Adaptive Control of a Farm Tractor with Varying Yaw Properties
Accounting for Actuator Dynamics and Nonlinearities
Except where reference is made to the 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 classi ed information.
J. Benton Derrick
Certi cate of Approval:
George T. Flowers
Professor
Mechanical Engineering
David M. Bevly, Chair
Associate Professor
Mechanical Engineering
John Y. Hung
Professor
Electrical and Computer Engineering
Joe F. Pittman
Interim Dean
Graduate School
Adaptive Control of a Farm Tractor with Varying Yaw Properties
Accounting for Actuator Dynamics and Nonlinearities
J. Benton Derrick
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Ful llment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
May 10, 2008
Adaptive Control of a Farm Tractor with Varying Yaw Properties
Accounting for Actuator Dynamics and Nonlinearities
J. Benton Derrick
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 Author
Date of Graduation
iii
Vita
J. Benton Derrick was born in Rome, GA, on April 24, 1984. He is the second
child of David and Rebecca and brother to Jennifer. He was raised in Centre,
AL, and attended Centre Elementary School, Centre Middle School, and Cherokee
County High School. Upon graduation from high school in 2002, He attended
Auburn University and completed his bachelor of mechanical engineering degree
in the spring of 2006. He then accepted a graduate research assistant position
in the Global Positioning System and Vehicle Dynamics Laboratory the following
summer to work on a masters of science degree with specialization in dynamics
and controls. Benton was wed to Vanessa Attaway in January of 2007.
iv
Thesis Abstract
Adaptive Control of a Farm Tractor with Varying Yaw Properties
Accounting for Actuator Dynamics and Nonlinearities
J. Benton Derrick
Master of Science, May 10, 2008
(B.M.E., Auburn University, 2006)
112 Typed Pages
Directed by David M. Bevly
Two adaptive control algorithms for the automatic steering control of a farm
tractor with varying hitch forces are developed. Tractors can be con gured many
di erent implements, and implements interact with the soil in various ways. These
variations cause the yaw dynamics to change with respect to di erent implements
and soil conditions; therefore, this thesis uses a model reference adaptive (MRAC)
control law to compensate for di erent implement con gurations. Models are de-
scribed and analyzed for the steering actuator, yaw rate plant, and lateral position
plant. It is shown that the DC gain of the steering angle to yaw rate transfer func-
tion is the model parameter that changes the most with hitch loading. In order to
develop the adaptive control algorithm, a cascaded controller is rst implemented
with three feedback loops containing the steering angle, yaw rate, and lateral po-
sition measurements. Controllers are designed for each subsystem, and root locus
analysis is used to describe the stability and performance characteristics.
v
Two MRAC algorithms are derived to compensate the loop gain and feed-
forward gain of the yaw rate controller to account for changes in the yaw rate
plant. The two algorithms are named the model reference adaptive control loop
gain adaptation (MRAC-LGA) algorithm and the model reference adaptive control
feed-forward gain adaptation (MRAC-FGA) algorithm. Simulations are presented
that show that the algorithms perform poorly due to neglected steering actua-
tor properties. Both algorithms are modi ed to account for the steering actuator
properties, and more simulations are presented that demonstrate satisfactory per-
formance. Experimental results are presented for the LGA algorithm, and issues
with experimental implementation are discussed. Next, experimental results are
presented for the FGA algorithm that show improved performance over the LGA
algorithm. Finally, experimental tests further validate that the FGA algorithm
improves lateral error performance versus a xed-gain controller.
vi
Acknowledgments
I would rst like to thank God for my strength and wisdom that he gives me.
I am truly blessed with all of the opportunity that has be a orded me. I next
thank my wife Vanessa for her love, patience, support and allowing me to put our
life on hold while I pursue my education. I also would like to thank my parents for
their love and support throughout my entire life. I would not be where I am today
if it was not for them. Thanks goes to Professor David Bevly for his guidance
and assistance during my tenure in the GAVLAB and for giving me a chance to
earn my master?s degree while working on exciting projects. I would nally like to
thank all of the members of the GAVLAB for their assistance and comradery over
the last couple of years.
vii
Style manual or journal used Journal of Approximation Theory (together with
the style known as \aums"). Bibliography follows van Leunen?s A Handbook for
Scholars.
Computer software used The document preparation package TEX (speci cally
LATEX) together with the departmental style- le aums.sty.
viii
Table of Contents
List of Figures xi
List of Tables xiv
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Background and Prior Work . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 System Models and Control Architecture 8
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Steering Actuator Model . . . . . . . . . . . . . . . . . . . . 9
2.2.2 Dynamic Yaw Model . . . . . . . . . . . . . . . . . . . . . . 12
2.2.3 Lateral Position Model . . . . . . . . . . . . . . . . . . . . . 19
2.3 Control Structure and Design . . . . . . . . . . . . . . . . . . . . . 21
2.3.1 Control Structure . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.2 Steering Actuator Controller Design . . . . . . . . . . . . . . 22
2.3.3 Yaw Rate Controller Design . . . . . . . . . . . . . . . . . . 24
2.3.4 Lateral Position Controller Design . . . . . . . . . . . . . . . 26
2.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 28
3 Adaptive Control by Compensating Yaw Rate Loop Gain 30
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Adaptive Control Techniques . . . . . . . . . . . . . . . . . . . . . . 31
3.3 MRAC-LGA Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.1 Adaptation System Architecture . . . . . . . . . . . . . . . . 34
3.3.2 LGA Algorithm Derivation . . . . . . . . . . . . . . . . . . . 35
3.4 Simulations of the LGA Algorithm . . . . . . . . . . . . . . . . . . 38
3.4.1 Simulation Results with Initial LGA Algorithm . . . . . . . 39
3.4.2 LGA Algorithm Modi cations . . . . . . . . . . . . . . . . . 41
3.4.3 Simulation Results with Modi ed LGA Algorithm . . . . . . 44
3.5 Experimental Testing of the Modi ed LGA Algorithm . . . . . . . . 47
3.5.1 Step Input Testing . . . . . . . . . . . . . . . . . . . . . . . 47
ix
3.5.2 LGA Experimental Implementation Issues . . . . . . . . . . 50
3.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 51
4 Adaptive Control by Compensating Yaw Rate Feed-Forward
Gain 53
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 New Algorithm Requirements . . . . . . . . . . . . . . . . . . . . . 54
4.3 MRAC-FGA Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3.1 Adaptation System Architecture . . . . . . . . . . . . . . . . 55
4.3.2 FGA Algorithm Derivation . . . . . . . . . . . . . . . . . . . 57
4.4 Simulations of the FGA Algorithm . . . . . . . . . . . . . . . . . . 61
4.4.1 Simulation Results with Initial FGA Algorithm . . . . . . . 61
4.4.2 FGA Algorithm Modi cations . . . . . . . . . . . . . . . . . 63
4.4.3 Simulations Results of Modi ed FGA Algorithm . . . . . . . 65
4.5 Experimental Testing of the FGA Algorithm . . . . . . . . . . . . . 67
4.5.1 Step Input Testing . . . . . . . . . . . . . . . . . . . . . . . 68
4.5.2 Lateral Error Testing . . . . . . . . . . . . . . . . . . . . . . 71
4.5.3 Lateral Error Testing With Changing Implement Position . . 74
4.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 78
5 Conclusions 80
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . 81
Bibliography 83
Appendices 86
A Nomenclature 87
B Experimental Setup 90
x
List of Figures
1.1 Field Bedded for Planting with Automatically Steered Tractor . . . 1
1.2 Lateral Position Response with Implement Lifted Out of Ground . . 3
2.1 Steady State Slew Rate vs. Input Command . . . . . . . . . . . . . 10
2.2 Steering Actuator Model . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Bicycle Model with Augmented Hitch Force . . . . . . . . . . . . . 13
2.4 Step Response of Yaw Model with Varying C h . . . . . . . . . . . 17
2.5 DC Gain of Yaw Model vs. C h . . . . . . . . . . . . . . . . . . . . 18
2.6 Primary Pole of Yaw Model vs. C h . . . . . . . . . . . . . . . . . . 18
2.7 Secondary Pole of Yaw Model vs. C h . . . . . . . . . . . . . . . . . 19
2.8 Lateral Position Schematic . . . . . . . . . . . . . . . . . . . . . . . 20
2.9 Cascaded Controller Block Diagram . . . . . . . . . . . . . . . . . . 21
2.10 Steering Actuator Root Locus and Closed-Loop Pole-Zero Locations
for kp = 3:84 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.11 Yaw Rate Root Locus and Closed-Loop Pole-Zero Locations for
kpr = 0:30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.12 Lateral Position Root Locus and Closed-Loop Pole-Zero Locations . 28
3.1 MRAC System Block Diagram . . . . . . . . . . . . . . . . . . . . . 34
3.2 Simulated Adaptation Gain and Yaw Rate Response with Initial
LGA Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Simulated Steering Actuator Response with Initial LGA Algorithm 41
xi
3.4 MRAC-LGA Closed-Loop Reference Model . . . . . . . . . . . . . . 42
3.5 MRAC-LGA Total System Block Diagram . . . . . . . . . . . . . . 43
3.6 Simulated Adaptation Gain and Yaw Rate Response with Modi ed
LGA Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.7 Simulated Steering Actuator Response with Modi ed LGA Algorithm 46
3.8 Experimental Lateral Position Response (Left: With Implement,
Right: Without Implement) . . . . . . . . . . . . . . . . . . . . . . 47
3.9 Experimental Adaptation Gain and Yaw Rate Response with Mod-
i ed LGA Algorithm (Top: With Implement, Bottom: Without
Implement) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.10 Experimental Steering Actuator Response with Modi ed LGA Al-
gorithm (Top: With Implement, Bottom: Without Implement) . . . 50
4.1 Cascaded Control Block Diagram with Feed-Forward Control . . . . 55
4.2 MRAC System Block Diagram with Feed-Forward Control . . . . . 57
4.3 Adaptation Gain and Yaw Rate Response with Initial FGA Algorithm 62
4.4 Simulated Steering Actuator Response with Initial FGA Algorithm 63
4.5 MRAC-FGA Closed-Loop Reference Model . . . . . . . . . . . . . . 64
4.6 MRAC-FGA Total System Block Diagram . . . . . . . . . . . . . . 65
4.7 Simulated Adaptation Gain and Yaw Rate Response with Modi ed
FGA Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.8 Simulated Steering Actuator Response with Modi ed FGA Algorithm 67
4.9 Experimental Lateral Position Response (Left: With Implement,
Right: Without Implement) . . . . . . . . . . . . . . . . . . . . . . 68
4.10 Experimental Adaptation Gain and Yaw Rate Response with Mod-
i ed FGA Algorithm (Top: With Implement, Bottom: Without
Implement) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
xii
4.11 Experiemental Steering Actuator Response with Modi ed FGA Al-
gorithm (Top: With Implement, Bottom: Without Implement) . . . 70
4.12 Lateral Position Response of Experimental Line Tracking with Im-
plement Lifted Out of Ground at the Speci ed Point in Time with
a Fixed Gain Controller . . . . . . . . . . . . . . . . . . . . . . . . 75
4.13 Lateral Position Response of Experimental Line Tracking with Im-
plement Lifted Out of Ground at the Speci ed Point in Time with
an Adaptive Controller . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.14 Adaptation Gain Response of Experimental Line Tracking with Im-
plement Lifted Out of Ground at the Speci ed Point in Time with
the MRAC-FGA Adaptive Controller . . . . . . . . . . . . . . . . . 76
B.1 John Deere 8420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
B.2 StarFire DGPS Receiver . . . . . . . . . . . . . . . . . . . . . . . . 91
B.3 Bosch IMU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
B.4 Steering Angle Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . 93
B.5 Versalogic PC104 Computer . . . . . . . . . . . . . . . . . . . . . . 94
B.6 Experimental Setup Block Diagram . . . . . . . . . . . . . . . . . . 95
B.7 Experimental Lateral Position Calculation . . . . . . . . . . . . . . 97
xiii
List of Tables
2.1 Steady-State Slew Rate vs. Input Command Fits . . . . . . . . . . 10
2.2 Inverse Mapping of Steady-State Slew Rate to Input Command . . 11
2.3 Steering Actuator Model Parameters . . . . . . . . . . . . . . . . . 12
2.4 Yaw Rate Model Parameters . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Steering Actuator Controller Closed-Loop Pole Properties . . . . . . 24
2.6 Yaw Rate Controller Closed-Loop Pole Properties . . . . . . . . . . 26
2.7 Lateral Position Controller Coe cients . . . . . . . . . . . . . . . . 27
2.8 Lateral Position Controller Closed-Loop Pole Properties . . . . . . . 28
3.1 Parameters Used in LGA Simulations . . . . . . . . . . . . . . . . . 39
4.1 Parameters Used in FGA Simulations . . . . . . . . . . . . . . . . . 61
4.2 Experimental Statistics with Implement While Adapting . . . . . . 72
4.3 Experimental Statistics with Implement and Fixed Gain . . . . . . 72
4.4 Experimental Statistics without Implement While Adapting . . . . 73
4.5 Experimental Statistics without Implement and Fixed Gain . . . . . 73
4.6 Experimental Statistics for Test with Changing Implement Position
with Fixed Gain Controller (The left two columns correspond to the
parts of the runs with the implement, and the right two columns
correspond to the parts of the runs with the implement out of the
ground.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
xiv
4.7 Experimental Statistics for Test with Changing Implement Position
with Adaptive Controller (The left two columns correspond to the
parts of the runs with the implement, and the right two columns
correspond to the parts of the runs with the implement out of the
ground.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
A.1 Nomenclature Table of Variables Used in Thesis Part I . . . . . . . 88
A.2 Nomenclature Table of Variables Used in Thesis Part II . . . . . . . 89
xv
Chapter 1
Introduction
1.1 Motivation
Automatically steered farm tractors have become popular in the last few years
due to the advantages that they bring farmers. Advances in Global Positioning
System (GPS) technology has created a means to measure the position of a user
within 2 cm [Montgomery, 1996]. This accuracy has allowed precision farming
to ourish. Many problems such as overlap, driver inexperience, poor visibility,
in-ground irrigation destruction, and crop destruction can be addressed with auto-
matically steered farm tractors. Figure 1.1 shows a eld that has been plowed using
an automatically steered farm tractor. Notice that the rows are nearly perfectly
straight.
