MODELING AND VALIDATION OF HITCHED LOADING
EFFECTS ON TRACTOR YAW DYNAMICS
Except where reference is made to work of others, the work described in this thesis is my
own or was done in collaboration with my advisory committee. This thesis does not
include proprietary or classified information
Paul James Pearson
Certificate of Approval:
Robert Jackson
Professor
Mechanical Engineering
David M. Bevly, Chair
Assistant Professor
Mechanical Engineering
Randy L. Raper
Agricultural Engineer and Head
Scientist, USDA/ARS National Soil
Dynamics Lab
Biosystems Engineering
Joe F. Pittman
Interim Dean
Graduate School
MODELING AND VALIDATION OF HITCHED LOADING
EFFECTS ON TRACTOR YAW DYNAMICS
Paul James Pearson
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Masters of Science
Auburn, Alabama
May 10, 2007
iii
MODELING AND VALIDATION OF HITCHED LOADING
EFFECTS ON TRACTOR YAW DYNAMICS
Paul James Pearson
Permission is granted to Auburn University to make copies of this thesis at
its discretion, upon the request of individuals or institutions and
at their expense. The author reserves all publication rights.
Signature of the Author
Date of Graduation
iv
THESIS ABSTRACT
MODELING AND VALIDATION OF HITCHED LOADING
EFFECTS ON TRACTOR YAW DYNAMICS
Paul James Pearson
Master of Science, May 10, 2007
(B.M.E., Auburn University, 2004)
113 Typed Pages
Directed by David M. Bevly
This thesis develops a yaw dynamic model for a farm tractor with a hitched
implement, which can be used to understand the effect of tractor handling characteristics
for design applications as well as for new automated steering control systems. A model is
found in which hitched implement conditions can be accounted for, and an improvement
in yaw rate tracking prediction in both steady state and dynamic conditions is seen vs.
traditional models. This model is termed the ?3-wheeled? Front and Hitch Relaxation
Length (?3-wheeled? FHRL) Model. Experimental data from a hitch force dynamometer
are used to validate the way the hitched implement forces are derived in the ?3-wheeled?
FHRL Model and to determine if differential hitch forces can be ignored. Steady state
and dynamic chirp data taken for a variety of implements at varying depths and speeds
v
are used to quantify the variation in the hitch parameter and to find the front and hitch
relaxation length values. Finally, a model which accounts for four-wheel drive forces is
derived, and experiments are taken which provide a preliminary look into the effect of
four-wheel drive traction forces on the yaw dynamics of the tractor.
In comparisons with other traditional models, the ?3-wheeled? FHRL Model is
shown to be superior in its steady state yaw rate tracking ability with an RMS error of
.245 deg/s vs. 1.96-2.07 deg/s for other models at a certain depth and also superior in its
dynamic tracking ability with an RMS error of .675 deg/s vs. .748-1.37 deg/s for the
other models. The experimental results from the hitch force dynamometer show that the
implement performs according to the linear tire model and that the moment caused by
differential forces at the hitch can be ignored. The hitch parameter,
h
C
?
, ranges from 452-
3385 N/deg for various implements and depths tested in this thesis. The front tire
relaxation length is found to be .37 m and the hitch relaxation length is found to be .4 m.
The four-wheel drive experiments show that using four-wheel drive provided an increase
in yaw rate from 9-21%, depending on the implement depth and speed.
vi
ACKNOWLEDGEMENTS
I would first of all like to thank God for His wisdom and understanding. I would
also like to thank Dr. Bevly and all of the current and past members of the GavLab- Matt
Heffernan, Will Travis, Rob Daily, Evan Gartley, Randy Whitehead, Warren Flenniken,
Rusty Anderson, Christof Hamm, John Wall, Kenny Lambert, Dustin Edwards, Wei
Huang, Mike Newlin, Benton Derrick, Ben Clark, Matt Lashley, and David Hodo for
their help, support, and companionship throughout the years.
I thank Deere & Co. for their support and donation of the use of the Deere 8420
and 8520 during this project. I also thank USDA and Dr. Randy Raper for the use of their
dynamometer. Thanks to Eric Schwabb and Dexter LaGrand. Thanks go to Dr. Jim
Bannon, and Bobby Durbin and all the folks at the EV Smith Research Station for their
help and use of land for experiments. I would like to thank the gang back home for all of
their support and love too- Mom, Dad, Jr., Sam, Katie, Sunny, Cody, Sissy, and Chad.
vii
Style of Journal Used:
ASME Journal of Dynamic Systems, Measurement, and Control
Computer Software Used:
Microsoft Word 2003
viii
TABLE OF CONTENTS
LIST OF FIGURES....................................................................................................... xi
LIST OF TABLES........................................................................................................ xiv
1. INTRODUCTION
1.1 Motivation.................................................................................................. 1
1.2 Background................................................................................................ 3
1.3 Purpose of Thesis and Contribution........................................................... 3
1.4 Outline of Thesis........................................................................................ 5
2. ANALYTICAL MODELING OF THE TRACTOR
2.1 Introduction................................................................................................ 6
2.2 The General Diagram................................................................................. 6
2.3 The ?3 Wheeled? Bicycle Model............................................................... 9
2.4 Other Models.............................................................................................. 15
2.4.1 Neutral Steer Model....................................................................... 15
2.4.2 Kinematic Model........................................................................... 17
2.4.3 Bicycle Model?............................................................................ 18
2.4.4 Front Tire Relaxation Length Model............................................. 18
2.4.5 ?3-wheeled? Front Tire Relaxation Length Model....................... 19
2.4.6 ?3-wheeled? Front and Rear Tire Relaxation Length Model........ 20
2.4.7 ?3-wheeled? Front, Rear, and Hitch Relaxation Length Model.... 21
2.4.8 ?3-wheeled? Front and Hitch Relaxation Length Model.............. 21
2.5 Model Comparisons................................................................................... 22
2.6 Conclusions................................................................................................ 29
ix
3. HITCH MODELING
3.1 Introduction................................................................................................. 31
3.2 Hitch Modeling. ......................................................................................... 31
3.3 Data Collection........................................................................................... 36
3.4 Hitch Model Validation.............................................................................. 42
3.5 Differential Hitch Forces?........................................................................ 47
3.6 Conclusions????................................................................................. 49
4. SOLVING OF PARAMETERS WITH SYSTEM IDENTIFICATION
4.1 Introduction................................................................................................ 50
4.2 Data Collection????........................................................................... 50
4.3 DC Gain Response..................................................................................... 53
4.3.1 Empirical DC Gain......................................................................... 53
4.3.2
Solving of
h
C
?
.................................................................................
56
4.4 Dynamic Response: Empirical Modeling................................................... 60
4.5 Conclusions................................................................................................ 74
5. MODELING OF FOUR-WHEEL DRIVE EFFECTS
5.1 Introduction................................................................................................ 76
5.2 Modeling Front Axle Drive Forces............................................................ 76
5.3 Effects of Using Four-Wheel Drive on Yaw Rate..................................... 79
5.4 Conclusions................................................................................................ 84
x
6. CONCLUSIONS
6.1 Summary................................................................................................... 85
6.2 Recommendation for Future Work........................................................... 87
REFERENCES.............................................................................................................. 88
APPENDICES............................................................................................................... 90
A Experimental and Data Acquisition Setup................................................. 91
B Model Parameter Values............................................................................ 96
xi
LIST OF FIGURES
1.1 Field Furrowed Using GPS Guidance............................................................... 2
1.2 Tractor Pulling a Cultivator............................................................................... 2
2.1 The General Diagram Depicting a 4WD Tractor with a Hitched Implement.... 7
2.2 ?3-wheeled? Bicycle Model Which Takes Into Account Front Traction
Forces................................................................................................................. 12
2.3 The ?3-wheeled? Bicycle Model FBD.............................................................. 12
2.4 The Traditional Bicycle Model.......................................................................... 16
2.5 Steering Angle Profile for Figure 2.6................................................................ 23
2.6 Yaw Rate Comparisons, 18 Inches Deep, 4 mph?????????.......... 24
2.7 Yaw Rate Comparisons, 12 Inches Deep, 4 mph.............................................. 24
2.8 Yaw Rate Comparisons, 8 Inches Deep, 4 mph............................................... 25
2.9 Dynamic Steering Maneuver??????????????????? 26
2.10 Yaw Rate Comparisons, Chirp 18 Inches Deep................................................ 27
2.11 Yaw Rate Comparisons, Chirp 18 Inches Deep................................................ 28
2.12 Yaw Rate Comparisons, Chirp 18 Inches Deep................................................ 28
3.1 How the Tire Model Relates to an Implement................................................... 32
3.2 Tire Schematic, courtesy of Gillespie [12]........................................................ 33
3.3 Tire Curve, courtesy of Gillespie [12]............................................................... 33
3.4 The Hitched Implement Schematic................................................................... 34
3.5 Steering Trajectory............................................................................................ 37
3.6 Tractor Position Trajectory............................................................................... 38
3.7 Diagram of Load Cells Measuring the Side Forces on USDA?s Dyno.............
39
3.8 Roll Moment Effect in F
yh
Calculations............................................................ 40
3.9 Lateral Force and Velocity taken from the Dyno.............................................. 41
xii
3.10
yh
F vs. Slip Angle at the Hitch for 6? Depth @ 4mph (individual run)??... 42
3.11
yh
F vs. Slip Angle at the Hitch for 12? Depth @ 4mph (individual run)......... 43
3.12
yh
F vs. Slip Angle at the Hitch for 18? Depth @ 4mph (individual run)?..... 43
3.13
yh
F vs. Slip Angle at the Hitch for 6? depth @ 4mph (runs combined).......... 44
3.14
yh
F vs. Slip Angle at the Hitch for 12? depth @ 4mph (runs combined)?.... 44
3.15
yh
F vs. Slip Angle at the Hitch for 18? depth @ 4mph (runs combined)........ 45
3.16
yh
F vs. Slip Angle at the Hitch for all depths @ 4mph.................................... 45
3.17
Correlation of Draft Force,
xh
F with
h
C
?
.........................................................
47
3.18 Yaw Moment from
yh
F about CG vs. Moment from Diff Forces at 18?......... 48
3.19 Yaw Moment from
yh
F about CG vs. Moment from Diff Forces at 12?......... 49
4.1 Steering Angle Profile and Yaw Rate Response for DC Gain Experiment....... 52
4.2 Steering Angle Profile and Yaw Rate Response for Dynamic Chirp
Experiment......................................................................................................... 53
4.3 Least Squares Fit of the Steady State Yaw Rate................................................ 54
4.4 Empirically Determined DC Gain for All Depths, 8420 with Four Shank
Ripper................................................................................................................. 55
4.5 Empirically Determined DC Gain for All Implements at All Depths, 8520..... 55
4.6 Empirical and Solved DC Gain Comparison for the 4 Shank Ripper on the
8420 at Various Depths......................................................................................
57
4.7 Empirical and Solved DC Gain Comparison for the 5 Shank Ripper on the
8520 at Various Depths...................................................................................... 58
4.8 Empirical and Solved DC Gain Comparison for the Bedder out of the
Ground, the Bedder in the Ground, and the Cultivator on the
8520................................................................................................................... 58
4.9 ETFE of the Cultivator at 1 mph....................................................................... 60
4.10 ETFE of the 4 Shank Ripper with 4
th
order Box Jenkins.................................. 62
4.11 DC Gain, Natural Frequency, and Damping Ratio for the 4 Shank Ripper at
4? of depth on the 8420, 4.=
f
? m, 002.=
h
? m.............................................
63
4.12 DC Gain, Natural Frequency, and Damping Ratio for the 4 Shank Ripper at
8? of depth on the 8420, 4.=
f
? m, 002.=
h
? m.............................................
