ON-LINE ESTIMATION OF IMPLEMENT DYNAMICS FOR ADAPTIVE 
STEERING CONTROL OF FARM TRACTORS 
 
 
 
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 
 
 
 
 
 
 
 
 
Evan R. Gartley 
 
 
 
 
Certificate of Approval: 
 
 
 
George T. Flowers 
Professor 
Mechanical Engineering 
 
 
 
 
 David M. Bevly, Chair 
Assistant Professor 
Mechanical Engineering 
John Y. Hung 
Associate Professor 
Electrical and Computer Engineering 
 Stephen L. McFarland 
Dean 
Graduate School 
 
 
 
ON-LINE ESTIMATION OF IMPLEMENT DYNAMICS FOR ADAPTIVE 
STEERING CONTROL OF FARM TRACTORS 
 
 
 
Evan Robert Gartley 
 
 
 
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 
December 16, 2005 
 
 iii 
ON-LINE ESTIMATION OF IMPLEMENT DYNAMICS FOR ADAPTIVE 
STEERING CONTROL OF FARM TRACTORS 
 
 
 
 
 
 
Evan Robert Gartley 
 
 
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
 
ON-LINE ESTIMATION OF IMPLEMENT DYNAMICS FOR ADAPTIVE 
STEERING CONTROL OF FARM TRACTORS 
 
Evan Robert Gartley 
 
Master of Science, December 16, 2005 
 (B.M.E., Auburn University, 2003) 
 
135 Typed Pages 
 
Directed by David M. Bevly 
 
 
 
An adaptive control technique for the control of a farm tractor during low levels 
of excitation and at low velocities is presented.  Results of a set of system identification 
experiments are compared to previous tractor models.  A cascaded controller is then 
designed for the feedback of steer angle, yaw rate, and lateral position baed on the new 
tractor model.  An on-line analysis of the data is used to determine if enough excitation is 
available for adaptation.  A cascaded Kalman Filter is presented to estimate the slope of 
the DC gain of the steer angle to yaw rate transfer function, MDC, with respect to velocity.  
An estimator also provides faster updates of position.  From the on-line estimate of MDC, 
the controller gains are scheduled based on a lookup table of predetermined values that 
were calculated from system identification tests. 
 
v 
The sensitivity of the controller to model simplifications, incorrect velocities, and 
MDC estimate errors are investigated.  The accuracy of the estimated MDC due to 
neglected dynamics and the rate of convergence is shown.  A simulation is used to show 
the errors that can be induced in the position estimator by the GPS delay.  The yaw rate 
estimator is designed for fixed point math using a square root covariance filter.  
Experimental and simulation results are provided which show the validity of the MDC 
estimate.  Finally, experimental results which show that the accuracy changes little as a 
result of hitch loading and velocity are presented and discussed. 
 
vi 
ACKNOWLEDGEMENTS 
 
 The author would like to thank the John Deere Corporation for providing a tractor 
for this research and Andy Rekow for providing assistance with the tractor.  Thanks are 
expressed to the AU Agriculture Department for supplying land and an implement for 
testing.  Thanks are also due to the entire GPS and Vehicle Dynamics Laboratory for 
their help. 
 
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.................................................................................................. xv 
 
1. INTRODUCTION  
1.1 Motivation............................................................................................. 1 
1.2 Background........................................................................................... 2 
1.3 Outline of Thesis................................................................................... 3 
1.3 Contributions......................................................................................... 5 
 
2. MODELING AND SYSTEM INENTIFICATION  
2.1 Introduction............................................................................................ 6 
2.2 Steering Servo Dynamics....................................................................... 6 
2.3 Previous Tractor Yaw Dynamics Models.............................................. 9 
2.4 8420 Tractor Yaw Dynamics................................................................. 13 
2.5 Summary and Conclusion...................................................................... 21 
 
 
 
ix 
3. CASCADED CONTROLLER DESIGN AND ANALYSIS  
3.1 Introduction............................................................................................ 23 
3.2 Steering Servo Controller...................................................................... 24 
3.2 Yaw Rate Controller.............................................................................. 26 
3.3 Closed Loop Yaw Rate Sensitivity........................................................ 31 
3.4 Lateral Position Controller..................................................................... 38 
3.5 Closed Loop Lateral Position Sensitivity.............................................. 43 
3.6 Summary and Conclusion...................................................................... 46 
 
4. STATE AND PARAMETER ESTIMATION  
4.1 Introduction........................................................................................... 48 
4.2 Extended Kalman Filter......................................................................... 48 
4.3 Design of Yaw Rate Dual Estimator..................................................... 53 
4.4 Sensitivity of Estimation to Neglected Dynamics................................. 59 
4.5 Rate of Convergence............................................................................. 70 
4.6 Design of Position Estimator................................................................. 73 
4.7 Effects of GPS Delay............................................................................. 79 
4.8 Summary and Conclusion...................................................................... 85 
 
5. EXPERIMENTAL IMPLEMENTATION  
5.1 Introduction............................................................................................ 86 
5.2 Trajectory Design.................................................................................. 86 
 
x 
5.3 Results.................................................................................................... 89 
5.4 Summary and Conclusion...................................................................... 94 
 
6. FIXED POINT IMPLEMENTATION OF YAW RATE ESTIMATOR  
6.1 Introduction............................................................................................ 95 
6.2 Square Root Covariance Filter............................................................... 95 
6.3 Fixed Point Math................................................................................... 97 
6.4 Estimator Model.................................................................................... 98 
6.5 Comparison Between Fixed and Floating Point Implementation.......... 100 
6.6 Summary and Conclusion...................................................................... 108 
 
8. CONCLUSIONS  
8.1 Summary................................................................................................ 110 
8.2 Recommendation for Future Work........................................................ 112 
 
REFERENCES....................................................................................................... 114 
 
APPENDIX  
A Experimental Setup................................................................................ 118 
 
 
xi 
LIST OF FIGURES
 
 
2.1 Steady state steering rate versus command input....................................... 7 
2.2 ETFE of steering servo dynamics from input to slew rate......................... 9 
2.3 Bicycle model augmented with a hitch force and moment........................ 10 
2.4 Comparison of ETFE and Box Jenkins Fits............................................... 14 
2.5 Experimental fits of DC gain...................................................................... 15 
2.6 Experimental fits of the dominant poles..................................................... 17 
2.7 Experimental fits of the secondary poles.................................................... 18 
2.8 Experimental fits of the imaginary zeros.................................................... 19 
2.9 Experimental fits of the right half plane zero time constant....................... 20 
3.1 Block diagram of the controller system...................................................... 23 
3.2 Block diagram of steering servo system..................................................... 24 
3.3 1?s  versus velocity for each yaw model...................................................... 29 
3.4 1?r  versus velocity for each model............................................................... 30 
3.5 Sensitivity of yaw rate closed loop for no implement................................ 32 
3.6 Root locus of closed loop yaw rate............................................................. 33 
3.7 Sensitivity of yaw rate closed loop with an implement.............................. 34 
3.8 Sensitivity with no implement to too low MDC........................................... 36 
 
xii 
3.9 Sensitivity with no implement with 2 mph control gains........................... 37 
3.10 Sensitivity with no implement with 5 mph control gains........................... 38 
3.11 Controller values for various velocities and hitch loadings....................... 41 
3.12 Controller values for various velocities and hitch loadings....................... 42 
3.13 Sensitivity of lateral error closed loop for no implement........................... 44 
3.14 Sensitivity of lateral error closed loop with an implement......................... 45 
4.1 Windowed auto-correlation of steer angle.................................................. 56 
4.2 Estimate of MDC and yaw rate.................................................................... 58 
4.3 Mean MDC versus input frequency using the input of ( )wtAsin=? ......... 61 
4.4 Magnitude and phase difference between yaw rate and steer angle times 
DC gain for ( )t3sin5=? ............................................................................ 
 
62 
4.5 Estimate of MDC with an input of ( )t3sin5=? .......................................... 63 
4.6 Standard deviation of estimated MDC versus input frequency.................... 64 
4.7 Bode magnitudes divided by velocity for no implement............................ 65 
4.8 Bode magnitudes divided by velocity for various hitch loadings at 3 
mph............................................................................................................. 
 
66 
4.9 Mean MDC versus input frequency for second order model....................... 68 
4.10 Frequency response of the actual system and second order estimator 
model.......................................................................................................... 
 
69 
4.11 Standard deviation of MDC for improved estimator.................................... 70 
4.12 Convergence from different offsets............................................................ 71 
4.13 Time to converge within 99.9 % of the correct value................................ 72 
 
xiii 
4.14 Estimated standard deviation for MDC........................................................ 73 
4.15 Delay in GPS messages.............................................................................. 76 
4.16 Comparison of measured and estimated course from experimental test.... 78 
4.17 North error.................................................................................................. 80 
4.18 East error..................................................................................................... 82 
4.19 Heading Error............................................................................................. 83 
4.20 Comparison of covariances for an estimator with no delay and estimator 
that is compensated for the delay............................................................... 
 
84 
5.1 Tracking a straight line............................................................................... 87 
5.2 Curved trajectory to the straight line.......................................................... 88 
5.3 Estimation of MDC during experimental tests............................................. 90 
5.4 Lateral error versus velocity....................................................................... 91 
5.5 Lateral error without an implement............................................................ 92 
5.6 Lateral error with a four shank ripper at a twelve inch depth..................... 93 
6.1 Windowed analysis of experimental data to determine when to adapt...... 101 
6.2 Comparison of the estimation of the slope of the dc gain equation with 
respect to velocity for fixed and floating point using experimental data... 
 
102 
6.3 Comparison of the estimation of the yaw rate bias in fixed and floating 
point using experimental data..................................................................... 
 
103 
6.4 Comparison of the estimation of yaw rate in fixed and floating point 
using experimental data.............................................................................. 
 
 
104 
 
xiv 
6.5 Comparison of the estimation of yaw rate in fixed and floating point for 
simulated data............................................................................................. 
 
105 
6.6 Comparison of the estimation of the yaw rate bias in fixed and floating 
point using simulated data.......................................................................... 
 
106 
6.7 Comparison of the estimation of the yaw rate dc gain slope with respect 
to velocity in fixed and floating point for simulated data........................... 
 
107 
6.8 Windowed analysis of experimental data to determine when to adapt...... 108 
A.1 John Deere 8420 Tractor............................................................................ 118 
A.2 Versalogic PC/104 CPU [Versalogic, 2005a]............................................ 119 
A.3 Versalogic PC/104 data acquisition board [Versalogic, 2005b]................. 119 
A.4 Versalogic VL-ENCL-4 Ruggedized Enclosure [Versalogic, 2005c]........ 119 
A.5 Bosch IMU................................................................................................. 120 
A.6 Steer angle sensor....................................................................................... 120 
A.7 Starfire GPS reciever.................................................................................. 120 
 
xv 
LIST OF TABLES 
 
 
2.1 Steering servo input transformation............................................................. 8 
2.2 DC Gain Linear Fits..................................................................................... 16 
2.3 Tractor Model Parameters........................................................................... 21 
3.1 Inverse mapping of steady state steering rate versus command value........ 25 
3.2 Yaw rate controller values........................................................................... 30 
3.3 Gain margin for yaw rate............................................................................. 35 
3.4 Yaw rate controller values........................................................................... 42 
3.5 Gain margin for lateral position................................................................... 46 
3.6 Phase margin for lateral position................................................................. 46 
4.1 Discrete noise values used for the yaw rate dual estimator......................... 59 
4.2 Square root of covariance of MDC provided by the estimator...................... 65 
4.3 Discrete noise values used for the position estimator.................................. 79 
5.1 Statistics of lateral error............................................................................... 94 
 
 
1 
CHAPTER 1
INTRODUCTION 
1.1 Motivation 
Farm tractor models vary widely due to the many configurations possible and the 
different ground conditions encountered.  The controller must give an adequate response 
to a system in which towed or hitched implements of varying sizes can be used at a wide 
range of depths.  By automating tracking operations, costs such as row overlaps can be 
reduced. Also, less trained operators can be used without a reduction in accuracy and 
tractor operations can be performed during low visibility. 
Due to the various implement loadings that are possible, a controller designed for a 
time-invariant vehicle model has the potential to either have a poor response or go 
unstable.   Therefore, it is desired to have a controller that can compensate for the change 
in dynamics.  It is also desired to not add more sensors to the tractor and as such only 
sensors that can be found on tractors that currently have an option for a controlled 
response were used. Complicating the problem, desired trajectories consist of straight line 
segments, which lead to low levels of excitation.  Also, the implement depth is not set 
until the straight line trajectory begins.  Therefore, fits from when the tractor is lining up 
for the controlled run are not applicable for the controller design. 
 
2 
The algorithms designed must also be able to operate on a low cost microcontroller.   
The microcontroller that is considered for this research is only capable of fixed point 
math. 
 
1.2 Background 
 A number of companies currently produce tractor control systems for tracking 
straight lines, such as John Deere and Beeline Technologies, which use GPS for 
navigation.  Work at Stanford developed simple tractor models and control algorithms 
that produced sub inch accuracy of a tractors position using carrier-phase differential 
GPS [O?Conner, 1996].  Further research allowed for the tracking of additional 
trajectories [Bell, 1999].  Additionally, combines have been controlled with GPS 
[Cordesses, 1999]. 
 Before the use of GPS as a navigation system, computer vision was used for the 
control of tractors [Reid, 1987].  Research into blending the navigation of GPS and vision 
has shown improvements in position accuracy [Zhang, 1999].  A comparison of the 
blending of GPS with a magnetometer against using only GPS has also been shown 
[Benson, 1998]. 
 System identifications of tractor dynamics between steer angle and yaw rate have 
been conducted for input frequencies up to one hertz, for controller design [Bevly, 2001; 
Stombaugh, 1998].  Additionally a model of a tractor tire has been developed which 
accounts for faulty circularity through the modification of a Pacejka Similarity Method 
tire model and has been verified with measured tire data [Bohler, 1999]. 
 
3 
 Numerous tire-soil interaction models exist [Saarilahti, 2002].  However, they are 
currently not used for the design of controllers for the lateral dynamics of tractors.   
Tractor models used for the design of tractor controllers have included kinematic models 
[O?Conner, 1997].  Kinematic models have also been able to model the dynamics 
between an object being towed and a vehicle [Svestka, 1995; Hingwe, 2000].  These 
kinematic models for the angle between the implement and tractor have been used to 
control the position of the implement [Bevly, 2001].  Controllers have also been designed 
for tractors modeled using a first order lag and a bicycle model [O?Conner, 1997].  A 
bicycle model with tire relaxations has been used to control a tractor at high velocities 
[Bevly, 2001]. 
 Online adaptation of a tractor model has been investigated using an Extended 
Kalman Filter / LMS algorithm [Rekow, 2001].  That work identified the parameters 
defined in [O?Conner, 1997].  An adaptive controller has also been designed for a 
tractor?s steering servo to account for a changing dead band and dynamics [Wu, 2001].  
Additionally, the slope of the DC gain with respect to velocity of the steer angle to yaw 
rate transfer function has been previously estimated along with a steering bias [Bell, 
1999]. 
 
