hence, improving the accuracy of the estimation algorithm. VDC
systems depend on vehicle parameters to obtain the control law. The results from this study
can be used for real-time model update to improve the controller efficiency.
Master of Science Thesis
Shelav Jain
68
6-2
Conclusions and Recommendations
Recommendations
The following section presents recommendations that can be performed to extended the results obtained in this study.
Improvement of cornering stiffness estimation
As discussed, the developed algorithm does not give satisfactory results when lateral forces
are low or when system information is not adequate. The accuracy of the results can be
improved in these conditions. Also, the reason for divergence of the estimates in steady state
maneuvers in UKF without sideslip angle as measurement needs to be investigated.
Complexity of the vehicle and tire model
In any model based estimation process accuracy of the algorithm depends on the accuracy of
the system model. In this case, vehicle model can be extended to include the roll and pitch
dynamics. To capture the dynamic behavior of the tire, the transient tire model as described
in the chapter 2 can be implemented in the algorithm. As the tire forces are considered as
states in the UKF designed with Four-Wheel vehicle model (FWVM), the transient tire model
can be incorporated provided the relaxation time constant is known.
Estimation of vehicle parameters
Vehicle’s inertial parameters like vehicle mass and moment of inertia change often in the actual
operating conditions. In controller design, accurate information of these vehicle parameters
is important. The algorithm developed in this study can form the basis of estimation of these
parameters.
Determination of lateral force saturation
For VDC systems, information of nonlinear operating region of tires is important. This
information can be obtained using Instantaneous Cornering Stiffness (ICS), as explained in
section 1-1. The algorithm presented in this study can be extended to obtain the ICS.
Shelav Jain
Master of Science Thesis
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Master of Science Thesis
Glossary
List of Acronyms
ACC
Adaptive Cruise Control
ABS
Anti-Lock Braking System
TCS
Traction Control System
ESC
Electronic Stability Control
EKF
Extended Kalman Filter
COG
Centre of Gravity
DOF
Degree of Freedom
ADAS
Advanced Driver Assistance Systems
GRV
Gaussian Random Variable
UT
Unscented Transformation
UKF
Unscented Kalman Filter
FWVM
Four-Wheel vehicle model
EV
Electric Vehicle
VDC
Vehicle Dynamics Control
ESP
Electronic Stability Program
AFS
Active Front Steering
ARS
Active Rear Steering
4WAS
Four-Wheel Active Steering
DYC
Direct Yaw Moment Control
Master of Science Thesis
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76
Glossary
ICE
Internal Combustion Engine
ICS
Instantaneous Cornering Stiffness
GPS
Global Positioning System
RMS
Root mean squared
Shelav Jain
Master of Science Thesis