63
4.13 DC Gain, Natural Frequency, and Damping Ratio for the 4 Shank Ripper at
12? of depth on the 8420, 4.=
f
? m, 002.=
h
? m...........................................
64
4.14 DC Gain, Natural Frequency, and Damping Ratio for the Bedder out of the
Ground on the 8520, 9.=
f
? m, 002.=
h
? m...................................................
64
xiii
4.15 DC Gain, Natural Frequency, and Damping Ratio for the Cultivator on the
8520, 9.=
f
? m, 002.=
h
? m...........................................................................
65
4.16 DC Gain, Natural Frequency, and Damping Ratio for the Bedder on the
8520, 9.=
f
? m, 002.=
h
? m...........................................................................
65
4.17 DC Gain, Natural Frequency, and Damping Ratio for the 5 Shank Ripper at
10? depth on the 8520, 9.=
f
? m, 002.=
h
? m...............................................
66
4.18 DC Gain, Natural Frequency, and Damping Ratio for the 5 Shank Ripper at
15? depth on the 8520, 9.=
f
? m, 002.=
h
? m...............................................
66
4.19 DC Gain, Natural Frequency, and Damping Ratio for the 5 Shank Ripper at
20? depth on the 8520, 9.=
f
? m, 002.=
h
? m...............................................
67
4.20
?3-wheeled? FHRL Model for the Cultivator on the 8520, 9.=
f
? m,
002.=
h
? m.......................................................................................................
68
4.21
?3-wheeled? FHRL Model for the Cultivator on the 8520, 37.=
f
? m,
4.=
h
? m...........................................................................................................
69
4.22 DC Gain, Natural Frequency, and Damping Ratio for the 4 Shank Ripper at
4? of depth on the 8420, 37.=
f
? m, 4.=
h
? m............................................
70
4.23 DC Gain, Natural Frequency, and Damping Ratio for the 4 Shank Ripper at
8? of depth on the 8420, 37.=
f
? m, 4.=
h
? m............................................
70
4.24 DC Gain, Natural Frequency, and Damping Ratio for the 4 Shank Ripper at
12? of depth on the 8420, 37.=
f
? m, 4.=
h
? m..........................................
71
4.25 DC Gain, Natural Frequency, and Damping Ratio for the Bedder out of the
Ground on the 8520, 37.=
f
? m, 4.=
h
? m..................................................
71
4.26 DC Gain, Natural Frequency, and Damping Ratio for the Cultivator on the
8520, 37.=
f
? m, 4.=
h
? m..........................................................................
72
4.27 DC Gain, Natural Frequency, and Damping Ratio for the Bedder on the
8520, 37.=
f
? m, 4.=
h
? m..........................................................................
72
4.28 DC Gain, Natural Frequency, and Damping Ratio for the 5 Shank Ripper at
10? depth on the 8520, 37.=
f
? m, 4.=
h
? m...............................................
73
4.29 DC Gain, Natural Frequency, and Damping Ratio for the 5 Shank Ripper at
15? depth on the 8520, 37.=
f
? m, 4.=
h
? m...............................................
73
4.30 DC Gain, Natural Frequency, and Damping Ratio for the 5 Shank Ripper at
20? depth on the 8520, 37.=
f
? m, 4.=
h
? m...............................................
74
5.1 ?3-wheeled? 4-WD Bicycle Model Schematic.................................................. 77
5.2 Tire Force Schematic......................................................................................... 78
5.3 6 Shank Paratil Attached to the 8420................................................................. 80
5.4 Representative Individual Data Run with Linear Fit, 2WD 9? Depth............... 81
xiv
5.5 Comparison of 4WD vs. 2WD Yaw Rates per Steering Angle, 9? Depth, 4
mph.................................................................................................................... 82
5.6 Comparison of 4WD vs. 2WD Yaw Rates per Steering Angle, 13? Depth, 4
mph................................................................................................................. 82
5.7 Comparison of 4WD vs. 2WD Yaw Rates per Steering Angle, 17? Depth,
1.5 mph.............................................................................................................. 83
A.1 Experimental Test Tractor- John Deere 8420.................................................... 92
A.2 Versalogic Data Acquisition Computer............................................................. 93
A.3 Steering Angle Sensor....................................................................................... 94
A.4 Inertial Measurement Unit................................................................................. 94
A.5 Starfire GPS Receiver........................................................................................ 95
A.6 Hitch Force Dynamometer................................................................................. 95
xv
LIST OF TABLES
2.1 The General Diagram Parameters?????................................................ 8
2.2 The ?3-wheeled? Bicycle Model Specific Parameters.................................... 13
2.3 RMS Errors (deg/s) of Models at Different Depths, Static.............................. 25
2.4 RMS Errors (deg/s) of Models, Dynamic Response........................................ 29
3.1
Empirically Determined
h
C
?
Values on the 8420 with a Deere 955 4 Shank
Ripper............................................................................................................... 46
4.1 Data Collection, 8420....................................................................................... 51
4.2 Data Collection, 8520....................................................................................... 51
4.3
Empirically Determined
h
C
?
Values for all the Implements...........................
59
4.4 Front and Hitch Relaxation Length Values...................................................... 67
5.1 Values from 4WD Analysis............................................................................. 83
B.1 The ?3 Wheeled? Bicycle Model Parameters.................................................. 97
B.2
h
C
?
Values for all the Implements..................................................................
98
1
CHAPTER 1
INTRODUCTION
1.1 Motivation
Agriculture is the backbone of our modern society. Our stores are filled with
items produced from crops grown in this country and elsewhere around the world.
Needless to say, farming is important and so are the tools used in the process. Tractors
are one of the necessary tools used in farming, and as technology has increased over the
years, so has the level of technology in tractors. Modern tractors are starting to use GPS
tracking and automated metering systems that allow farmers to be more efficient not only
with the time it takes to do a job but also with wasting less product and raw material. As
GPS tracking systems become more competitive and advanced, better understanding of
the tractor and implement behavior is important in enabling the tracking systems to be
more accurate and have better overall performance. Figure 1.1 shows an image of a field
furrowed using GPS guidance.
2
Figure 1.1: Field Furrowed Using GPS Guidance
Gaining a better understanding of tractor and implement behavior through
mathematical modeling is the motivation behind the research in this thesis. A more
accurate system model naturally creates a more accurate control system that is based on
that model. Figure 1.2 shows a tractor pulling a hitched cultivator.
Figure 1.2: Tractor Pulling a Cultivator
3
1.2 Background
Much yaw dynamic modeling of tractors has been done in the past for the purpose
of tractor control. Many of the models are based on the Traditional Bicycle (TB) Model
form. For example, a simplification of the TB model was used by the researchers in [1] to
form a simple kinematic model. Also, Rekow [2] used assumptions to form a simpler first
order model called the Neutral Steer (NS) Model. Alternately, O?Connor [3] looked at a
model used by Ellis [4] that neglects lateral tire dynamics and also a similar bicycle
model that uses Wong?s tire model [5]; both of these models assume constant forward
velocity. Owen and Bernard used a TB Model with added front and rear tire relaxation
lengths for studying a tractor-loader-backhoe [6], while Bevly found that just adding
front tire relaxation lengths was adequate for his modeling [7].
Additionally, tractor-implement models have been looked at by different
researchers. A tractor towed-implement dynamic model was developed by Bell [8]. Bevly
developed a tractor towed-implement model based on the simple kinematic model [9].
Feng developed a tractor towed-implement model by adding a towed implement model to
the TB model [10]. A tractor hitched implement model developed by O?Connor used
Wong?s tire model to describe the hitch dynamics [3]. The O?Connor model is similar to
the ?3-wheeled? Bicycle Model developed in this thesis in that the hitched implement
forces are generated using a tire model; however, the derivations of the two models are
different. For example, the O?Connor model is a five state model, where the ?3-wheeled?
Bicycle Model developed in this thesis is a two state model. Bukta collected data on a
tractor with and without hitched implements to determine the effects of hitch free-play on
the tractor?s dynamics [11].
4
1.3 Purpose of Thesis and Contribution
The purpose of this thesis is to derive a yaw dynamic model which can capture
and quantify the effects that a hitched implement such as a ripper imposes on the tractor.
Deriving such a model will provide a more accurate model which can potentially improve
a controller using this model.
In this thesis, a number of both traditional and non-traditional models are derived
and then compared for yaw rate tracking ability using experimental data as a basis. A
model is found in which hitched implement conditions can be accounted for, and a great
improvement in yaw rate tracking ability in both steady state and dynamic conditions is
seen vs. traditional models.
Experiments are taken where implement forces are recorded using a hitch force
dynamometer. The data are used to validate the way the hitched implement forces are
derived in the new hitched implement yaw dynamic model. The experiments show that
the way the implement forces are derived is reasonably correct.
Steady state and dynamic chirp data are taken for a variety of implements at
varying depths and speeds. The steady state data are used to quantify the variation in the
hitch parameter from implement to implement and depth to depth for the chosen hitched
implement yaw dynamic model. The dynamic chirp data are used to solve for other
unknown parameters of the new model. Once these parameters are known, a time and
frequency analysis is done on the new hitched implement yaw dynamic model to gain
understanding of its characteristics.
5
A model which accounts for four-wheel drive forces is also derived in this thesis.
Experiments are taken which provide a preliminary look into the effect of four-wheel
drive traction forces on the yaw dynamics of a tractor. A definite difference in using two-
wheel drive vs. four-wheel drive is shown.
1.4 Outline of Thesis
Chapter 2 presents a general model and then derivations of several new and
traditional analytical yaw dynamic models for a tractor. The models are then compared
for yaw rate tracking ability, and a model which best predicts the yaw rate of a tractor
with a hitched implement is chosen. Chapter 3 presents a validation of the hitched
implement lateral force equation. In Chapter 4, steady state experiments are used to
quantify the variations of the hitched implement parameter in the new hitched implement
yaw dynamic model for various implements at varying depths. Dynamic chirp
experiments are used to solve for other parameters of the new hitched implement yaw
dynamic model. Chapter 5 presents a derivation of a model which accounts for the four-
wheel drive traction forces present in the yaw dynamics of a four-wheel drive tractor.
Experiments are used to show how much the yaw rate is actually affected in a four-wheel
drive vs. non four-wheel drive setup.
6
CHAPTER 2
ANALYTICAL MODELING OF THE TRACTOR
2.1 Introduction
In this chapter, a variation of the bicycle model, termed the ?3 wheeled? Bicycle
Model is developed. This model takes into account the effects of a hitched implement.
This model with added front and hitch relaxation lengths is then compared with various
models used by other researchers in vehicle applications. It is shown that this model can
most accurately represent the dynamics of a tractor with a hitched implement.
2.2 The General Diagram
The general diagram from which all the models in this chapter can be derived is
shown in Figure 2.1. The vectors and angles in this figure show a positive sign
convention, which associates positive forces with negative slip angles. The diagram
represents a 4WD tractor with a hitched implement. In this general diagram, the hitched
implement is modeled as two tires at the hitch designating that the left and right sides of
the implement may have differing conditions. Table 2.1 describes the variables found in
Figure 2.1. Equations (2.1-2.3) represent the dynamic equations of motion for the tractor,
which are derived from the general diagram by summing all the forces and moments
7
acting on the tractor. Equation (2.2) takes into account the lateral acceleration from total
vehicle sideslip through the )cos(???? rVm term.
xmFFFFFFF
rTfrlTflxhrxhlTrrTrlx
&&=?+?+??+=
?
)cos()cos( ??
(2.1)
)sin()sin(
rTfrlTflyrryrlyhryhlyy
FFFFFFamF ?? ?+?++++=?=
?
)cos()cos()cos( ??? ???+=?+?+ rVmVmFF
yryfrlyfl
&
(2.2)
aFaFaFaFM
ryfrlyflrTfrlTflCG
??+??+??+??=
?