1.3 Outline of Thesis 
 In Chapter 2, system identification of the steering servo is performed and a model 
is presented which accounts for its nonlinearities.  A review of previous yaw dynamic 
models for tractors is also covered.  The results of open loop system identification 
experiments of a tractor with and without an implement are given.  The similarities and 
 
4 
differences between the previously used models and the open loop system identification 
fits are discussed. It is justified that the slope of the DC gain with respect to velocity 
(MDC) of the steer angle to yaw rate transfer function can be estimated rather than the DC 
gain. Finally, the model of the tractor used for the design of the controllers in this thesis 
is given. 
 In Chapter 3, the design of the controllers for the steering servo, yaw rate, and 
lateral position are detailed.  The method of determining the controller gains using the 
estimation of the parameter MDC is discussed.  The sensitivity of the controllers to the 
model simplifications in Chapter 2, incorrect velocities, and an incorrect MDC estimation 
are shown. 
 Chapter 4 develops a method for estimating the yaw rate and MDC for the 
adaptation of the controller.  A method of determining when there exists sufficient 
excitation to estimate MDC is given.   The rate of convergence upon initial determination 
of when to adapt is shown.   The effect of the neglected dynamics on the estimation is 
displayed and a method of correcting for some of these errors is discussed.  The rate of 
the convergence of the estimated parameter due to the forgetting factor in the estimator is 
shown.  The design of the position estimator for providing faster updates of position is 
also detailed.  The output equations for the estimator which account for the GPS delay are 
justified through the use of a simulation.  Also, approximations are given for the errors 
that are induced from not including the GPS delay in the estimator, which are validated in 
simulation. 
 In Chapter 5, the design of the trajectories that were used for the tractor controller 
in this thesis are discussed.  Experimental results showing the accuracy of the estimation 
 
5 
algorithms is given.  It is shown that the tracking response using the estimation and 
adaptive control methods developed in this thesis do not vary significantly. 
 The estimator for the yaw rate and MDC estimation is designed using fixed point 
calculations in Chapter 6.  A comparison of estimators using fixed and floating point 
math are displayed for both experimental and simulation data.  Finally, in Chapter 7, the 
conclusions and future work are given. 
 
1.4 Contributions 
A summary of the contributions provided in this thesis are listed below. 
? A model for a John Deere 8420 tractor was developed using system 
identifications experiments 
? A controller was designed which compensated for changes in velocity and hitch 
loading.  Analysis showed that not accounting for changes in the velocity and 
hitch loading could cause the tractor controller to be unstable. 
? An estimator was designed to estimate the effect of the hitch loading on the 
tractors dynamics.  The accuracy and the amount of time for convergence was 
provided. 
? A position estimator was designed which accounted for the delay in the receipt of 
the GPS message. 
? A fixed point estimator was developed for estimating hitch loading through the 
parameter MDC, in order to allow implementation of the algorithm on low cost 
microcontrollers. 
 
6 
CHAPTER 2
MODELING AND SYSTEM IDENTIFICATION 
2.1 Introduction 
 In this chapter the modeling of the hydraulic steering servo and the tractors yaw 
rate dynamics will be discussed.  While a number of different models have been 
presented for the yaw rate dynamics of a tractor, few have provided justifications of their 
designs through the use of system identification techniques, instead relying on controlled 
responses to determine which model gives the best results.  A review of past models as 
well as their similarities to system identifications of a John Deere 8420 tractor is 
presented.  While the past models do not adequately describe the yaw rate dynamics that 
of the 8420 tractor, it is shown that one of the models shows promise in describing the 
DC gain of the steer angle to yaw rate transfer function.  The DC gain equation of this 
model will be used as part of the justification for only estimating the slope of the DC gain 
(MDC).   This will also be justified with the use of system identification results.  Finally, 
the model that is used in the design of the controllers, which is based on the system 
identification results, is presented. 
 
2.2 Steering Servo Dynamics 
Based on earlier research [Bell, 1999], the steering servo dynamics were assumed 
to be a nonlinear system described using an input transformation and a transfer function 
 
7 
which included an integrator.  An integrator is necessary since the steering servo is 
hydraulic with an input of flow rate and an output of steer angle.  To identify the input 
transformation, constant inputs were commanded to the tractor resulting in the steer angle 
moving at a constant rate as shown below in Figure 2.1. 
 
Figure 2.1: Steady state steering rate versus command input. 
 
The steer angle measurement was differentiated with a center difference equation 
to obtain the steer angle rate, slew rate, for a wide range of inputs in order to produce the 
data in Figure 2.1.  The input transformation was then modeled with a dead-band, 
saturations, and nonlinear equations with the values given in Table 2.1, where u is the 
 
8 
value provided to the tractor.  The fitted model is also compared to the identified values 
in Figure 2.1. 
 
Table 2.1: Steering servo input transformation 
Steady State Slew Rate ( DC?& ) Input Command (u) 
36.0?=DC?&  598<u  
835.100324.06295.1 2 ?+??= uueDC?&  866598 <? u  
0=DC?&  1055866 <? u  
213.1003111.06859.1 2 +??= uueDC?&  13251055 <? u  
36.0=DC?&  1325?u  
 
To obtain the transfer function of the steering servo, a chirp input was given to the 
tractor while accounting for the input transformation, which is detailed in Chapter 3.  The 
output was then differentiated with a center difference equation and an empirical transfer 
function estimate (ETFE) [Ljung, 1999] was formed, Figure 2.2, which resembles a 
system which has two poles.  A Box-Jenkins model [Ljung, 1999] was used to fit the 
system identification data. 
 
 
9 
10?2 10?1 100 101 102
?50
0
50
Magnitude (dB)
10?2 10?1 100 101 102
?300
?200
?100
0
Phase (deg)
Frequency (rad/sec)  
Figure 2.2: ETFE of steering servo dynamics from input to slew rate. 
 
The Box-Jenkins model was then combined with the integrator to produce the 
discrete transfer function describing the steering servo dynamics (Equation (2.1)) at 100 
hertz. 
69916985892732.084973304358720.215076318465988.2
9022540063552511.0
23 ?+?= zzzu
?  (2.1) 
 
2.3 Previous Tractor Yaw Dynamic Models 
The schematic displayed in Figure 2.3 is a common vehicle dynamics model 
known as the bicycle model [Wong, 1978], that has been augmented with the addition of 
 
10 
a hitch force and moment caused by an implement.  The use of a towed implement would 
have no moment, while a hitched implement would have a force and moment.  The 
bicycle model lumps the inner and outer tires and together and assumes that there is no 
weight transfer.  The bicycle model also relates the forces at the tires to the velocities of 
the tire, through slip angles.  The additional force that has been added for the implement 
will also be modeled as being a function of the slip angle at the hitch. 
 
Figure 2.3: Bicycle model augmented with a hitch force and moment. 
  
As can be seen in Figure 2.3, the force on the front tire, rear tire, and hitch are 
defined as Ff, Fr, and Fh respectively.  The distance from the center of gravity to the front 
tire and rear tire are defined as a and b, and the distance from the rear tire to the hitch is 
Vy 
Vx 
Mh 
Fh 
Fr 
Ff 
?f 
?r 
r 
a 
b 
c 
? 
? 
N 
E 
? 
 
11 
c.  The front, rear, and hitch slip angles are defined as ?f, ?r, and ?h respectively.  Also 
depicted in the figure are the longitudinal velocity (Vx), lateral velocity (Vy), yaw rate (r), 
and steer angle (?).  The heading angle (?) is the angle between the North direction (N) 
and the tractors longitudinal direction.  The side slip (?) is the angle between the 
longitudinal and lateral velocities.  The course angle (?), the angle provided by GPS, is 
the heading angle (?) plus the side slip (?).  The dynamics of the bicycle model in a state 
space representation are shown below. 
( ) ( )( )
( )( ) ( )( ) ??
?
??????
??????
?
?
?
?
?
?
?
?
?
?
?
?
+?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
+++??++
??++++?
=?
?
?
??
?
z
f
f
y
xz
frh
xz
frh
x
x
frh
x
frh
y
I
aC
m
C
r
V
VI
CaCbCcb
VI
aCbCCcb
VmV aCbCCcbmV CCC
r
V
2
2
22
22
222&
&  
(2.2) 
The front, rear, and hitch cornering stiffnesses are C?f, C?r, and C?h respectively.  The 
cornering stiffnesses represent the resistance to turning and are a factor in computing the 
forces from the slip angles.  The mass of the tractor is m and the mass moment of inertia 
about the vertical axis is Iz. 
The bicycle model has two poles which decrease in bandwidth as velocity 
increases.  Additionally, the bandwidth of the bicycle model goes to infinity as velocity 
goes to zero.  However, it will be shown that the poles of the system identification model 
do not go to infinity as velocity decreases.  Therefore this is not a good model for low 
velocities.  Another model that has been used assumes that the lateral velocity is 
negligible which causes the bicycle model to collapse into a first order model.  These 
models also have too few poles to describe what was obtained through system 
identification tests shown in the next section.  It is believed, however, that the bicycle 
 
12 
model can provide an adequate description of how the DC gain changes with hitch 
loading [Pearson, 2005].  The DC gain equation for the bicycle model is shown below. 
( ) ( )[ ]
( ) ( ) ( )( ) 2222 5.0 xfrhhfhrrf
xrhf
ss
ss
VaCbCCcbmCCcbaCCcCCba
VCbaCcbaCr
?????????
???
? ?++++++++
++++=  (2.3) 
This equation can be condensed to Equation (2.4), where K2, the understeer 
[Gillespie, 1992] component, is typically much smaller than K1.   
2
21 x
x
VKK
Vr
+=?  (2.4) 
K2 being much smaller causes the steady state relationship between steer angle and rate to 
be linear at low velocities with an intercept of zero and begin curving at higher velocities.  
As will be seen from experimental data, the velocity range of interest does indeed 
produce linear fits.  Therefore, the term K2 can be neglected in the estimation scheme. 
A bicycle model with tire relaxations for a tractor without an implement has been 
used to improve the bicycle model for low velocities [Owen, 1982].  The relaxation 
length causes a lag in the production of the slip angle generating the tire force.  The 
addition of the tire relaxation lengths to the bicycle model adds two poles to the system 
and maintains the same DC gain equation.  In Equation (2.5), the bicycle model with 
front and rear tire relaxation lengths (?f and ?r) is presented with the hitch modeled as a 
cornering stiffness. 
 
13 
( )
( ) ( )
?
???
???
????
?
????
????
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
???
?+?+
?+?
=
??
?
?
?
?
?
??
?
?
?
?
?
0
0
0
01
01
2222
2222
2
f
x
r
f
y
r
x
rr
f
x
ff
z
r
z
f
xz
h
xz
h
rf
x
x
h
x
h
r
f
y
Vr
V
Vb
Va
I
bC
I
aC
VI
Ccb
VI
Ccb
m
C
m
CV
mV
Ccb
mV
C
r
V
&
&
&
&
 (2.5) 
This model has four poles and two zeros, with the dominant poles no longer going to 
infinity as velocity decreases.  The addition of relaxation lengths has been used to control 
farm tractors at high velocities [Bevly, 2001].  In that research it was assumed that the 
four poles and two zeros from the model were of similar bandwidth.  Therefore the 
resulting frequency response contained only one resonant peak. 
There are relatively few system identifications of farm tractors in the literature 
and none were found which provided more then one hertz of excitation.  Therefore, no 
previous work could be compared to the system identifications for the dynamics found 
above one hertz. 
 
2.4 8420 Tractor Yaw Dynamics 
For the system identification of the tractor, chirp inputs were used on a road with 
no implement and in a field with a four shank ripper at three different depths.  In Figure 
2.4, a comparison of the ETFE and Box Jenkins fits shows that the transfer function from 
steer angle to yaw rate had four poles and three zeros. 
The location of the four poles is clearly seen by the two resonant peaks in Figure 
2.4.  The three zeros are found between the two sets of poles, and one of the zeros is non-
 
14 
minimum phase which is why the second resonant peak is so high and the phase goes to   
-270 degrees. 
10?1 100 101 102
?20
0
20
Magnitude (dB)
10?1 100 101 102
?300
?200
?100
0
Phase (deg)
Frequency (rad/sec)
ETFE
Box Jenkins Fit
ETFE
Box Jenkins Fit
 
Figure 2.4: Comparison of ETFE and Box Jenkins Fits. 
 
From the comparison of an ETFE and Box Jenkins Fit of experimental data the 
tractor was found to have four poles and three zeros, which means that the model in 
Equation (2.5) is not even adequate for describing the system. 
It may be possible that the poles and zeros of the bicycle model with relaxation 
lengths are not lumped into the first resonance of Figure 2.4 and there is only a slight 
adjustment needed to this model to add a zero without majorly changing the DC 
characteristics of the model.  Because the focus of this work is on the control of the 
 
15 
tractor, system identification fits provide adequate information, without fitting them to a 
known model. 
 In Figure 2.5, a comparison of the DC gain of the identified transfer functions 
shows that the DC gain is linear through the velocities of interest. 
0.5 1 1.5 2 2.50.2
0.3
0.4
0.5
0.6
0.7
0.8
DC Gain
Velocity (m/s)
No Implement
Ripper at 4" Depth
Ripper at 8" Depth
Ripper at 12" Depth
 
Figure 2.5: Experimental fits of DC gain. 
 
 Table 2.2 provides fits of the DC gains identified, which produced intercepts that 
were close to zero.  This provides further evidence that the understeer of the tractor does 
not affect the linearity of the DC gain in the velocities of interest.  Therefore the 
estimator introduced in Chapter 4 can simply estimate the slope of the DC gain to 
characterize the tractors steady state handling on-line. 
 
16 
Table 2.2: DC Gain Linear Fits 
DC Gain = Slope*Velocity 
 Baseline 4? depth 8? depth 12? depth 
Slope 0.334 0.284 0.258 0.227 
DC Gain = Slope*Velocity + Intercept 
 Baseline 4? depth 8?depth 12? depth 
Slope 0.329 0.298 0.259 0.231 
Intercept 0.00800 0.0222 0.00153 0.00616 
 
As seen in Figure 2.6, the dominant poles of the system, did not contain as clear 
of trends as the DC gain.  Linear fits for the natural frequency with no implement appear 
to decrease slightly with velocity while, with the ripper they appear to increase with 
velocity.  However, with the ripper the fits are not as clean.  For this reason, the natural 
frequency was chosen to be constant at each depth regardless of velocity since this is a 
good approximation for the tractor with no implement. 
 
