Dynamic Simulation Modeling and Control
of the Odyssey III Autonomous Underwater Vehicle
by
Mark E. Rentschler
B.S., Mechanical Engineering
University of Nebraska, Lincoln, NE (2001)
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
June 2003
JUL 0 8 2003
0 2003 Mark E. Rentschler. All rights reserved.
LIBRARIES
The author hereby grants to MIT permission to reproduce and to distribute publicly paper and
electronic copies of this thesis document in whole or in part.
Author ..................................
Certified by.....
1'
Certified by.........
C ertified by ....
Accepted by ..........................
!
Department pf Mechanical Engineering
8 May 2003
Chr'yssostomos Chryssostomidis
y
Doherty Professor of Ocean Science and Engineering, MIT
Thesis Co-supervisor
.....-.....------..--.--------------------Franz S. Hover
Principal Research Engineer, MIT
Thesis Co-supervisor
...-..-.--....-------------------Nicholas M. Patrikalakis
Kawasaki Professor of Engineering
Professor of Ocean and Mechanical Engineering, MIT
Thesis Reader
----------------------------------Ain A. Sonin
Professor of Mechanical Engineering, MIT
Chairperson, Committee on Graduate Students
BARKER
2
Dynamic Simulation Modeling and Control
of the Odyssey III Autonomous Underwater Vehicle
by
Mark E. Rentschler
Submitted to the Department of Mechanical Engineering
on May 8 2003, in Partial Fulfillment of the
Requirements for the Degree of Master of Science in
Mechanical Engineering
Abstract
Dynamic modeling and control of the Bluefin Odyssey III class vehicle "Caribou," operated by
the MIT Sea Grant AUV Laboratory is addressed. Focus is on demonstrating a simple forward design
procedure for the flight control system, which can be carried out quickly and routinely to maximize
vehicle effectiveness. In many situations, the control loops are tuned heuristically in the field; frequent
retuning is necessitated by the inevitable changes in vehicle components, layout, and geometry. Our
paradigm here is that 1) a prototype controller is developed, based on an initial model, 2) this controller is
then used to perform a very compact set of runs designed to identify the vehicle dynamic response, and 3)
a revised, precision controller based on this improved model is implemented for the real mission.
We first developed a hydrodynamic model of the vehicle from theory and benchtop laboratory
tests; no data from prior field tests with this vehicle was used. Body added mass approximations were
included as well as lift and hydrostatic forces and moments. Inertial properties were approximated by
assuming the vehicle density was that of water. Caribou's tailcone assembly consists of a doublegimbaled thrust-vectoring duct, with significant positioning dynamics and a non-traditional
hydrodynamics. We carefully tested this tailcone's response behavior through laboratory tests, and
created a low-order model. Using the tailcone model and the vehicle's initial hydrodynamic model, we
developed a conservative controller design from basic principles. The control system consisted of a
heading controller, pitch controller, and depth controller; the pitch control loop was nested inside the
depth control loop. This control system was successfully tested in the field: the vehicle was controllable
within several degrees of heading and approximately one-half meter of depth, on the first-pass design.
System identification tests were then completed with the preliminary controller to gain a better
understanding of the complete hydrodynamics of the vehicle, and in order to develop a precision
controller based on the improved model. The resulting data provided a full-system linear model of the
vehicle, and led to a successful controller redesign, with improved performance about five times that of
the initial controller.
Thesis Co-supervisor: Chryssostomos Chryssostomidis
Title: Doherty Professor of Ocean Science and Engineering, MIT
Thesis Co-supervisor: Franz S. Hover
Title: Principal Research Engineer, MIT
Thesis Reader: Nicholas M. Patrikalakis
Title: Kawasaki Professor of Engineering, Professor of Ocean and Mechanical Engineering, MIT
3
Biographical Note
The author completed his Bachelor of Science degree in Mechanical Engineering at the University of
Nebraska in May, 2001. He graduated with Highest Distinction, and is also a graduate of the University
of Nebraska Honors Program. His undergraduate thesis work included the design, construction and
testing of several robotic highway safety markers. This research project has gathered momentum at the
University of Nebraska since completion of his undergraduate thesis work. The author is a 2001 recipient
of the Tau Beta Pi Centennial Graduate Fellowship, and a 2001-2003 recipient of a National Defense
Science and Engineering Graduate Fellowship.
4
Acknowledgments
The completion of this work has been made possible by the direction, support, and guidance from many
individuals for whom I owe much gratitude.
At MIT, I would like to first thank my advisor Professor Chryssostomos Chryssostomidis, for
giving me the opportunity to work on such an interesting project. I would like to thank Dr. Franz Hover
for helping me get started, encouraging me along the way, and allowing me to figure things out on my
own. I would like to thank Professor Nicholas Patrikalakis for reading this thesis on top of an already
busy schedule. I would also like to thank Professor Sanjay Sarma and Dr. Vivek Sujan for encouraging
me to explore my options, and Professor John Leonard for introducing me to the MIT Sea Grant Program.