)cos()cos()sin()sin( ????
2
)sin(
2
)sin(
222
)cos(
2
)cos(
t
F
t
F
t
F
t
F
t
F
t
F
ryfrlyflTrrTrlrTfrlTfl
??+??????+?????+ ????
rILFLFbFbFcbFcbF
zxhrxhlyrryrlyhryhl
&?=?+???+??+??+??
11
)()(
(2.3)
Where
z
I is the mass moment of inertia of the tractor and implement.
Figure 2.1: The General Diagram Depicting a 4WD Tractor with a Hitched Implement
8
Table 2.1: The General Diagram Parameters
yfl
F &
yfr
F Front tire lateral forces for the left and right side of the vehicle,
respectively
yrl
F &
yrr
F Rear tire lateral forces
yhl
F &
yhr
F
Implement lateral forces
Tfl
F &
Tfr
F
Front tire traction forces
Trl
F &
Trr
F
Rear tire traction forces
xhl
F
&
xhr
F
Implement draft forces
l
? &
r
? Steering angles
fl
? &
fr
?
Slip angles for the front tires
rl
? &
rr
?
Slip angles for the rear tires
hl
? &
hr
? Slip angles for the hitched implement
fl
V
&
fr
V
Front tire velocities
rl
V &
rr
V Rear tire velocities
hl
V &
hr
V Hitch velocities
V Vehicle velocity
x
V Vehicle forward velocity
y
V
Vehicle lateral velocity
? Vehicle sideslip angle
r Yaw rate
a Distance from center of front wheels to center of gravity
b Distance from center of rear wheels to center of gravity
c Distance from hitch to center of rear wheels
t Track width of the tractor
1
L Distance between centers of force action on implement
9
2.3 The ?3 Wheeled? Bicycle Model
The ?3-wheeled? Bicycle Model is developed from Figure 2.1 by making the
following simplifications. It is natural to make as many simplifications as possible
because this leads to a less computationally intensive and/or linear model.
The tire sideslip angle is defined as the angle between the longitudinal axis of the
tire and the velocity vector at the tire. Therefore, the tire slip angles can be calculated
from the longitudinal and lateral velocity at the tire as shown in Equations (2.4-2.9).
?
?
?
?
?
?
?
?
?
?
?
?
??
??
=
?
rtV
rbV
x
y
rl
2
1
tan
1
?
(2.4)
?
?
?
?
?
?
?
?
?
?
?
?
?+
??
=
?
rtV
rbV
x
y
rr
2
1
tan
1
?
(2.5)
l
x
y
fl
rtV
raV
?? ?
?
?
?
?
?
?
?
?
?
?
?
?
??
?+
=
?
2
1
tan
1
(2.6)
r
x
y
fr
rtV
raV
?? ?
?
?
?
?
?
?
?
?
?
?
?
?
?+
?+
=
?
2
1
tan
1
(2.7)
?
?
?
?
?
?
?
?
?
?
?
?
??
?+?
=
?
rtV
rcbV
x
y
hl
2
1
)(
tan
1
?
(2.8)
?
?
?
?
?
?
?
?
?
?
?
?
?+
?+?
=
?
rtV
rcbV
x
y
hr
2
1
)(
tan
1
?
(2.9)
If
x
V >>
rt?
2
1
, and assuming left and right steering angles are the same, =
l
?
r
? , then
10
?
fl
? ??? ?
?
?
?
?
?
?
?
? ?+
==
?
x
y
ffr
V
raV
1
tan
(2.10)
?
rl
?
?
?
?
?
?
?
?
?
??
==
?
x
y
rrr
V
rbV
1
tan??
(2.11)
?
hl
?
?
?
?
?
?
?
?
? ?+?
==
?
x
y
hhr
V
rcbV )(
tan
1
??
(2.12)
Also, if vehicle sideslip angle,? is assumed to be small, then
?? ?
?+
=
x
y
f
V
raV
(2.13)
x
y
r
V
rbV ??
=?
(2.14)
x
y
h
V
rcbV ?+?
=
)(
?
(2.15)
Assuming the same tires are on the left and right sides and that there is no weight
transfer from left to right enables
yfl
F
tireyfyfr
FF
_
=?
(2.16)
yrl
F
tireyryrr
FF
_
=?
(2.17)
so that
tireyfyf
FF
_
2?=
(2.18)
tireyryr
FF
_
2?=
(2.19)
The left and right tire traction forces are assumed to be the same.
11
tireTfTfrTfl
FFF
_
=?
(2.20)
tireTrTrrTrl
FFF
_
=?
(2.21)
tireTftrac
FF
_
2?=
(2.22)
tireTrTr
FF
_
2?=
(2.23)
The left and right draft forces are also assumed to be the same.
sidexhxhlxhr
FFF
_
=?
(2.24)
This allows any moment from the difference in the forces, shown below in Equation
(2.25), to be dropped out.
11
LFLFM
xhlxhrdiff
???=
(2.25)
Validation of this assumption is given in Chapter 3. Also, the total implement draft force
becomes that defined by Equation (2.26)
sidexhxh
FF
_
2?=
(2.26)
The left and right implement lateral forces are also assumed to be the same, and the total
lateral implement force becomes that shown in Equation (2.28)
sideyhyhlyhr
FFF
_
=?
(2.27)
sideyhyh
FF
_
2?=
(2.28)
12
Tractor mass and center of gravity location are assumed constant. Because
diff
M
is ignored and yaw modeling, not longitudinal modeling, is being done, the summation of
forces in the longitudinal axis is not needed and is left out. If the front traction forces are
not neglected, a FBD for the ?3-wheeled? Bicycle Model which takes into account four-
wheel drive effects can be created and is shown in Figure 2.2. This model is specifically
dealt with in Chapter 5.
Figure 2.2: ?3-wheeled? Bicycle Model Which Takes Into Account Front Traction Forces
However, neglecting the four-wheel drive (front axle) traction forces and using
the other simplifications allow Figure 2.3 to be used as the FBD of the ?3-wheeled?
Bicycle Model.
Figure 2.3: The ?3-wheeled? Bicycle Model FBD
13
Table 2.2: The ?3-wheeled? Bicycle Model Specific Parameters
h
?
Hitch side slip angle
yh
F
Lateral force at the hitch
h
C
?
Hitch cornering stiffness
r
? Tractor rear tire side slip angle
f
?
Tractor front tire side slip angle
r
C
?
Tractor rear tire cornering stiffness, per axle
f
C
?
Tractor front tire cornering stiffness, per axle
? Steering angle
yr
F
Lateral force on the rear tractor tire
yf
F
Lateral force on the front tractor tire
r
V
Velocity of the rear tire
f
V
Velocity of the front tire
Summing the forces in the lateral and vertical axes of the tractor yields Equations (2.29-
2.30).
)cos()cos( ?? ???+=?++=?=
?
rVmVmFFFamF
yyfyryhyy
&
(2.29)
rIbFcbFaFM
zyryhyfCG
&?=??+????=
?
)()cos(?
(2.30)
The small ? assumption causes 1)cos( ?? and assuming small steering angles
allows 1)cos( ?? , so that Equations (2.29-2.30) become Equations (2.31-2.32).
14
yfyryhyy
FFFamrVmVm ++=?=??+
&
(2.31)
)( cbFbFaFrI
yhyryfz
+?????=? &
(2.32)
Equations (2.33-2.35) represent the forces at the tires and are derived from the linear tire
model given in Equation (3.3).
ffyf
CF ?
?
??=
(2.33)
rryr
CF ?
?
??=
(2.34)
hhyh
CF ?
?
??=
(2.35)
Where
tireff
CC
_
2
??
?=
(2.36)
tirerr
CC
_
2
??
?=
(2.37)
Substituting Equations (2.33-2.35) into Equations (2.31-2.32) yields Equations (2.38-
2.39) shown below.
ffrrhhy
CCCrVmVm ???
???
??????=??+
&
(2.38)
aCbCcbCrI
ffrrhhz
?????++??=? ???
???
)(&
(2.39)
Using the small ? assumption to say VV
x
? , substituting Equations (2.13-2.15) into
Equations (2.38-2.39), and organizing the resulting equations into state space form yields
the state space form of the ?3-wheeled? Bicycle Model, which is given in Equation
(2.40).
15
( ) ( )( )
() ()
?
?
?
??????
??????
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?+?+?+?
?
???+?+
?
?
???+?+
?
++?
=
?
?
?
?
?
?
z
f
f
y
z
frh
z
frh
frhfrh
y
I
Ca
m
C
r
V
VI
CaCbCcb
VI
CaCbCcb
V
Vm
CaCbCcb
Vm
CCC
r
V
222
&
&
(2.40)
This state space model can be transformed using a Laplace transformation to yield
the transfer function given in Equation (2.41). This transfer function has an input of
steering angle and an output of yaw rate.
?
?
?
?
?
?
+
??
+?
?
?
?
?
?
+
?
+?
?+?
+?
=
1
2
2
132322
21
)(
)(
)(
C
mV
CCC
s
V
C
mV
IC
sI
mV
CaCCC
saC
s
sr
z
z
ff
f
??
?
?
(2.41)
Where
() )(
1 frh
CaCbCcbC
???
???+?+=
( )
frh
CCCC
???
++=
2
() )(
222
3
aCbCCcbC
frh ???
+++=
(2.42)
2.4 Other Bicycle Models
This section lists alternative models used in vehicle dynamics. These other models
have been developed so that the ?3-wheeled? Bicycle Model can be compared to models
used in previous research.
2.4.1 The Traditional Bicycle Model
The Traditional Bicycle Model is a more rudimentary form of the ?3-wheeled?
Bicycle Model [12]. It can be also viewed in the sense that the ?3-wheeled? Bicycle
Model is a more complicated version of the Traditional Bicycle Model. Both models are
16
derived in the same manner from Figure 2.1 except that the Bicycle Model neglects
implement effects in any axis. Figure 2.4 is the free body diagram for the Traditional
Bicycle Model. Equation (2.43) is the resulting state space form for this model.
( ) ( )
()()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?+??
?
???
?
?
???
?
+?
=
?
?
?
?
?
?
z
af
af
y
z
afar
z
afar
afarafar
y
I
Ca
m
C
r
V
VI
CaCb
VI
CaCb
V
Vm
CaCb
Vm
CC
r
V
22
&
&
(2.43)
Likewise, Equation (2.44) is the transfer function form of Equation (2.43) derived from a
Laplace transformation.
?
?
?
?
?
?
+
??
+?
?
?
?
?
?
+
?
+?
?+?
+?
=
1
2
2
132322
21
)(
)(
)(
C
mV
CCC
s
V
C
mV
IC
sI
mV
CaCCC
saC
s
sr
z
z
ff
f
??
?
?
(2.44)
Where
)(
1 afar
CaCbC ???=
( )
afar
CCC +=
2
)(
22
3
aCbCC
fr ??
+=
(2.45)
Figure 2.4: The Traditional Bicycle Model
?
f
?
r
?
yf
F
yr
F
r
y
V
x
V
f
V
r
V
a b
V
17
2.4.2 Neutral Steer Bicycle Model
The Neutral Steer Bicycle Model is a special case of the Traditional Bicycle
Model where the understeer gradient is equal to zero [2]. A neutral steer vehicle is
defined as a vehicle whose ratio of the weight on the front wheels divided by the
cornering stiffness of the front tire is equal to the ratio of the weight on the rear wheels
divided by the cornering stiffness of the rear tire. A oversteer vehicle tends to spin out in
cornering while an understeer vehicle tends to plow in cornering and a neutral steer
vehicle does neither. The understeer gradient is defined in Equation (2.46).
r
r
f
f
us
C
W
C
W
k
??
?=
(2.46)
where
f
W and
r
W are the weights at the front and rear axles, respectively. Setting the
understeer gradient to zero yields
r
r
f
f
C
W
C
W
??