17 
0.5 1 1.5 2 2.58
10
12
14
16
Natural Frequency 
 (rad/s)
0.5 1 1.5 2 2.50
0.5
1
Damping Ratio
Velocity (m/s)  
Figure 2.6: Experimental fits of the dominant poles.  (x = baseline tractor, o = 4 shank 
ripper at 4? depth, + = 4 shank ripper at 8? depth, * = 4 shank ripper at 12? depth) 
 
With an implement the dominant poles have considerably more damping; 
however, it was not clear how the damping changes between hitch loadings.  Therefore, 
for the design of the controller, the damping was chosen as a linear fit for no implement 
and a separate linear fit with an implement that was the same for all depths. 
The remainder of the poles and zeros are displayed in Figures 2.7-2.9.  Due to the 
noise in the fits with an implement it is not clear as to how the natural frequency and 
damping changes with implement depth.  However most of these vary only slightly from 
what is obtained with no implement.  Since these values will have a lesser effect on the 
 
18 
design of the controller, as they are at higher natural frequencies, they were modeled as 
the same as with no implement in the design of the controller. 
0.5 1 1.5 2 2.542
43
44
45
46
47
Natural Frequency 
 (rad/s)
0.5 1 1.5 2 2.50.08
0.1
0.12
0.14
0.16
Damping Ratio
Velocity (m/s)  
Figure 2.7: Experimental fits of the secondary poles.  (x = baseline tractor, o = 4 shank 
ripper at 4? depth, + = 4 shank ripper at 8? depth, * = 4 shank ripper at 12? depth) 
 
 
 
19 
0.5 1 1.5 2 2.514
16
18
20
22
Natural Frequency 
 (rad/s)
0.5 1 1.5 2 2.50.2
0.4
0.6
0.8
1
Damping Ratio
Velocity (m/s)  
Figure 2.8: Experimental fits of the imaginary zeros.  (x = baseline tractor, o = 4 shank 
ripper at 4? depth, + = 4 shank ripper at 8? depth, * = 4 shank ripper at 12? depth) 
 
 
20 
0.5 1 1.5 2 2.5
?11
?10
?9
?8
?7
?6
?5
Time Constant
Velocity (m/s)  
Figure 2.9: Experimental fits of the right half plane zero time constant.  (x = baseline 
tractor, o = 4 shank ripper at 4? depth, + = 4 shank ripper at 8? depth, * = 4 shank ripper 
at 12? depth) 
 
 A summary of values used to model the tractor are given in Table 2.3.  These 
values describe the continuous poles that were used in the design of the controllers in 
Chapter 3.   
 
 
 
 
 
21 
Table 2.3: Tractor Model Parameters 
 No 
Implement 
Ripper at 4? 
Depth 
Ripper at 8? 
Depth 
Ripper at 
12? Depth 
MDC 0.335 0.285 0.258 0.227 
Dominant pole 
frequency 
9 11 13 15 
Dominant pole 
damping 
0.064Vx 
+0.10496 
0.1719Vx 
+0.1396 
  
Secondary pole 
frequency 
0.46858Vx 
42.17 
   
Secondary pole 
damping 
0.013764Vx 
+0.078253 
   
Imaginary zero 
frequency 
1.1276Vx 
14.784 
   
Imaginary zero 
damping 
-0.065439Vx 
+0.86505 
   
Non-minimum 
phase  zero time 
constant 
1.532Vx2 
-7.5414Vx 
-1.1705 
   
 
2.5 Summary and Conclusions 
 A model of the steering servo was found which uses an input transformation, to 
account for its nonlinearities, and a transfer function.  It was shown that previous tractor 
 
22 
models were inadequate in describing the tractor used in this research.  Therefore a model 
based on system identification fits was developed.  It was found that the DC gain from 
steer angle to yaw rate could simply be determined through the use of the slope of the DC 
gain.  Finally, a table of the tractor model parameters was presented for use in the 
controller design. 
 
23 
CHAPTER 3
CASCADED CONTROLLER DESIGN AND ANALYSIS 
3.1 Introduction 
This chapter discusses the design of the cascaded controllers using the dynamics 
identified in Chapter 2.  Also, the sensitivity of the closed loop systems to the model 
simplifications in Chapter 2 are discussed along with the effects the incorrect velocity and 
MDC estimate in the design of the controller. 
A block diagram of the cascaded controllers [Franklin, 2002] is displayed in 
Figure 3.1, where Gc1(z), Gc2(z), and Gc3(z) are the controllers for the steering servo, yaw 
rate, and lateral position.  Gp1(z), Gp2(z), and Gp3(z) are the open loop dynamics of the 
steering servo, yaw rate, and lateral position.  The values being fed back to the controllers 
are the steer angle, yaw rate, and lateral position. 
 
Figure 3.1: Block diagram of the controller system 
 
Gc1(z) Gc2(z) Gc3(z) Gp1(z) Gp2(z) Gp3(z) 
y 
r 
? 
+ + + 
- - - 
 
24 
3.2 Steering Servo Controller 
The input to the tractor is the flow rate of the steering servo, which as discussed in 
Chapter 2 has been modeled as a third order system with input nonlinearities.  To 
compensate for these nonlinearities, a transformation of the controller output has been 
used as shown in the block diagram in Figure 3.2.  This transformation is generated by 
taking the inverse of the mapping of the steady state steering rate versus the command 
input shown previously in Figure 2.1.   
Figure 3.2: Block diagram of steering servo system 
 
The equations for the inverse mapping of the steering servo nonlinearities are 
given in Table 3.1.  The inverse mapping effectively linearizes the system such that 
classical design techniques can be used for the design of the steering servo controller. 
 
 
 
 
 
Gc1(z) Gp1(z) + - ? u 
Lookup 
Table 
u?  
Nonlinearities 
 
25 
Table 3.1: Inverse mapping of steady state steering rate versus command value 
Input to plant Range of u?  for function 
u = 598 u?  < -0.36 
u = 518.7 2?u +920.2u? +864.4 u?  >= -0.36 & u?  < 0 
u = -887.9 2?u +1045u? +1059 u?  >= 0 & u?  < 0.36 
u = 1325 u?  >= 0.36 
 
Since the input frequency to the tractor is at 50 hertz and the identified transfer 
function for the steering servo given previously in Equation (2.1) is at 100 hertz, a 
discrete conversion [Franklin, 1998] was done leading to Equation (3.1). 
( )
( ) 32213 32 azazaz
bzb
zu
z
+++
+=?  (3.1) 
The discrete conversion adds a zero to the transfer function that is located close to minus 
one inside of the discrete unit circle. 
A constant proportional control law was used for the steering servo control loop.  
This results in a closed loop with three poles.  The decision on modeling the controller as 
a proportional gain was made because the steering dynamics already was an integrator in 
the plant and it reduced the number of poles in the design of the yaw rate and position 
controllers.  Equation (3.2) gives the polynomials of the closed loop denominator used in 
the controller design. 
( ) ( )( )2121 fzfzezzB +++=  (3.2) 
 
26 
The first order polynomial contains the pole that can be placed by the controller.  The 
second order polynomial contains the poles that can not be placed because the controller 
only allows the placement of one variable. 
 The system of equations that must be solved to place the desired pole is shown in 
Equation (3.3), where s0 is the proportional controller gain.   
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
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??
?
3
2
11
2
1
0
13
12
0
1
010
a
a
ae
f
f
s
eb
eb  (3.3) 
Also, solving for this equation allows for the set of poles that have not been placed to be 
checked to make sure that they don?t interfere with the desired closed loop dominant 
pole.  The closed loop system in Equation (3.4) then can then be used in to the design of 
the yaw rate and position controllers. 
( )
( )
( )
( )32032213
320
bzbsazazaz
bzbs
z
z
des +++++
+=
?
?  (3.4) 
 
3.3 Yaw Rate Controller 
 A set of yaw rate controllers were designed for each of the yaw rate models given 
previously in Table 2.3, as a function of velocity.  Fourth order curve fits of each set were 
then found so that upon the estimation of MDC it could be determined what controller 
values to use through interpolation.  In order to design the individual points for the fit, the 
damping and natural frequency of the poles and zeros were calculated from Table 2.3 
based on the velocity for each value of MDC.  This led to a transfer function that when 
converted to 50 Hz led to Equation (3.5) which could be used to calculate the controller 
values for that velocity and MDC. 
 
27 
( )
( ) 4322314
43
2
2
3
1
azazazaz
bzbzbzb
z
zr
++++
+++=
?  (3.5) 
The transfer function from steer angle to yaw rate (Equation (3.5)) was then 
multiplied by the closed loop steering servo (Equation (3.4)) producing the open loop 
yaw rate system, shown below in Equation (3.6). 
( )
( )
( )
( )
76
2
5
3
4
4
3
5
2
6
1
7
76
2
5
3
4
4
3
43
2
2
3
1
4
43
2
2
3
1
32032
2
1
3
320
???????
?????
azazazazazazaz
bzbzbzbzb
azazazaz
bzbzbzb
bzbsazazaz
bzbs
z
zr
des
+++++++
++++=
++++
+++?
+++++
+=
?  
(3.6) 
The yaw rate controller was chosen to be a lag controller [Ogata, 1998] consisting 
of a pole and no zero as shown below. 
1
1
?
?
rz
s
er
des
+=
?  (3.7) 
This was primarily done to mitigate the effect of the second resonant peak that was 
identified in Figure 2.3.  It was also done in order to prevent a non-minimum phase zero 
being placed during the design process, which tended to occur when designing for larger 
values of MDC.  This would have caused complications in the design of the position 
controller. 
 Since the controller has two design variables, the location of two closed-loop 
poles can be selected.  Equation (3.8) shows the closed loop denominator, which contains 
the second order polynomial of the poles that can be selected and the sixth order 
polynomial of the remaining closed loop poles. 
( ) ( )( )65423342516212 ??????? fzffzzfzfzfzezezzB ++++++++=  (3.8) 
 
28 
 By calculating the denominator from Equations (3.6) and (3.7) and setting that 
equal to Equation (3.8), leads to the set of equations in Equation (3.9) to be solved for to 
obtain the controller values. 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
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=
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??
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?00000??
??0000??
1??000??
01??00??
001??0??
0001??0?
00001?0?
00000101
7
6
5
4
3
22
11
6
5
4
3
2
1
1
1
277
1266
1255
1244
1233
122
11
a
a
a
a
a
ae
ae
f
f
f
f
f
f
s
r
eba
eeba
eeba
eeba
eeba
eea
ea
 (3.9) 
Solving for Equation (3.9) can also determine if any of the remaining poles would cause 
the system to be unstable. 
The controller values were calculated for various velocities for each of the yaw 
rate models in Table 2.3.  The trends in the controller values could then be seen as a 
function of hitch loading.  Figures 3.3 and 3.4 show how the controller values varied for 
the various yaw rate models. 
 
29 
 
Figure 3.3: 1?s  versus velocity for each yaw model 
 
 As seen in Figure 3.3 the variation due to the implement simply shifts the curve 
down.  However, in Figure 3.4 there is a more significant difference in the trend of the 
curves.  Also, there is a less of a percentage difference between the curves in Figure 3.4 
than there is in Figure 3.3. 
 As stated previously, fourth order fits were used for of the curves.  Since each of 
these curves has a known MDC value related to it, the controller gains could then be 
calculated through interpolation using an estimated value of MDC.  In Table 3.2, the curve 
fits used for the interpolations are listed according to the yaw rate controller variable. 
 
30 
 
Figure 3.4: 1?r  versus velocity for each model 
 
 
Table 3.2: Yaw rate controller values (variable 43223140 gVgVgVgVg ++++= ) 
Values for 0?s  
MDC 0g  1g  2g  3g  4g  
0.355 0.016544 -0.12928 0.39308 -0.59242 0.45384 
0.285 0.020274 -0.15792 0.47963 -0.7158 0.53252 
0.258 0.02307 -0.17972 0.54481 -0.81169 0.60811 
0.227 0.02681 -0.20882 0.63224 -0.94108 0.70793 
 
31 
Values for 0?r  
MDC 0g  1g  2g  3g  4g  
0.355 0.00069355 -0.0054523 0.017887 -0.021717 -0.90949 
0.285 0.0010538 -0.0066872 0.021571 -0.019729 -0.9006 
0.258 0.00093468 -0.006382 0.020994 -0.020493 -0.89685 
0.227 0.00088336 -0.0063049 0.020818 -0.021387 -0.89455 
 
3.4 Closed Loop Yaw Rate Sensitivity 
To examine the how the accuracy of the MDC estimate (developed in Chapter 4) 
will effect the phase and gain margins of the yaw rate controller, a frequency domain 
analysis using the system identification models and the previously described controller 
adaptation was conducted.  The values of MDC were assumed to be known exactly and 
within ten percent of the actual value. 
 Figure 3.5 shows the results of the bodes that were obtained with no implement.  
In the figure, the closed loop poles are at 4 rad/s which is before the first resonant peak 
that can be seen in the magnitude graph.  Recall that if this resonant peak crossed 0 dB 
then the system is unstable.  Fortunately, it decreases as velocity increases, ensuring the 
system is stable with increasing velocity. 
 
32 
10?1 100 101 102
?100
?50
0
Magnitude (dB)
10?1 100 101 102
?800
?600
?400
?200
0
Phase (deg)
Frequency (rad/sec)  
Figure 3.5: Sensitivity of yaw rate closed loop for no implement 
 
 The cause of the resonant peak can be seen more clearly in the root locus of the 
closed loop system.  Figure 3.6 demonstrates that after the dominant poles there are a set 
of poles which have little damping.  Recall that the yaw controller selected could only 
specify the location of two poles.  Therefore these additional poles are not constrained to 
a specified location. 
Velocity 
Increasing 
 
33 
?1 ?0.5 0 0.5 1 1.5?1
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
0.8
1
Pole?Zero Map
Real Axis
Imaginary Axis
 
Figure 3.6: Root locus of closed loop yaw rate. 
 
 Figure 3.7 shows the closed loop bode for the tractor with an implement at all of 
the depths.  Note that the resonant peak with the implement is not nearly as pronounced 
as in Figure 3.5.  It looks like it could exist on a couple of lines, but these are believed to 
be bad fits, since that Box-Jenkins fit did not match the ETFE well.  The DC portion of 
the bode also looks thicker then that of the frequency response without an implement.  
This is most likely due to the fact that the fits are better for the system identification tests 
without an implement. 
 
34 
10?1 100 101 102
?100
?50
0
Magnitude (dB)
10?1 100 101 102
?800
?600
?400
?200
0
Phase (deg)
Frequency (rad/sec)  
Figure 3.7: Sensitivity of yaw rate closed loop with an implement 
  
For all of the cases tested varying MDC the controller was found to be stable.  
Table 3.3 shows the maximum, minimum, mean, and standard deviation of gain margin 
for the various hitch loadings.  As seen in the table, the mean gain margin appears 
increase as the implement hitch loading increases.  This is most likely caused by the 
unconstrained set of closed loop poles near the first resonant peak increasing in damping 
ratio as the hitch loading is increased.  Note that all the cases investigated had infinite 
phase margin. 
 
 
 
35 
Table 3.3: Gain margin (dB) for yaw rate 
 Maximum Minimum Mean ? 
No Implement 8.65 3.77 6.67 1.52 
Ripper 4? Depth 14.09 8.93 11.63 1.40 
Ripper 8? Depth 15.60 7.80 11.18 1.90 
Ripper 12? Depth 15.00 8.89 12.44 1.51 
 
 Figure 3.8 shows that for a low value of MDC used in the design of the control law 
design would cause the closed loop yaw rate to be unstable.  Additionally, values of MDC 
significantly greater than the actual value in the controller design can lead to degraded 
performance.   
 