And finally, I would like to thank all of the folks at Sea Grant for welcoming me, supporting me, and, at
times, braving the weather with me, especially Rob, Sam, and Jim. Thank you.
I would like to thank the National Defense Science and Engineering Graduate Fellowship
program for providing me with the means to pursue graduate education. I would also like to thank the
Office of Naval Research (N00014-98-1-0814, N00014-02-C-0202) and the National Science Foundation
(OPP-9910290) ALTEX project, as well as the Sea Grant College Program (NA16RG2255) for their
resources and assistance with this work.
At home, I would like to thank Dr. Shane Farritor for his encouragement and advice during the
past few years. I want to thank my entire family for their confidence, support and love during all the days
of my life. And finally, I would like to thank Lindsey for absolutely everything.
MARK E. RENTSCHLER
Cambridge,Massachusetts
5
6
Contents
Chapter 1...................................................................................................................................................
Introduction..........................................................................................................................................
1.1
Background ...........................................................................................................................
15
15
15
1.2
1.3
1.4
1.5
1.6
M otivation.............................................................................................................................16
Research Platform Caribou...............................................................................................
Sim ulation M odel Developm ent........................................................................................
System Identification........................................................................................................
Controller Developm ent....................................................................................................
16
17
18
18
1.7
Thesis Outline.......................................................................................................................
18
Chapter 2...................................................................................................................................................
The Odyssey III Class AUV Caribou .................................................................................................
2.1
Vehicle Profile......................................................................................................................
19
19
19
2.2
Caribou Specifications ......................................................................................................
2.3
Ring Fin Propeller.................................................................................................................
21
2.4
Coordinate System s ..............................................................................................................
21
2.5
Vehicle W eight and Buoyancy ........................................................................................
22
2.6
Centers of Buoyancy and Gravity ....................................................................................
22
2.7
Inertia Tensor........................................................................................................................
22
Chapter 3...................................................................................................................................................
Governing Equations of Motion..........................................................................................................
20
23
23
3.1
Body-Fram e Coordinate System ......................................................................................
23
3.2
3.3
3.4
Vehicle Kinem atics...............................................................................................................
Vehicle Rigid-Body Dynam ics ........................................................................................
Vehicle M echanics................................................................................................................
23
25
26
3.5
3.6
The Double-Gim baled Duct Thruster ...............................................................................
Duct Angle due to Vehicle Roll........................................................................................
27
28
Chapter 4..................................................................................................................................................
Derivation of Nonlinear Coefficients.............................................................................................
4.1
Hydrostatics..........................................................................................................................
4.2
Hydrodynam ic Damping....................................................................................................
33
33
33
34
7
4.3
Added M ass ..........................................................................................................................
36
4.4
4.5
Body Lift and M oment......................................................................................................
Duct Thruster System ........................................................................................................
39
40
Chapter 5...................................................................................................................................................
Linearized M odel..................................................................................................................................
49
49
5.1
5.2
5.3
Linearizing the Vehicle Equations of M otion....................................................................
Vehicle Linearized Kinem atics........................................................................................
Vehicle Linearized Rigid-Body Dynam ics ........................................................................
49
49
51
5.4
Vehicle Linearized M echanics...........................................................................................
51
5.5
5.6
5.7
Yaw Plane Linearized Coefficients ......................................................................................
Tabulated Linear Yaw Plane Coefficients ............................................................................
Pitch Plane Linearization ...................................................................................................
52
61
67
5.8
Depth M odel .........................................................................................................................
71
Chapter 6...................................................................................................................................................
79
Vehicle Sim ulation................................................................................................................................
79
6.1
6.2
6.3
6.4
Complete Nonlinear Equations of M otion........................................................................
Nonlinear Simulation States and M atrices........................................................................
Linear Sim ulation States and M atrices ..............................................................................
Computer Simulation............................................................................................................
Chapter 7...................................................................................................................................................
Tailcone Testing and M odeling...........................................................................................................
7.1
7.2
7.3
Experim ental Setup...............................................................................................................
Experim ental Results ........................................................................................................
Tailcone Actuator M odel...................................................................................................
Chapter 8...................................................................................................................................................
Initial Controller D esign......................................................................................................................
8.1
8.2
8.3
Heading Controller................................................................................................................
Pitch Controller...................................................................................................................
Depth Controller .................................................................................................................
Chapter 9.................................................................................................................................................
System Identification..........................................................................................................................
9.1
9.2
9.3
9.4
9.5
System Identification Process.............................................................................................