=
(2.47)
f
W and
r
W in Equation (2.47) can be substituted to get
rf
C
agm
C
bgm
??
??
=
??
(2.48)
This in turn yields
0=???
araf
CbCa
(2.49)
18
Substituting Equation (2.49) into Equation (2.43) allows simplifications to be made and
gives the transfer function form of the Neutral Steer Bicycle Model, shown below.
VI
aCbC
s
I
aC
s
sr
z
fr
z
f
22
)(
)(
??
?
? +
+
=
(2.50)
As can be seen, this model is a first order model and is simpler that the Traditional
Bicycle Model, given the simulated vehicle is approximately neutral steer.
2.4.3 Kinematic Model
The Kinematic Model is also a special case of the Traditional Bicycle Model as it
neglects vehicle and wheel sideslip and assumes a constant forward velocity. It provides a
purely kinematic relationship from steering angle to yaw rate by assuming that the yaw
rate is directly proportional to the steering angle for a slow moving vehicle [1]. Equation
(2.51) represents the transfer function form of the Kinematic Model.
L
V
s
sr
=
)(
)(
?
(2.51)
Where baL += , the wheelbase length.
2.4.4 Front Tire Relaxation Length Model
The Front Tire Relaxation Length (FRL) model is a more complicated version of
the Traditional Bicycle Model. As the name denotes, this model has front tire relaxation
lengths added to the bicycle model. The tire relaxation length,? , is the amount a tire
must roll in order to generate the steady state slip angle,
0
? , at the tire [7].
The equation describing the tire relaxation length is a first order model and is
shown in Equation (2.52).
19
()
ffo
f
x
f
V
??
?
? ??=&
(2.52)
Where
f
? is the front tire relaxation length. Incorporating this equation into the
equations of motion for the bicycle model and arranging them into the state space form of
the front tire relaxation length model yields Equation (2.53).
?
?
?
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
??
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
f
f
y
fff
z
af
z
ar
z
ar
af
arar
f
y
V
r
V
Va
I
Ca
VI
Cb
VI
Cb
m
C
V
Vm
Cb
Vm
C
r
V
0
0
1
2
&
&
&
(2.53)
The first use of this particular model was seen by Bevly in [9]. He showed that only a
front tire relaxation length was adequate for his modeling, whereas the authors in [6] used
a model with front and rear relaxation lengths. Bevly also showed that the Front Tire
Relaxation Length Model was necessary to adequately describe the tractor?s handling
dynamics where a Traditional Bicycle Model was not adequate.
2.4.5 ?3 Wheeled? Front Tire Relaxation Length Model
The ?3 Wheeled? Front Tire Relaxation Length Model (?3-wheeled? FRL Model)
is similar to the FRL Model in that a front tire relaxation length is added to the ?3-
wheeled? Bicycle Model instead of the Traditional Bicycle Model. The equations of
motion for the ?3-wheeled? Front Tire Relaxation Length Model placed in state space
form are shown in Equation (2.54).
20
()( )( )
() ()
?
?
?
???
?
?
????
?
????
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?+?+?
?
?+?+
?
?
?
?+?+
?
+?
=
?
?
?
?
?
?
?
?
?
?
f
f
y
fff
z
f
z
rh
z
rh
f
rhrh
f
y
V
r
V
Va
I
Ca
VI
CbCcb
VI
CbCcb
m
C
V
Vm
CbCcb
Vm
CC
r
V
0
0
1
22
&
&
&
(2.54)
2.4.6 ?3 Wheeled? Front and Rear Tire Relaxation Length Model
The model shown below in Equation (2.55) is the ?3-wheeled? Bicycle Model
with added front and rear tire relaxation lengths (?3-wheeled? FRRL Model). This model
and the next model derived are for investigating if only a front tire relaxation length is
adequate as shown by [9]. However, it may be the case where front relaxation lengths are
not adequate to describe the dynamics, but a model with front plus rear or hitch
relaxations is adequate. The equation describing the rear tire relaxation length is shown
below.
()
rro
r
x
r
V
??
?
? ??=&
(2.55)
Where
r
? is the rear tire relaxation length. Both Equation (2.52) and Equation (2.55) are
incorporated into Equations (2.38-39) to develop the state space model for the ?3-
wheeled? FRRL Model shown in Equation (2.56).
(2.56)
( ) ()( )
() ()
?
?
?
?
???
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
??
?
?+?
?
?+
?
?
?
?
?+
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
0
0
0
0
1
0
1
2
f
r
f
y
rrr
fff
z
ar
z
af
z
ah
z
ah
ar
afhaha
r
f
y
V
r
V
Vb
Va
I
Cb
I
Ca
VI
Ccb
VI
Ccb
m
C
m
C
V
Vm
Ccb
Vm
C
r
V
&
&
&
&
21
2.4.7 ?3 Wheeled? Front, Rear, and Hitch Relaxation Length Model
The ?3-wheeled? Bicycle Model with added front and rear tire and hitch
relaxation lengths (?3-wheeled? FRHRL Model) is similar to the ?3-wheeled? FRRL
Model except hitch relaxation lengths are also added. The equation describing the hitch
relaxation length,
h
? , is shown below.
()
hho
h
x
h
V
??
?
? ??=&
(2.57)
Equation (2.52), Equation (2.55), and Equation (2.57) are incorporated with the
?3-wheeled? Bicycle Model?s equations, and arranging in state space yields the state
space form of the ?3-wheeled? FRHRL Model, shown in Equation (2.58) below.
(2.58)
2.4.8 ?3 Wheeled? Front and Hitch Relaxation Length Model
The ?3-wheeled? Bicycle Model with added front and hitch relaxations is similar
to the ?3-wheeled? FRHRL Model except the rear relaxation lengths are left out.
Integrating the respective relaxation lengths with Equations (2.38-39) of the ?3-wheeled?
Bicycle Model yields Equation (2.59).
?
?
?
?
?
???
???
???
?
?
? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?+?
??
?
?+?
??
??
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
0
0
0
0
00
)(1
00
1
00
1
)(
00
0
f
h
r
f
y
hhh
rrr
fff
z
ah
z
ar
z
af
ahar
af
h
r
f
y
V
r
V
Vcb
Vb
Va
I
Ccb
I
Cb
I
Ca
m
C
m
C
m
C
V
r
V
&
&
&
&
&
22
(2.59)
2.5 Model Comparisons
All of the models in the previous section are compared against yaw rate data taken
on a Deere 8420 with single rear wheels and a four shank ripper rigidly hitched to the
rear. A full description of the tractor, sensors, and data acquisition system is given in
Appendix A. Appendix B gives the model parameter values used in the simulation of
these models. The steering angle profile for the yaw rate data is illustrated in Figure 2.5
and shows that the steering maneuver creates a series of steady state yaw rate steps. The
speed for this maneuver was four miles per hour. It should be noted that during a steady
state maneuver, relaxation length terms drop out, the FRL Model breaks down into the
Traditional Bicycle Model, and the ?3-wheeled? FRL, FRRL, FRHRL, and FHRL
Models break down into the ?3-wheeled? Bicycle Model. Therefore, the FRL and the ?3-
wheeled? FRL, FRRL, FRHRL, and FHRL Models are left out of this particular
comparison. Figures 2.6-2.8 show the tracking response of the remaining models at
depths of 18, 12, and 8 inches, respectively. As can be seen, the ?3-wheeled? Bicycle
Model has the best yaw rate tracking response. This visual inspection is also backed by
the fact that the ?3-wheeled? Bicycle Model has the lowest RMS errors at each depth,
shown in Table 2.3 [13]. Note that with decreasing depth, the errors of the Traditional,
( ) ( )
()()
()
()
?
?
?
?
???
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?+?
?
+?
??
?
??
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
0
0
0
0
1
0
1
2
f
h
f
y
hhh
fff
z
ah
z
af
z
ra
z
ra
ah
afrara
h
f
y
V
r
V
Vcb
Va
I
cbC
I
Ca
VI
bC
VI
bC
m
C
m
C
V
Vm
bC
Vm
C
r
V
&
&
&
&
23
Neutral Steer, and Kinematic Bicycle Models decrease. Following this trend, a property
of the ?3-wheeled? Bicycle Model is that it collapses to the Traditional Bicycle Model in
the event that there is no implement, which is the case where the hitch cornering stiffness
is zero.
Figure 2.5: Steering Angle Profile for Figure 2.6
24
Figure 2.6: Yaw Rate Comparisons, 18 Inches Deep, 4 mph
Figure 2.7: Yaw Rate Comparisons, 12 Inches Deep, 4 mph
25
Figure 2.8: Yaw Rate Comparisons, 8 Inches Deep, 4 mph
Table 2.3: RMS Errors (deg/s) of Models at Different Depths, Static
Depth 18? 12? 8?
?3-wheeled? Bicycle Model .24499 .4515 .49637
Traditional Bicycle Model 1.9634 1.1505 .52977
Neutral Steer Bicycle Model 1.8116 1.0508 .55075
Kinematic Model 2.0736 1.2389 .6319
Since the Kinematic, Traditional, and Neutral Steer Bicycle and FRL Models do
not have adequate steady state tracking response while using an implement at depth, they
are thrown out of the study at this point. It is repeated that the focus of this research is to
capture and study the hitched loading effects with a given model.
26
The models left to study are the ?3-wheeled? Bicycle Model, the ?3-wheeled?
FRL Model, the ?3-wheeled? FRRL Model, the ?3-wheeled? FHRL Model, and the ?3-
wheeled? FRHRL Model. These models can be differentiated with a dynamic steering
maneuver. The dynamic steering maneuver chosen for this comparison is a chirp steering
input shown in Figure 2.9. This maneuver also uses a four shank ripper at a depth of 18
inches at a speed of four miles per hour.
Figure 2.9: Dynamic Steering Maneuver
As a single figure would have too many models to compare at once, the
comparison is broken down over several plots, Figures 2.10-2.12. Figure 2.10 is a
comparison of the ?3-wheeled? FRL Model and the ?3-wheeled? FRRL Model. As
shown by Bevly in [9], the two models have approximately the same response; therefore,
the ?3-wheeled? FRRL Model is thrown out of the study at this point since it is the more
complex of the two. Figure 2.11 is a comparison of the ?3-wheeled? FHRL Model and
the ?3-wheeled? FRHRL Model. Similar to the FRL and FRRL Models of Figure 2.10,
these two models have approximately the same response, so the ?3-wheeled? FRHRL
Model is also thrown out of the study at this point. Figure 2.12 is a comparison of the
27
?3-wheeled? Bicycle Model, the ?3-wheeled? FRL Model, and the ?3-wheeled? FHRL
Model. As can be seen, the model that includes hitch relaxation lengths has the best
response, especially in the higher frequencies. This visual inspection is also backed by
the ?3-wheeled? FHRL Model having the lowest RMS error, given in Table 2.4.
Figure 2.10: Yaw Rate Comparisons, Chirp 18 Inches Deep
28
Figure 2.11: Yaw Rate Comparisons, Chirp 18 Inches Deep
Figure 2.12: Yaw Rate Comparisons, Chirp 18 Inches Deep
29
Table 2.4: RMS Errors (deg/s) of Models, Dynamic Response
?3-wheeled? FHRL Model .67457
?3-wheeled? FRL Model .74815
?3-wheeled? Bicycle Model 1.3691
2.6 Conclusions
A number of different analytical mathematical models have been derived to show
their state-space form. All of these models are based on a general diagram, and making
different assumptions leads to each particular model. The ?3-wheeled? Bicycle Model
format is a unique derivation that is basically a bicycle model that includes hitched
implement effects. A number of models can be derived by adding front tire, rear tire, or
hitch relaxation lengths to the Bicycle Model or the ?3-wheeled? Bicycle Model.