Figure 1.1: Field Bedded for Planting with Automatically Steered Tractor
Farm tractors can be out tted with a variety of attachments that can change
the forces that are applied to the hitch. Additionally, di erent soil conditions can
1
be encountered that will make the same implement have di erent force proper-
ties. Because the hitch force response changes, the yaw rate response changes as
well. It is desired that the response of the tractor be as consistent as possible. A
xed-gain controller designed for a time-invariant plant has the potential for poor
performance and unstable characteristics.
On current John Deere production models, there is a sensitivity parameter
that has to be eld tuned by the user. The adjustment parameter e ectively
scales the steering angle of the tractor by a gain. There are many problems with
this method. First, the sensitivity parameter is often poorly tuned. The poorly
tuned gain causes oscillation of the tractor or undue lateral position error. Since
the sensitivity parameter is a xed value, the tractor will often exhibit unstable
behavior while the implement is out of the ground when the system is tuned for
big implements. Also, there is no way for the sensitivity parameter to be adjusted
to account for changing soil conditions other than by the operator stopping the
tractor and manually changing the gain. An example of this behavior is shown in
Figure 1.2. The tractor was started with the implement deep in the ground, and
then the implement was lifted up at the speci ed point in time. Notice that there
is signi cantly more lateral error after the implement is lifted our of the ground.
For all these reasons, having a xed sensitivity parameter is not a good solution.
2
Figure 1.2: Lateral Position Response with Implement Lifted Out of Ground
It is desired to make the sensitivity parameter automatically adjust to di erent
implements, depths, and soil conditions by a direct adaptive control approach. The
focus of this thesis will be on developing an adaptive control algorithm to directly
adjust the sensitivity parameter of the yaw rate controller.
1.2 Background and Prior Work
Work has been previously completed on modeling, control, and estimation of
farm tractor dynamics with implements. The rst work of modeling a farm tractor
with an implement was [O?Conner, 1997]. This work took the Wong bicycle model
and augmented the implement forces onto the rear of the tractor [Wong, 1978].
The implement forces were modeled as an extra axle behind the rear axle, and
3
the lateral axle force is proportional to the slip angle between the velocity vector
and the heading angle of the axle. There have been numerous implement-soil
mechanics models developed, but these models only predict the longitudinal and
vertical forces [R. Berntsen and Aasen, 2006], [Godwin and O?Dogherty, 2007],
[Rosa and Wulfsohn, 1999], [Sahu and Raheman, 2006]. Since the lateral forces of
the implement are what in uence the yaw rate dynamics, the bicycle model with
the third axle is the best model available to predict yaw dynamics of a tractor with
a hitched implement. This model was shown to be valid for implements attached to
the tractor using the three-point hitch [Pearson, 2007]. Simulations have also been
complemented on tractor-implement interactions for implements that are attached
to the draw bar hitch of the tractor [Pota, 2007].
In the more general eld of modeling and controlling farm tractors, many
other works have been completed. Work has been completed on modeling high-
speed ground vehicles using tire relaxation lengths [Bevly, 2001]. A farm tractor
has been controlled using only a single carrier phase di erential global positioning
system (CP-DGPS) receiver [Thuilot and Berducat, 2002]. Also, a farm tractor
has been controlled using vision based navigation [Zolton, 1998].
Because the implement forces cause the yaw rate dynamics to vary, there have
been studies completed to adapt the lateral position controller to account for this
variation. One work used estimation algorithms such as least mean squares and the
Kalman lter to estimate yaw model parameters [Rekow, 2001]. Another work used
an extended Kalman lter to estimate the slope of the DC gain of the steering angle
4
to yaw rate transfer function with respect to velocity [Gartley and Bevly, 2005] .
In both cases, the estimated parameters were used to indirectly adapt the lateral
position controller.
There have been other research completed on adaptive control of ground ve-
hicles. A model reference adaptive control (MRAC) system has been implemented
on a vehicle to create an active steering control system [Fukao, 2001]. Also, a vehi-
cle guidance controller has been adapted using a form of MRAC [Hessburg, 1995].
There has also been a gain scheduling approach to active steering using a form of
the bicycle model [Baslamish, 2007].
MRAC systems rst came into existence during the middle of the 20th cen-
tury. The original theory was applied to controlling airplanes that have varying
loads and aerodynamic properties [Whitaker and Kezer, 1958]. The fundamental
idea behind MRAC systems is to create a controller update law that will adjust
controller parameters so that the plant output matches a desired model output
[Astrom and Wittenmark, 1995]. The MIT rule was the rst approach to MRAC,
and it was widely accepted for its straight-forward application.
5
1.3 Contributions
This thesis presents a novel application of model reference adaptive control
theory on an automatically steered farm tractor. In development of this control
application, the following speci c tasks were performed:
A cascaded controller was designed to control the lateral position of the farm
tractor.
A MRAC system was designed to compensate for yaw plant variations by
adapting the loop gain of the yaw rate controller called the MRAC-LGA
algorithm.
A MRAC system was designed to compensate for yaw plant variations by
adapting the feed-forward yaw rate controller called the MRAC-FGA algo-
rithm.
Both MRAC algorithms were modi ed to account for steering actuator dy-
namics and nonlinearities.
The MRAC system was implemented experimentally and was shown to out-
perform a xed-gain controller.
1.4 Outline of Thesis
In Chapter 2, models are described and analyzed for the steering actuator,
yaw rate plant, and lateral position plant. It is shown that the DC gain of the
6
steering angle to yaw rate transfer function is the model parameter that changes the
most with hitch loading. A cascaded controller is implemented with three feedback
loops containing the steering angle, yaw rate, and lateral position measurements.
Controllers are designed for each subsystem, and root locus analysis is used to
describe the stability and performance characteristics.
A MRAC algorithm is developed in Chapter 3 to adapt the loop gain of the
yaw rate control system to account for the changing yaw rate plant parameters.
Simulation results are presented to show the performance of the MRAC system,
and modi cations are made to the algorithm to account for the steering actuator
dynamics and nonlinearities. Finally, the algorithm is implemented on a John
Deere 8420, and experimental results are presented. Some issues are presented
describing the short comings of the algorithm.
In Chapter 4, another MRAC algorithm is designed to address the issues that
came from adapting the loop gain. A feed-forward yaw rate controller is added
to the system, and this will be adapted to account for changing yaw dynamics.
Simulations are again presented that show the performance of the new algorithm.
Experimental results are presented that show the performance under a step input.
Finally, the lateral error characteristics of the algorithm are compared to a xed-
gain con guration.
7
Chapter 2
System Models and Control Architecture
2.1 Introduction
This chapter presents the system models, controller architecture, and con-
troller design used in this thesis. Three di erent subsystems are combined to
regulate the lateral position of the farm tractor: the steering actuator, yaw rate
plant, and lateral position plant. A model and controller is presented and analyzed
for each subsystem. The steering actuator model is derived from system identi ca-
tion experiments performed previously by [Gartley, 2005]. The steering actuator
subsystem contains nonlinearities that are accounted for by an inversion lookup
table. A linear model of the actuator is then presented combining the lookup
table and dynamic model. The presented yaw rate dynamic model is from work
by [Pearson and Bevly, 2005] and [O?Conner, 1997]. The yaw rate dynamic model
is a derivation of a bicycle model augmented with a third axle at the rear of the
tractor. Analysis is performed on the model in simulation which describes how the
model varies with hitch loading to provide basis for adaptive control. A kinematic
model of the lateral position plant is then presented. Finally, the controllers of the
subsystems are presented and analyzed.
8
2.2 System Modeling
2.2.1 Steering Actuator Model
The steering actuator on the John Deere 8420 is a hydraulic cylinder that is
controlled through the controller area network (CAN) bus on the farm tractor.
The hydraulic valve of the steering actuator is commanded at 50 Hz by sending
a counts value that corresponds to a desired ow. The ow through the valve is
proportional to the steering angle slew rate. Previous research has been completed
on identifying the steering actuator dynamics and nonlinearities [Gartley, 2005].
Figure 2.1 shows the input to output characteristics of the steering actuator. As
can be seen, there is deadband, saturations, and nonlinearities. The curve ts of
each region of the steering calibration are shown in Table 2.1.