36 
10?1 100 101 102
?100
?50
0
Magnitude (dB)
10?1 100 101 102
?800
?600
?400
?200
0
Phase (deg)
Frequency (rad/sec)  
Figure 3.8: Sensitivity with no implement to too low MDC. 
 
Additionally, if the controller does not account for the changes in velocity it can 
cause the yaw rate controller to be unstable.  Figure 3.9 shows the results of the controller 
being designed for 2 mph for all of the system identification models without an 
implement.  Note that the magnitude crosses 0 dB after there has been -180 degrees of 
phase change for a number of models, indicating that the closed loop system is unstable.  
This instability was also seen using the system identification models with an implement. 
 
 
37 
10?1 100 101 102
?100
?50
0
Magnitude (dB)
10?1 100 101 102
?800
?600
?400
?200
0
Phase (deg)
Frequency (rad/sec)  
Figure 3.9: Sensitivity with no implement with 2 mph control gains. 
 
When the controller was designed for a constant velocity of 5 mph the closed loop 
systems were stable for all hitch loadings identified as seen in Figure 3.10.  However the 
performance of the controller was highly degraded since the system was at a lower 
bandwidth. 
 
38 
10?1 100 101 102
?100
?50
0
Magnitude (dB)
10?1 100 101 102
?800
?600
?400
?200
0
Phase (deg)
Frequency (rad/sec)  
Figure 3.10: Sensitivity with no implement with 5 mph control gains. 
 
3.5 Lateral Position Controller 
The transfer function for lateral position used in the controller design was 
obtained from Equations (3.10) and (3.11).   
( )?? sinVy =&  (3.10) 
?? && += r  (3.11) 
In Equation (3.10), the lateral velocity is defined as the total velocity times the sine of the 
course angle (which is taken with respect to the desired longitudinal direction).  Equation 
(3.11) shows that the course angle is the integrated yaw rate plus side slip. 
 
39 
Neglecting the vehicle sideslip and linearizing about a small course angle, the 
Laplace transform of Equations (3.10) and (3.11) can be combined to produce the 
approximation shown below for lateral position. 
rsVy 2??  (3.12) 
By discretizing the above equation, the following discrete transfer function between yaw 
rate and lateral position is obtained. 
( )
( ) zzz
bzb
zr
zy
+?
+=
22
21
((
 (3.13) 
The above transfer function can then be combined with the closed loop yaw rate transfer 
function, forming the open loop transfer function for lateral position shown below. 
( )
( ) 109283746556473829110
109
2
8
3
7
4
6
5
5 ~~~~~~~~~~
~~~~~~
azazazazazazazazazaz
bzbzbzbzbzb
zr
zy
des ++++++++++
+++++=  (3.14) 
 A simple lead control law with on zero and one pole was used for the lateral error 
controller as seen below. 
( )
( ) 1
10 ~
~~
rz
szs
ze
zr
y
des
+
+=  (3.15) 
Since there are three parameters in the controller, three poles of the closed loop 
system could be selected.  Equation (3.16) shows the polynomials of the selected poles 
and the polynomial for the remaining poles of the closed loop transfer function 
denominator.   
( ) ( )( )87263544536271832213 ~~~~~~~~~~~~ fzfzfzfzfzfzfzfzezezezzB +++++++++++=  (3.16) 
The controller values can then solved for using Equation (3.17), which uses the values 
from Equations (3.14-3.16). 
 
40 
?
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=
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0
~
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~~
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~
~0000000~0~
~~000000~~~
~~~00000~~~
1~~~0000~~~
01~~~000~~~
001~~~00~~~
0001~~~00~~
00001~~~00~
000001~~00~
0000001~00~
00000001001
10
9
8
7
6
5
4
33
22
11
8
7
6
5
4
3
2
1
1
0
1
31010
239109
123898
123787
123676
123565
12354
1233
122
11
a
a
a
a
a
a
a
ae
ae
ae
f
f
f
f
f
f
f
f
s
s
r
eba
eebba
eeebba
eeebba
eeebba
eeebba
eeeba
eeea
eea
ea
 (3.17) 
Once the controller gains were determined for various velocities for each of the 
yaw rate models, the trends in the controller values as a function of hitch loading could 
be studied.  Figures 3.11 and 3.12 show how the controller values varied gains vary with 
velocity and implement loading. 
 
41 
 
Figure 3.11: Controller values for various velocities and hitch loadings. 
 
Like the yaw rate controller, fourth order fits were made for the controller gain 
curves shown in Figure 3.11 and 3.12.  Each of these curves has a known MDC value 
related to it, the controller gains could again be calculated through interpolation using an 
estimated value of MDC.  In Table 3.4, the curve fits used for the interpolations are listed 
according to the position controller variable. 
 
42 
 
Figure 3.12: Controller values for various velocities and hitch loadings. 
 
 
Table 3.4: Yaw rate controller values (variable 43223140 gVgVgVgVg ++++= ) 
Values for 0~s  
MDC 0g  1g  2g  3g  4g  
0.355 8.0185 -58.726 161.18 -200.3 106.83 
0.285 12.479 -89.378 242.36 -295.36 156.02 
0.258 12.108 -88.024 240.09 -294.62 155.17 
0.227 11.843 -86.358 235.71 -289.88 152.3 
 
43 
Values for 1~s  
MDC 0g  1g  2g  3g  4g  
0.355 -7.9577 58.281 -159.95 198.77 -106 
0.285 -12.382 88.687 -240.48 293.07 -154.8 
0.258 -12.014 87.34 -238.22 292.32 -153.94 
0.227 -11.75 85.684 -233.87 287.6 -151.09 
Values for 1~r  
MDC 0g  1g  2g  3g  4g  
0.355 0.011359 -0.080389 0.22571 -0.27599 -0.7263 
0.285 0.028812 -0.17191 0.4395 -0.48022 -0.61013 
0.258 0.019994 -0.13449 0.3703 -0.43411 -0.62486 
0.227 0.017956 -0.12553 0.34986 -0.41878 -0.63368 
 
3.6 Closed Loop Lateral Position Sensitivity 
To examine the how the accuracy of the MDC estimate effects the phase and gain 
margins of the position loop, a bode analysis using the system identification models and 
the previously described controller adaptation was again conducted.  The values of MDC 
were again assumed to be known exactly and within ten percent of the actual value.  
 Figure 3.13 shows the results of the bode plots that were obtained with.  Note that 
in the figure the closed loop poles are at 1 rad/s. 
 
44 
10?1 100 101 102
?200
?100
0
Magnitude (dB)
10?1 100 101 102
?1000
?500
0
Phase (deg)
Frequency (rad/sec)  
Figure 3.13: Sensitivity of lateral position closed loop for no implement 
 
 Figure 3.14 shows the results of the bode analysis with the varying MDC when an 
implement is used.  Unlike the results from the yaw rate analysis, the change in MDC has 
very little change on the closed loop frequency response (with or without an implement). 
 
45 
10?1 100 101 102
?200
?100
0
Magnitude (dB)
10?1 100 101 102
?1000
?500
0
Phase (deg)
Frequency (rad/sec)  
Figure 3.14: Sensitivity of lateral position closed loop with an implement 
 
 The gain margins for Figure 3.13 and 3.14 are shown in Table 3.5.  Like the yaw 
rate controller, the gain margin for the position controller increases with hitch loading 
(which improves the stability of the controller). 
 
 
 
 
 
 
 
46 
Table 3.5: Gain margin (dB) for lateral position 
 Maximum Minimum Mean ? 
No Implement 3.504 2.383 2.989 0.283 
Ripper 4? Depth 3.654 1.434 2.469 0.638 
Ripper 8? Depth 5.382 0.673 3.019 1.109 
Ripper 12? Depth 5.158 2.563 3.745 0.726 
 
The phase margins for Figure 3.13 and 3.14 are shown in Table 3.6, which also 
shows that the phase margin increases with hitch loading. 
 
Table 3.6: Phase margin (deg) for lateral position 
 Maximum Minimum Mean ? 
No Implement 30.963 20.332 26.221 3.436 
Ripper 4? Depth 30.053 12.926 21.423 5.235 
Ripper 8? Depth 38.202 7.421 26.594 7.693 
Ripper 12? Depth 42.268 28.401 34.874 4.710 
 
3.7 Summary and Conclusion 
 The controller for the steering servo using a transformation for the nonlinearities 
was designed.  The yaw rate and position controllers were also designed using the system 
identification models from Chapter 2.  Information was also provided on how the 
controller gains would be adapted through the estimation of MDC.  The method involves 
interpolating the control gain curve fits for various values of MDC.  The sensitivity of the 
 
47 
closed loop system was investigated for the values of MDC within plus and minus ten 
percent.  It was also shown that the controllers could become unstable if designed with a 
velocity that was less than the actual value or for a value of MDC that was less than the 
actual value of MDC.  Additionally, when the controllers were designed with too high of a 
velocity or too large of a value of MDC, they had a degraded performance. 
 
48 
CHAPTER 4
STATE AND PARAMETER ESTIMATION 
4.1 Introduction 
 This chapter describes the design of the estimator used to identify the slope of the 
DC gain with respect to longitudinal velocity (MDC) of the steer angle to yaw rate transfer 
function.  The errors associated with neglecting dynamics in the yaw rate transfer 
function will be examined and an improvement in the estimator design is given.  The rate 
of convergence of the estimator due to the forgetting factor on MDC is also investigated.  
A description of the position estimator is detailed and a justification of its output 
equations is produced. 
 
4.2 Extended Kalman Filter 
 The Extended Kalman Filter [Stengal, 1994; Bryson, 1975; Maybeck, 1979b; 
Gelb, 1992] is a first order Taylor series expansion of the Kalman Filter for nonlinear 
systems.  The state and measurement equations are shown in Equations (4.1) and (4.2). 
( )twuxfx ,,,=&  (4.1) 
( )tvuxhy ,,,=  (4.2) 
The values passed to the state function are the previous estimate of the states, 
inputs, disturbances, and time.  The values passed to the measurement function include 
the current estimate of the states, inputs, measurement noise, and time.  The 
 
49 
nonlinearities for the yaw rate estimator occur from a state being multiplied by the input.  
In the linearization, the second order partial derivatives in the Taylor series expansion 
and higher are zero.  Hence the estimator is not sub-optimal because an Extended Kalman 
Filter is being used.   The estimator is sub-optimal because an approximated system is 
being used.  The position estimator contains states times the sine and cosine of another 
state.  This results in higher order partial derivatives in its Taylor series expansion which 
are non-zero.  Therefore, a sub-optimal estimation is caused by the use of an Extended 
Kalman Filter.  However, a state times the sine or cosine of another state is not a large 
nonlinearity for an estimator and an Extended Kalman Filter will provide a decent 
estimate, especially since it is updated at 100 hertz.  An iterated Extended Kalman Filter 
[Gelb, 1992] could also provide the same improvement that a faster update gives.  The 
position estimator is sub-optimal do to the approximation necessary for the use of an 
Extended Kalman Filter and an approximated system being used.  Neglecting the higher 
order terms of a system that is modeled perfectly causes the estimate to be biased and the 
estimate of the covariance to be smaller then the correct value.  The covariance being less 
than the actual value can lead to filter divergence.  However, this can be corrected by 
designing the estimator with extra process noise. 
Second order Taylor expansion Kalman Filters [Maybeck, 1979b] such as the 
Truncated Second Order Filter and the Gaussian Second Order Filter use Hessian 
matrices for bias correction and improvement of the covariance propagations.  However, 
these estimators are much more computationally intensive then the Extended Kalman 
Filter.  Most of the added computation is in the covariance propagations, and not in the 
 
50 
bias correction terms.  The bias correction terms from these estimators can easily be used 
in an Extended Kalman Filter [Maybeck, 1979b]. 
Other Kalman Filters that would provide a better approximation include 
statistically linearized filters such as the Unscented Kalman Filter [Wan, 2000; Haykin, 
2001].  The Unscented Kalman Filter provides a third order approximation for Gaussian 
noise systems and second order approximations for all other noise systems.  It takes 
fewer computations than the second order Taylor series filters.  This estimator does not 
require that the states or measurements have a derivative.  Additionally, the Unscented 
Kalman Filter can be extremely helpful in debugging other estimators because it doesn?t 
require derivative information. 
The Extended Kalman Filter was chosen since the majority of the errors in the 
estimator are caused by model approximation and not by the format of the estimator.  
Also, the Extended Kalman Filter was chosen so that the yaw rate estimator could be 
implemented in fixed point as discussed in Chapter 6, for which a simple estimator was 
desired.   
In the Extended Kalman Filter, the Jacobian matrices defined in Equations (4.3-
4.5) are linearized with the state estimates from the time update. 
( )
x
twuxfH
?
?= ,,,  (4.3) 
( )
w
twuxfG
?
?= ,,,  (4.4) 
( )
v
tvuxhU
?
?= ,,,  (4.5) 
 
51 
The covariance matrices for the Extended Kalman Filter are given in Equations 
(4.6) and (4.7), where Q is semi-definite and R is positive definite. 
[ ]wwEQ ,=  (4.6) 
[ ]vvER ,=  (4.7) 
The Kalman Filter gain (K) is computed with Equation (4.8), where U is typically an 
identity matrix.   
( ) 1?+= TTT URUHPHPHK  (4.8) 
Sequential processing is frequently used so that only a scalar value is inverted.  The 
Kalman Filter gain is then used for updating the state estimates with the current 
measurements as shown below. 
( )( )tuxhyKxx kkkkkkk ,0,,1|1| ?? ?+=  (4.9) 
As a result of the measurement update the covariance decreases by Equation (4.10), 
where I is the identity matrix. 
( ) 1| ??= kkk PKHIP  (4.10) 
 Equations (4.8-4.10) are typically referred to as the measurement update since 
they only need to occur if a measurement is received.  Equations (4.11) and (4.12) are 
typically referred to as the time update since they propagate the estimates to the next time 
step. 
( )tuxfx kkkk ,0,, 111| ??? =&  (4.11) 
TT
kkkk GQGFPFPP ++= ??? 111|&  (4.12) 
The time update for an Extended Kalman Filter can either be in discrete time or in 
continuous time.  Equations (4.11) and (4.12) show the continuous state and covariance 
 
52 
propagations, which is how they are used for the position estimator.  Note that the time 
update can be performed before a measurement is received. 
 The covariance propagation in the time update (4.12) increases the uncertainty in 
the states.  The Jacobian matrices in Equation (4.12) are linearizied with the state 
propagations from Equation (4.11) and the two equations are ideally integrated together 
so that the most accurate Jacobian matrices available are used. 
The yaw rate estimator used in Section 4.3 has a discrete time update, and 
Equation (4.12) is replaced with Equation (4.13). 
d
T
dkdkk QFPFP += ?? 11|  (4.13) 
Additionally, a discrete state equation replaces Equation (4.11). 
The Extended Kalman Filter reduces to the Kalman Filter if the state and 
measurement equations are linear.  Furthermore, the Extended Kalman Filter reduces to a 
Recursive Least Squares estimator [Stengel, 1994] if the system is linear, discrete and 
Equations (4.14) and (4.15) are used for the time update. 
01| =?kkx&  (4.14) 
11| ?? = kkk PP  (4.15) 
The Extended Kalman Filter can reduces to a Recursive Least Squares [Astrom, 1989] 
with a forgetting factor by using Equations (4.14) and (4.13) using the following 
equation, and setting Fd to an identity matrix. 
11
1
???
??
?
? ?=
kd PQ ?  (4.16) 
 