Results from the Field.........................................................................................................
M odel Adjustm ents.............................................................................................................
Stability and Verification of the Improved M odel..............................................................
M odel Comparisons............................................................................................................
Chapter 10...............................................................................................................................................
8
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100
102
105
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105
106
107
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116
119
Controller Redesign ...........................................................................................................................
Root Locus of M odels and Controllers ...............................................................................
10.1
Controller Gains at Different Thrust Levels .......................................................................
10.2
Chapter 11...............................................................................................................................................
119
119
125
127
Closed-Loop Controller Com parisons..............................................................................................
127
11.1
Tailcone Problem s ..............................................................................................................
127
11.2
11.3
Controller Comparisons......................................................................................................
Large Coupled H eading and Depth Changes under Control...............................................
129
133
11.4
Shallow Im Depth M ission.................................................................................................
134
Chapter 12...............................................................................................................................................
Conclusions and Future W ork..........................................................................................................
135
135
Bibliography............................................................................................................................................
137
Appendix A ..............................................................................................................................................
139
Param eter Definitions ........................................................................................................................
Appendix B ..............................................................................................................................................
Root Locus Plots for V arious Models and Controllers ...................................................................
Initial M odel A and Initial Controller A l ...........................................................................
B. 1
Improved M odel B and Initial Controller A l .....................................................................
B.2
Improved M odel B and Improved Controller B..................................................................
B.3
Improved M odel Cl_5 and Initial Controller A l ...............................................................
B.4
Improved Model C1_5 and Heuristically Tuned Controller A2.........................................
B.5
Improved M odel Cl_5 and Improved Controller B............................................................
B.6
Improved M odel C2_0 and Im proved Controller B............................................................
B.7
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9
10
List of Figures and Diagrams
Figure 1.1: The Caribou autonomous underwater vehicle in short configuration .................................
Figure 1.2: System Identification modeling and control procedure .....................................................
Figure 2.1: Profile of Caribou's prescribed shape .................................................................................
Figure 2.2: Ring fin duct thruster arrangement on Caribou..................................................................
Figure 2.3: Inertial and body frame coordinate systems........................................................................
Figure 3.1: Body frame, elevator frame and rudder frame ...................................................................
Figure 3.2: Vehicle roll with respect to the body frame ........................................................................
Figure 3.3: Desired rudder position in the inertial frame .....................................................................
Figure 3.4: Desired elevator position....................................................................................................
Figure 4.1: Caribou's duct fin elevator and rudder system....................................................................
Figure 4.2: Duct coordinate frame.............................................................................................................
Figure 4.3: Caribou's propulsion system ...............................................................................................
Figure 5.1: Ring coordinate frame in yaw plane ...................................................................................
Figure 5.2: Duct coordinate frame in pitch plane .................................................................................
Figure 7.1: Closed loop system .........................................................................................................
Figure 7.2: Experim ental setup ..................................................................................................................
Figure 7.3: Experiment results for rudder with a Is' order model...........................................................
Figure 7.4: Experiment results for elevator with a 1st order model.....................................................
Figure 7.5: Experiment results for rudder with the 2 nd order model......................................................
Figure 7.6: Experiment results for rudder with the 2 nd order model and time delay .............................
Figure 7.7: Pole-Zero plot for the rudder dynamic model......................................................................
Figure 7.8: Experiment results for elevator with the 2nd order model ...................................................
Figure 7.9: Experiment results for elevator with the 2nd order model and time delay...........................
Figure 7.10: Pole-Zero plot for the elevator dynamic model .................................................................
Figure 8.1: H eading control diagram ....................................................................................................
Figure 8.2: Root Locus for heading system without tailcone dynamics...............................................
Figure 8.3: Root Locus plot for the heading system with tailcone dynamics.........................................
Figure 8.4: Pitch control diagram ............................................................................................................
Figure 8.5: Root Locus for pitch system without tailcone dynamics ......................................................
Figure 8.6: Root Locus plot for the pitch system with tailcone dynamics ..............................................
Figure 8.7: D epth and pitch control diagram ...........................................................................................
Figure 8.8: Root Locus plot for the depth-pitch system with tailcone dynamics ....................................
Figure 9.1: Rudder step response mission for commanded angles of -10, 10, -15, and 15 degrees ........
Figure 9.2: Elevator step response mission for commanded angles of 3, and 5 degrees.........................
Figure 9.3: Typical usable rudder step response mission ...................................................................
Figure 9.4: Model adjustment simulation process...................................................................................
Figure 9.5: Typical usable elevator step response mission.................................................................
16
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11
Figure 9.6: Closed-loop straight run with heuristically tuned controller.................................................
Figure 9.7: Yaw model improvements, 10' rudder angle........................................................................