9
Figure 2.1: Steady State Slew Rate vs. Input Command
Table 2.1: Steady-State Slew Rate vs. Input Command Fits
Input Command ( input) Steady State Slew Rate ( _ (rad/s))
input < 598 _ = 0:36
598 input < 866 _ = 0:000001295 2input + 0:00324 input 1:835
866 input < 1055 _ = 0
1055 input < 1325 _ = 0:000001859 2input 0:003111 input + 1:213
input 1325 _ = 0:36
To counter the deadband and nonlinearities, an inverse lookup table was cre-
ated by [Gartley, 2005] so that the input to output characteristics of the steering
actuator are linear with the exception of saturation. The lookup table functions
are listed in Table 2.2. Using the look-up table, the controller speci es a desired
10
Table 2.2: Inverse Mapping of Steady-State Slew Rate to Input Command
Intermediate Input Cmd (^ input) Input Command ( input)
^ input < 0:36 input = 598
0:36 ^ input < 0 input = 518:7 ^ 2input + 920:2 ^ input + 864:4
0 ^ input < 0:36 input = 887:9 ^ 2input + 1045 ^ input + 1059
^ input 0:36 input = 1325
slew rate (^ input) for the steering actuator and the proper count value is sent across
the CAN. This makes the DC gain of the e ective actuator input to steering slew
rate transfer function equal to unity.
The entire steering actuator model from input to steering angle is represented
by the schematic in Figure 2.2. The steering actuator controller commands a
desired slew rate (^ input) which is converted to the steering actuator input ( input) by
inverting the nonlinearities. Next, the signal input goes through the nonlinearities
and model, and the slew rate ( _ ) gets integrated into the steering angle ( ).
Figure 2.2: Steering Actuator Model
Through system identi cation experiments conducted by [Gartley, 2005], the
parameters seen in Table 2.3 were identi ed. The identi ed model is second order
with no zeros as represented in Equation (2.1).
GP =
_ (s)
^ input(s) =
!2n
s2 + 2 !n s+!2n (2.1)
11
Table 2.3: Steering Actuator Model Parameters
Parameter Value
!n 28:425 rad=s
0:633
max 32 deg
_ max 20:6 deg/s
2.2.2 Dynamic Yaw Model
Previous studies have been done on modeling the e ects of lateral hitch forces
on yaw rate dynamics. A model has been proposed in which the farm implement
is modeled as a third axle behind the tractor [O?Conner, 1997]. Another study has
been conducted that has shown its validity for implements attached to the three-
point hitch through experimentation [Pearson and Bevly, 2005]. Studies have also
created implement models using soil mechanics that predict the longitudinal and
vertical forces, but these studies lack the ability to predict the lateral forces needed
to turn the implement [R. Berntsen and Aasen, 2006],
[Godwin and O?Dogherty, 2007], [Rosa and Wulfsohn, 1999],
[Sahu and Raheman, 2006]. Therefore, this research uses the model in which the
implement is modeled as a tire to predict the e ects of the implement on yaw
dynamics.
The bicycle model of a vehicle lumps the inner and outer wheels of each
axle into a single wheel. Therefore, the model inherently neglects weight transfer
between the inner and outer wheels. A schematic of the tractor modeled as a
12
Figure 2.3: Bicycle Model with Augmented Hitch Force
bicycle is shown in Figure 2.3. Notice that the hitch force is modeled as an extra
axle behind the rear axle. The lateral tire force at each axle (Ff;Fr;Fh) is a
function of the slip angles ( f; r; h) at each axle [Gillespie, 1992]. The slip
angle is de ned as the angle between the velocity vector and the heading of each
axle. The linearized bicycle model assumes the steering and slip angles are small,
and the lateral tire force is proportional to the slip angles at the respective axle.
The proportionality constant is called the cornering sti ness and is denoted by
C f;C r;C h. The cornering sti ness is a function of normal force on the tire,
but the normal force in this thesis is assumed to be constant. The hitch cornering
sti ness ,C h, is the parameter that changes with di erent implements. In reality,
the lateral tire force reaches a point of saturation, but it will be assumed that the
tires on the tractor stay in the linear region. The lateral tire force equations are
represented by Equation (2.2).
13
Ff = C f f
Fr = C r r
Fh = C h h
(2.2)
The lengths from the center of gravity to the front and rear axle are a and b,
and the length from the rear axle to the hitch is c. The variables Vy and Vx are the
lateral and longitudinal velocities at the center of gravity, is the steering angle of
the front axle, and r is the yaw rate of the tractor. Using Newton?s equations, the
lateral forces and moments about the center of gravity are represented by Equation
(2.3).
PF
y = m ay = Ff cos +Fr +Fh
PM
CG = Izz _r = a Ff cos b Fr (b+c) Fh
(2.3)
By assuming that the tractor is a rigid body, a kinematic relationship between ve-
locity, yaw rate, steering angle, and slip angles can be formed. These relationships
are shown in Equation (2.4).
14
f = tan 1(Vy+r aVx )
r = tan 1(Vy r bVx )
h = tan 1(Vy r (b+c)Vx )
(2.4)
The following equations can be combined to form a transfer function from steering
angle to yaw rate. By linearizing assuming small slip and steering angles, the linear
model shown in Equation (2.5) can be formed.
GPr = r(s) (s) = n1 s+n0d
2 s2 +d1 s+d0
(2.5)
Where:
n0 = C fC1+aC fC2mVx
n1 = aC f
d0 = C2C3 C21mV2
x
+C1
d1 = C2IzzmVx + C3Vx
d2 = Izz
(2.6)
And:
15
C1 = ((b+c) C h +b C r a C f)
C2 = (C h +C r +C f)
C3 = ((b+c)2C h +b2C r +a2C f)
(2.7)
The yaw model parameters of the tractor are shown in Table 2.4.
Table 2.4: Yaw Rate Model Parameters
Parameter Value
a 1:00 m
b 2:00 m
c 2:19 m
Izz 18500 kg m2
m 11340 kg
C f 2400 N=deg
C r 5000 N=deg
C h 0 4000 N=deg
Vx 2 m=s
In this research, the yaw dynamics with respect to the cornering sti ness of
the hitch is of primary interest. This is due to the face that once it is known
how the lateral hitch force a ects the yaw rate response, an adaptive controller
can be designed to compensate for this variation. It has been shown empirically
that C h can range from 0 N/deg with no implement to 4000 N/deg for a heavy
implement [Pearson, 2007]. To investigate the di erences in dynamic behavior
caused by varying lateral hitch forces, a step input simulation was completed using
MATLAB. The results of the simulation are shown in Figure 2.4. Notice that all
16
of the runs settle between 0.4 and 0.6 seconds, but the steady-state value varies
signi cantly.
Figure 2.4: Step Response of Yaw Model with Varying C h
A plot is shown in Figure 2.5 of the DC gain of the yaw rate transfer function
verses C h. It can be seen by looking at the black dotted line that C h = 600 N/deg
is the value of C h that results in the median DC gain value. This will be de ned
as the nominal value for the DC gain. Later in this chapter, this value of C h will
be used to nominally tune the yaw rate controller.
17
Figure 2.5: DC Gain of Yaw Model vs. C h
It can also be observed in Figure 2.6 that the e ect of C h on the primary
pole is very small. Recall in Figure 2.4 this change in the dynamics was barely
noticeable in the previously shown step response plots.
Figure 2.6: Primary Pole of Yaw Model vs. C h
It is shown in Figure 2.7 that the secondary pole changes more than three-fold
over the range of C h. However, since the secondary pole is 5-15 times faster that
18
the primary pole, the e ect of secondary pole on the yaw rate response is negligible.
From the previous plots, it can be concluded that the DC gain of the steering
angle to yaw rate transfer function is the parameter that changes the response the
most with respect to variable hitch loading. Therefore, the transient dynamics
can reasonably be assumed to be constant with respect to C h. Because of these
properties, an adaptive controller will be developed in this thesis to compensate
only for the change in DC gain of the steering angle to yaw rate transfer function.
Figure 2.7: Secondary Pole of Yaw Model vs. C h
2.2.3 Lateral Position Model
A kinematic relationship between yaw rate and lateral position is used as the
lateral position plant model. A diagram is shown in Figure 2.8 that describes the
relationship. Lateral position (y) can be described by Equation (2.8) where is
the side slip angle at the center of gravity, is the course angle of the velocity
19
vector with respect to the desired longitudinal direction, and V is the magnitude
of velocity.
_y = V sin( )
_ = r + _
(2.8)
Figure 2.8: Lateral Position Schematic
Lateral velocity is the total velocity times the sine of the course angle, and the
course angle is de ned as the integrated yaw rate plus side slip angle. Neglecting
side slip and linearizing assuming small angles yields the transfer function from
yaw rate to lateral position shown in Equation (2.9).
20
GPy = y(s)r(s) = Vs2 (2.9)
2.3 Control Structure and Design
2.3.1 Control Structure
In order to regulate the lateral position of the farm tractor, a controller sys-
tem must be designed to regulate the three subsystems described in the previous
sections. A cascaded controller is advantageous because the three subsystems
can be individually controlled using classical control techniques. Also, a cascaded
controller allows more than one measurement to be fed back to produce a more
accurate response [Levine, 1996]. A diagram of the cascaded controller network is
shown in Figure 2.9.
Figure 2.9: Cascaded Controller Block Diagram
The boxes labeled GP , GPr, and GPl are the steering actuator, yaw rate,
and lateral position plants, respectively. The boxes labeled GC , GCr, and GCl
are the steering actuator, yaw rate, and lateral position controllers. The three
measurements being fed back are the steering angle ( ), yaw rate (r), and lateral
position (y). Note that all of the analysis in the following sections is performed at
V = 2 m/s.
21
2.3.2 Steering Actuator Controller Design
As was discussed in a Section 2.2.1, the steering actuator dynamics are second
order with an integrator. Because the steering actuator has a pure integrator, the
system is type one. This means that there is zero steady state error in a step
response [Dorf and Bishop, 2005]. The system type combined with the system
being well damped allows a proportional control law to be used. The control law
is described by Equation (2.10) with the parameters being described in Equation
(2.11).
GC = input(s)
err(s)
= kp (2.10)
kp = 3:84 (2.11)
Equation (2.12) shows the closed-loop transfer function of the steering actua-
tor dynamics.
G CL = (s)
des(s)
= kp !
2
n
s3 + 2 !n s2 +!2n s+kp !2n (2.12)
To analyze the closed loop performance and stability, a root locus analysis
was performed. The root locus plot can be seen in Figure 2.10. The closed-loop
poles are denoted in the gure by the triangles. Tuning of kp was performed
so that the steering actuator would have the highest possible bandwidth without
22
oscillation. The location, natural frequency, and damping ratios for each of the
three closed-loop poles are listed in Table 2.5.
Figure 2.10: Steering Actuator Root Locus and Closed-Loop Pole-Zero Locations
for kp = 3:84
23
Table 2.5: Steering Actuator Controller Closed-Loop Pole Properties
Pole Location Damping Ratio Frequency (rad/s)
15:6465 + 20:4036i 0:609 25:7123
15:6465 20:4036i 0:609 25:7123
4:6930 1 4:6930
2.3.3 Yaw Rate Controller Design
The yaw rate plant has two poles that are over-damped. In steady state, the
desired yaw rate will be approximately equal to zero because the tractor will be
tracking a straight line. Because of these two properties, a proportional control
law will also be used to regulate the yaw rate. The equation of the control law
is shown in Equation (2.13), and the controller coe cient values are shown in
Equation (2.14).
GCr = desfb(s)r
err(s)
= kpr (2.13)
kpr = 0:30 (2.14)
Equation (2.15) shows the closed-loop transfer function of the yaw rate dynamics.