Based on steady state responses, the Traditional Bicycle, Neutral Steer Bicycle,
Kinematic, and FRL Models are shown to have inadequate yaw rate tracking ability and
therefore are not considered further for the purposes of this research. The ?3-wheeled?
Bicycle, ?3-wheeled? FRL, ?3-wheeled? FRRL, ?3-wheeled? FRHRL, and ?3-wheeled?
FHRL Models have been shown to have excellent yaw rate tracking in steady state.
Based on dynamic responses, the ?3-wheeled? FRL and ?3-wheeled? FRRL
Models have approximately the same response, similar to that shown by [9]. Also, the ?3-
wheeled? FRHRL and ?3-wheeled? FHRL Models have approximately the same
response. Therefore, the ?3-wheeled? FRL and ?3-wheeled? FHRL Models are chosen
over the ?3-wheeled? FRRL and ?3-wheeled? FRHRL Models, respectively.
30
Finally, the ?3-wheeled? Bicycle, ?3-wheeled? FRL, and ?3-wheeled? FHRL
Models are compared against each other with the same dynamic steering maneuver, and it
is shown that the ?3-wheeled? FHRL Model provides the best static and dynamic yaw
rate tracking response.
31
CHAPTER 3
HITCH MODELING
3.1 Introduction
This chapter validates the assumption of the implement forces being able to be
modeled by the linear tire model. The validation is accomplished through taking force
data from a hitch force dynomometer provided by the USDA-ARS Soil Dynamics
Laboratory and comparing it to the sideslip of the implement to verify that the
relationship between the two is as assumed. This chapter also shows that neglecting
diff
M
, as done by the ?3-wheeled? Bicycle Model derivation, is a valid assumption.
3.2 Hitch Modeling
The following figure is a graphical representation of how a tire compares to an
implement at the hitch.
32
Figure 3.1: How the Tire Model Relates to an Implement
According to the linear tire model given in Equation (3.1), a force affecting the
yaw dynamics of a vehicle is present only when the tire is experiencing a slip angle [12].
?
?
??= CF
y
(3.1)
Slip angle is defined as the angle between the lateral and longitudinal velocities of the
body in motion, shown in Equation (3.2).
?
?
?
?
?
?
?
?
=
?
xbody
ybody
body
V
V
1
tan?
(3.2)
Figure 3.2 below shows a schematic of a tire with the associated lateral force. As
can be seen, a deformation of the contact patch causes a slip angle, ? , and a lateral force,
y
F . The
y
F and ? of Figure 3.2 are governed by the tire curve shown in Figure 3.3.
y
F peaks and becomes non-linear when it reaches a certain slip angle. This peak is
determined by ground conditions. Also, within small ? , the
y
F is linear with the slope
of the line defined by
?
C .
?
C is dependent on tire characteristics such as tread, wall
stiffness, tire pressure, etc.
33
Figure 3.2: Tire Schematic, courtesy of Gillespie [12]
Figure 3.3: Tire Curve, courtesy of Gillespie [12]
Likewise, the ?3-wheeled? Bicycle Model assumes that with an implement, in the
same manner as a tire, the lateral force is proportional to the slip angle of the implement
so that Equation (3.1) becomes Equation (3.3).
34
hhyh
CF ?
?
??=
(3.3)
Section 3.4 shows that this relationship holds and the assumption is valid.
Additionally, the longitudinal forces on the implement are assumed to be proportional to
the implement?s longitudinal velocity.
Figure 3.4 shows the schematic used to analyze the implement forces. Recall
from Chapter 2 that the implement left and right lateral forces and slip angles are lumped
together. This schematic demonstrates how the forces generated on the hitched
implement affect the center of gravity of the tractor. Figure 3.4 does not assume that the
left and right draft forces are the same. It should be noted, that for the ?3-wheeled?
Bicycle Model in Chapter 2, left and right draft forces are assumed to be the same, and
therefore,
diff
M
= 0, an assumption validated in Section 3.5.
Figure 3.4: The Hitched Implement Schematic
The FBD in Figure 3.4 is used to analyze the effect that the implement?s forces
and moments have on the equations of motion when forming the ?3-wheeled? Bicycle
35
Model. Summing the moments of the implement about the center of gravity of the tractor
results in Equation (3.4), below.
( ) LFLFcbFM
xhlxhryhtCGimplemen
???++?=
(3.4)
Where:
hhyh
CF ?
?
??=
(3.5)
hrxxhr
VCF ?=
(3.6)
hlxxhl
VCF ?=
(3.7)
Note that
x
C is a draft coefficient and is assumed to be equal for both sides.
Substituting Equations (3.5-3.7) into Equation (3.4) yields:
( ) LVCLVCcbCM
hlxhrxhhtCGimplemen
?????++???= ?
?
(3.8)
Where:
LrVV
xhr
??=
(3.9)
LrVV
xhl
?+=
(3.10)
Substituting Equations (3.9-3.10) into Equation (3.8) yields:
( ) ( ) ( )[ ]LrVLrVLCcbCM
xxxhhtCGimplemen
?+?????++???= ?
?
( )
2
2 LrCcbC
xhh
???+???= ?
?
(3.11)
The term
diff
M
=
2
2 LrC
x
??? from Equation (3.11) represents the moment
caused by the difference between the left and right longitudinal forces,
xhl
F
and
xhr
F
.
There will always be a longitudinal draft force,
xh
F
, as long as the implement is in the
36
ground. However, if the left and right draft forces are not significantly different, they will
not affect the tractor?s yaw dynamics.
Equations (3.12-3.13) summarize the effect of the implement forces on the tractor
if
diff
M
is ignored and
yh
F
can be modeled as a tire.
hhyh
CF ?
?
??=
(3.12)
( )cbCM
hhtCGimplemen
+???= ?
?
(3.13)
3.3 Data Collection
Figures 3.5 and 3.6 show the steering profile and test trajectory for the data
collection experiments used to validate the implement model. The experiments are
designed such that distinct lateral forces can be recorded for distinct implement slip
angles. In order to validate Equation (3.12), the lateral forces and also the slip angles
must be recorded so that the relationship between the two can be determined. It should be
noted here that
h
? was calculated from the recorded yaw rate based on the kinematics of
the tractor. The equation used to calculate
h
? is shown in Equation (3.14) below.
( )
?
?
?
?
?
?
?
? ??+?
=
?
x
x
h
V
Vcbr )sin(
tan
1
?
?
(3.14)
Where () )sin(???+?
x
Vcbr is the lateral velocity at the hitch,
y
V with )sin(??
x
V
being subtracted out as the lateral velocity caused by the total sideslip of the vehicle. The
? was calculated by subtracting the GPS course measurement from the heading by
integrating the yaw rate measurement shown in Equation (3.15) below.
37
?
?= r
GPS
??
(3.15)
The experiments are taken at varying depths for the purpose of defining the
different implement cornering stiffnesses,
h
C
?
, and are limited to one speed due to data
collection time constraints. Data was collected at 4 mph at depths of 6?, 12?, and 18? on a
Deere 955 four-shank ripper. Multiple data runs were collected at each depth with the
ripper. Appendix A details the experimental setup for these data collection runs.
Figure 3.5: Steering Trajectory
38
Figure 3.6: Tractor Position Trajectory
Figure 3.7 shows the configuration of the two load cells used to calculate the
lateral force,
yh
F
. They are referred to as SU4 and SL5, respectively and are 20,000 lb
cells that are positive in compression.
39
Figure 3.7: Diagram of Load Cells Measuring the Side Forces on USDA?s Dyno
The
yh
F
shown in Figure 3.7 is in the positive direction for the ?3-wheeled?
Bicycle Model, with SL5 in tension and SU4 in compression. The roll moment on the
dynomometer caused by the implement must be taken into consideration when using the
data from these two load cells. This is due to the fact that the roll moment imparts a force
into each of the cells. For a positive
yh
F
, the SU4 data will be positive and the SL5 data
will be negative. A roll moment decreases the positive SU4 value and increases the
negative SL5 value by the same amount. The lateral force,
yh
F
, is calculated according to
Equation (3.16). The roll effect force cancels out because what was subtracted from SU4
has been added to SU5.
54 SLSUF
yh
?=
(3.16)
40
Figure 3.8 is a graphical demonstration of how the roll moment cancels out when
the measured
yh
F
is calculated.
Figure 3.8: Roll Moment Effect in
yh
F
Calculations
Figure 3.9 shows the measured
yh
F
data obtained from the dynomometer on a
particular data run. The lateral forces, as seen in the graph, increase with an increase in
steering angle.
41
Figure 3.9: Lateral Force and Velocity taken from the Dyno
Slip angle data is also used for the hitch model validation. The slip angle for the
implement is calculated using the GPS data and yaw gyro data according to Equations
(3.17-3.20).
?
?= r
GPS
??
(3.17)
( )?sin?=VV
y
(3.18)
( )cbrVV
yyh
+?+=
(3.19)
?
?
?
?
?
?
?
?
=
?
x
yh
h
V
V
1
tan?
(3.20)
Where
GPS
? is the GPS course measurement.
42
3.4 Hitch Model Validation
The collected force and slip angle data are compared in this section to determine
if their relationship can be approximated by the linear tire model. Figures 3.10-3.12 show
individual data runs of
yh
F
vs. slip angle for each depth at the test speed of 4 mph. Plots
of the combined data runs fitted with a linear fit, for each depth respectively, are shown
in Figures 3.13-3.15. Although the data runs show a high dispersion, a linear relationship
in the data can still be seen. Figure 3.16 shows the data for each depth plotted together.
Based on the linear fits of Figure 3.16, the slope
h
C
?
increases with the depth of the
implement as one would expect. This is due to the fact that as the depth increases, there is
more resistance to turning, meaning that more lateral force is created for a given
h
? per
each depth respectively, and an increasing
h
C
?
.
Figure 3.10:
yh
F
vs. Slip Angle at the Hitch for 6? Depth @ 4mph (individual run)
43
Figure 3.11:
yh
F
vs. Slip Angle at the Hitch for 12? Depth @ 4mph (individual run)
Figure 3.12:
yh
F
vs. Slip Angle at the Hitch for 18? Depth @ 4mph (individual run)
44
Figure 3.13:
yh
F
vs. Slip Angle at the Hitch for 6? depth @ 4mph (runs combined)
Figure 3.14:
yh
F
vs. Slip Angle at the Hitch for 12? depth @ 4mph (runs combined)
45
Figure 3.15:
yh
F
vs. Slip Angle at the Hitch for 18? depth @ 4mph (runs combined)
Figure 3.16:
yh
F
vs. Slip Angle at the Hitch for all depths @ 4mph
46
Table 3.1 lists the values of
h
C
?
obtained for each depth using the dynamometer.
It should be noted that these are empirically determined cornering stiffnesses strictly
based on the slope of the fitted
yh
F
line of Figure 3.16.
Table 3.1: Empirically Determined
h
C
?
Values on the 8420 with a
Deere 955 4 Shank Ripper
Depth, inches
h
C
?
, N/deg
6 534
12 937
18 1647
Comparing the lines of Figure 3.16 with Figure 3.3, the linear tire curve, it can be
seen that the hitch forces do not peak, even at rather high slip angles like the tire forces
do. This is most likely due to the fact that an implement has parts that actually stick into
the ground instead of riding on top like a tire. Also, where relaxation lengths are
concerned, a small relaxation length at the implement could occur due to slop in the 3
point hitch and/or from deformation of the dirt due to the geometry of the tine in the
ground.
On a different note, the implement draft force,
xh
F
, can be seen to be a function
of
h
C
?
, as shown in Figure 3.17 below. Moreover, it appears that there is a linear
relationship between the two.