Figure 9.8: Yaw model improvements, 150 rudder angle........................................................................
Figure 9.9: Pitch model improvements, 50 elevator angle.......................................................................
Figure 9.10: Pitch model improvements, 15' elevator angle...................................................................
Figure 10.1: Root Locus plot for the heading system, model Cl_5 and controller B .............................
Figure 10.2: Root Locus plot for the pitch system, model Cl_5 and controller B..................................
Figure 10.3: Root Locus plot for the depth system, model Cl_5 and controller B.................................
Figure 10.4: Field test at 40%, 60% and 80% thrust with controller B...................................................
Figure 11.1: Desired rudder and actual rudder simulation responses......................................................
Figure 11.2: Simulation straight run with rudder correction problem.....................................................
Figure 11.3: Controlled straight run at 4m depth and -80' heading, controller Al.................................
Figure 11.4: Controlled straight run at 3m depth and -80' heading, controller B...................................
Figure 11.5: Controlled heading change from -80' to -60' to -80' at 4m depth, controller Al..............
Figure 11.6: Controlled heading change from -80' to -60' to -80' at 4m depth, controller B................
Figure 11.7: Controlled depth change from 4m to 5m to 4m at -80' heading, controller Al..................
Figure 11.8: Controlled depth change from 4m to 5m to 4m at -80' heading, controller B....................
Figure 11.9: Controlled coupled heading and depth change, controller B ..............................................
Figure 11.10: Controlled shallow run at im depth and -80' heading, controller B...............
Figure B. 1: Root Locus plot for the heading system, model A and controller Al ..................................
Figure B.2: Root Locus plot for the heading system, model A and controller Al ..................................
Figure B.3: Root Locus plot for the heading system, model A and controller Al ..................................
Figure B.4: Root Locus plot for the heading system, model B and controller Al ..................................
Figure B.5: Root Locus plot for the pitch system, model B and controller Al .......................................
Figure B.6: Root Locus plot for the depth system, model B and controller Al ......................................
Figure B.7: Root Locus plot for the heading system, model B and controller B.....................................
Figure B.8: Root Locus plot for the pitch system, model B and controller B .........................................
Figure B.9: Root Locus plot for the depth system, model B and controller B ........................................
Figure B. 10: Root Locus plot for the heading system, model C1_5 and controller A1 ..........................
Figure B. 11: Root Locus plot for the pitch system, model Cl_5 and controller Al ...............................
Figure B. 12: Root Locus plot for the depth system, model Cl_5 and controller Al ..............................
Figure B. 13: Root Locus plot for the pitch system, model C1_5 and controller A2...............................
Figure B. 14: Root Locus plot for the depth system, model Cl 5 and controller A2 ..............................
Figure B.15: Root Locus plot for the heading system, model Cl_5 and controller B.............................
Figure B.16: Root Locus plot for the pitch system, model Cl_5 and controller B .................................
Figure B. 17: Root Locus plot for the depth system, model C1 5 and controller B ................................
Figure B. 18: Root Locus plot for the heading system, model C2_0 and controller B.............................
Figure B. 19: Root Locus plot for the pitch system, model C2_0 and controller B .................................
Figure B.20: Root Locus plot for the depth system, model C2_0 and controller B ................................
12
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List of Tables
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
T able
T able
Table
T able
Table
Table
Table
2.1: Odyssey III class specifications (Bluefin Robotics Corp.).................................................
2.2: Center of mass and volume with respect to body-frame origin...........................................
2.3: Inertial properties with respect to body-frame origin ..........................................................
3.1: Variables used in duct thruster angle analysis......................................................................
4.1: Axial added mass parameter cx [18]......................................................................................
4.2: Short Caribou configuration non-linear force coefficients.................................................
4.3: Short Caribou configuration non-linear moment coefficients .............................................
4.4: Extended (1.05m) Caribou configuration non-linear force coefficients...............................
4.5: Extended (1.05m) Caribou configuration non-linear moment coefficients..........................
5.1: Short Caribou configuration linear force coefficients, U=1.5m/s ........................................
5.2: Short Caribou configuration linear moment coefficients, U=1.5m/s ...................................
5.3: Extended (1.05m) Caribou configuration linear force coefficients, U=1.5m/s ....................
5.4: Extended (1.05m) Caribou configuration linear moment coefficients, U=1.5m/s ...............
5.5: Extended (1.05m) Caribou configuration linear force coefficients, U=1.3m/s ....................
5.6: Extended (1.05m) Caribou configuration linear moment coefficients, U=1.3m/s ...............
5.7: Short Caribou configuration linear force coefficients, U=1.5m/s ........................................
5.8: Short Caribou configuration linear moment coefficients, U=1.5m/s ...................................