GrCL = r(s)r
des(s)
= kprG CL(n1s+n0)d
2s2 + (d1 +n1kprG CL)s+d0 +n0kprG CL
(2.15)
24
The term G CL is the closed loop dynamics of the steering actuator.
In order to analyze the dynamics and stability of the closed-loop behavior,
a root locus analysis is again used. While designing the yaw rate controller, the
inner-loop steering actuator dynamics are included in the analysis because the
bandwidths of the two systems are similar. The root locus plot is shown in Figure
2.11, where the closed loop poles are noted by triangles. The ve closed-loop pole
locations, damping ratios, and frequencies are shown in Table 2.6. The slower
poles dominate the response.
Figure 2.11: Yaw Rate Root Locus and Closed-Loop Pole-Zero Locations for kpr =
0:30
25
Table 2.6: Yaw Rate Controller Closed-Loop Pole Properties
Pole Location Damping Ratio Frequency (rad/s)
7:7062 + 0:7552i 0:995 7:7431
7:7062 0:7552i 0:995 7:7431
15:7899 + 20:1817i 0:616 25:6246
15:7899 20:1817i 0:616 25:6246
60:2030 1 60:2030
2.3.4 Lateral Position Controller Design
The lateral position controller is chosen to be a standard proportional-integral-
derivative (PID) controller. Since the two open-loop poles of the lateral position
plant are at the origin, there needs to be some damping that will be supplied
by the derivative term of the controller. Also, the integral term will o set any
steady-state error cause by calibration errors or unlevel terrain. The equation of
the controller is shown in Equation (2.16), and the coe cients are described in
Table 2.7.
GCy = rdes(s)y
err(s)
= kpy(1 + kiys +kdy s) (2.16)
The inner-loop yaw rate dynamics can be neglected since they are many times
faster that the lateral position dynamics. Only the DC gain of the inner-loop
transfer function will have to be included. Equation (2.17) describes the closed-
loop lateral position dynamics neglecting the yaw rate dynamics.
26
GyCL = y(s)y
des(s)
= V DCyaw(kpy s+kdy s
2 +kiy)
V DCyaw(kpy s+kdy s2 +kiy) +s3 (2.17)
As can be seen, the lateral position dynamics are a function of velocity and the
DC gain of the closed-loop yaw rate transfer function. Since it is desired that
the tractor?s lateral position response be the same for all implements, it is critical
that DCyaw remain nearly constant. This is the reason the yaw rate controller
will be adapted. The value of DCyaw is to be determined in a later chapter,
so the controller values shown in Table 2.7 are represented as a function of this
variable. By adjusting the proportional controller coe cient by the inverse of
DCyaw, the reference yaw rate is scaled so that the DC gain of the yaw rate
response is e ectively unity.
Table 2.7: Lateral Position Controller Coe cients
Coe cient Value
kpy 0:10=DCyaw
kpd 2:50
kpi 0:01
Root locus is again used to analyze the controller as shown in Figure 2.12.
The closed-loop poles are again marked by the triangles. The closed-loop pole
locations, damping ratio, and frequencies are listed in Table 2.8.
27
Figure 2.12: Lateral Position Root Locus and Closed-Loop Pole-Zero Locations
Table 2.8: Lateral Position Controller Closed-Loop Pole Properties
Pole Location Damping Ratio Frequency (rad/s)
0:2449 + 0:3674i 0:555 0:4415
0:2449 0:3672i 0:555 0:4415
0:0103 1 0:0103
2.4 Summary and Conclusions
This chapter has discussed the system models and control architecture that
a ect the lateral position control of the John Deere 8420 farm tractor. The steering
actuator model was presented including a lookup table that makes the input to
28
output characteristics linear. A dynamic yaw rate model was discussed that allows
the lateral hitch force to be time varying. The change in the yaw rate response
with respect to hitch loading was shown to be dominated by the DC gain of the
steering angle to yaw rate transfer function. Next, a kinematic model of the lateral
position plant was presented. Finally, the control architecture including design of
gains was presented using root locus analysis.
29
Chapter 3
Adaptive Control by Compensating Yaw Rate Loop Gain
3.1 Introduction
This chapter presents a model reference adaptive control (MRAC) algorithm
used to compensate for the change in the DC gain of the steering angle to yaw
rate transfer function due to changing hitch forces. The algorithm accomplishes
this by adapting the loop gain of the closed-loop yaw rate control system. Di er-
ent adaptive control methods exist, and a brief review is completed that describes
di erent methods considered for this research and why a MRAC was chosen. The
adaptation system is presented with a series of block diagrams, and the adaptation
algorithm is derived. A preliminary simulation is presented that shows the per-
formance of the MRAC system while neglecting the inner-loop steering actuator
properties. The MRAC system is modi ed to include the steering actuator prop-
erties, and more simulation results are presented that demonstrate the improved
performance of the adaptation system. Finally, experimental results are presented
that prove the algorithm can be implemented on a real system. The performance
of the algorithm is discussed including shortcomings and bene ts.
30
3.2 Adaptive Control Techniques
Adaptive control techniques are advantageous because controllers can be ad-
justed to match changes in the plant they are trying to regulate. There are several
techniques in the literature, and a list of methods that were considered in this re-
search includes indirect self-tuning, direct self-tuning, gain scheduling, and MRAC.
A brief discussion of each will be presented, and reasons will be given to justify
the decision to implement a MRAC system. A more in depth description of each
method can be found in [Astrom and Wittenmark, 1995].
Direct and indirect self-tuning regulators have been around for decades and
are similar. The rst paper presenting the self tuning idea came out in the middle
of the twentieth century [Kalman, 1958]. Indirect self-tuning regulators rely on es-
timation techniques such as recursive least squares to estimate plant parameters.
Once the plant parameters are known, methods such as pole placement are used
to design the controller. Direct self-tuning regulators, on the other hand, estimate
the controller parameters directly with no intermediate estimation of model pa-
rameters. This is accomplished by a reparameterization of the controller design
equations into the system model. The goal of this research is to design a direct
method of adapting the yaw rate controller; therefore, the indirect self-tuning reg-
ulator will not be investigated any further. A direct method is desired so that there
are no complications associated with pole placement. A direct self-tuning would
work for a general yaw rate plant, but the steering actuator has saturations that
31
cannot be accounted for using a linear estimation technique. Therefore, a direct
self-tuning regulator cannot ideally be implemented on the farm tractor.
Gain scheduling is advantageous when it is known how an auxiliary system
parameter a ects the system performance. In the case of the farm tractor, the
type of implement would be the auxiliary variable. A series of o -line system
identi cation experiments could be completed to document how the implement
a ects the yaw dynamics, and a schedule of controller gains could be formed to
compensate for di erent implements. For this solution to be viable, however, every
implement the farmer uses would have to be identi ed. While this is doable, it does
not take into account varying soil conditions and depth of the implement. Because
soil moisture and soil type plays an important role in implement-soil mechanics,
neglecting this factor would not be a good solution.
A MRAC system compares the outputs of a model and plant, and through an
adaptation mechanism adjusts the controller so that the di erence in the outputs
go to zero. There are two methods to produce the adaptation algorithm: gradient
techniques and stability techniques. The gradient technique uses a cost function of
the error between model and plant. The adaptation law is formed by moving the
controller parameters in the direction of the negative gradient of the cost function
with respect to the controller parameters. This method is appealing because the
algorithm is straight forward to derive. The problem is that general stability has
not been proven for this method. Stability of a time invariant plant using the
gradient technique has been proven, but this cannot be applied to the tractor
32
system since the yaw rate plant is time varying [Mareels, 1989]. The stability
techniques generally use a Lyapunov function that consists of the states of the
system and mismatch functions. The mismatch functions are equal to zero if
the plant perfectly matches the model. The adaptation algorithm is formed by
making the time rate of change of the Lyapunov function negative semi-de nite.
The problem with applying this method to the farm tractor is that the mismatch
functions cannot be formed due to approximations that will be described in later
sections. For these reasons, the gradient technique is used in this thesis to adapt the
yaw rate controller. Issues with proving stability will be resolved with experimental
testing. This method is advantageous because a good model is known that predicts
how the yaw rate dynamics change with hitch loading. Therefore, the closed-loop
yaw rate model will be tuned to a desired con guration, and the plant yaw rate
controller will be adjusted using a gradient MRAC system so that the plant output
matches the model.
3.3 MRAC-LGA Algorithm
This section presents the layout and update law of the MRAC loop gain
adaptation algorithm. A schematic is shown that describes all of the di erent
components that are required to implement a MRAC system. The value of the
adaptation gain to drive the plant dynamics to match the model dynamics is
calculated. Finally, the update law is derived representing the time rate of change
33
of the adaptation gain with respect to the reference yaw rate, plant output, and
model output.
3.3.1 Adaptation System Architecture
A MRAC system consists of a plant, model, controller, and adaptation mech-
anism. The goal of the system is to adjust the controller so that the output of the
plant matches the output of the model with respect to the same reference input.
Figure 3.1 shows the layout of the system.
Figure 3.1: MRAC System Block Diagram
As can be seen in the gure, the desired yaw rate is sent through the model
dynamics and plant dynamics. The model dynamics is the area encircled by a
dashed line and labeled \Closed-Loop Reference Yaw Model." The model dynamics
consists of the yaw rate model, a xed-gain yaw rate controller, and a feed-back
34
loop. The area encircled by a dashed line and labeled \Yaw Rate Controller" is the
transfer function GCr described in Chapter 2. The yaw rate controller consists of a
xed proportional gain multiplied by an adaptation gain K. The two outputs are
compared, and the di erence (e) is used as the input in the adaptation mechanism
along with the plant output and reference yaw rate.
3.3.2 LGA Algorithm Derivation
The purpose of the adaptation law is to make the closed-loop yaw dynamics
constant while in the presence of the varying hitch cornering sti ness. The desired
yaw dynamics are represented by the closed-loop reference yaw model, and the
adaptation law is used to match the plant output to the output of the reference
model. This is accomplished when the model and plant are equal. As was discussed
in Chapter 2, the DC gain of the steering angle to yaw rate transfer function is
the parameter that dominates the change in yaw dynamics with respect to hitch
loading. Therefore, the proportional control law will be adjusted to o set any
changes in the DC gain of the steering angle to yaw rate transfer function.
Equation (3.1) represents the closed-loop plant and closed-loop reference model.
GrCL = KkprkDCtracG1 +Kk
prkDCtracG
= kprkDCmodG1 +k
prkDCmodG
(3.1)
The left transfer function represents the actual closed-loop yaw dynamics, and
the right transfer function represents the reference model?s closed-loop dynamics.
As can be seen, the yaw rate plant, GPr, has been replaced by kDCtrac times G.
35
The variable kDCtrac is the DC gain of the actual yaw rate transfer function, and G
represents the yaw rate transfer function dynamics. The variable kDCmod represents
the DC gain of the reference model yaw rate transfer function. It is assumed that
the dynamics, G, of the steering angle to yaw rate transfer function are constant
and the only variable that changes with hitch loading is kDCtrac. For this equation
to be equal on both sides, K times kDCtrac must be equal to kDCmod. Therefore,
Equation (3.2) represents the value that the adaptation gain must be in order to
keep the closed-loop yaw dynamics equal for all hitch loadings.