47
Figure 3.17: Correlation of Draft Force,
xh
F
with
h
C
?
for the 4 Shank Ripper on
the 8420
3.5 Differential Hitch Forces
The data used to validate the
yh
F
model can also be used to look at the moment
created from differential hitch forces. Figure 3.18 below shows the moment from the
lateral force about the tractor?s center of gravity,
CG
M
from Equation (3.13), compared
with the moment caused by the differential forces,
diff
M
from a dyno run at a depth of
18? on the four shank ripper. Figure 3.19 shows a run at 12?. As can be seen in the
figures, the moment acting upon the tractor is dominated by the moment from the lateral
force. The data are similar for all runs at 18? and for all runs at 12?. Based on these
48
results, the moment from the differential forces at the hitch is negligible compared to the
effect of the lateral force moment, and neglecting
diff
M
in the ?3-wheeled? Bicycle
Model structure will not significantly affect its performance. This validation is also true
for the ?3-wheeled? FHRL Model because it uses the ?3-wheeled? Bicycle Model in its
structure.
Figure 3.18: Yaw Moment from
yh
F
about CG vs. Moment from Diff Forces at 18?
49
Figure 3.19: Yaw Moment from
yh
F
about CG vs. Moment from Diff Forces at 12?
3.6 Conclusions
This chapter provides a more in depth derivation of the hitched implement model.
It has been shown through the lateral force vs. slip angle data that the implement can be
modeled as a linear tire. It has also been shown through experimental data that the
moments due to differential longitudinal loading on the implement are negligible
compared to the lateral implement forces. Therefore, this chapter has shown that the ?3-
wheeled? bicycle model, developed in the previous chapter, accurately represents the
hitched implement and makes reasonable assumptions about the hitched implement
model.
50
CHAPTER 4
SOLVING OF PARAMETERS WITH SYSTEM IDENTIFICATION
4.1 Introduction
This chapter validates the models chosen in the previous chapter against the real
tractor. Two types of data have been collected on a variety of implements at varying
depths and speeds for this purpose: DC gain data and dynamic chirp data. The DC gain
data is used in conjunction with the ?3-wheeled? Bicycle Model to solve for the ranges in
h
C
?
with each implement and depth. The chirp data is used to create a system
identification model in order to compare the dynamics of the ?3-wheeled? Front and
Hitch Relaxation Length (FHRL) Model and validate that it captures the critical dynamic
responses of the real system. The front and hitch relaxation lengths are also solved for
using the dynamic response data.
4.2 Data Collection
A John Deere 8420 tractor with single rear wheels and also a Deere 8520 with
dual rear wheels were used to gather the DC gain and dynamic data. The experimental
setup for these experiments is detailed in Appendix A. Table 4.1 lists the depths and
speeds for data collected on a four shank ripper with the 8420. Table 4.2 lists the depths,
51
speeds, and implements for the data that was taken on the 8520. The implements used
with the 8520 were a 5 shank ripper, an 18 bottom cultivator, an 11 row bedder, and the
bedder out of the ground. The difference in using a single vs. dual wheel tractor for
experiments is that the rear tire cornering stiffness is doubled because the dual rear wheel
tractor has twice as many tires.
Table 4.1: Data Collection, 8420
Implement: 4 Shank Ripper Speed, mph
Depths: 4, 8, 12? 2.5 3 3.5 4 4.5 5 5.5
Table 4.2: Data Collection, 8520
Implement Depth at Given Speed
1 mph 2.5 mph 4 mph 5.5 mph
Bedder Out of Gnd -- -- -- --
11 Row Bedder 9? 9? 9? 9?
18 Bottom Cultivator 9? 9? 9? 9?
5 Shank Ripper 10, 15, 20? 10, 15, 20? 10, 15, 20? 10, 15, 20?
The DC gain experiments consisted of a series of steady state step steering inputs
of increasing magnitude to create a series of steady state step yaw outputs. Using these
experiments, the DC gain from steady state steer angle to steady state yaw rate is
identified empirically. Figure 4.1 shows the steering angle and resulting yaw rate of a
typical DC gain run at a speed of 2.5 mph.
52
Figure 4.1: Steering Angle Profile and Yaw Rate Response for DC Gain Experiment
While a steady state step steering input is useful for identifying the steady state
characteristics, a chirp steering input is useful for identifying the dynamic response of the
tractor [14]. A chirp steering input consists of applying a sine wave of increasing
frequency to the steering wheels. Figure 4.2 shows a typical chirp steering angle profile
and resulting yaw rate for the dynamic response experiments. As can be seen, the steering
angle is not a true chirp signal at the wheels, which does affect the system identification.
This is most likely due to the steering servo and hydraulic system on the tractor not being
able to react fast enough to create a true chirp signal at high frequencies.
53
15 20 25 30 35 40 45 50
-20
-10
0
10
20
time, s
s
t
ee
r
i
n
g
an
gl
e,
de
g
15 20 25 30 35 40 45 50
-5
0
5
time, s
y
a
w
r
a
te
, d
e
g
/
s
e
c
Figure 4.2: Steering Angle Profile and Yaw Rate Response for Dynamic Chirp
Experiment
4.3 DC Gain Response
4.3.1 Empirical DC Gain
The steering angle input and the yaw rate output from the steady state data is used
to find the empirical DC gain of the system. Least squares fits are performed to determine
the empirical DC gain of the system at each depth over the range of speed at that depth
for each implement [15]. Equation (4.2) models the measured yaw rate as a function of
DC gain times steering angle, a gyro bias, and white noise.
noiserGr
biasDCmeas
++?= ?
(4.2)
54
Note that Equation (4.2) is only valid in steady state. The DC gain and gyro bias were
identified using recorded steering angles and yaw rates for each configuration and speed.
After the DC gain is identified, it can be validated by comparing the fit to the
measured response. Figure 4.3 shows a typical result for this validation. The fit from
Figure 4.3 has an RMS error of .1726 deg/sec, where the sensor noise is the largest
contributor to the magnitude of the RMS error.
Figure 4.3: Least Squares Fit of the Steady State Yaw Rate
The empirical DC gain fits obtained for the 8420 and four shank ripper at three
different depths are shown in Figure 4.4. Figure 5.5 shows the empirical DC gain fits for
all the implements, depths, and speeds with the 8520.
55
Figure 4.4: Empirically Determined DC Gain for All Depths, 8420 with Four Shank
Ripper
Figure 4.5: Empirically Determined DC Gain for All Implements at All Depths, 8520
56
4.3.2 Solving of
h
C
?
Once the empirical DC gains for each implement, depth, and velocity are found,
the
h
C
?
which best fits empirical DC gain values is found. Equation (4.2) is an analytical
equation for the DC gain and is derived from Equation (2.33) of the ?3-wheeled? Bicycle
Model.
()
)(
)()(
12
2
123
21
C
mV
CCC
mV
CaCCC
Gdc
ff
+
??
?+?
=
??
(4.2)
Where C
1
, C
2
, and C
3
were defined previously in Equation (2.34), but are shown below
for reference.
() )(
1 frh
CaCbCcbC
???
???+?+=
( )
frh
CCCC
???
++=
2
() )(
222
3
aCbCCcbC
frh ???
+++=
The above equation for DC gain is only a function of
h
C
?
and velocity since the
other parameters are already known. Therefore,
h
C
?
can be solved for by numerically
minimizing Equation (4.3).
2/1
2
1
))()(_(
1
?
?
?
?
?
?
??=
?
N
h
CGdcNempiricalGdc
N
E
?
(4.3)
Figure 4.6 shows the results of the minimization for the four shank ripper used on
the 8420. The values obtained for
h
C
?
at each depth are also shown: 451.65 N/deg at four
57
inches, 1002.8 N/deg at eight inches, and 1719.3 N/deg at twelve inches of depth. Figure
4.7 and 4.8 show the results from the implements used on the 8520. The
h
C
?
values
obtained on the five shank ripper are: 887.3 N/deg at ten inches, 1025 N/deg at fifteen
inches, and 3385.3 N/deg at twenty inches of depth. The value obtained for the bedder
out of the ground is .983 N/deg, which is approximately 0, as should be in the case where
there is no cornering stiffness from the implement. The value obtained for the eighteen
tine cultivator is 639.8 N/deg, and the value for the eleven row bedder is 951.2 N/deg.
The large jump in
h
C
?
at the deepest depth in Figure 4.7 may be due to variation in soil
compaction.
Figure 4.6: Empirical and Solved DC Gain Comparison for the 4 Shank Ripper on the
8420 at Various Depths
58
Figure 4.7: Empirical and Solved DC Gain Comparison for the 5 Shank Ripper on the
8520 at Various Depths
Figure 4.8: Empirical and Solved DC Gain Comparison for the Bedder out of the Ground,
the Bedder in the Ground, and the Cultivator on the 8520.
59
Based on Figures 4.6-4.8, a trend in the value of the hitch cornering stiffness can
be seen. As would be expected, the cornering stiffness increases with the size of the
implement and also with the depth of each implement respectively. Additionally,
h
C
?
increases with the draft load as shown by Figure 3.17. This trend corresponds to an
implement providing a greater resistance to turning the deeper that it is in the ground and
also corresponds to an implement with more or larger tines in the ground being harder to
turn.
Table 4.3 summarizes the values of
h
C
?
obtained for all of the implements and
their respective depths.
Table 4.3: Empirically Determined
h
C
?
Values for all the Implements
Implement Depth, inches
h
C
?
, N/deg
8420, 4 Shank Ripper 4 451.65
8420, 4 Shank Ripper 8 1002.80
8420, 4 Shank Ripper 12 1719.30
8520, Bedder Out of Gnd 0 0.98 ~ 0.00
8520, Cultivator 9 639.83
8520, Bedder 9 951.24
8520, 5 Shank Ripper 10 887.33
8520, 5 Shank Ripper 15 1025.00
8520, 5 Shank Ripper 20 3385.30
60
4.4 Dynamic Response: Empirical Modeling
The interpretation and results from using the dynamic chirp input and output
responses of Figure 4.2 and from the implements and depths of Tables 4.1 and 4.2 are
discussed in this section.
Experiments using a chirp input are useful for determining the dynamic
characteristics of a system since the chirp signal excites a range of frequencies. This
allows the magnitude and phase shift of the output vs. the input over that range of
frequencies to be analyzed. This can be accomplished by taking an Empirical Transfer
Function Estimate (ETFE) of the data sets using Matlab?s ?etfe( )? command [14]. Figure
4.9 shows an ETFE for the cultivator at a speed of one mile per hour. The large jump in
noise in the graph past 100 rad/s is due to lack of input excitation frequency. This ETFE
represents the general shape and form for all of the data sets recorded.
Figure 4.9: ETFE of the Cultivator at 1 mph
61
Since the ETFE is a purely empirical model, a system identification model is
fitted for each ETFE. For this research, a Box Jenkins model was chosen for the system
identification model [16]. )(sG of Equation (4.4) represents the real tractor system,
while the Box Jenkins models represents )(
?
sG , the approximated system.
)(
?
)(
)(
sG
s
sr
=
?
(4.4)
Figure 4.10 shows the ETFE of the four shank ripper with a fourth order Box
Jenkins identified model plotted alongside. Fourth order Box Jenkins models were found
to best fit the data sets in Tables 4.1 and 4.2. The first resonant peak of Figure 4.10
represents the dominant dynamics of the tractor (which is of most interest); it is uncertain
what dynamics are creating the second resonance. Both test tractors had an independent
front suspension which may be causing this second resonance.
62
10
0
10
1
10
2
10
-1
10
0
A
m
pl
i
t
ude
10
1
10
2
-300
-200
-100
0
P
has
e (
degr
ees
)
Frequency (rad/s)
ETFE
BJ
Figure 4.10: ETFE of the 4 Shank Ripper with 4
th
order Box Jenkins
Figures 4.11-4.19 show plots of the empirical DC gains, natural frequencies, and
damping ratios vs. velocity of the tractor-implement combination calculated from the Box
Jenkins Models. It should be noted that the natural frequencies and damping ratios are
those associated with the first resonant peaks. These figures also show the analytical
values attained by adjusting the front and hitch relaxation lengths of the ?3-wheeled?