 
 
53 
4.3 Design of Yaw Rate Dual Estimator 
 A dual estimator simultaneously estimates clean states and model parameters.  A 
clean estimate of yaw rate is needed for the position estimator.  An estimate of MDC is 
needed to adapt the steering controller.  The state equations used for the design of the 
Extended Kalman Filter are shown in Equation (4.17). 
( )( )
?
?
?
?
?
?
?
?
?
?
+
+
++
=
?
?
?
?
?
?
?
?
?
?
=
DC
bias
MDC
rbias
VxDC
DC
bias
wM
wr
wwVM
M
r
r
x
??
 (4.17) 
The states in the discrete estimator are yaw rate, yaw rate bias, and MDC.  Note 
that the yaw rate depends on the current measurement of steer angle.  Therefore, the time 
update can not be calculated before a measurement is received.  The velocity used in the 
estimator is from GPS which is available at 5 Hz.  Although GPS provides velocity in the 
direction of course, a small angle approximation allows GPS velocity to be used for the 
longitudinal velocity. 
 The Jacobian of the state equations with respect to the states (4.18) shows that the 
rate at which MDC converges will depend on the velocity and the steer angle. 
?
?
?
?
?
?
?
?
?
?
=
100
010
00 ?x
d
V
F  (4.18) 
The measurement in the Kalman filter is the yaw rate gyro which includes the yaw rate 
plus its bias, which is reflected in the Jacobian of the measurement equation with respect 
to the states shown below. 
[ ]011=H  (4.19) 
 
54 
However, the yaw rate gyro bias will only be estimated when MDC can be 
estimated, otherwise the bias would be corrupted with scale factor errors.  Therefore, 
Equation (4.20) will be used when there is too little excitation. 
[ ]001=H  (4.20) 
The bias could be estimated under low excitation by augmenting the estimator with a 
heading state and using the approximation that course angle equals heading angle at low 
levels of excitation [Bevly, 2001].  However, in Chapter 6 the estimator is designed in 
fixed point where only a limited number of states can be estimated.   
 Since a discrete covariance time update is used, the discrete process noise matrix 
was approximated using Equation (4.21). 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
==
100
010
00
00
000
000
000
000
1000
0100
00
2
2
2
2
1
?
?
?
?
?? ?
DC
xDC
M
r
V
DCxDC
T
ddd
M
VMMVM
GQGQ
DC
bias
 (4.21) 
In Equation (4.21), the process noise on the yaw rate state was computed to be a 
function of the DC gain times the yaw rate disturbance covariance ( ?? ) plus the noise 
covariance for the velocity ( V? ) times MDC and the steer angle measurement.  The sensor 
noise on the steer angle measurement was considered negligible.  The yaw rate bias and 
MDC were both modeled as random walks with covariances 
biasr
?  and 
DCM
? , which will 
determine how fast the estimates will react to changes in the states after the estimator has 
settled.  The measurement noise for the estimator was simply found with Equation (4.22). 
[ ]2rR ?=  (4.22) 
 
55 
Even though there is a clear relationship between the yaw rate and steer angle, it 
can not be estimated continuously, because too little excitation will cause the estimate to 
fit disturbances.  A method of determining how much excitation exists in the system is 
therefore needed.  Auto-correlation matrices of varying sizes can be used to determine 
how many parameters can be estimated [Astrom, 1995].  The largest auto-correlation 
matrix that is of full rank corresponds to the order of excitation there is in the system.  
The order of excitation then corresponds to how many parameters can be estimated.  
Since only one parameter is desired to be estimated in this thesis, only a one by one auto-
correlation matrix would have to be of full rank, which is always of full rank if it is non 
zero.  It is possible to determine when there is enough excitation by evaluating larger 
auto-correlation matrices.  However, sensor noise may make it appear as if there is more 
excitation than actually exists.  Therefore, the magnitude of a windowed auto-correlation 
was used to determine the degree of excitation in the system.  This indicates that there is 
a current trend in the steer angle, as opposed to a single large measurement, which makes 
it likely that MDC can be estimated for some period of time.  With this method there is a 
possibility that the window analysis may determine there is enough excitation for 
adaptation for the period defined by the time in the window, even though current 
excitation is quite low.  However, the rate of convergence decreases, as seen in Equation 
(4.18), as the magnitude of the steer angle decreases.  Therefore, little convergence would 
occur towards an incorrect value in this case.  The main goal of the determination of 
when to adapt is to prevent a prolonged estimation during low levels of excitation. 
Figure 4.1 shows the value of the auto-correlation of the steer angle over a one 
second window for an experimental test.  When this value is large enough that a fixed 
 
56 
relationship can be seen between the steer angle and yaw rate, it is determined that it is 
safe to adapt the system.  A threshold value is used for this determination which should 
depend on the maximum expected disturbance in the yaw rate and the amount of sensor 
noise on the steer angle.  As can be seen from Figure 4.1, a reliable time to adapt usually 
occurs at the beginning of the controlled run when the controller makes a step input to the 
tracked line.   The analysis also allows for an improved estimate if the tracking degrades 
as is also shown Figure 4.1 (at time = 60 seconds). 
20 40 60 80 1000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Time (s)
Windowed Auto?Correlation of Steer Angle
Auto?Correlation
Threshold
 
Figure 4.1: Windowed auto-correlation of steer angle. 
 
 When there is adequate excitation to accurately estimate MDC, the covariance 
associated with MDC is reset 0.0025 in order to inject uncertainty into the system and 
 
57 
allow the estimate to converge quickly.  When there is too little excitation in the system, 
the Kalman gain associated with MDC is over-ridden with zero and the noise covariance 
for MDC and the yaw rate bias is set to zero.  This is numerically equivalent to changing 
(4.18) to (4.23) and changing (4.21) to (4.24), where 
DCM
P  is the estimated covariance of 
MDC. 
?
?
?
?
?
?
?
?
?
?
=
100
010
000
dF  (4.23) 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
00
100
010
00
00
0000
00000
00000
0000
0000
01000
00100
00 2
2
?
??
?
??
?
x
DC
xDC
M
VxDCxDC
d
V
M
VM
P
VMVM
Q
DC
 (4.24) 
 Figure 4.2 shows the results of MDC being estimated in a simulation with a steady-
state input.  The estimator was told to start estimating MDC at time = 200 seconds and 
stop at time = 400 seconds.  The actual value of MDC was also decreased by ten percent at 
time = 300 seconds without the covariance being reset.  As seen, the estimate of MDC 
almost immediately reaches the correct value due to the covariance being reinitialized at 
time = 200 seconds.  The estimate takes longer to converge to the new value at time = 
300 seconds because only the forgetting factor is causing the estimate to move.  This rate 
of convergence is investigated more in Section 4.5.   System identification tests found 
that the disturbances were best described as second or third order transfer functions.  
However, the process disturbances are modeled as white noise in the estimator.  In order 
for all of the process disturbances to propagate through the estimator, the value of the 
 
58 
disturbance covariance ( ?? ) would have to be large.  Unfortunately, this would result in 
the MDC estimate taking a long time to converge.  Therefore, the disturbance covariance 
is kept small to allow for faster convergence at the cost of filtering out some of the 
process disturbances.  Because the disturbances were filtered, adaptively changing the 
noise covariance for MDC was not investigated. 
200 250 300 350 400
0.02
0.025
0.03
Yaw Rate (rad/s)
200 250 300 350 4000.2
0.25
0.3
0.35
0.4
Time (s)
M D
C
Estimate
? Bounds
Estimate
? Bounds
 
Figure 4.2: Estimate of MDC and yaw rate. 
  
 The nominal noise values used in the design of the Extended Kalman Filter are 
listed in Table 4.1.   
 
 
59 
Table 4.1 Discrete noise values used for the yaw rate dual estimator 
Parameter Value Units Frequency 
V?  0.05 meters/second 5 Hz 
??  8.73x10
-4 radians 100 Hz 
biasr
?  1x10-7 radians/second 100 Hz 
DCM
?  1x10-4 1/meters 100 Hz 
r?  5.23x10
-3 radians/second 100 Hz 
 
4.4 Sensitivity of Estimation to Neglected Dynamics 
 To examine how the accuracy of the estimate is affected by neglecting the poles 
and zeros in the yaw rate transfer function, a simulation was designed to see what kind of 
errors this would cause.  The transfer function used to simulate the disturbance free plant 
and the disturbances (shown in Equations (4.25) and (4.26), respectively) were taken 
from a Box Jenkins fit of experimental data at 2 mph with no implement. 
0.8994213.522835z-5.343205z3.718368z-z
0.2393940.765003z-0.814621zz 0.288557-
234
23
++
++=
?
r  (4.25) 
 These transfer functions represented a worse case scenario, because they 
contained the largest magnitude and phase change over the low frequencies found from 
system identification tests. 
0.9444991.880024z-z
1.0021912.002189zz
2
2
+
++=
e
rdist  (4.26) 
 The standard deviation of the noise generating the disturbances was also obtained 
from the Box Jenkins fit, and is displayed below with the units of radians. 
 
60 
13869622000.0=e?  (4.27) 
Sine waves at a number of different frequencies and amplitudes were tested as inputs to 
determine the resulting mean and standard deviations of the estimates after the estimator 
had settled.  The simulation assumed that there was always enough excitation to estimate 
MDC and was run long enough to determine a clear trend in the data. 
 Figure 4.3 shows the actual value of MDC with the average value of the MDC 
versus frequency with input amplitudes of 5, 10, and 15 degrees.  Also shown in the 
figure is the magnitude of the gain in Equation (4.25) divided by velocity, which is 
labeled as (Bode Magnitude)/V.  Recall that MDC is the magnitude of the DC gain divided 
by velocity and is therefore represented as a constant value.  The magnitude of gain in 
Equation (4.25) divided by velocity corresponds to the value of MDC that should be 
estimated if there is no phase change.  As can be seen, the estimated values are close to 
the bode curve at less than 1 rad/sec but estimates lower than it at higher frequencies.  
The estimates became worse at higher frequencies. It should be noted that the tractor 
dynamics rarely operate in these high frequencies.  As will be shown later in this section, 
these errors can be reduced using a more accurate model in the estimator. 
 
61 
0.5 1 1.5 2 2.5 30.36
0.37
0.38
0.39
0.4
0.41
0.42
0.43
0.44
0.45
Frequency (rad/s)
M D
C
Amplitude = 5 deg.
Amplitude = 10 deg.
Amplitude = 15 deg.
Actual MDC
(Bode Magnitude)/V
 
Figure 4.3: Mean MDC versus input frequency using the input of ( )wtAsin=? . 
 
 Figure 4.4 contains the clean yaw rate, the steer angle times the DC gain of 
Equation (4.25), and the estimated yaw rate for a 3 rad/s simulation with a steer angle 
input of 5 degrees.  As seen in the figure, there also exists a phase difference in addition 
to the magnitude difference.  However the magnitude error is the main source of error 
shown in Figure 4.3.  This is because the difference in the phase between the actual 
system and estimator model is minimal (compared to the differences in magnitude)  as is 
shown later in Figure 4.10.   
 
62 
50 52 54 56 58 60
?0.03
?0.02
?0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
Time (s)
Yaw Rate (rad/s)
Actual
Steer Angle * DC gain
Estimate
 
Figure 4.4: Magnitude and phase difference between yaw rate and steer angle times DC 
gain for ( )t3sin5=?  (degrees). 
 
 Figure 4.5 shows the estimated MDC for the simulated run shown in Figure 4.4.   
As seen in the figure, the estimate contains some of the 3 rad/s sine wave due to the 
forgetting factor placed on the estimate. 
 
 
63 
50 52 54 56 58 600.425
0.43
0.435
0.44
0.445
0.45
0.455
Time (s)
Estimated   M
DC
 
Figure 4.5: Estimate of MDC with an input of ( )t3sin5=?  (degrees). 
 
 In Figure 4.6 shows the standard deviation of the estimated MDC for each velocity 
and input amplitude, which should contain the same standard deviation regardless of 
frequency.  The Jacobian of the state equations with respect to the states (4.18) shows 
that the estimate will be less uncertain when the input is larger.  However, this is not 
reflected in Figure 4.6.  This may because due to the fact that at higher amplitudes more 
of the sine wave is incorporated into its estimate. 
 
 
64 
0 0.5 1 1.5 2 2.5 33.6
3.8
4
4.2
4.4
4.6
4.8
5 x 10
?3
?
Frequency (rad/s)
5 degrees (?est = 0.0033)
10 degrees (?est = 0.0026)
15 degrees (?est = 0.0022)
 
Figure 4.6: Standard deviation of estimated MDC versus input frequency. 
 
 The Kalman Filter provides an estimate of the covariances of the states.  The 
square root of the covariance once the estimator has converged for the MDC state is 
contained in Table 4.2 for an input amplitude of 1 rad/s.  Note that the values in Figure 
4.6 are greater than the values in Table 4.2, because as mentioned previously the 
forgetting factor causes some of the sine wave to be incorporated into the estimate. 
 
 
 
 
 
65 
Table 4.2: Square root of covariance of MDC provided by the estimator 
Sine Wave Amplitude Standard Deviation 
5 degrees 3.11x10-3 
10 degrees 2.18x10-3 
15 degrees 1.77x10-3 
 
 Using the bode magnitudes from the system identification tests divided by 
velocity for the upper bounds on the MDC estimate, Figure 4.7 shows that with no 
implement the errors decrease as velocity increases.  Since the value should be the same 
as at zero frequency.  Similarly Figure 4.8 shows that the error decreases with hitch 
loading. 
0
1
2
3
0
1
2
3
0.3
0.35
0.4
0.45
Frequency (rad/s)Velocity (m/s)
M D
C Bounds
 
Figure 4.7: Bode magnitudes divided by velocity for no implement. 
 
66 
0.5 1 1.5 2 2.5 3
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
Frequency (rad/s)
M D
C Bounds
No Implement
Ripper at 4 inches
Ripper at 8 inches
Ripper at 12 inches
 
Figure 4.8: Bode magnitudes divided by velocity for various hitch loadings at 3 mph. 
 
 The estimation of MDC can be improved through the use of the dominant poles of 
the system.  Equation (4.28) shows the transfer function that was used in an improved 
estimator. 
( )
0.9706041.962266z
0.9706041.9622661
2 +?
+?=
z
VMr xDC
?  (4.28) 
This requires that an additional intermediate state (?) be added to the estimator, which is 
shown in Equation (4.29). 
 