Kmatch = kDCmodk
DCtrac
(3.2)
As was previously discussed, the gradient MRAC approach will be used to
derive the adaptation law. The MIT Rule is an approach to gradient based MRAC
[Whitaker and Kezer, 1958]. In order to derive the adaptation law, a cost function
of the error between the model and plant must be formed. The cost function and
error de nition for the tractor are shown in Equation (3.3) where r is yaw rate.
e = rmod r
J = 12e2
(3.3)
The MIT Rule creates the adaptation law by moving the adaptation gain in the
negative direction of the cost function gradient with respect to the adaptation gain
as seen in Equation (3.4).
36
dK
dt =
@J
@K (3.4)
The variable is called the adaptation algorithm gain, and it changes the
speed that the algorithm converges. This parameter is eld tuned so that the
convergence rate is suitable and the algorithm has a smooth response. If the gain
is too high, there may be convergence issues due to noise. By using the chain rule,
Equation (3.4) can be reduced to Equation (3.5)
dK
dt = e
@e
@K (3.5)
Since e = rmod r and rmod does not contain terms with K, the adaptation law
can be represented by Equation (3.6).
dK
dt = e
@r
@K (3.6)
To apply the previous equation to the system, an equation must be formed to
represent the yaw rate plant output (r). The closed-loop yaw rate transfer function
of the plant is shown in Equation (3.7).
GrCL = r(s)r
des(s)
= KkprGPr1 +Kk
prGPr
(3.7)
By taking the inverse Laplace transform, the plant output di erential equation
shown in Equation (3.8) can be derived.
37
r = 1d
0 +kprKn0
(kprK(n1( _rdes _r) +n0rdes) d1 _r d2 r) (3.8)
The ni and di coe cients come from Equation (2.5). By applying the formula
shown in Equation (3.6), the adaptation law in Equation (3.9) is formed.
dK
dt = (n1d0( _rdes _r) +n0(d0rdes +d1 _r +d2 r)) e
= kpr(d0+kprKn0)2
(3.9)
As can be seen in the previous equation, the adaptation law is a function of the
unknown parameter C h since the ni and di coe cients are a function of that
variable. The unknown values can either be absorbed into the adaptation algorithm
gain, , or approximated by using the nominal C h parameter from Chapter 2.
Since this algorithms varies the loop gain of the yaw rate control system, the
algorithm will be called the Loop Gain Adaptation (LGA) system.
3.4 Simulations of the LGA Algorithm
In this section, simulation results re presented that describe the adaptation
algorithm?s performance. Table 3.1 list the values of all parameters used in the
simulations.
38
Table 3.1: Parameters Used in LGA Simulations
Parameter Value
a 1:00 m
b 2:00 m
c 2:19 m
Izz 18500 kg m2
m 11340 kg
C f 2400 N=deg
C r 5000 N=deg
C h;mod 600 N=deg
C h;trac 4000 N=deg
Vx 2 m=s
kpr 0:30
kp 3:84
200
3.4.1 Simulation Results with Initial LGA Algorithm
To test performance of the adaptation law, a MATLAB simulation was de-
veloped. A reference yaw rate signal was fed in to the system shown previously
in Figure 3.1. The LGA is e ectively estimating the DC gain of the yaw rate
plant. In order for the estimate to converge, there has to be at least a constant,
non-zero input [Astrom and Wittenmark, 1995]. To further insure convergence, a
single frequency was used for the reference signal, and the adaptation parameter
should reach Kmatch in a nite amount of time.
The yaw rate plant includes the steering actuator properties. To force the
steering actuator to become saturated, a cosine function was used as the reference
signal. The cosine signal creates a large initial yaw rate error so that the steering
actuator experiences saturation. The yaw rate reference model uses the nominal
39
value of C h = 600 N/deg and the yaw rate plant was simulated with C h = 4000
N/deg. A nominal velocity of 2 m/s was used in all simulations.
The adaptation gain and yaw rate responses are shown in Figure 3.2. Note that
there is signi cant overshoot in the adaptation gain response, and the adaptation
gain never converges to the desired value. Because the adaptation gain is not at
the right value, the yaw rate of the tractor does not match the reference model.
The steering angle and steering angle rate responses are shown in Figure 3.3. It
can be noted that the steering angle of the model reaches the set point instantly,
and there is a lag in the response of the plant steering angle. This is because there
are no steering actuator properties included into the reference model.
Figure 3.2: Simulated Adaptation Gain and Yaw Rate Response with Initial LGA
Algorithm
40
Figure 3.3: Simulated Steering Actuator Response with Initial LGA Algorithm
Additionally, a good bit of the lag in the steering angle response can be ac-
counted to the steering angle rate of the plant becoming initially saturated as well
as the dynamics associated with the actuator. Since there is instant steering angle
into the yaw rate reference model, the adaptation gain becomes large to compen-
sate for the di erence in output between the reference model and plant. From this
simulation, it can be determined that the adaptation mechanism will not function
satisfactorily due to neglected actuator dynamics.
3.4.2 LGA Algorithm Modi cations
In order to properly account for the steering actuator, the inner-loop steering
actuator dynamics must be included into the algorithm. However, deriving a new
41
algorithm that takes the steering actuator into account results in a higher order
solution. A higher order algorithm would require higher order derivatives of the
outputs. This is not practical because high order derivatives of yaw rate are not
measurable, and di erentiation is too noisy.
However, if the gradient formed in Equation (3.9) could be used as an approx-
imate solution, the inner-loop steering actuator dynamics can be included into the
reference model while using the same algorithm derived by neglecting the steering
actuator. This would allow the same order adaptation algorithm to be used as
before, yet allowing the model to more closely depict the actual system. The new
reference model block diagram is shown in Figure 3.4.
Figure 3.4: MRAC-LGA Closed-Loop Reference Model
The new adaptation and controller system is created by combining the systems
shown in Figures 2.9, 3.1, and 3.4. A block diagram of this total system in shown
in Figure 3.5. As can be seen, the lateral position controller commands a desired
yaw rate based o of the lateral position error. The desired yaw rate is fed into the
yaw rate plant, yaw rate model, and adaptation mechanism. Inside the closed-loop
yaw rate plant and model are the steering actuator dynamics.
42
Figure 3.5: MRAC-LGA Total System Block Diagram
The steering actuator saturations are also put into the reference model. The
steering angle and steering angle rate of the reference model are limited to the
same values listed in Table 2.3. This adds another depth of delity to the model
and will mitigate the overshoot seen previously in Figure 3.2. The model steering
actuator will follow Equations (3.10) and (3.11).
modmax = max (3.10)
_ mod
max =
_ max (3.11)
During periods when the steering actuator is saturated, the gradient of the
system shown in Equation (3.9) is not a good approximation for the system. This
is due to the strong nonlinearities caused by the saturations. To account for this,
43
the adaptation gain K will be held constant during periods of saturations using
the constraints provided in Equation (3.12).
dK
dt =
8>
><
>>:
dK
dt for j
_ measj< _ max j measj< max
0 for j_ measj _ max j measj max
(3.12)
Although the system is not adapting when the steering actuator is saturated, the
algorithm can be implemented on a system that experiences actuator saturations.
3.4.3 Simulation Results with Modi ed LGA Algorithm
To show that the approximated gradient algorithm works for the total system,
another MATLAB simulation was created using the same reference signal as the
simulation of the initial LGA algorithm. As was done in the previous simulation,
the sensors were assumed to be noise and bias free. The hitch cornering sti ness for
the reference model was again set to C h = 600 N/deg and the yaw rate plant was
simulated with C h = 4000 N/deg. As was the case in the initial LGA simulation,
the plant was modeled with the steering actuator dynamics and nonlinearities so
that the simulated plant closely matches the actual tractor system. The major
di erence is that the reference model also had the steering actuator dynamics and
nonlinearities.
The adaptation gain and yaw rate response from the simulation are shown in
Figure 3.6. Notice that the large overshoot is mitigated as compared to Figure
3.2, and the yaw rate output from the plant converge to the same value when the
44
Figure 3.6: Simulated Adaptation Gain and Yaw Rate Response with Modi ed
LGA Algorithm
adaptation gain reaches the desired value. This is the desired performance of the
MRAC system. The steering actuator response can be seen in Figure 3.7.
45
Figure 3.7: Simulated Steering Actuator Response with Modi ed LGA Algorithm
Observe in the steering angle response that the slew rate of the plant and
reference model are initially saturated. This causes additional lag in the steering
angle response on top of the dynamics. It can be seen in Figure 3.6 that the
adaptation gain is held constant at a value of one during the slew rate saturation
period. Also notice that there is more steering angle being applied to the plant
than the model. This is because there is more steering angle needed to turn
the bigger implement. This simulation shows that the modi ed LGA algorithm
can be implemented on a system that has actuator saturations and non-negligible
dynamics. Also, the simulation shows that the gradient calculated from the system
with neglected steering actuator dynamics can be implemented on a system that
has actuator saturations.
46
3.5 Experimental Testing of the Modi ed LGA Algorithm
To further test the LGA algorithm, it was implemented on a John Deere
8420. A detailed description on the experimental setup is presented in Appendix
B. It is desired that the tractor track straight paths. To test the algorithm in
real conditions, the tractor was initialized 2 meters o of the desired path creating
a step input into the system. Two di erent experiments were conducted. The
rst was a step input with an implement, and the second was a step input with
no implement. The implement was a four shank ripper set at about 0.25 meters
depth. To combat gyroscopic sensor noise, a 5 Hertz second order Butterworth
lter was implemented on the yaw rate measurement.
3.5.1 Step Input Testing
Figure 3.8 shows the lateral position response for the two runs showing the
maneuver that the tractor is performing. The left plot is the results with the
implement, and the right plot is the results with no implement.
Figure 3.8: Experimental Lateral Position Response (Left: With Implement, Right:
Without Implement)
47
The yaw rate and adaptation gain responses for each experiment are shown
in Figure 3.9. The top plots correspond to the experiment with an implement,
while the bottom plots correspond to the experiment without an implement. As
can be seen in the adaptation gain responses, the adaptation gain (K) increases in
the run with an implement and decreases in the run without an implement. The
desired adaptation gain was calculated using Equation (3.2) and the real DC gain
was found from steady-state steering tests. The gain increases as expected due to
more lateral hitch forces being applied to the tractor. The gain decreases in the
case with no implement because there are no lateral hitch forces being applied to
the tractor.
48
Figure 3.9: Experimental Adaptation Gain and Yaw Rate Response with Modi ed
LGA Algorithm (Top: With Implement, Bottom: Without Implement)
Notice in Figure 3.10 that the slew rate is initially saturated as it was in the
previously discussed simulations. It can be noted that the adaptation gain was
frozen for the term of the saturation as can be seen in the previous gure. The
algorithm does not create a large adaptation gain overshoot, and it converges to
approximately the correct value. A possible reason the adaptation gain does not
exactly reach the desired value is a persistence of excitation issue. This is due to
the fact that once the tractor is on the desired path, there is very little yaw rate
49
output to drive the algorithm. This is an issue that is inherent to farm applications
since the tractor is designed to track straight paths.
Figure 3.10: Experimental Steering Actuator Response with Modi ed LGA Algo-
rithm (Top: With Implement, Bottom: Without Implement)
3.5.2 LGA Experimental Implementation Issues
During the experimental implementation of the modi ed LGA algorithm, sev-
eral issues surfaced. First, note that the adaptation gain uctuates a good deal
before settling out in both runs. A probable reason for this is due to the noise on
the r term in the adaptation law. The _r and r terms of the adaptation law are
50
calculated using a backwards-di erence numerical di erentiation technique, which
injects additional noise into the algorithm. The higher order derivatives cannot be
measured, so numerical di erentiation techniques must be used. The adaptation
algorithm gain, , could be lowered, but this would increase the convergence time.