FHRL Model. Recall, however, that the DC gain values are not a function of the
relaxation lengths. The DC gain values are for reference only and are the values shown
for the empirical gains in the previous section.
63
Figure 4.11: DC Gain, Natural Frequency, and Damping Ratio for the 4 Shank Ripper at
4? of depth on the 8420, 4.=
f
? m, 002.=
h
? m
Figure 4.12: DC Gain, Natural Frequency, and Damping Ratio for the 4 Shank Ripper at
8? of depth on the 8420, 4.=
f
? m, 002.=
h
? m
64
Figure 4.13: DC Gain, Natural Frequency, and Damping Ratio for the 4 Shank Ripper at
12? of depth on the 8420, 4.=
f
? m, 002.=
h
? m
Figure 4.14: DC Gain, Natural Frequency, and Damping Ratio for the Bedder out of the
Ground on the 8520, 9.=
f
? m, 002.=
h
? m
65
Figure 4.15: DC Gain, Natural Frequency, and Damping Ratio for the Cultivator on the
8520, 9.=
f
? m, 002.=
h
? m
Figure 4.16: DC Gain, Natural Frequency, and Damping Ratio for the Bedder on the 8520,
9.=
f
? m, 002.=
h
? m
66
Figure 4.17: DC Gain, Natural Frequency, and Damping Ratio for the 5 Shank Ripper at
10? depth on the 8520, 9.=
f
? m, 002.=
h
? m
Figure 4.18: DC Gain, Natural Frequency, and Damping Ratio for the 5 Shank Ripper at
15? depth on the 8520, 9.=
f
? m, 002.=
h
? m
67
Figure 4.19: DC Gain, Natural Frequency, and Damping Ratio for the 5 Shank Ripper at
20? depth on the 8520, 9.=
f
? m, 002.=
h
? m
The resulting values for the front and hitch relaxation lengths are summarized in
Table 4.4, below. The value of the front relaxation length for the four shank ripper at all
depths on the 8420 was a lower value than that needed for the 8520 at all its implements
and depths.
Table 4.4: Front and Hitch Relaxation Length Values
Tractor, All Implements Front Relaxation Length
Value (m)
Hitch Relaxation
Length Value (m)
8420 .9 .002
8520 .4 .002
68
Although the relaxation lengths given in Table 4.3 match the experimental natural
frequency, they do not capture the experimental trend in damping ratio. Additionally,
these relaxation length values led to an RMS error (between experimental and model yaw
rates) of 1.92 deg/s when analyzed in the same manner as Figure 2.12. Figure 4.20 shows
the Bode velocity response of the ?3-wheeled? FHRL Model at these relaxation length
values, and it can be seen that the damping decreases with velocity, which is opposite of
what Figures 4.11-4.19 denote.
Figure 4.20: ?3-wheeled? FHRL Model for the Cultivator on the 8520, 9.=
f
? m,
002.=
h
? m
This problem was remedied by using the relaxation length values that were used
to obtain an RMS error of .67457 for Figure 2.12. These values are 37.=
f
? m and
4.=
h
? m. Using these values in the ?3-wheeled? FHRL Model also produces damping
69
that increases with velocity, seen in the Bode plots of Figure 4.21. Shown in Figures
4.22-30 are the DC gain, natural frequency, and damping ratio comparisons for all of the
data sets. As seen in these figures, using values of .37 and .4 m for the front and hitch
relaxation lengths yields natural frequencies that are relatively close to the values from
the experimental fits.
Figure 4.21: ?3-wheeled? FHRL Model for the Cultivator on the 8520, 37.=
f
? m,
4.=
h
? m
70
Figure 4.22: DC Gain, Natural Frequency, and Damping Ratio for the 4 Shank Ripper at
4? of depth on the 8420, 37.=
f
? m, 4.=
h
? m
Figure 4.23: DC Gain, Natural Frequency, and Damping Ratio for the 4 Shank Ripper at
8? of depth on the 8420, 37.=
f
? m, 4.=
h
? m
71
Figure 4.24: DC Gain, Natural Frequency, and Damping Ratio for the 4 Shank Ripper at
12? of depth on the 8420, 37.=
f
? m, 4.=
h
? m
Figure 4.25: DC Gain, Natural Frequency, and Damping Ratio for the Bedder out of the
Ground on the 8520, 37.=
f
? m, 4.=
h
? m
72
Figure 4.26: DC Gain, Natural Frequency, and Damping Ratio for the Cultivator on the
8520, 37.=
f
? m, 4.=
h
? m
Figure 4.27: DC Gain, Natural Frequency, and Damping Ratio for the Bedder on the 8520,
37.=
f
? m, 4.=
h
? m
73
Figure 4.28: DC Gain, Natural Frequency, and Damping Ratio for the 5 Shank Ripper at
10? depth on the 8520, 37.=
f
? m, 4.=
h
? m
Figure 4.29: DC Gain, Natural Frequency, and Damping Ratio for the 5 Shank Ripper at
15? depth on the 8520, 37.=
f
? m, 4.=
h
? m
74
Figure 4.30: DC Gain, Natural Frequency, and Damping Ratio for the 5 Shank Ripper at
20? depth on the 8520, 37.=
f
? m, 4.=
h
? m
4.5 Conclusions
In this chapter, steady state data were used to solve for the empirical DC gains of
the tractor on a number of implements at a variety of depths and speeds. A minimization
function was used to find
h
C
?
for each implement at each depth. Dynamic chirp data was
used to find the ETFE estimates of each data set, and then fourth order Box Jenkins
models were fitted to the ETFE?s. The Box Jenkins models were characterized and the
DC gains, natural frequencies, and damping ratios were plotted for each implement and
depth. Initial values for the front and hitch relaxation lengths were found to be inadequate
in modeling the tractor. Therefore, values of front and hitch relaxation lengths were
75
selected that produced similar damping characteristics as the Box Jenkins models and that
also gave the least RMS error values when compared in a dynamic yaw rate tracking
scenario as seen in Chapter 2. The values for the front and hitch relaxation lengths were
found to be 37.=
f
? m and 4.=
h
? m, and the values of
h
C
?
for the various
implements ranged from 0 N/deg to 3385 N/deg.
76
CHAPTER 5
MODELING OF FOUR-WHEEL DRIVE EFFECTS
5.1 Introduction
A tractor?s towing ability is greatly increased by the addition of a driven front
axle. However, since the front axle is usually the steered axle, traction forces affect the
tractor?s yaw dynamics. In this chapter, a lateral model is derived which takes into
account the front axle traction forces of a four-wheel drive tractor. Data has been
collected on a tractor both with 4WD on and off. The difference in the respective yaw
rates is shown.
5.2 Modeling Front Axle Drive Forces
This section derives the ?3-wheeled? Bicycle Model where the four-wheel drive
traction forces are not ignored and takes up where Chapter 2 left off. The FBD for the ?3-
wheeled? Bicycle Model with four-wheel drive traction forces of Figure 2.2 is shown
again below as Figure 5.1. The ?3-wheeled? Bicycle Model with four-wheel drive
traction forces is now termed the ?3-wheeled? 4-WD Bicycle Model (3W4BM).
77
Figure 5.1: ?3-wheeled? 4-WD Bicycle Model Schematic
Summing the forces in the Y and Z axes of the tractor yields Equations (5.1-5.2).
)cos()sin()cos( ??? ???+=?+?++=?=
?
rVmVmFFFFamF
ytracyfyryhyy
&
(5.1)
rIaFbFcbFaFM
ztracyryhyfCG
&?=??+??+????=
?
)sin()()cos( ??
(5.2)
The small ? assumption causes 1)cos( ?? and assuming small steering angles
allows 1)cos( ?? , so that Equations (5.1-5.2) become Equations (5.3-5.4).
??+++=?=??+
tracyfyryhyy
FFFFamrVmVm
&
(5.3)
aFcbFbFaFrI
tracyhyryfz
??++?????=? ?)(&
(5.4)
trac
F is defined below in Equation (5.5)
=
trac
F SlipC
x
%?
(5.5)
Where Slip% is the percent of slip along the longitudinal axis of the tire and
x
C is the
longitudinal tire stiffness.
x
C is dependent on the particular tire design, but not on tire-
soil conditions. A diagram of the tire forces can be seen in Figure 5.2 below. The tractive
78
forces of a tire are highly dependent on the tire-soil interaction which affects the Slip%.
It should be noted that Equation (5.5) is valid only below a slip of approximately 10%.
Figure 5.2: Tire Force Schematic
Substituting Equations (2.33-2.35) into Equations (5.3-5.4) yields Equations (5.5-5.6).
????
???
?+??????=??+
tracffrrhhy
FCCCrVmVm
&
(5.5)
aFaCbCcbCrI
tracffrrhhz
??+?????++??=? ????
???
)(&
(5.6)
Using the small ? assumption to say VV
x
? , substituting Equations (2.13-2.15) into
Equations (5.5-5.6) and organizing the resulting equations into state space form yields
Equations (5.7-5.8).
( )
?
???????
?
?
?
?
?
?
?
?
?
+
+
?
?
?
?
?
?
?
?
?
++?
?+
?
?
?
?
?
?
?
?
?
?
???++?
?=
m
CF
Vm
CCC
VV
Vm
aCbCcbC
rV
ftracfrh
y
frh
y
)(
&
(5.7)
()
?
???????
?
?
?
?
?
?
?
?
? ?+?
+
?
?
?
?
?
?
?
?
?
???++?
?+
?
?
?
?
?
?
?
?
?
?+?++??
?=
z
ftrac
z
frh
y
z
frh
I
aCaF
VI
aCbCcbC
V
VI
aCbCcbC
rr
)()(
222
&
(5.8)
79
Rearranging Equations (5.7-5.8) and assuming
trac
F is a steady state value yields the
state-space form of the ?3-wheeled? 4-WD Bicycle Model, shown in Equation (5.9).
( )
()
?
?
?
??????
??????
?
?
?
?
?
?
?
?
?
?
?
?
?
?+?
+
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?+?++??
?
???++?
?
?
???++?
?
++?
=
?
?
?
?
?
?
z
ftrac
ftrac
y
z
frh
z
frh
frhfrh
y
I
aCaF
m
CF
r
V
VI
aCbCcbC
VI
aCbCcbC
V
Vm
aCbCcbC
Vm
CCC
r
V
222
)()(
)(
&
&
(5.9)
The transfer function form of Equation (5.9) is shown in Equation (5.10), below.
( ) ( ) ( ) ( )( )
Den
aCbCcbCCCCaCFsaCaFVm
s
sr
frhfrhftracftrac
???++?+++??++??+???
=
????????
?
)(
)(
)(
(5.10)
Where
( ) ( )( ) +??+?++??++++???= saCbCcbCmCCCsVmIDen
frhfrhz
2222
)(
??????
()( )+?+?++??++
222
)( aCbCcbCCCC
frhfrh ??????
( )
2
)()( aCbCcbCaCbCcbCVm
frhfrh
???++?+???++???
??????
(5.11)
5.3 Effects of Using Four-Wheel Drive on Yaw Rate
Data from a typical setup using a hitched implement is recorded with and without
using the four-wheel drive to determine the effect on yaw rate of the traction forces in
nominal conditions. Data was taken on the 8420 with a six shank Paratil at depths of 9,
13, and 17 inches at speeds of 4 mph and 1.5 mph at a depth of 17 inches. The Paratil can
be seen in Figure 5.3.