5.9: Extended (1.05m) Caribou configuration linear force coefficients, U=1.5m/s ....................
5.10: Extended (1.05m) Caribou configuration linear moment coefficients, U=1.5m/s .............
5.11: Extended (1.05m) Caribou configuration linear force coefficients, U=1.3m/s ...................
5.12: Extended (1.05m) Caribou configuration linear moment coefficients, U=1.3m/s .............
8.1: Initial controller gains based on the initial model ..................................................................
9.1: Y aw initial acceleration rates .................................................................................................
9.2: Pitch initial acceleration rates.................................................................................................
9.3: Original and adjusted models A, B, and C .............................................................................
9.4: Yaw plane model information................................................................................................
9.5: Pitch plane model information ..........................................................................................
9.6: Rate error squared sensitivity to parameter changes ..............................................................
9.7: Stability indicators..................................................................................................................
9.8: T urning rates and radii............................................................................................................
9.9: Pitch plane transfer functions .................................................................................................
9.10: M odel improvem ents............................................................................................................
10.1: Controller gains used in the field..........................................................................................
10.2: Closed loop poles for model A, model B and model C ........................................................
11.1: C losed-loop improvem ents...................................................................................................
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Chapter 1
Introduction
1.1
Background
Developments made in autonomous underwater vehicle (AUV) related technologies have enabled AUVs
to move out of the research laboratory and into commercial, military, and scientific areas.
The commercial use of AUVs centers on the gas and oil industry. The Hugin AUV, developed by
Kongsberg Simrad, is used by C & C Technologies as its workhorse for high-resolution seabed mapping
and imaging for commercial mapping and oil pipeline surveying. The Racal Survey Group Ltd plans to
use Bluefin AUVs for shallow and deep water surveys for the oil and gas industry, as well as cable
inspection, dredging, and alluvial mining.
Military use of AUVs focuses on surveillance, minesweeping and mine countermeasure work.
Lockheed Martin Corp. has developed a Remote Minehunting System (RMS) for surveillance,
minesweeping and reconnaissance. The CETUS TM AUV, whose prototype was developed by the MIT
Sea Grant AUV lab, has also been used by Lockheed Martin for mine countermeasures. The Naval
Postgraduate School in Monteray, CA, works with a focus on clandestine mine countermeasures, using
the ARIES AUV. The Naval Oceanographic Office AUV Program, which started with a large AUV
developed and tested at Draper Labs, has focused efforts on multiplying the effectiveness of
oceanographic survey ships, by allowing survey of larger areas in less time than with the ship alone.
The scientific community continues to make advancements in AUV usage. MBARI uses AUVs
and ROVs for high-resolution mapping of the deep ocean floor, as well as mapping salinity, temperature,
oxygen, fluorescence, backscatter and pH over a full annual cycle. MBARI has also teamed with MIT
and several other partners to develop an AUV for Arctic research. This Atlantic Layer Tracking
Experiment (ALTEX) plans to reach unprecedented endurance and the ability to relay data through ice to
satellite receivers. Quantitative survey of hydrothermal plumes and other near-bottom surveys in rugged
seafloor terrain have been completed by the Woods Hole Oceanographic Institution, using the ABE
vehicle. The Alfred Wegener Institute for Polar and Marine Research has planned a 4-D mapping of a
methane plume above a cold seep at the Norwegian continental margin. The REMUS vehicle, operated at
Woods Hole, as well as Ocean Explorer, operated at Florida Atlantic University, have both demonstrated
the capability to dock autonomously with a base station. These scientific and research pursuits
demonstrate that the usage and capabilities of AUVs are continuing to grow.
15
1.2
Motivation
Continual improvement in all areas of the performance of autonomous underwater vehicles (AUVs) is
needed to enhance the developments made in long-range oceanographic surveys, shallow-water mine
reconnaissance and countermeasures, and procedures in autonomous docking as well as other tasks [1].
One aspect of this performance improvement is maneuvering, which can be achieved by improving the
vehicle's control system. In order to develop a precise control system, a good controls platform is needed
for testing and research purposes. In this case, that platform is a finely tuned dynamic model of the
vehicle. Previous work in dynamic modeling of AUVs includes simulation model verification done
through field tests [2] as well as theoretical and empirical methods in addition to tow tank results [3].
Recent work done on AUV control has included gains obtained using partial model matching methods
[4], as well as sliding mode control based on estimated coefficients [5], in addition to fuzzy sliding mode
control systems [6].
The Odyssey III AUV used for this work is highly maneuverable and has a variable configuration
in length and payload. It can operate at a range of speeds, and it has a novel propulsor and control surface
that consists of a double-gimbaled ring finned duct thruster, which allows for vectored thrusting. These
attributes make the modeling and control of substantial interest and challenge. Odyssey III is also a
leading design in current AUVs, so this analysis is extremely relevant. The dynamic simulation modeling
and control of the Odyssey III AUV through system identification tests is addressed in this work.