This is not practical since there is a limited amount of excitation, and the system
must respond during short periods when there is su cient excitation. An avenue
for future work would be to investigate an algorithm in which the higher order
terms are not necessary.
By looking at Figures 3.9 and 3.10, it can be seen that there is some oscillation
in the steering angle and yaw rate responses with the implement. A probable reason
for this is due to the loop gain being increased since the oscillation does not occur
when the loop gain is lower. The higher loop gain overdrives the system and causes
oscillation. A new algorithm needs to be developed to change the magnitude of
the steering angle into the yaw rate plant without increasing the loop gain of the
closed-loop system. Therefore, a new algorithm that addresses these issues will be
investigated in the next chapter.
3.6 Summary and Conclusions
This chapter has presented a MRAC system to account for a changing DC
gain of the steering angle to yaw rate transfer function by varying the loop gain
of the closed-loop system. Various adaptive control theories were presented, and
reasons were given for selecting a MRAC system for this problem. The MRAC
51
system architecture was discussed including the elements required to implement
the algorithm. A controller variable was selected that would make the plant out-
put match the model output, and the value of the controller variable was shown
to be a function of the plant and reference model. The MRAC algorithm was
then calculated using the MIT rule gradient approach. The algorithm was tested
in simulation yielding unsatisfactory results due to neglected steering actuator
properties. The algorithm was then modi ed to included the inner-loop steering
actuator dynamics, and more simulation results showed that the algorithm gives
satisfactory results. The algorithm was next implemented on the real system, and
experimental data was presented. Issues were presented including a uctuating
adaptation response and system oscillations. Probable causes for the unsatisfac-
tory results were discussed, and suggestions for improvement were presented.
52
Chapter 4
Adaptive Control by Compensating Yaw Rate Feed-Forward Gain
4.1 Introduction
After discovering unsatisfactory experimental results with the algorithm de-
veloped and analyzed in Chapter 3, a new algorithm was created. Since the DC
gain of GPr is the parameter that changes with hitch loading, the amount of in-
puted steering angle is the factor that must be compensated. This chapter will
discuss the development of a new control architecture that will allow a feed-forward
yaw rate controller to be used. By using a feed-forward control law, the DC gain
of the closed-loop yaw response will be unity compared with the non-unity value
of the previous controller. To compensate for changes is the plant DC gain, the
feed-forward control coe cient will be adapted. This new system architecture will
be discussed, and the new algorithm will be derived in this chapter. Simulations
will be presented that will describe the performance of the new algorithm. Modi -
cations to the algorithm will be made so that the adaptation algorithm can handle
the steering actuator dynamics and nonlinearities. Finally, experimental imple-
mentation results are presented showing improved performance compared to the
previous algorithm and compared to a xed-gain control law.
53
4.2 New Algorithm Requirements
As was discussed in the previous chapter, adapting the loop gain was not
a practical solution. The previously derived adaptation algorithm requires high-
order derivatives that must be obtained using numerical di erentiation techniques.
This causes a good deal of additional noise to be injected into the algorithm. This
increased noise causes the adaptation gain response to uctuate erratically before
settling out. Although the adaptation algorithm gain ( ) could be lowered, this is
not practical since a relatively fast gain response is desired. This is because there
is a limited amount of time that the tractor has enough excitation to drive the
algorithm. Therefore, it is desired to have an algorithm that does not require the
higher order derivatives and has a quick adaptation rate.
Secondly, increasing the loop gain of the yaw rate control system caused os-
cillations in the yaw rate and steering angle responses. It is believed that a higher
loop gain was overdriving the system, and therefore the steering angle tended to
oscillate. Therefore, an algorithm that does not increase the loop gain of the yaw
rate control system is desired. Since the DC gain of the steering angle to yaw
rate transfer function is changing with hitch loading, the amount of steering angle
input is the parameter that should be compensated.
If a feed-forward yaw rate controller is implemented, the controller could be
adapted to adjust the amount of steering angle put into the system. By adding
a feed-forward controller, the DC gain of the closed-loop yaw rate control system
will be unity. This is advantageous since a lower lateral position controller gain
54
can be used as seen in Table 2.7. In this chapter, this approach is investigated in
order to derive an improved adaptation scheme for the tractor.
4.3 MRAC-FGA Algorithm
This section presents the system architecture and algorithm derivation. A
new control law is presented that adds a feed-forward component to the yaw rate
controller. The MRAC system block diagram is presented, and the changes are
discussed. Finally, the MRAC algorithm to adapt the feed-forward controller is
derived.
4.3.1 Adaptation System Architecture
As was mentioned above, a feed-forward yaw rate controller will be added
to the system. The feed-forward controller takes in the desired yaw rate from the
lateral position controller and commands a desired steering angle. Figure 4.1 shows
the diagram of the cascaded controller with the feed-forward yaw rate controller.
The feed-forward controller that is added is GCrff.
Figure 4.1: Cascaded Control Block Diagram with Feed-Forward Control
55
The transfer function for the feed-forward control law is shown in Equation (4.1).
GCrff = desff(s)r
des(s)
= kff (4.1)
Recall that the yaw rate feedback controller is
GCr = desfb(s)r
err(s)
= kpr (4.2)
By putting the two controllers together, the desired steering angle is
des(s) = desfb(s) + desff(s) (4.3)
The feed-forward controller coe cient in Equation (4.1) is kff. This variable will
be equal to the inverse of the DC gain of GPr as shown in Equation (4.4).
kff = 1k
DCmod
(4.4)
Recall that kDCmod is the DC gain of the model steering angle to yaw rate transfer
function, and it is calculated using the nominal C h parameter shown in Chapter
2.
The MRAC system block diagram is shown in Figure 4.2. Notice that the
feed-forward yaw rate controller has been added to the system. The closed-loop
model now consists of the steering angle to yaw rate transfer function, feedback
gain, feed-forward gain, and feedback loop. The closed-loop plant consists of these
56
same elements and an adaptation gain K that has been added to the feed-forward
controller. The adaptation gain will be used to compensate for changes in the yaw
rate plant.
Figure 4.2: MRAC System Block Diagram with Feed-Forward Control
4.3.2 FGA Algorithm Derivation
The purpose of the algorithm is to adjust the feed-forward gain so the closed-
loop yaw rate dynamics have a DC gain of one. The transfer functions for the
closed-loop dynamics of the plant and model are shown in Equation (4.5).
57
GrCL = (kpr +kffK)kDCtracG1 +k
prkDCtracG
= (kpr +kff)kDCmodG1 +k
prkDCmodG
(4.5)
By letting G = 1, the desired steady-state transfer functions are shown below.
GrCLSS = (kpr +kffK)kDCtrac1 +k
prkDCtrac
= (kpr +kff)kDCmod1 +k
prkDCmod
= 1 (4.6)
Since both steady state closed-loop systems are desired to be equal to one, the
adaptation gain K can be calculated in terms of the DC gain of the plant and
model so that the closed-loop plant equation is equal to unity. The steady-state
equation for the closed-loop plant is seen in Equation (4.7).
GrCLSS = kprkDCtrac +kffKkDCtrac1 +k
prkDCtrac
= 1 (4.7)
It can be seen that if kff times K times kDCtrac is equal to 1, Equation (4.7) is
satis ed. Therefore, the matching condition is shown in Equation (4.8)
Kmatch = kDCmodk
DCtrac
(4.8)
since kff is equal to the inverse of kDCtrac as shown in Equation (4.4).
The MIT rule is going to be used again to calculate the adaptation update
law. Recall that the cost function and adaptation error de nition are shown in
Equation (4.9).
58
e = rmod r
J = 12e2
(4.9)
The MIT rule gradient equations are the same as before and are seen in Equation
(4.10).
dK
dt =
@J
@K = e
@e
@K = e
@r
@K (4.10)
Recall that the gradient calculation is simpli ed because the plant di erential
equation is the only term that contains the adaptation gain K. The transfer
function of the closed-loop plant dynamics are shown in Equation (4.11).
GrCL = (kpr +kffK)GPr1 +k
prGPr
(4.11)
By taking the inverse Laplace transform, the di erential equation of the plant yaw
rate can be written as Equation (4.12).
59
r = ( n1 _rdes + n0rdes r _r d2 r)
Where:
= 1d0+n0kpr
= kpr +kffK
= d0 +n0kpr
= d1 +n1kpr
(4.12)
By applying the MIT rule in Equation (4.10) to the previous equation, the update
law is as follows:
dK
dt =
kff
d0+n0kpr(n1 _rdes +n0rdes) e (4.13)
The new algorithm will be called the Feed-Forward Gain Adaptation (FGA)
algorithm. Notice that the new adaptation algorithm is not a function of r or _r.
This should provide a cleaner adaptation gain response as compared to the previous
algorithm. In the new FGA algorithm, the only signal that has to be numerically
di erentiated is the reference yaw rate. This results in acceptable performance
since the signal that is di erentiated is coming from a controller instead of a sensor.
The controller acts as a lter and provides a smooth signal for di erentiation. As
was the case in the previous algorithm, the ni and di coe cients are a function of
the unknown parameter C h. The unknown parameters are either absorbed into
60
the adaptation algorithm gain or approximated using the nominal value of C h
discussed in Chapter 2.
4.4 Simulations of the FGA Algorithm
This section provides simulation analysis of the new algorithm. Table 4.1
presents the values used in the simulations.
Table 4.1: Parameters Used in FGA Simulations
Parameter Value
a 1:00 m
b 2:00 m
c 2:19 m
Izz 18500 kg m2
m 11340 kg
C f 2400 N=deg
C r 5000 N=deg
C h;mod 600 N=deg
C h;trac 4000 N=deg
Vx 2 m=s
kpr 0:30
kp 3:84
200
4.4.1 Simulation Results with Initial FGA Algorithm
In order to analyze the feed-forward adaptation algorithm in the same way
as the algorithm described in Chapter 3, a simulation is completed in which the
inner-loop steering actuator is neglected in the model. The simulated tractor will
again contain the steering actuator properties so that the simulation will be a
61
high- delity as possible. The reference yaw rate signal will be a cosine function. A
cosine signal was again chosen so that there is enough excitation for the adaptation
gain to reach its true value and to provide an initially large yaw rate error. The
large initial error causes the steering actuator to become saturated. Figure 4.3
shows the adaptation gain and yaw rate response of the simulation.
Figure 4.3: Adaptation Gain and Yaw Rate Response with Initial FGA Algorithm
As can be seen, the new adaptation gain response exhibits the same unwanted
overshoot as the algorithm in Chapter 3. Also, the gain response does not converge
to the desired value. This is again contributed to the neglected steering actuator
properties. Since the adaptation gain does not converge to the desired value, the
yaw rate outputs from the plant and reference model do not converge.
62
The steering actuator response is shown in Figure 4.4. It can be seen that
the steering angle of the reference model reaches the setpoint instantly while the
plant steering actuator has dynamics and saturation. As was the case in Chapter
3, the neglected steering actuator properties in the reference model are the reason
the algorithm exhibits poor performance.