80
Figure 5.3: 6 Shank Paratil Attached to the 8420
Figure 5.4 shows a typical data set where yaw rate has been plotted vs. steering
angle at the wheel. Figure 5.4 has been fitted with a linear fit and is representative of the
other data runs at the respective depths and speeds. Figure 5.5 shows the results of
comparing yaw rates at 4 mph and 9 inches of depth for both the 4WD on and off. The
blue shaded area represents the difference in yaw rate between the two options. A
summary of the difference in yaw rate slope vs. steering angle is given in Table 5.1.
There is approximately an 8.8% difference in yaw rate between having the 4WD on vs.
off at 9 inches depth and 4 mph with the Paratil. Figure 5.6 compares yaw rates with and
without 4WD on the Paratil at a depth of 13 inches and a speed of 4mph. There is a
difference of 20.6% between the respective yaw rates. Figure 5.7 compares yaw rates at a
81
depth of 17 inches and a speed of 1.5 mph. The forward velocity on this data set was
traction limited by the tractor in 2WD mode. The difference in yaw rates for 17 inches of
depth is 15.4%. Intuition would lead one to think that there should be more difference in
yaw rates as the depth increases such as for the depths of 9 and 13 inches. One reason for
17 inches being a 15.4% difference and 13 inches being a 20.6% difference is that the
velocity at 17 inches is only 1.5 mph instead of 4 mph.
Figure 5.4: Representative Individual Data Run with Linear Fit, 2WD 9? Depth
82
Figure 5.5: Comparison of 4WD vs. 2WD Yaw Rates per Steering Angle, 9? Depth,
4 mph
Figure 5.6: Comparison of 4WD vs. 2WD Yaw Rates per Steering Angle, 13? Depth,
4 mph
83
Figure 5.7: Comparison of 4WD vs. 2WD Yaw Rates per Steering Angle, 17? Depth,
1.5 mph
Table 5.1: Values from 4WD Analysis
Depth,
inches
Speed, mph Yaw Rate
Slope, 1/s
% Difference in 4WD
ON vs. OFF
% Diff/ Velocity
9 4 .34 4WD
.31 2WD
8.8 2.20
13 4 .34 4WD
.27 2WD
20.6 5.15
17 1.5 .13 4WD
.11 2WD
15.4 10.27
84
5.4 Conclusions
A model which takes into account the front axle traction forces has been
developed and termed the ?3-wheeled? 4-WD Bicycle Model. Data were collected on a
Paratil at three depths and two speeds, both with and without the four-wheel drive. This
data shows that under typical conditions, using four-wheel drive does significantly affect
the yaw dynamics of the tractor. Using four-wheel drive provided an increase in yaw rate
from 9-21%, depending on the depth and speed.
85
CHAPTER 6
CONCLUSIONS
6.1 Summary
A number of mathematical models for the dynamics of a tractor with a hitched
implement has been developed and compared for accuracy. A model has been developed
which has the ability to capture changing implement conditions through a hitch cornering
stiffness term. The model used to capture the hitched implement forces has been verified
to be reasonably correct. The hitch cornering stiffness term has been solved for through a
minimization for a variety of implements at varying depths. A model has also been
developed which can take into account the effects of using four-wheel drive on the yaw
dynamics. A summary of each chapter given in this thesis is provided below.
In Chapter 2, a general diagram is shown from which many tractor vehicle models
can be developed. The ?3-wheeled? Bicycle Model is developed which can account for
changing hitched implement conditions. A model is also developed where front and hitch
relaxation lengths are added to the ?3-wheeled? Bicycle Model (FHRL Model). Also a
number of models used in previous research are derived for a comparison. It is shown
that the FHRL Model breaks down into the ?3-wheeled? Bicycle Model under steady
state conditions. Additionally, under steady state and dynamic steering maneuvers, the
FHRL model provides the most accurate yaw rate tracking ability.
86
In Chapter 3, the Linear Tire Model used in modeling the hitch forces of the
FHRL Model is verified. Analysis of experimental data shows that the lateral hitch force
vs. slip angle of the implement is relatively linear and can be represented by the Linear
Tire Model.
In Chapter 4, steady state experiments using various implements at various depths
are used to derive empirical DC gain data. This empirical DC gain data are used to solve
for the hitch cornering stiffness term for the varying implements at their varying depths.
It is shown that the trends in the hitch cornering stiffness values behave as expected when
related to real world behavior. Dynamic steering experiments taken with the various
implements and their varying depths are used to derive empirical system identification
models for the implements at each respective depth. The system identification models are
used to find the front tire and hitch relaxation lengths of the FHRL Model.
In Chapter 5, a ?3-wheeled? 4-WD Bicycle Model is developed which takes into
account front axle traction forces in a four-wheel drive tractor. Experimental data show
that under typical conditions, using four-wheel drive does significantly affect the yaw
dynamics of the tractor. Using four-wheel drive provided an increase in yaw rate from 9-
21% for a 6 shank Paratil, depending on depth and speed.
6.2 Recommendations for Future Work
Varying ground conditions such as ground moisture, type and compaction also
affect the amount of lateral force an implement generates. These varying conditions need
to be studied to find their effect on the implement model. The work in this thesis only
87
considers hitched implement conditions. Research could also be conducted to include
towed implements, articulated tractors, skid steer, and even rear steer tractors.
The four-wheel drive modeling should be researched further. It was shown that
using four-wheel drive increases the yaw rate at a given steering angle by a significant
amount. It needs to be determined whether this difference in yaw rate is captured by the
analytical model. It could be the case where physically using four-wheel drive on the
tractor reduces the slip losses enough so that the actual yaw rate and the predicted yaw
rate from a non four-wheel drive model match since slip is neglected in the model.
88
REFERENCES
[1] O?Connor, M., Bell, T., Elkaim, G., and Parkinson, B. ?Automatic
Steering of Farm Vehicles Using GPS,? in Proc. 3rd Int. Conf.
Precision Farming, June 1996 pp. 767?777.
[2] Rekow, A., System Identification, Adaptive Control, and Formation Driving of
Farm Tractors, Ph.D. Dissertation, Stanford University, March 2001.
[3] O?Connor, M.L., Carrier-Phase Differential GPS for Automatic Control of Land
Vehicles, Ph.D. Dissertation, Stanford University, December 1997.
[4] Wong, J.Y. Theory of Ground Vehicles. Wiley, New York, 1978.
[5] Ellis, J.R. Vehicle Dynamics. London Business Books Ltd., 1969.
[6] Owen, R.H., and Bernard, J.E., ?Directional Dynamics of a Tractor-Loader-
Backhoe,? Vehicle System Dynamics, Vol. 11, 1982, pp. 251-265.
[7] Bevly, D. M., Gerdes, J. C., and Parkinson, B., 2002, ??A New Yaw Dynamic
Model for Improved High Speed Control of a Farm Tractor,?? J. Dyn.
Syst., Meas., Control, 124(4), pp. 659?667.
[8] Bell, T., Precision Robotic Control of Agricultural Vehicles on Realistic Farm
Trajectories, Ph.D. Dissertation, Stanford University, June 1999.
[9] Bevly, D.M., High Speed, Dead Reckoning, and Towed Implement Control for
Automatically Steered Farm Tractors Using GPS, Ph.D. Dissertation,
Stanford University, August 2001.
[10] Feng , L., He, Y., and Zhang, Q., ?Dynamic Trajectory Model of a Tractor-
Implement System for Automated Navigation Applications,?
Proceedings of Automation Technology for Off-road Equipment, MI,
USA, pp.243-254, 2004.
89
[11] Bukta, A.J., 1998, ?Nonlinear Dynamics of Traveling Tractor-Implement System
Generated by Free Play in the Linkage,? J. Japanese Society of Agric.
Machinery 60(4): 45-53.
[12] Gillespie, T., 1992, Fundamentals of Vehicle Dynamics, Society of Automotive
Engineers, Inc., Warrendale, PA.
[13] Figiola, R.S., Beasley, D.E., Theory and Design for Mechanical Measurements, 3
rd
ed., John Wiley & Sons, Inc., 2000.
[14] Ljung, L., 1987 System Identification: Theory for the User, PTR Prentice Hall,
Inc., Englewood Cliffs, NJ.
[15] Stengel, R. F., Optimal Control and Estimation, Dover Publications, Inc., New
York, NY.
[16] Bosch, P. P. J., Klauw, A. C., Modeling, Identification, and Simulation of
Dynamical Systems, CRC Press, London.
90
APPENDICES
91
APPENDIX A
Experimental and Data Acquisition Setup
A.1 Introduction
Appendix A contains information about the physical setup of the experimental
tractors used for this research and the data acquisition setup.
92
A.2 Experimental and Data Acquisition Setup
Two tractors were used to take data. The first tractor, shown in Figure A.1, was a
John Deere 8420 with single rear wheels. The second tractor was a Deere 8520 with
duals, of which a photo is not shown. Both tractors had the independent front suspension
setup.
Figure A.1: Experimental Test Tractor- John Deere 8420
A data acquisition computer was used to record data from an inertial measurement
unit (IMU), a steering angle sensor, and GPS data. The data acquisition computer is
shown in Figure A.2. It is a Versalogic PC-104 stack computer with a Bobcat processor
and data acquisition card enclosed in a Versatainer ruggedized enclosure. The steering
angle sensor is shown in Figure A.3 and is a linear potentiometer. The IMU is a 6 DOF
93
system created from 3 Bosch automotive grade sensors. The Bosch sensors sense both
yaw rate and acceleration. Figure A.4 shows how the sensors are arranged in a custom
fabricated box to create the IMU which senses yaw, pitch, and roll, and acceleration in
each respective axis. The GPS data was gathered on a Starfire GPS unit shown in Figure
A.5.
Also, for the experiments done in conjunction with the USDA-ARS National Soil
Dynamics Laboratory, a hitch force dynamometer was used. The dynamometer is shown
in Figure A.6.
Figure A.2: Versalogic Data Acquisition Computer
94
Figure A.3: Steering Angle Sensor
Figure A.4: Inertial Measurement Unit
95
Figure A.5: Starfire GPS Receiver
Figure A.6: Hitch Force Dynamometer
96
APPENDIX B
Model Parameter Values
B.1 Introduction
Appendix B details the values of the set parameters used in the aforementioned
models and also summarizes the values obtained for the parameters whose values were
solved for.
97
B.2 Model Parameter Values
Table B.1: The ?3 Wheeled? Bicycle Model Parameters
a 1.00 m
b 2.00 m
c 2.19 m
t 1.20 m
z
I
18500 kgm
2
m 25,000 lb
r
C
?
5000 N/deg (singles, per axle)
r
C
?
10,000 N/deg (duals, per axle)
f
C
?
2,400 N/deg (per axle)
h
C
?
Values shown below in Table B.2
f
? .34 m
h
? .40 m
The values for a, b,
z
I ,
r
C
?
, and
f
C
?
were taken from previous research on a
similar setup from Bevly [7]. The values for c and t were obtained from physical
measurements of the tractor. The value for m was obtained from the shipping information
of the tractor.
98
Table B.2:
h
C
?
Values for all the Implements
Implement Depth, inches
h
C
?
, N/deg
8420, Deere 955 4 Shank Ripper** 6 534.00
8420, Deere 955 4 Shank Ripper** 12 937.00
8420, Deere 955 4 Shank Ripper** 18 1647.00
8420, 4 Shank Ripper 4 451.65
8420, 4 Shank Ripper 8 1002.80
8420, 4 Shank Ripper 12 1719.30
8520, Bedder Out of Gnd 0 0.98 ~ 0.00
8520, Cultivator 9 639.83
8520, Bedder 9 951.24
8520, 5 Shank Ripper 10 887.33
8520, 5 Shank Ripper 15 1025.00
8520, 5 Shank Ripper 20 3385.30
**Note: These values were obtained in a different manner from the rest.
The
h
C
?
was determined directly from the
yh
F
vs.
h
? plot instead of a
minimization based on empirically determined DC gains.