Previous work in system identification of AUNV systems has been done using neural network identifiers
[7] and neurofuzzy identification techniques [8]. The work presented here focuses on development of
dynamic models and control systems from first principles. The system identification tests and simulation
process form a forward design process.
1.3
Research Platform Caribou
The autonomous underwater vehicle (AUNV) platform for this research was an Odyssey III class vehicle,
Caribou. This AUV was built by Bluefin Robotics Corporation, and is one of several AUVs at MIT Sea
Grant. Caribou represents the culmination of significant development efforts. Featuring a modular hull
design and the latest evolution of AUV navigation, propulsion and power systems, Caribou provides
significant new autonomous underwater surveying capabilities and flexibility for various scientific needs.
The operating system, which Caribou uses to operate, is called MOOS (Mission Oriented Operating
Suite). This operating system is composed of C++ classes and it uses a publish-and-subscribe protocol.
[9]
Figure 1.1: The Caribou autonomous underwater vehicle in short configuration
Caribou's modular payload section approach allows a single vehicle to support widely different missions.
The short configuration was designed to accommodate the Edgetech(R) FS-AU side-scan sonar and subbottom profiler, while additional payload sections can accommodate numerous other scientific packages,
as well as additional battery power. Caribou is equipped with state-of-the-art sensors, which allow it to
16
collect high-quality data. Caribou will be used in the near future by the AUV Lab at MIT Sea Grant for
archaeological remote sensing [10], multi-static acoustic modeling [11], fisheries resource studies and
development of concurrent mapping and localization techniques [12]. Figure 1.2, shows the process used
to develop an accurate model and control system for the AUV, as described in this thesis.
Initial
Dynamic
Model
lILI
Initial
Conservative
Model Adjustment
Procedure to fit Model
Responses to the
AUV Test Responses
Revised
Controller
Dynamic
Open-loop
System
Identification
Tests
Rvs
Revised
Precision
Controller
Model
Figure 1.2: System Identification modeling and control procedure
1.4
Simulation Model Development
Described in this thesis is the development and verification of a simulation package for the motion of an
autonomous underwater vehicle in six degrees of freedom, or less. This simulation model is used to
adjust the model coefficients so that the simulation results closely match the results found from field tests,
as shown in Figure 1.2. The external forces and moments resulting from the vehicle hydrostatics,
hydrodynamic lift and drag, added mass, and the control input of the vehicle's ducted thruster are all
defined in terms of specific coefficients for this model [28]. This thesis describes the derivation of these
coefficients in detail. Substantial work was performed in adjusting and verifying these coefficients in
order to validate the vehicle's model results with the results from field experiments with the vehicle.
Nonlinear equations were used to determine the vehicle's coefficients, and rigid-body dynamics.
A linearized model was also derived and the simulation has the capability of using either model. While
the vehicle is inherently nonlinear, the nonlinear simulations provide more realistic simulations, but for
control purposes, the linear model provides a sufficient platform, while maintaining less complexity. The
simulation package operates on the numeric integration of the equations of motion, using Matlab
integration software. The simulation output was checked with various data collected from field
experiments. The simulator was shown to not only accurately model the vehicle motion in six degrees of
freedom, but also in the independent three degrees of freedom yaw plane and pitch plane, due to the
decoupled nature of the system.
In order to simplify the challenge of completely modeling an autonomous underwater vehicle, it
was necessary to make several key assumptions on which to base the simulation model development. The
following assumptions about the vehicle and the environment were made:
*
The vehicle is a rigid body of constant mass, i.e. the vehicle mass and mass-distribution does
not change during operation.
17
1.5
"
The vehicle is deeply submerged in a homogenous, unbounded fluid, that is, the vehicle is
located far from the free surface (no surface effects, i.e. no sea waves or vehicle wave-making
loads), walls, and bottom.
*
The vehicle does not experience memory effects, i.e. the simulator neglects the effects of the
vehicle passing through its own wake.
*
The vehicle does not experience underwater turbulence
System Identification
System identification tests were performed in the field to help fit the model response to the responses seen
in the field, by adjusting model coefficients and parameters, as shown in Figure 1.2. This allows a more
precise controller to be designed as the dynamic model is adjusted to more closely model the AUV. The
system identification tests used were simple step response tests. The AUV was roughly controlled to a
desired depth and heading angle using a controller that was designed from the original dynamic model.
At this controlled state, the heading controller was turned off, and the rudder was commanded to a desired
deflection angle for a short response time. Likewise, the pitch plane system was examined by turning the
depth and pitch controllers off and setting the elevator to a prescribed angle for a short response time.