Figure 4.4: Simulated Steering Actuator Response with Initial FGA Algorithm
4.4.2 FGA Algorithm Modi cations
To address the poor performance of the algorithm, the steering actuator prop-
erties are again added to the reference model. Figure 4.5 shows the closed-loop
reference model that replaces the existing reference model. As can be seen from
63
the gure, the inner-loop steering actuator dynamics have been added to the sys-
tem. The same algorithm shown in Equation (4.13) will be used to adapt the
controller. This is because including the actuator dynamics into the algorithm
derivation would produce a higher-order update law. A higher-order update law
would require derivatives of the plant yaw rate output which was shown in Chap-
ter 3 to be impractical. Therefore, using the algorithm in Equation (4.13) derived
while neglecting the steering actuator dynamics will be an approximation for the
system since the true gradient would include the steering actuator dynamics.
Figure 4.5: MRAC-FGA Closed-Loop Reference Model
Combining Figures 4.5, 4.1, and 4.2, the total system block diagram is repre-
sented in Figure 4.6. The top portion represents the closed-loop reference model,
and it is comprised of the yaw rate model, steering actuator model, and controllers
for each subsystem. The desired yaw rate being fed into the plant and model comes
from the lateral position controller.
64
Figure 4.6: MRAC-FGA Total System Block Diagram
As was the case in the algorithm presented in Chapter 3, the steering actuator
saturations will be included into the reference model. Also, the adaptation gain
will be held constant during periods of saturation because the large nonlinearities
caused by the saturation prevents the gradient from being valid for the system.
4.4.3 Simulations Results of Modi ed FGA Algorithm
A simulation was created to test the modi ed FGA algorithm. As was the
case for the initial FGA algorithm, the yaw rate reference signal will be a cosine
function. The simulated plant is the same as before, and it includes the steering
actuator properties. Figure 4.7 shows the adaptation gain and yaw rate response
of the modi ed FGA algorithm.
65
Figure 4.7: Simulated Adaptation Gain and Yaw Rate Response with Modi ed
FGA Algorithm
The adaptation gain overshoot is mitigated, and it converges to the correct
value to match the plant and reference model yaw rates. As can be seen in Figure
4.8, the steering actuator of the plant initially saturates. By forcing the reference
model steering actuator to saturate the same way as the plant actuator, the closed-
loop yaw rate reference model performs like the plant system.
66
Figure 4.8: Simulated Steering Actuator Response with Modi ed FGA Algorithm
The previous simulation results have shown that the modi ed FGA algorithm
will be a good solution for the system. By including the steering actuator properties
into the reference model, the adaptation gain response no longer exhibits the large
overshoot seen the the previous simulations.
4.5 Experimental Testing of the FGA Algorithm
The algorithm was next implemented on the John Deere 8420 to determine
if the FGA algorithm performs satisfactorily on the experimental platform. A
description of the experimental setup is detailed in Appendix B. Two di erent
types of experiments are presented: a step input test and a steady-state lateral
position test. The step input test is used to show how the algorithm performs as
67
compared to the algorithm presented in Chapter 3. On the other hand, the steady-
state lateral position experiment will display the system?s performance compared
to a xed gain controller. The desired path in both tests will be a straight line,
and the speed of the tractor is approximately 2 m/s.
4.5.1 Step Input Testing
The step input test is started with an initial lateral error of approximately
2 meters. The system was tested with an implement and without an implement.
Figure 4.9 shows the lateral position response of the tractor. The left plot is
the response with an implement, and the right plot is the response without an
implement.
Figure 4.9: Experimental Lateral Position Response (Left: With Implement, Right:
Without Implement)
The adaptation gain and yaw rate response for each experiment are shown
in Figure 4.10. The top plots represent the experiment with an implement, and
the bottom plots represent the experiment without an implement. The desired
adaptation gain was found by using steady-state turns to experimentally determine
68
the DC gain of the steering angle to yaw rate transfer function and applying it
to Equation (4.8). As can be seen, the adaptation gain response has a smoother
response as compared with the experiments shown in Figure 3.9. This is attributed
to the modi ed FGA algorithm not having to use numerical di erentiation on the
yaw rate measurement.
Figure 4.10: Experimental Adaptation Gain and Yaw Rate Response with Modi ed
FGA Algorithm (Top: With Implement, Bottom: Without Implement)
The steering actuator response is shown in Figure 4.11. Notice that the slew
rate is saturated just as it was in the simulations. Recall that the FGA algorithm
was developed to mitigate the oscillation experienced when the adaptation gain
69
increases. It can be seen in Figures 4.10 and 4.11 that there is no oscillation at
the higher gains. This shows that the algorithm provides improved performance
compared to the adaptation algorithm that changes the loop gain of the yaw rate
control system.
Figure 4.11: Experiemental Steering Actuator Response with Modi ed FGA Al-
gorithm (Top: With Implement, Bottom: Without Implement)
70
4.5.2 Lateral Error Testing
The lateral error experiments are used to show that the modi ed FGA algo-
rithm improves performance over a xed-gain controller. Four di erent con gura-
tions are tested:
1. Adapting with an Implement
2. Fixed Gain with an Implement
3. Adapting without an Implement
4. Fixed Gain without an Implement
Seven runs are made with each con guration. Each run consists of starting the
tractor o the path by about 2 meters and allowing the controller steer the tractor
to the line. The tractor is kept on the line for 40-50 seconds, and this part of
the run is where the experimental statistics are calculated. The experimental
statistics that are calculated are the mean error (x), error standard deviation ( ),
and average adaptation gain. The results from setup 1 are shown in Table 4.2.
71
Table 4.2: Experimental Statistics with Implement While Adapting
Run x (m) (m) Average Adaptation Gain
1 0:065564 0:039947 1:061393
2 0:040891 0:047243 1:000751
3 0:084655 0:045661 1:213300
4 0:054450 0:065198 1:214141
5 0:080246 0:077658 1:115439
6 0:041295 0:045255 1:118982
7 0:040319 0:048844 1:176321
Avg 0:027635 0:052830 1:128618
The results from setup 2 are shown in Table 4.3. The xed gain controller was
identical to the adaptive controller, but the adaptation gain was held at K = 1.
This is the nominal tuning for the system. As can be seen from the tables, the
adaptive controller had a 13.28% lower standard deviation in the lateral error.
Table 4.3: Experimental Statistics with Implement and Fixed Gain
Run x (m) (m)
1 0:050265 0:047430
2 0:012640 0:078276
3 0:038548 0:047353
4 0:027680 0:047296
5 0:005753 0:063803
6 0:025262 0:058420
7 0:026896 0:076351
Avg 0:017392 0:059847
The next two con gurations are without an implement. Table 4.4 shows the
results from setup 3 that was adapting on-line. Table 4.5 shows the results from
72
setup 4. This experiment was without an implement and had a xed-gain con-
troller. As can be seen from looking at the results with no implement, the adaptive
controller has a 12.81% smaller standard deviation that the xed-gain controller.
Table 4.4: Experimental Statistics without Implement While Adapting
Run x (m) (m) Average Adaptation Gain
1 0:018037 0:065402 0:916123
2 0:007931 0:068065 0:938451
3 0:002076 0:043496 0:735783
4 0:056243 0:046521 0:935534
5 0:028691 0:043412 0:893810
6 0:054315 0:044221 0:906893
7 0:005463 0:061379 0:823834
Avg 0:022413 0:053214 0:878633
Table 4.5: Experimental Statistics without Implement and Fixed Gain
Run x (m) (m)
1 0:047593 0:041622
2 0:045640 0:077849
3 0:061712 0:058706
4 0:009557 0:083034
5 0:041637 0:083213
6 0:011104 0:039432
7 0:004683 0:036362
Avg 0:017392 0:060031
It can be observed from the following experiment that the adaptive controller
out-performs the xed-gain controller. By adapting the yaw rate controller, a more
precise response can be achieved in the presence of changing tractor loads verses
a nominally-tuned controller.
73
4.5.3 Lateral Error Testing With Changing Implement Position
As was mentioned in the motivation section of Chapter 1, one of the biggest
problems with the automatically steered tractors is that the sensitivity gain that
is set by the user does not adjust when the implement is lifted out of the ground.
This causes poor performance due to oscillations in the lateral position. To show
that the MRAC-FGA algorithm addresses this problem, an experiment was set up
in which the tractor starts down a path with the implement in the ground and
then the implement is lifted out of the ground to simulate the tractor crossing
a waterway. Due to wet eld conditions, the implement was modeled on the
tractor by a reduced yaw rate controller gain. Since the yaw rate plant DC gain is
what uctuates the most with hitch loading, this is an accurate realization of the
implement. The implement has a cornering sti ness of C h = 3000 N/deg. Several
runs were conducted using a xed-gain controller tuned to the speci c implement,
and several runs were conducted using the MRAC-FGA algorithm to adjust the
controller on-line. The lateral position response of one run with the xed-gain
controller is shown in Figure 4.12 . The point at which the implement is taken out
of the ground is denoted by the dashed black line. As can be seen, the tractor has
signi cantly more lateral error in the second half of the run that the rst half. This
is due to the controller gain being set too high for the implement con guration.
74
Figure 4.12: Lateral Position Response of Experimental Line Tracking with Im-
plement Lifted Out of Ground at the Speci ed Point in Time with a Fixed Gain
Controller
The lateral position response is shown in Figure 4.13 for a run with the MRAC-
FGA algorithm adapting on-line. Notice that the lateral position response is the
same on the left and right of the dashed black line. This is because the MRAC-FGA
algorithm adjusts the controller gain to the changing implement depth.
Figure 4.13: Lateral Position Response of Experimental Line Tracking with Im-
plement Lifted Out of Ground at the Speci ed Point in Time with an Adaptive
Controller
The adaptation gain response is shown in Figure 4.14. It can be seen that
the adaptation gain increases to adjust the controller to the implement. After the
implement is lifted out of the ground, the adaptation gain slowly decreases so that
75
Figure 4.14: Adaptation Gain Response of Experimental Line Tracking with Im-
plement Lifted Out of Ground at the Speci ed Point in Time with the MRAC-FGA
Adaptive Controller
the yaw rate output matches the reference model yaw rate output. The reason the
adaptation gain moves slowly is because there is not much excitation as compared
to the beginning of the run. Nevertheless, the response is improved when compared
to the xed-gain controller.
In order to quantify the improvement, ve runs were made with the xed-
gain controller, and ve runs were made with the adaptive controller. The results
from the xed-gain controller runs are shown in Table 4.6 and the results from
the adaptive controller runs are shown in Table 4.7. As can be seen in Table 4.6,
the average standard deviation for the xed-gain controller with the implement is
0:068136 meters, and the average standard deviation with the implement out of
the ground is 0:074769 meters. There is an increase in lateral error due to the
controller being poorly tuned.
76
Table 4.6: Experimental Statistics for Test with Changing Implement Position
with Fixed Gain Controller (The left two columns correspond to the parts of the
runs with the implement, and the right two columns correspond to the parts of
the runs with the implement out of the ground.)
Run x (m) Imp. (m) Imp x(m) No Imp. (m) No Imp
1 0:013588 0:069492 0:029255 0:077616
2 0:043811 0:066799 0:025351 0:073999
3 0:031344 0:062622 0:011652 0:072500
4 0:048415 0:061910 0:054524 0:064842
5 0:062653 0:079858 0:040065 0:084889
Avg 0:039962 0:068136 0:032169 0:074769
It is shown in Table 4.7 that the average standard deviation with the imple-
ment is 0:068333 meters, and the average standard deviation with the implement
lifted out of the ground is 0:054716 meters. The standard deviation for the xed-
gain and adaptive controller with the implement in the ground are almost identical.