67 
( ) ( )( )
?
?
?
?
?
?
?
?
?
?
?
?
+
+
+++?+
+
=
?
?
?
?
?
?
?
?
?
?
?
?
=
DCMDC
biasrbias
VxDC
DC
bias
wM
wr
wwVM
M
r
r
x ??
?
? 970604.096226.110.970604r-
1.962266r
 (4.29) 
The Jacobian of the measurement equations with respect to the states then changes to 
Equation (4.30) when the bias can be estimated. 
[ ]0101=H  (4.30) 
The computation of the discrete covariance matrix is performed with Equation (4.31) 
using the same approximation as in Equation (4.21) and using the same value of Q1. 
( ) ( )
?
?
?
?
?
?
?
?
?
?
?
? +?+?
=
1000
0100
0000
00970604.096226.11970604.096226.11 ?DCxDC
d
MVM
G  (4.31) 
 The simulation was then run again with the new estimator.  As seen in Figure 4.9 
the difference between the estimated gain and the Bode magnitude at 3 rad/sec is now 
half that of what was obtained using the previous estimator.  The (Bode Magnitude)/V 
value shown in the figure was obtained through Equation (4.32).   
???
?
???
? ??=?
DC
ModelOrderSecondModelFullestsys M
V
Mag
V
Mag
V
Mag  (4.32) 
This corrects the Bode magnitude from Equation (4.25) for the magnitude improvement 
obtained from using the second order model in the estimator.  As can be seen in Figure 
4.9, the estimates are now closer to the Bode magnitude, which implies that the phase 
errors are having less of an effect then what was shown in Figure 4.3. 
 
68 
0.5 1 1.5 2 2.5 30.36
0.365
0.37
0.375
0.38
0.385
0.39
0.395
0.4
0.405
0.41
Frequency (rad/s)
M D
C
Amplitude = 5 deg.
Amplitude = 10 deg.
Amplitude = 15 deg.
Actual MDC
(Bode Magnitude)/V
 
Figure 4.9: Mean MDC versus input frequency for second order model. 
 
 Figure 4.10 shows the frequency response of the simulated system and the second 
order system in the estimator for the frequencies used in the simulation.  As seen in the 
figure, the dominant poles do not account for all of the magnitude and phase change in 
the simulated system.  This is most likely caused by the set of zeros after the dominant 
poles having little damping themselves.  Therefore, it may be possible to design an 
improved estimator by adding more dynamics or by possibly by modifying the identified 
dominant poles to have less damping such that the magnitudes of the actual model and 
estimator model match more closely.   
 
69 
10?1 100 101
?10
?5
0
5
10
Magnitude (dB)
10?1 100 101
?150
?100
?50
0
Phase (deg)
Frequency (rad/sec)
Actual Model
Second Order Model
 
Figure 4.10: Frequency response of the actual system and second order estimator 
model. 
 
 Figure 4.11 shows the standard deviation of the simulations with the improved 
estimator.  Note that the estimates now have the same standard deviations regardless of 
the frequency and that the effects of the estimate incorporating the sine wave have been 
reduced. 
 
 
70 
0.5 1 1.5 2 2.5 32.8
3
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
x 10?3
Frequency (rad/s)
?
5 degrees (?est = 0.014)
10 degrees (?est = 0.0095)
15 degrees (?est = 0.0078)
 
Figure 4.11: Standard deviation of MDC for improved estimator. 
 
4.5 Rate of Convergence 
 A simulation was conducted using steady state inputs to examine how long it 
takes for the estimator to converge on the correct value without using the covariance 
resetting, such that only the forgetting factor caused the convergence.  No noise was 
added to the simulated measurements passed to the estimator, but the assumed that the 
nominal noise characteristics.  This produces the same results as a monte carlo simulation 
since the measurement noise is zero mean.  Figure 4.12 shows the results from three 
different offsets, for which it can be seen that they all converge at the same amount of 
time.  This is understandable because the Extended Kalman Filter effectively acts as a 
 
71 
closed loop controller of the estimates and the value of MDC is effectively making a step 
input to the correct value. 
0 10 20 30 40 500.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
Time (s)
M D
C
Actual Value
Initially 60% of Actual Value
Initially 80% of Actual Value
Initially 90% of Actual Value
 
Figure 4.12: Convergence from different offsets. 
 
 While the amount of time to converge is independent of the offset, it is dependant 
on Vx? according to Equation (4.18).  Therefore, the simulation was conducted again for 
various Vx? with different noise values used for MDC (
DCM
? ).  The time to converge to 
within 99.9 % of the correct value was recorded.  As shown in Figure 4.13, MDC 
converges faster for larger values of 
DCM
?  and when Vx? is larger. 
 
 
72 
0.1
0.2
0.3
0.4
1
1.5
2
2.5
3x 10
?3
0
20
40
60
80
Vx? (m*rad/s)?
MDC
Time (s)
 
Figure 4.13: Time to converge within 99.9 % of the correct value. 
 
 The square root of the estimated covariances for the simulations in Figure 4.13 
was also recorded and is shown in Figure 4.14.  It can be seen that the estimates get 
nosier as the noise value on MDC is increased.  It can also be seen in Figure 4.14 that the 
estimator noise placed on the MDC state can be increased, allowing for faster convergence 
of the estimate, when ?xV  increases without sacrificing accuracy of the MDC estimate.  
However, recall that in Section 4.3 it was found that the actual accuracy of the estimate is 
likely much larger than the predicted accuracy shown in Figure 4.14. 
 
 
73 
0.1
0.2
0.3
0.4
1
1.5
2
2.5
3x 10
?3
0
1
2
3
4
5
x 10?3
Vx? (m*rad/s)? 
MDC
? E
sti
ma
te
 
Figure 4.14: Estimated standard deviation for MDC. 
 
4.5 Design of Position Estimator 
 The states in the position estimator are north, east, and course.  The input to the 
position estimator is the filtered yaw rate from the yaw rate estimator.  Unlike the yaw 
rate estimator which uses GPS velocity to approximate longitudinal velocity, the position 
estimator needs the velocity in the direction of the course angle.  Therefore, GPS 
provides the correct measurement of velocity for this estimator.  The calculation of north 
and east involves a coordinate transformation from the body fixed frame to the global 
frame, as can shown in Equation (4.31), which contains the continuous state equations for 
the estimator.   
 
74 
( ) ( )
( ) ( )
?
?
?
?
?
?
?
?
?
?
+
+
+
=
?
?
?
?
?
?
?
?
?
?
=
r
V
V
wr
wV
wV
E
N
x ?
?
?
?
?
sin
cos
&
&
&
&  (4.31) 
 Since there is no velocity perpendicular to the course direction, there is only one 
term in the north and east derivatives.  Note that yaw rate plus the derivative of side slip 
is the derivative of course ( ?? && += r ).  However, it will be shown that the course 
measurement provided by GPS is clean enough to compensate for neglecting the 
derivative of sideslip in the estimator. 
The Jacobian of the state equations with respect to the states is shown in Equation 
(4.32). 
( )
( )
?
?
?
?
?
?
?
?
?
? ?
=
000
cos00
sin00
?
?
?
?
V
V
F  (4.32) 
Since the errors in the state equations include sensor noise on the velocity and uncertainty 
in the yaw rate estimate passed to the position estimator, the Jacobian of the state 
equations with respect to the disturbance contains two columns as shown in Equation 
(4.33). 
( )
( )
?
?
?
?
?
?
?
?
?
?
=
10
0sin
0cos
?
?
G  (4.33) 
 The first column is associated with the velocity sensor noise and the second column is 
associated with the uncertainty in the yaw rate estimate. 
 
75 
 The continuous process noise covariance matrix shown in Equation (4.34) 
contains the covariance of the sensor noise for the velocity measurement ( V? ) and the 
covariance of the yaw rate that has been passed from the yaw rate estimator (
DCM
P ). 
???
?
???
?=
DCM
V
PQ 0
02?  (4.34) 
It should be noted that the yaw rate estimator provides a discrete covariance for its 
filtered yaw rate which must be converted to continuous before being used in Equation 
(4.34).  However, no coordinate transformation of the yaw rate covariance is necessary 
since roll and pitch angles have been assumed to be negligible. 
The GPS messages have significantly more computation time than the inertial 
measurements.  However, if this computational delay is known, it can be accounted for in 
the output equation of the estimator.  In order to obtain the amount of delay in the GPS 
output, an RTD GPS receiver with a pulse per second (PPS) output, a one hundred 
kilohertz clock (synchronized to the GPS PPS), and two counters were used.  Knowing 
that the pulse per second of the RTD GPS receiver and the Starfire measurement are 
syncronized, the one hundred kilohertz clock and the two counters were used to 
determine the lag between when the GPS measurement occurred and when the serial 
message arrived at the computer.  Figure 4.15 shows the results of one experimental test.   
The mean value of the delay in the figure is 0.0787 seconds.  It should be noted 
that some of the noise is most likely to be from the message passing between the serial 
driver and the logging program, which can generally be slow.  A more accurate way of 
getting this lag would be to monitor the serial port?s interrupt line. 
 
 
76 
0 20 40 60 80 1000.076
0.078
0.08
0.082
0.084
0.086
0.088
0.09
0.092
0.094
Time (s)
Delay (s)
 
Figure 4.15: Delay in GPS messages 
 
 The lag in the receipt of the GPS messages can then be accounted for in the 
measurement update of the estimator.  Equation (4.35) shows that a simple Eulers 
approximation can be used along with the estimates of the states to propagate the current 
estimates back to when the measurement should have been received. 
( )
( )
?
?
?
?
?
?
?
?
?
?
+?
+?
+?
=
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
? vrT
vTVE
vTVN
E
N
y
lag
Elag
Nlag
meas
meas
meas
sin
cos
 (4.35) 
Section 4.6 will provide further evidence that this compensation is needed.  In Equation 
(4.35) Tlag is the time that has elapsed between when the measurements should have been 
 
77 
received by the estimator and when they are used in the estimator.  While the time update 
of the position estimator is conducted at 100 hertz, the measurement update only occurs 
when a GPS measurement is available, which is at 5 hertz. 
 In the Jacobian of the measurement equations with respect to the states, Equation 
(4.36), it can be seen that a correction of the course angle is provided directly from the 
position measurements because of the delay. 
( )
( )
?
?
?
?
?
?
?
?
?
?
?=
100
cos10
sin01
lag
lag
TV
TV
H ?
?
?
?
 (4.36) 
The Jacobian of the measurement equations with respect to the measurement 
noises shown in Equation (4.37) is no longer an identity matrix, since there is now a 
direct contribution from the input of the estimator to the output.   
?
?
?
?
?
?
?
?
?
?
?
=
lagT
U
100
0010
0001
 (4.37) 
 The discrete covariance of yaw rate from the yaw rate estimator is then needed to 
be included in the measurement noise covariance matrix shown in Equation (4.38). 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
DCM
E
N
P
R
000
000
000
000
2
2
2
??
?
?
 (4.38) 
In the equation N?  and E?  are the noises on the position states, whose errors are 
dominated by its bias.  However, since the estimator doesn?t have a way to remove this 
bias, the estimator uses a noise value in order to filter some of the bias.  The value ??  is 
 
78 
the standard deviation of the course angle measurement which is a function of velocity 
and 
DCM
P  is the covariance passed from the yaw rate estimator. 
In Figure 4.16 a comparison of the measured and estimated course shows that 
despite the derivative of sideslip ( ?& ) being neglected in the position estimator, the GPS 
course measurement is accurate enough to provide corrections. 
10 20 30 40 50 60 70 80 90 1003.85
3.9
3.95
4
4.05
4.1
4.15
4.2
Time (s)
Course (rad)
Measured
Estimated
 
Figure 4.16: Comparison of measured and estimated course from experimental test. 
 
 The discrete noise values used in the Extended Kalman Filter for the position 
estimator are listed in Table 4.3. These values and their continuous counterparts were 
used in Equations (4.34) and (4.38). 
 
 
79 
Table 4.3: Discrete noise values used for the position estimator 
Parameter Value Units Frequency 
V?  0.05 meters/second 5 Hz 
N?  0.02 meters 5 Hz 
E?  0.02 meters 5 Hz 
??  
V
05.0  radians 5 Hz 
 
4.6 Effects of GPS Delay 
 To test the effects of the GPS delay on the position estimator, the simulation in 
Section 4.4 was again used.  The input used was a sinusoidal steer angle with a 5 degree 
amplitude at 2 mph, driving in open loop, in the direction of north.  Three simulations 
were run to compare the effect of the GPS delay.  The first simulation had no GPS delay, 
the second simulation had a 0.08 second delay with no compensation in the estimator, 
and the third simulation had a 0.08 second delay which was compensated for in the 
estimator.  The estimators used the nominal sensor and input noise specifications, but 
noiseless values were provided to the estimators so that a clearer picture of the errors 
could be obtained.  Additionally, no side slip was included in the simulation and hence 
course reduced to heading. 
 Figure 4.17 shows the result for the error in the north position estimate for the 
three simulations.  Without compensating for the delay the vehicle estimate is behind the 
true location.  However for this work, longitudinal accuracy is not a degrading factor.  
Some of the cosine wave can also be seen in the uncompensated system, which is due to 
 
80 
the lateral effects from the tractor not always heading north.  The estimates for the 
compensated system are slightly ahead of the true position, likely due to integration 
errors. 
0 50 100 150 200?0.08
?0.07
?0.06
?0.05
?0.04
?0.03
?0.02
?0.01
0
0.01
Time (s)
Notrh Error (m)
No Delay
Compensating For Delay
Not Compensating For Delay
 
Figure 4.17: North error 
  
From Equation (4.35) it can be seen that the longitudinal error of the vehicle can 
be approximated with Equation (4.39).   
lagxallongitudin TVe ?=  (4.39) 
For the simulation, the value computed from (4.39) was 0704.0?  meters, which 
corresponds well with what is seen in Figure 4.17.  Equation (4.40) approximates the 
longitudinal error with respect to the desired path, where err?  is the difference between 
 
81 
the heading of the desired path and the tractor heading during the GPS delay, and err?  is 
the angle between the desired path and the course angle.   
( ) ( )[ ] ( ) lagerrlagerryerrxpathtoallongitudin TVTVVe ??? ? cossincos ?=??=  (4.40) 
Note that the above equation is an approximation since the sensor noise covariances in 
the estimator will provide some filtering and there is an interaction of errors.   
 Figure 4.18 shows that the error in the east position from the uncompensated 
system can produce errors that would affect the tractors lateral error tracking 
performance.  If the measurement delay is not compensated, the GPS measurement of the 
tractor could be to the left of the desired path when the tractor is actually to the right of 
the desired path.  This would cause the controller to move the tractor in the wrong 
direction leading to instability.  The error shown in Figure 4.18 is due to the effect of the 
longitudinal error entering into the East error at small heading angles which affects the 
lateral error of the tractor since it is not always heading parallel to the line it is tracking.  
Equation (4.41) provides an estimate of the lateral error in the estimator due to the GPS 
measurement delay. 
( ) ( )[ ] ( ) lagerrlagerryerrxpathtolateral TVTVVe ??? ? sincossin ?=+?=  (4.41) 
For the simulation in Figure 4.18, the maximum value of err?  was 0.115 radians which 
resulted in a maximum lateral error of 0.008 meters.  This corresponds very well to the 
predicted lateral error of 0.00808 using Equation (4.41). 
The lateral errors induced from this measurement latency can also be important if 
the tractor is oscillating on the verge of instability, since it could cause the tractor to go 
unstable.  Recall that measurement latency can be modeled as a loss of phase in the 
 
82 
system which reduces the phase margin [Franklin, 2002].  The compensated system still 
contains some of the cosine wave but its errors are much more manageable. 
0 50 100 150 200?0.01
?0.005
0
0.005
0.01
0.015
Time (s)
East Error (m)
No Delay
Compensating For Delay
Not Compensating For Delay
 
Figure 4.18: East error 
 Figure 4.19 shows that the heading error produces errors that appear similar to 
that of the error in the lateral direction from the vehicle.  Equation (4.42) approximates 
the heading estimation error due to the measurement latency. 
lagcourse rTe ?=  (4.42) 
For the simulation in Figure 4.19 the maximum heading error using Equation (4.42) is 
0.00225 radians which corresponded well with the results in Figure 4.19. 
 