During the rudder step tests, the pitch and depth controllers were left on, and during the elevator step
tests, the heading controller was left on. These simple step response test results were then used to adjust
the dynamic model coefficients and parameters. This approach proved to worked well and resulted in a
much more accurate dynamic model, and a much improved control system.
1.6
Controller Development
The controller development described in this thesis is based heavily on experimentally obtained data that
was used to drive system identification, as outlined in Figure 1.2. Based on an initial linear model of the
AUV, a conservative initial controller was developed using Root Locus methods. This controller was
then used to perform a very compact set of system identification tests designed to identify the vehicle
dynamic response. Using the data from these tests, the dynamic model was improved, and a revised,
precision controller was developed based on this updated model. This precision controller showed
significant improvements when used during closed-loop maneuvers, as compared to the same tests done
with the initial controller as shown in Chapter 11. This work shows that the simple, forward design from
basic principles allowed us to design an accurate control system without spending more than a few hours
in the water.
1.7
Thesis Outline
The discussion of this work begins with a closer looks at the AUV, Caribou, and then moves on to the
equations of motion, the nonlinear coefficients as well as the linearized models and the vehicle
simulation. The testing and modeling of the tailcone system is presented next, followed by the initial
controller design. Then the system identification process is addressed as well as the redesign of the
control system. Finally, the results of field tests using the initial controller and the redesigned controller
are explained as well as the conclusions and future work to be addressed.
18
Chapter 2
The Odyssey III Class AUV Caribou
Prior to calculating the hydrodynamic and hydrostatic coefficients of the vehicle, several characteristics of
the vehicle needed to be determined: vehicle hull profile, mass distribution, and buoyancy properties.
2.1
Vehicle Profile
The hull shape of the Odyssey III vehicle is based on a Series 58, Model 4154 Gertler polynomial [13]
defined below, with a length of 84in (2.13m) and a maximum diameter of 21in (0.53m). The origin of
this polynomial is at the nose of the vehicle, with the x-axis along the horizontal plane of the vehicle. The
y-axis is in the vertical plane, and describes the radial magnitude at the corresponding position on the xaxis as shown in Figure 2.1.
y2 = alx+ a2x 2 + a 3x 3 + a 4 x 4 + a5 x 5 + a6 x 6
where
a,
(2.1)
1.000000
=+
a 2 =+
2.1496653
a 3 = - 17.773496
a 4 =+36.716580
a 5 = - 33.511285
a6 =+11.418548
and
x
=
L
(2.2)
Therefore, x is a non-dimensional ratio of position X with respect to L, where L is the total length of the
streamlined vehicle (i.e. 2.13m) and X is the longitudinal position from the nose of the vehicle.
19
Y
to x
Figure 2.1: Profile of Caribou's prescribed shape
2.2
Caribou Specifications
The Odyssey III Class AUV was designed with a modular approach in mind. This means that additional
payload sections could be added to the mid-section of the vehicle for various mission tasks. As
prescribed by the hull profile in section 2.1, the maximum radius of 0.267m occurs at a point, x=0.40
from the vehicle nose using the Gertler equation. For the nominal hull length of 2.13m (84in), this point
is at 0.884m from the nose. Caribou's short configuration (i.e. without additional payloads), includes a
standard mid-body extension of 0.377m, for an overall streamlined body length of 2.51 lm. With the
addition of the ring fin, the standard body length is 2.58m. Additional cylindrical payload sections can be
added. A typical payload is approximately lm in length and 0.58m in diameter.
These additional payloads, of course, alter not only the physical specifications of the vehicle, but
also the vehicle coefficients, inertial terms and control capabilities. Table 2.1 below lists parameters for
the short configuration, i.e. no additional payloads.
Vehicle Configuration
Short Base Vehicle
Extended Vehicle for Tests
Length:
2.58 m (102 inches)
3.63 m (144 inches)
Maximum Diameter:
0.533 m (21 inches)
Mass in air:
-250 kg (550 lbs)
-350 kg (770 lbs)
Mass in water:
-400 kg (880 lbs)
-650 kg (1400 lbs)
Buoyancy:
-+2.2 kg (+51bs.)
Maximum Depth:
4500 m
Operation Depth:
3000 m
Survey Speed:
3-4 knots (1.5-2 m/s)
Survey Endurance:
20 hours (at 3 knots)
Batteries:
Lithium Polymer
Line Keeping:
2 meters
Altitude Keeping:
1 meter
Payload:
Designed to accommodate various sonar,
oceanographic systems in modular sections.
camera,
Table 2.1: Odyssey III class specifications (Bluefin Robotics Corp.)