This shows that the MRAC-FGA algorithm properly adjusts the controller to the
implement. The di erence between average standard deviations of the xed-gain
and adaptive controllers with the implement out of the ground is 0:020053 me-
ters. The MRAC-FGA adaptive controller improves the lateral position tracking
by 26.6% when the implement is lifed out of the ground.
77
Table 4.7: Experimental Statistics for Test with Changing Implement Position
with Adaptive Controller (The left two columns correspond to the parts of the
runs with the implement, and the right two columns correspond to the parts of
the runs with the implement out of the ground.)
Run x (m) Imp. (m) Imp x(m) No Imp. (m) No Imp
1 0:021700 0:071580 0:048268 0:061464
2 0:027487 0:067878 0:025821 0:057126
3 0:051233 0:065987 0:052516 0:056507
4 0:060673 0:066517 0:034550 0:049674
5 0:032346 0:069704 0:036121 0:048809
Avg 0:038688 0:068333 0:039455 0:054716
4.6 Summary and Conclusions
This chapter presented a MRAC-FGA algorithm that addressed the issues
found with the MRAC-LGA algorithm presented in Chapter 3. The requirements
of the new algorithm were discussed as well as the desired performance character-
istics. A new controller was proposed that added a feed-forward controller to the
yaw rate controller, and a new adaptation structure was discussed to adapt this
new controller to changes in hitch loading. The new algorithm, called the FGA
algorithm, was developed using the same gradient technique used in Chapter 3.
Simulations were presented that showed the new algorithm must take the steering
actuator properties into account in similar fashion as the LGA algorithm. More
simulated results were presented that displayed satisfactory performance of the
modi ed FGA algorithm. Experimental results were presented that demonstrated
satisfactory performance. Finally, the performance of the adaptive controller was
78
compared to a xed-gain controller, and the adaptive controller was shown to be
superior.
79
Chapter 5
Conclusions
5.1 Summary
This thesis has presented two adaptive control algorithms that adjust the
control law to suit changes in yaw rate plant parameters. Chapter 2 presented the
models and controllers for the steering actuator plant, yaw rate plant, and lateral
position plant. The steering actuator plant model was shown to be second order
with a pure integrator. A nonlinearity lookup table was presented that linearized
the system with the exception of slew rate saturation. The yaw rate plant model
was derived from a bicycle representation of the tractor. The hitched implement
was modeled as a third axle behind the rear axle. Analysis was completed on the
yaw rate plant model that showed the DC gain was the parameter that changed
the most with hitch loading. This analysis provided a basis to why an adaptive
control law was needed. Next, the lateral position plant model was presented.
Finally, controller architecture, design, and analysis were presented for each of the
three subsystems.
Chapter 3 presented the rst algorithm to adapt the yaw rate controller to
the changing yaw rate plant. A brief review of several adaptive control techniques
was completed, and reasons were given to justify the use of a MRAC. The MRAC
system architecture was presented, and the MRAC system update law was de-
rived using the MIT rule. A simulation was presented that showed the algorithm?s
80
performance was insu cient due to neglected steering actuator properties. The
MRAC-LGA algorithm was then modi ed to include the steering actuator prop-
erties, and more simulated results showed improved performance. Finally, the
algorithm was implemented on a John Deere 8420, and experimental results were
presented. Some issues with experimental implementation were discovered, and a
list of desirable characteristics for a new algorithm was formulated.
Chapter 4 presented a new algorithm to adapt the yaw rate controller that
required the addition of a feed-forward control law. The new MRAC system ar-
chitecture was presented, and the feed-forward adaptive control law was derived.
Simulations were presented that showed the new MRAC-FGA algorithm again per-
formed poorly due to neglected steering actuator properties. The algorithm was
then modi ed in the same way as the previously discussed algorithm, and sim-
ulated results showed ideal performance. The feed-forward adaptive control law
was implemented on the John Deere 8420, and the algorithm was experimentally
shown to be superior than the algorithm presented in Chapter 3. More experimen-
tal results were presented that showed the feed-forward algorithm out-performed
a xed-gain controller.
5.2 Recommendations for Future Work
Both MRAC algorithms presented in this thesis required the adaptation gain
to be xed when the steering actuator is saturated. When the steering actuator
is saturated, there is generally a good deal of yaw rate excitation. Since adaptive
81
control routines require persistently excited conditions, it would be desirable to
have an algorithm that can function when the steering actuator is saturated. This
would allow a quicker adaptation response that could lead to a more accurate
adaptation system.
All of the experimental testing presented in this thesis was performed with an
implement attached to the three-point hitch. Since many implements are attached
to the tractor using the trailer tongue, more testing of this algorithm should be
completed with these types of implements. Also, yaw rate model validation needs
to be completed with this arrangement since the model presented in Chapter 2 has
only be validated for hitched implements.
The testing of the algorithms in this thesis were completed using only one
velocity. More development of the algorithms should be completed to expand the
capabilities for a variety of di erent setups.
Finally, since analytical stability was not shown for the adaptation mechanism,
extensive simulations and experiments should be completed to ensure that the
algorithm is stable for all possible con gurations.
82
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85
Appendices
86
Appendix A
Nomenclature
87
Table A.1: Nomenclature Table of Variables Used in Thesis Part I
Variable De nition
a Distance From CG to Front Axle
b Distance From CG to Rear Axle
c Distance From Rear Axle to Hitch
Izz Mass Moment of Inertia About the CG
m Mass
C f Cornering Sti ness of Front Axle
C r Cornering Sti ness of Rear Axle
C h Cornering Sti ness of the Hitch
f Slip Angle at Front Axle
r Slip Angle at Rear Axle
h Slip Angle at Hitch
Ff Lateral Force at Front Axle
Fr Lateral Force at Rear Axle
Fh Lateral Force at Hitch
Vx Longitudinal Velocity
Vy Lateral Velocity
Tractor Steering Angle
mod Model Steering Angle
_ Tractor Steering Angle Rate
^ input Intermediate Steering Input Command
input Steering Input Command
des Desired Steering Angle
max Maximum Steering Angle
_ max Maximum Steering Angle Rate
r Tractor Yaw Rate
_r 1st Derivative of Tractor Yaw Rate
r 2nd Derivative of Tractor Yaw Rate
rmod Model Yaw Rate
rdes Desired Yaw Rate
y Lateral Position
ydes Desired Lateral Position
GP Steering Actuator Plant
GPr Yaw Rate Plant
GPl Lateral Position Plant
GC Steering Actuator Controller
GCr Yaw Rate Controller
GCl Lateral Position Controller
88
Table A.2: Nomenclature Table of Variables Used in Thesis Part II
Variable De nition
LGA Loop Gain Adaptation
FGA Feed-Forward Gain Adaptation
!n Natural Frequency of Steering Actuator
Damping Ratio of Steering Actuator
Side Slip Angle at the CG
Course Angle with Respect to Desired Direction
N North Direction
E East Direction
kp Steering Actuator Proportional Control Coe cient
kff Yaw Rate Feed-Forward Control Coe cient
kpr Yaw Rate Proportional Control Coe cient
kpy Lateral Position Proportional Control Coe cient
kdy Lateral Position Derivative Control Coe cient
kiy Lateral Position Integral Control Coe cient
DCyaw DC Gain of the Closed-Loop Yaw Rate Dynamics
GrCL Closed-Loop Yaw Rate Dynamics
K Adaptation Gain
G Constant Yaw Rate Dynamics
kDCtrac DC Gain of the Tractor Yaw Rate Transfer Function
kDCmod DC Gain of the Model Yaw Rate Transfer Function
Kmatch Adaptation Gain that Matches the Tractor to the Model
e Error Between the Tractor and Model Outputs
J Cost Function of the Adaptation Error
Adaptation Algorithm Gain
ni Numerator Coe cients of GPr
di Denominator Coe cients of GPr
Ci Intermediate Coe cients of GPr
89
Appendix B
Experimental Setup
This appendix provides a detailed description of the experimental setup used
to test the MRAC algorithms presented in this thesis. The test vehicle is a pro-
duction model John Deere 8420 out tted with a StarFire DGPS receiver and Au-
toTrac technology. The AutoTrac technology allows the steering actuator to be
commanded over the controller area network (CAN). A four shank ripper was used
to create the lateral hitch forces to test the algorithm. The tractor and attached
ripper can be seen in Figure B.1.
Figure B.1: John Deere 8420
90
The tractor is equipped with a StarFire GPS receiver to measure the position
in the east-north coordinate frame. A picture of the StarFire is shown in Figure
B.2. The receiver uses corrections from the StarFire system which eliminates a
majority of the errors caused by the atmosphere. This allows the position accuracy
to be 10 cm CEP. The StarFire outputs its messages at 5 Hertz.
Figure B.2: StarFire DGPS Receiver
91
A Bosch inertial measurement unit (IMU) is used to collect the yaw rate
measurement. A photograph of the IMU is shown in Figure B.3. The Bosch unit
is automotive grade MEMS and has an analog output. As can be seen in the gure,
the Bosch inertial sensors are packaged in sets with each containing a gyroscope
and accelerometer. Each package is mounted on an orthogonal axis inside a metal
case.
Figure B.3: Bosch IMU
92
A Novotechnik linear potentiometer is used to measure the steering angle. A
picture of the potentiometer attached to the tractor is shown in Figure B.4. The
output of the potentiometer is a voltage proportional to the linear displacement.
The sensor was calibrated using a series of steady-state steering inputs while mea-
suring the yaw rate output. The DC gain of the steering angle to yaw rate transfer
function is known, so a least squares t was used to calculate the calibration coef-
cients.
Figure B.4: Steering Angle Sensor
93
The Versalogic PC104 computer is at the center of the system. A picture
of the Versalogic is shown in Figure B.5. Inside is a CAN card, data acquisition
(DAQ) card, central processing unit (CPU), and hard drive. The CAN card is
used to send messages across the CAN bus of the tractor to command the steering
actuator. The DAQ card is used to convert the analog signals from the Bosch
IMU and steering angle sensor to a digital signal. The sample rate for the inertial
sensors and steering angle sensor is 50 Hertz. The StarFire GPS receiver outputs a
digital ASCII message over a RS232 connection. The Versalogic is running a QNX
real-time operating system, and the embedded software was written with the C++
language.
Figure B.5: Versalogic PC104 Computer
94
A schematic of the entire system is shown in Figure B.6. As can be seen, the
PC104 computer is at the center of the system. The sensor signals are fed into
the computer and a CAN output signal is sent out. The rate that the steering
actuator ow valve is commmaded is 50 Hertz. Once the ow valve is commanded,
the steering actuator is controlled. The steering angle feeds the yaw rate plant,
and the yaw rate feeds the lateral position plant. The steering angle, yaw rate,
and lateral position are measured and sent back to the PC104 computer.
Figure B.6: Experimental Setup Block Diagram
95
The straight path for the tractor was de ned by two points in the east-north
coordinate frame named A and B. A schematic of the path is shown in Figure
B.7. The angle from north to the desired path is labeled and is calculated by
Equation (B.1).
= tan 1
B
E AE
BN AN
(B.1)
The angle from point A to the tractor?s position is labeled . Its de nition is
shown in Equation (B.2).
= tan 1
Tr
E AE
TrN AN
(B.2)
The distance from point A to the tractor?s position is labeled h. Its de nition is
shown in Equation (B.3).
h =
q
(TrE AE)2 + (TrN AN)2 (B.3)
The lateral position is then de ned by Equation (B.4).
y = hsin( ) (B.4)
96
Figure B.7: Experimental Lateral Position Calculation
97