83 
0 50 100 150 200
?2
?1
0
1
2
3
4 x 10
?3
Time (s)
Heading Error (rad)
No Delay
Compensating For Delay
Not Compensating For Delay
 
Figure 4.19: Heading Error 
 
The covariance estimates of the estimators with no GPS measurement delay and 
with the delay compensation are shown in Figure 4.20.  As can be seen, the uncertainty in 
the compensated system is only slightly higher than without the delay.  Additionally, no 
lag can be seen in the covariance estimates between the two simulations.  The north 
estimate is also cleaner then east estimate.  This is because the uncertainty in course does 
not enter into the position state in the direction of course (as seen in Equation (4.32)), 
which is mostly in the direction of north in this simulation. 
 
84 
50 60 70 80 90 100
2
2.5
3
3.5
x 10?6
       North     Covariance       
50 60 70 80 90 100
6
8
10
x 10?5
       East      Covariance       
50 60 70 80 90 100
2
4
6
8
10
x 10?5
    Heading   Covariance    
Time (s)
Compensating For Delay
No Delay
 
Figure 4.20: Comparison of covariances for an estimator with no GPS measurement 
delay and an estimator that compensates for the delay. 
 
 
85 
4.7 Summary and Conclusions 
 Details on the design of an estimator to estimate the yaw rate, yaw rate bias, and 
MDC were given in this chapter.  A method was given that determined when to estimate 
the yaw rate bias and MDC so that disturbances did not dominant the estimation of MDC.  
It was shown that with the covariance resetting, the estimate of MDC would converge 
quickly.  A solution to estimating the bias during low levels of excitation was also 
explained and the effect of the neglected dynamics on the estimation was displayed. 
It was shown that most of the errors occurred because of magnitude errors 
between the estimator model and actual system and not phase errors.  It was also shown 
that the covariance estimate for the estimated MDC was lower than the actually accuracy 
of the estimated MDC.  The errors in the estimate of MDC caused by the magnitude errors 
were found to decrease as the velocity and hitch loading were increased.  A method of 
compensating the adaptation for these errors was found by approximating the pole 
locations.  The rate of convergence of MDC due to the forgetting factor was found to be 
independent of its offset and was instead simply a function of the longitudinal velocity 
times the steer angle.  It was shown that MDC would converge faster with higher 
amplitudes of excitation and with a larger forgetting factor on the estimated state.  
However, increasing the forgetting factor resulted in a noisier estimate of MDC. 
The design of the position estimator for providing faster updates of position was 
also detailed.  This estimator was designed to compensate for the delay in the receipt of a 
GPS message since it was shown that the effect of this lag could cause large position 
errors.  Equations were provided that approximated the errors induced by the GPS lag. 
 
86 
CHAPTER 5
EXPERIMENTAL IMPLEMENTATION 
5.1 Introduction 
The results of experimental tests using the controllers and estimators discussed in 
previous chapters are given in this chapter.  A John Deere 8420 tractor was used for the 
experimental tests.  A Bosch gyroscope was used to measure yaw rate and a linear 
potentiometer was used to measure the steer angle.  Both measurements were 
synchronized to GPS measurements using a GPS pulse per second (PPS).  The GPS 
measurements were provided by a Starfire GPS receiver.  The delay in the GPS messages 
was obtained using a ten kilohertz clock synchronized to the one hertz clock (PPS) as 
discussed in Chapter 4. 
 
5.2 Trajectory Design 
 In this thesis all tractor trajectories consist of straight line segments.  Figure 5.1 
displays a desired path in a North-East coordinate system.  The desired path is defined by 
a reference point (Nref,Eref) and a reference heading (?ref).  The point (N,E) is the location 
of the tractor.  (Ndes,Edes) is the desired point on the path which is defined by a line 
perpendicular to the path that goes to the point (N,E).  The tractor position can then be 
described as a longitudinal distance from the reference point and a lateral distance from 
the path. 
 
87 
Figure 5.1: Tracking a straight line. 
 
 Using the coordinate transformation given in Equation (5.1) the longitudinal and 
lateral distances can be found from the North-East coordinate system. 
( ) ( )
( ) ( ) ??
?
??
?
?
?
??
?
??
?
?=??
?
??
?
ref
ref
refref
refref
EE
NN
y
x
??
??
cossin
sincos  
(5.1) 
 It was shown in Chapter 4 that higher frequencies can cause the estimator to 
estimate the incorrect value of MDC.   Thus, it can be an advantageous to prevent the 
tractor from executing a step input by redrawing the path when the lateral distance is 
large and conducting a steady-state turn to the path.  Additionally, if the tractor is at too 
large of an offset it will approach the desired path perpendicular to it, creating an 
unsatisfactory response.  Therefore, the desired path is redefined using a circular path, 
which is tangential to the desired straight line trajectory, when the tractor is a large 
y 
x 
(Ndes,Edes) 
(N,E) 
(Nref,Eref) 
?ref 
North 
East 
Desired 
Path 
 
88 
distance away from the desired line.  A description of the circular path is shown in Figure 
5.2. 
 
Figure 5.2: Curved trajectory to the straight line. 
 
 The value z is the distance between the circular path and the straight line path.  
When it is determined that the offset from the desired straight line path is too large, the 
circular path is formed by setting the current lateral offset as shown in the equation 
below. 
yxinit =  (5.2) 
 The distance d that is desired for when the straight line path and the curved path 
converge is then computed as a function of the initial offset using Equation (5.3), where l 
is a value larger or equal to one. 
z 
d 
|R| 
(Ndes,Edes) 
x 
(Nref,Eref) 
(N,E) 
y 
 
89 
initinit lxd =  (5.3) 
As will be seen in the next few equations l sets the rate to which the tractor will (limited 
by the tractor dynamics) approach the line trajectory.  In this work l was chosen to be 4. 
The radius can then be calculated by solving for the hypotenuse of a triangle that 
takes into account the initial offsets as shown below. 
( ) ( )222 initinit xradiusdR ?+=  (5.4) 
Substituting Equation (5.3) into Equation (5.4) results in Equation (5.5), where the sign 
of R contains the information corresponding to which side of the straight line the circular 
path is on. 
initx
lradius
2
12 +=  (5.5) 
 Solving for Equation (5.4) with the current offsets instead of the initial offsets, the 
distance between the circular and straight path while the tractor is moving can be solved 
as shown below. 
( ) 22 dradiusradiussignradiusx ??=  (5.6) 
The lateral distance to the circular path can then be found using Equation (5.6) and the 
distance y. 
 
5.3 Results 
 To first validate the ability of the algorithms to provide correct estimates of MDC.  
Experimental test were conducted.  For these experiments, the tractor was commanded to 
track a straight line from an initial offset.  The estimator used an initial value of MDC set 
to 0.25.  Figure 5.3 shows the results of two tests.  The on-line experimentally estimated 
 
90 
MDC values have less than a five percent error from the true value identified in Chapter 2 
(MDC = 0.334). 
6 6.5 7 7.5 8 8.5 9 9.5 10
0.22
0.24
0.26
0.28
0.3
0.32
Time (s)
M D
C
 
Figure 5.3: Estimation of MDC during experimental tests. 
 
 Tests were also conducted to determine how the lateral error was affected by 
changes in velocity.  Figure 5.4 shows that the lateral error increases only slightly with 
increased velocity. 
 
91 
0.5 1 1.5 2 2.5?0.01
0
0.01
0.02
0.03
mean (m)
0.5 1 1.5 2 2.50.005
0.01
0.015
0.02
0.025
? (m)
Velocity (m/s)  
Figure 5.4: Lateral error versus velocity 
 
 The tracking performance of the controller without an implement is displayed in 
Figure 5.5, where five experimental results are shown.  As seen in the figure, the tracking 
performance appears to remain fairly constant for each of the tests. 
 
92 
20 30 40 50 60 70 80 90 100?0.08
?0.06
?0.04
?0.02
0
0.02
0.04
0.06
0.08
Time (s)
Lateral Position (m)
 
Figure 5.5: Lateral error without an implement. 
 
 In Figure 5.6 the tracking performance of the controller with a four shank ripper 
at a twelve inch depth is shown for five experimental tests.  As shown in the above 
figure, the lateral tracking accuracy degrades slightly with the implement. 
 
93 
20 30 40 50 60 70 80 90 100?0.08
?0.06
?0.04
?0.02
0
0.02
0.04
0.06
0.08
Time (s)
Lateral Position (m)
 
Figure 5.6: Lateral error with a four shank ripper at a twelve inch depth. 
 
 The corresponding mean and standard deviation of the experimental tests with 
and without an implement are shown in Table 5.1.  As can be seen in the table, the 
standard deviations of the controlled runs with an implement are larger than without an 
implement.  This is most likely due to the fact that the implement induces larger 
disturbances into the system.  However, the fact that the experimental control accuracy 
with and without an implement is fairly consistent shows that the adaptive control 
algorithm, based on on-line estimation of MDC, provides an effective means to handling 
the tractor dynamic variations. 
 
 
94 
Table 5.1: Statistics of lateral error at a constant velocity 
 No Implement Ripper at 12? depth 
Run Mean (m) ? (m) Mean (m) ? (m) 
1 0.00491 0.0191 0.00496 0.0187 
2 0.00639 0.0120 -0.0105 0.0167 
3 0.0109 0.0156 -0.00747 0.0221 
4 0.00454 0.0167 -0.00646 0.0228 
5 0.00453 0.0205 -0.0118 0.0210 
Average 0.00625 0.0168 -0.0252 0.0202 
 
5.4 Summary and Conclusions 
This chapter has detailed the experimental implementation of the estimation and 
control algorithms developed in this thesis.  The trajectories used for the tractor controller 
were discussed.  Experimental results showing the accuracy of the estimation algorithms 
were then provided.  Experimental tests demonstrated that the tracking response changed 
little with velocity.  The average standard deviations while varying velocities was found 
to be 0.016 m.  It was also shown that the standard deviation of the lateral position 
changed little with hitch loading with a standard deviation of 0.0168 m without an 
implement and 0.0202 m with a 4 shank ripper at a 12? depth while driving at 2 mph. 
 
 
95 
CHAPTER 6
FIXED POINT IMPLEMENTATION OF YAW RATE ESTIMATOR 
6.1 Introduction 
 Implementation of a Kalman filter is difficult in a fixed point microprocessor.  
The propagation of the covariance presents the largest challenge, since it can be poorly 
conditioned if small machine precision is used.  However, filters that propagate the 
square root of the covariance are more easily manageable, because they propagate a 
matrix that has the square root of the condition number of what the Kalman filter 
requires.  Examples of square root Kalman filters include the square root covariance filter 
and square root information filter [Anderson, 1979; Bierman, 1977; Maybeck 1979a].   
In this chapter, a square root covariance filter (SRCF) will be used to show that 
the Kalman filter from Chapter 4 can be implemented using fixed point math.  The 
estimator provides an estimate of the slope of the DC gain with respect to velocity of the 
transfer function between steer angle and yaw rate (MDC). 
 
6.2 Square Root Covariance Filter 
 The square root covariance filter is derived from the Kalman filter equations.  It 
uses orthogonal matrices, T1 and T2, shown in Equations (6.2) and (6.3) to solve for upper 
triangular matrices.   
( )111| , ??? = kkkk uxfx  (6.1) 
 
96 
??
?
??
?=
??
?
??
? ??
TT
TT
k
T
kk
LQ
FSTS
2/1
1
1
1|
0  (6.2) 
The process of splitting the matrix into an orthogonal and upper triangular matrix is 
known as QR decomposition and the most popular methods used for square root Kalman 
filters are the householder [Golub, 1989] and modified Gram-Schmidt methods.  The 
householder method is used in this work.  Equations (6.1) and (6.2) consist of the time 
update of the estimator.  Note that the state time update is the same as in the Kalman 
filter. 
 Equations (6.3-6.5) show the measurement update equations for the estimator. 
( )
???
?
???
?=
???
?
???
? +
??
T
kk
TT
kk
T
T
k
TTT
SHS
RT
S
KPHHR
1|1|
2/1
2
2/1 0
0
~
 (6.3) 
( ) 2/1~ ?+= PHHRKK T  (6.4) 
( )1|1| ?? ?+= kkkkk HxzKxx  (6.5) 
 The QR factorization displayed in Equation (6.3) returns terms that are used for both the 
covariance square root and the Kalman gain.  The state measurement update shown in 
(6.5) is the same as the Kalman filter state measurement update.  In Equation (6.4) it is 
evident that a matrix inverse is needed if there is more then one measurement.  However, 
sequential processing is used in square root filters to eliminate the need for a matrix 
inversion. 
 When compared with the Kalman filter the square root covariance filter takes 
more computation time [Bierman, 1977; Kaminski, 1971].  This is due to the fact that the 
SRCF needs more multiplications, additions, divisions, as well as square root 
calculations. 
 
97 
6.3 Fixed Point Math 
 For the fixed point implementation, additions and subtractions were performed as 
normal.  Multiplications were performed using Equation (6.6), where >> is the right shift 
operator and N is the number of bits to shift. 
Nab >>  (6.6) 
Divisions were carried out using Equation (6.7), where << is the left shift operator. 
( ) bNa /<<  (6.7) 
An overflow would result if the answer to the equations was larger then 215, since 
variables were defined as 16 bit signed integers. 
The householder method was used on Equation (6.8) as an example of the 
accuracy of the fixed point implementation of the algorithm. 
?
?
?
?
?
?
?
?
?
?
?
?
?
?=
412756119
1129645210
89623261435
15543156345
A  (6.8) 
The householder method reduces Equation (6.8) into A=QR, by doing a column wise 
reduction. 
 Equation (6.9) shows the result of the decomposition when floating point is used. 
?
?
?
?
?
?
?
?
?
?
?
?
?
???
=
6349.184000
4878.3523396.12100
2676.176926.8164332.2990
0321.324095.1254271.962026.607
R  (6.9) 
Performing the decomposition in fixed point results in errors build upon one another as 
each column is reduced as shown in Equation (6.10). 
 
98 
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
???
=
214002
33313034
18280129512
3012692605
R  (6.10) 
This implies that the first state will have the most accurate covariance estimate. 
 
6.4 Estimator Model 
 Since the first state will have the most accurate covariance estimate, the slope of 
the DC gain (MDC) was made the first state.  The yaw rate state in the estimator was set to 
be scaled by 9=N bits from the units of degrees/second.  Meaning that the yaw rate state 
would be yaw rate in degrees/second times 29.  The measured yaw rate, velocity, and 
steer angle were also scaled by 9 bits.  MDC was scaled by N bits plus 4=DCN  bits or 
( )DCNN +2 .  The bias was scaled by N bits plus 4=
biasN  bits or 
( )biasNN +2 .  MDC and the bias 
were scaled by extra bits in order to reduce the error in those states caused by the fact that 
MDC and the bias were smaller in magnitude then the yaw rate state. 
 Equation (6.11) shows the state time update equations, where the states are MDC, 
yaw rate, and the yaw rate bias. 
( )[ ]
1
1
1
?
?
?
=
>>>>>>=
=
kbiaskbias
kkDCkDCk
kDCkDC
rr
NNVNMr
MM
?  (6.11) 
The MDC and rbias values are simply the previously computed values.  The MDC state has 
to be shifted down in the yaw rate computation since it has a larger scaling in the 
estimator.  The order of the shifts can be changed, as long as it does not increase the risk 
of overflow. 
 
99 
 Equations (6.12-6.14) display the Jacobian matrices used in the square root 
covariance filter. 
( )
?
?
?
?
?
?
?
?
?
?
<<
+>>
<<
=
N
NNV
N
F DC
100
00
001
?  (6.12) 
( )[ ]biasNNNH ?<<<<= 110  (6.13) 
?
?
?
?
?
?
?
?
?
?
<<
<<
<<
=
N
N
N
L
100
010
001
 (6.14) 
Note that in F, the term related to the partial of the yaw rate with respect to MDC is shifted 
down due to the extra scaling on the parameter.  Additionally in H, the partial derivative 
with respect to the bias is shifted up less due to the extra scaling on the bias. 
 Equations (6.15) and (6.16) contain the square root of the noise covariances, 
which are displayed as a function of the extra scaling on the states. 
?
?
?
?
?
?
?
?
?
?
<<
<<
=
bias
DC
N
N
Q
600
01540
004
2/1  (6.15) 
[ ]682/1 =R  (6.16) 
The noise values had to be larger than the nominal values given in Chapter 4.  This was 
done do to the fact that even though the covariance may not converge to zero, the Kalman 
gain can become zero if the covariance is too small. 
 
 
 
 
100 
The initial covariance square root is shown in Equation (6.17). 
?
?
?
?
?
?
?
?
?
?
<<
=
biasN
S
46000
000
000
 (6.17) 
The initial value for the yaw rate covariance is set to zero, since the estimator will be 
initialized with the most recent yaw rate measurement.  The MDC covariance square root 
initial value is also set to zero because adaptation will not be performed until there is 
adequate excitation. 
 To determine when to estimate the parameter, Equation (6.18) was calculated 
recursively, where 6=wN  was chosen to prevent an overflow. 
( )? = ? >>= 19 0 2m wmk Nstatus ?  (6.18) 
The equation is simply an estimate of the covariance of the steer angle over a short period 
of time.  This period was chosen to be one second at a twenty hertz resolution. 
 The value of status was arbitrarily chosen to be 1920, therefore, if 1920<status  
then it was determined that there was insufficient excitation to estimate MDC.  Therefore 
in order to prevent the parameter from being estimated, the first element of the Kalman 
gain was set to zero ( ( ) 00,0 =K ).  Once there was adequate excitation to estimate MDC 
and MDC was not previously being estimated, then the covariance for MDC was reset 
to ( ) DCNS <<= 250,0  in order to inject some uncertainty into the state.   
 
6.5 Comparison between Fixed Point and Floating Point Implementation 
To test the performance of the fixed point estimator, experimental data was used 
during a straight line tracking test with an initial offset.  A floating point estimator was 
 
101 
also used for comparison.  Note that for parity of comparison, the nise values used in 
Equations (615-6.16) were used in both estimators.  Figures 6.1-6.4 show the comparison 
between the results using the fixed point versus floating point estimators.  Note that in the 
following figures the scaled fixed point vales have been rescaled back to more traditional 
values.  It can be seen in Figure 6.1 that there was enough excitation to estimate the 
parameter at the beginning of the experiment.  When the yaw rate moved to switch signs 
and was around zero, it was determined that there was not enough excitation and the 
estimation of MDC was stopped.  Once the yaw rate completed changing signs, the 
windowed analysis determined that there was adequate excitation to continue estimating 
MDC.   
0 5 10 15 20 25 30 35 400
5
10
15
20
25
30
35
Time (s)
Magnitude
Window Value
Threshold
 
Figure 6.1: Windowed analysis of experimental data to determine when to adapt. 
 
102 
The estimate of MDC using the fixed point estimator initially overshoots the 
estimate form the floating point estimator as seen in Figure 6.2.  Unfortunately the MDC 
estimate does not recover from the initial overshoot during the second opportunity for 
estimation. 
0 5 10 15 20 25 30 350.2
0.25
0.3
0.35
0.4
Time (s)
M D
C
Floating Point
Fixed Point
 
Figure 6.2: Comparison of the estimation of the slope of the dc gain equation with 
respect to velocity for fixed and floating point using experimental data 
 
Also, Figure 6.3 shows that there exists a slight difference in the estimated yaw 
gyro bias from the two estimators.  This results in a small difference in the yaw rate 
estimate seen in Figure 6.4.  The differences in the estimation of MDC can likely be 
attributed to the differences in the estimate of the yaw rate bias.  Note that if MDC had not 
 
103 
been made the first state, errors in the computation in the covariance could have caused 
the Kalman gain to go to zero even while the windowed analysis showed there was 
enough excitation.  This would cause the estimation to turn off, possibly before it 
converged to the correct value. 
0 5 10 15 20 25 30 35
?1
?0.8
?0.6
?0.4
?0.2
0
0.2
Time (s)
Bias (deg/s)
Floating Point
Fixed Point
 
Figure 6.3: Comparison of the estimation of the yaw rate bias in fixed and floating 
point using experimental data 
 
 
 
 
104 
0 5 10 15 20 25 30 35
?6
?4
?2
0
2
4
Yaw Rate (deg/s)
0 5 10 15 20 25 30 35
0
0.1
0.2
0.3
Time (s)
Difference (deg/s)
Floating Point
Fixed Point
 
Figure 6.4: Comparison of the estimation of yaw rate in fixed and floating point using 
experimental data 
 
  In order to test the repeatability of the estimator and whether the estimator was 
estimating the correct values, simulated data that was generated using a sine wave steer 
angle input and a second order disturbance model.  The results are given in Figures 6.5-
6.8.  As seen in the figures, the estimates for the fixed and floating point estimators both 
approach the correct value every time the parameter is estimated. 
 Recall that the process noise values in Equation (6.15) were set higher than the 
actual noise values in order to prevent the Kalman gain from going to zero.  This causes 
 
105 
the bias to incorporate a scale factor error effect (from the sine wave) due to MDC having 
not completely converged. 
 In Figure 6.7, it can be seen that the covariance resetting did not allow for the 
convergence on the correct value the first time it tried to estimate the parameter.  
However, resetting the covariance twice allowed the MDC estimate to approach the correct 
value.  Again this was due to the process noise values being set higher than the actual 
values.  Although, it took longer to converge with the larger noise covariances, the MDC 
estimate did converge to the correct value.  As shown in Figure 6.8, there were multiple 
opportunities where sufficient excitation existed for MDC to be estimated. 
0 5 10 15 20 25 30 35 40 45
?2
0
2
Yaw Rate (deg/s)
0 5 10 15 20 25 30 35 40 45
?0.05
0
0.05
Time (s)
Difference (deg/s)
Floating Point
Fixed Point
 
Figure 6.5: Comparison of the estimation of yaw rate in fixed and floating point for 
simulated data 
 
106 
0 5 10 15 20 25 30 35 40 45
?0.9
?0.8
?0.7
?0.6
?0.5
?0.4
?0.3
?0.2
?0.1
Time (s)
Bias (deg/s)
Floating Point
Fixed Point
Actual
 
Figure 6.6: Comparison of the estimation of the yaw rate bias in fixed and floating 
point using simulated data 
 
  
 
107 
0 5 10 15 20 25 30 35 40 450.2
0.25
0.3
0.35
Time (s)
M D
C
Floating Point
Fixed Point
Actual
 
Figure 6.7: Comparison of the estimation of the yaw rate dc gain slope with respect to 
velocity in fixed and floating point for simulated data 
  
 
 
108 
0 10 20 30 40 500
5
10
15
20
25
30
35
Time (s)
Magnitude
Window Value
Threshold
 
Figure 6.8: Windowed analysis of experimental data to determine when to adapt. 
 
6.6 Summary and Conclusions 
 A square root covariance filter was used to implement the yaw rate estimator 
using fixed point math.  The methods for the numerical operations were shown as well as 
the fact that the first state would have the most accurate covariance estimate.  The 
modifications of the Kalman filter matrixes were also given, where the noise covariances 
had to be made larger than necessary to prevent the Kalman gain from going to zero even 
when the covariance was not zero.  The algorithm was then compared using both a 
floating point estimator and fixed point estimator with experimental data.  Finally, it was 
 
109 
shown with simulated data that even with the increased noise values in the fixed point 
estimator that the correct values could be converged on. 
 
110 
CHAPTER 7
CONCLUSION 
7.1 Summary 
 In Chapter 2, system identification of the steering servo found that the steering 
servo could be modeled using a nonlinear input transformation plus a transfer function 
with an integrator.  A review of previous yaw dynamic models for tractors found that 
they did not match the dynamics of the 8420 tractor identified in this thesis using system 
identification tests.  From the system identification tests of the steer angle to yaw rate 
transfer function it was found that the system should be modeled as a fourth order model 
with a non-minimum phase zero.  It was found that the DC gain of the steer angle to yaw 
rate transfer function was linear and that only the slope of the DC gain with respect to 
velocity varied with velocity and implement loading.  A model of the tractor with various 
hitch loadings was identified for use in the controller design. 
Chapter 3 detailed the design of the cascaded controllers for the steering servo, 
yaw rate, and position of the tractor.  The method of determining the controller gains 
using parameter estimation was discussed.  The sensitivity of the controller to the model 
simplifications was given and it was shown that the stability margins of the controller 
improve with increasing hitch loading and velocity.  The sensitivity of the controller to 
the incorrect velocity was shown. It was found that the system could be unstable if 
designed for too low of a velocity and could have an undesirably slow response when 
 
111 
designed for too high of a velocity.  Similarly, it was found that an estimate of MDC that 
was too low could lead to instability and too high of an estimate could lead to a slow 
response. 
 In Chapter 4, the design of an estimator to estimate the yaw rate, yaw rate bias, 
and MDC was detailed.  It was shown that a windowed auto-correlation could be used to 
determine when to estimate MDC so that disturbances do not dominant the estimate.  It 
was shown that with the covariance resetting, the estimate of MDC would converge 
quickly.  The effect of the neglected dynamics on the estimation was displayed and it was 
concluded that most of the errors occurred because of magnitude errors between the 
estimator model and the true dynamics and not phase errors.  It was also shown that the 
covariance estimate for the estimated MDC was lower than it actually should be, making it 
unreliable the accuracy of the estimate.  The errors caused by neglecting dynamics were 
found to decrease as the velocity and hitch loading were increased.  A method of 
compensating the adaptation for these magnitude errors was found by approximating the 
pole locations of the tractor.  Even while augmenting the estimator with the dominant 
poles, the estimation still contained some errors.  However, this could be corrected by 
decreasing the damping ratio of the dominant poles or adding more dynamics to the 
estimator model.   The rate of convergence of MDC due to the forgetting factor was found 
to be independent of its offset and was instead simply a function of the longitudinal 
velocity times the steer angle.  It was shown that MDC would converge faster with higher 
amplitudes of excitation and with a larger forgetting factor on MDC.  However, the MDC 
estimate was shown to less accuracy as the forgetting factor was increased.  The design of 
the position estimator for providing higher updates of position was also detailed.  This 
 
112 
estimator was designed to compensate for the delay in the receipt of a GPS message.  It 
was shown that not accounting for this latency could cause large position errors. 
Analytical equations were developed that approximated the errors induced by the GPS 
lag. 
 Chapter 5 described the design of the trajectories used to control the tractor in this 
thesis.  Experimental results showing the repeatability of the estimation were given.  
Experimental tests showed that the tracking response changed little with velocity and 
hitch loading, validating the adaptive control and estimation algorithms in this thesis. 
In Chapter 6, the estimator for the yaw rate, yaw rate bias, and MDC was designed 
for fixed point implementation using a square root covariance filter.  A summary of the 
fixed point math used was given.  An example using a QR decomposition with fixed and 
floating point operations using the householder method was conducted.  It was shown 
that the first state would have the most accurate covariance estimate.  The scaling of the 
different states was detailed including differences from the estimator developed in 
Chapter 4.  A comparison of estimators designed using fixed and floating point math was 
displayed for both experimental and simulation data. 
 
7.2 Recommendation for Future Work 
An accurate parameter based or analytical model of a tractor needs to developed.  
Based on this model a more accurate controller could be designed, since approximations 
had to be made to the dynamics with the use of an implement. Controllers could also be 
designed for specific types of implements, with the use of user inputted information on 
the implement.  With the use of a user specified model, the estimate of MDC could more 
 
113 
accurately take into account higher dynamics. The first step would be the design of a 
model that matches the dynamics without an implement since from system identification 
tests they appear to be similar.  This may include investigating the effect of tire faulty 
circularity and suspension dynamics.  Then, models of various types of implements that 
take into account the hitch dynamics need to be developed. 
Recording the performance of the tractor could lead to improved performance. 
With the use of a user profile for each implement, the statistics of the use of that 
implement versus time and location could be stored.  A record of the performance over 
time for each recorded implement could allow for slight variations in the controller to test 
for improved performance.  While a record of the tractor position could reveal differences 
in the terrain for which different control strategies could be evaluated over time for that 
location. 
 
 
114 
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118 
APPENDIX A
EXPERIMENTAL SETUP 
  
A John Deere 8420 tractor with an independent link front suspension was used for 
experimental tests, which is shown in Figure A.1.  The tractor was interfaced to with the 
use of a Controller Area Network (CAN). 
 
Figure A.1: John Deere 8420 Tractor 
 
 
119 
 The algorithm computations were calculated with the use of a Versalogic Bobcat 
PC/104 CPU shown in Figure A.2. The analog sensors used were recorded with the use 
of a Versalogic PC/104 data acquisition board shown in Figure A.3. The PC/104 
components were stored inside of a PC/104 case from Versalogic shown in Figure A.4. 
  
Figure A.2: Versalogic PC/104 CPU 
[Versalogic, 2005a] 
Figure A.3: Versalogic PC/104 data 
acquisition board [Versalogic, 2005b] 
 
Figure A.4: Versalogic VL-ENCL-4 Ruggedized Enclosure [Versalogic, 2005c] 
 
 
120 
The Bosch IMU shown in Figure A.5 was used to obtain inertial measurements. 
The linear potentiometer shown in Figure A.6 was used for measurements of steer angle.  
GPS messages were obtain with the use of the Starfire GPS receiver shown in Figure A.7. 
  
Figure A.5: Bosch IMU Figure A.6: Steer angle sensor 
 
Figure A.7: Starfire GPS reciever