20
and
2.3
Ring Fin Propeller
The Caribou vehicle is equipped with a double-gimbaled vector duct thruster. The duct thruster's angle is
limited at ±15degrees in both the yaw and pitch plane, known typically as the rudder and elevator angle.
The propeller is confined to the duct for protection, as well as enhanced flow capabilities. The duct also
acts as a control surface. Thus, the Odyssey III class AUV does not need control fins to control the
vehicle motion because the duct and vector thrusting capability is sufficient to impose directional control.
Figure 2.2 shows the duct thruster arrangement.
Figure 2.2: Ring fin duct thruster arrangement on Caribou
2.4
Coordinate Systems
For this work, the body referenced coordinate frame is located along the symmetric axis of rotation at the
midpoint of the AUV. In this body coordinate system the x-axis is along the symmetric line proceeding
towards the nose of the vehicle. The y-axis extends towards the port side of the AUV, while the z-axis is
directed upwards (Figure 2.3). Body referenced velocity in the x direction, surge, is denoted by u.
Velocity in the body frame y and z directions is v, sway, and w, heave, respectively. Rotational velocity
about the x-axis, roll velocity, is denoted as p. Rotational velocity about the y-axis and z-axis is q, pitch
rate, and r, yaw rate, respectively. External body forces in the x, y and z direction are denoted as X, Y
and Z respectively, while external body moments in the x, y and z direction are K, M and N respectively.
The vehicle's angular orientation is described in the inertial frame of reference with Euler angles, W, 0,
and >, as described in Section 3.2.
Inertial-frame
coordinate system
SURGE: u, X
QLD
HEAVE: w, Z
YAW: r, N
-+5*
SWAY: v, Y
PITCH: q, M
xjf~
Body -frame
coordina te system
Figure 2.3: Inertial and body frame coordinate systems
21
2.5
Vehicle Weight and Buoyancy
The buoyancy force of Caribou depends strictly on the hull's geometry and is simply the weight of the
vehicle's displaced volume of water. For each separate hull configuration, this value stays fixed, because
there are no major changes in the vehicle's hull. However, the mass of the Caribou vehicle can change
between missions, depending on the type of batteries used in the vehicle, the electronics devices used, and
the amount of ballast added. The Caribou vehicle is typically ballasted with -2.2 kg (-5 lbs) of positive
buoyancy, so that in the event of power or computer failure, the AUV would eventually float to the
surface. Integration over the volume of the vehicle, and multiplication by the density of water and
acceleration of gravity determined the vehicle's buoyancy force, FB. The vehicle mass was then assumed
to be -2.2 kg (-22 N) less than the acting mass of the buoyancy. This value changes slightly during each
mission as the vehicle components change, however, the AUV is usually ballasted around +5 lbs buoyant.
2.6
Centers of Buoyancy and Gravity
For each separate hull configuration, the center of buoyancy stays fixed, however, the vehicle center of
mass can vary, as between missions it can be necessary to change the vehicle battery packs and re-ballast
the vehicle, in addition to changing payloads.
For this work, the origin of the body-coordinate frame is the symmetric middle of the vehicle.
The typical parameters for application points of the buoyancy force (center of volume, CV) and vehicle
weight (center of mass, CM) are shown in Table 2.2 for the short vehicle configuration, with respect to
the body-frame origin. The position of the center of volume and center of mass along the x-axis are
usually about the same and the center of volume and mass along the y-axis is generally zero as the AUV
is generally ballasted to achieve zero static pitch and roll. Like the buoyancy force, these parameters can
vary from mission to mission because the vehicle configuration changes.
Parameter
Units
YCM
Value
8.2
0.0
ZCM
-2.1
cm
XCV
8.2
cm
Ycv
0.0
ZCV
0.0
cm
cm
XCM
Cm
cm
Table 2.2: Center of mass and volume with respect to body-frame origin
2.7
Inertia Tensor
The vehicle inertia tensor is defined with respect to the body-frame origin at the vehicle's symmetric midsection. As the products of inertia Iy, Ixz, and IYZ are small compared to the moments of inertia Ixx, Iyy, and
Izz, and assuming that the vehicle has two axial planes of symmetry, we will assume that they are zero.
The inertial values were estimated based on the integration of a homogenous vehicle hull with an
average density of that of water. The estimated values are given in Table 2.3 for the short vehicle
configuration.
Parameter
Value
Units
IXX
12
kg-m 2
Iyy
133
kg-m2
Izz
133
kg-m2
Table 2.3: Inertial properties with respect to body-frame origin
22
Chapter 3
Governing Equations of Motion
In Chapter 3 the kinematics, dynamics, and mechanics of the Odyssey III vehicle and simulation model
are defined and explained. These derivations contribute to the initial dynamic model, which is the first
step in the controller design process as outlined in Figure 1.2.
3.1
Body-Frame Coordinate System
All future calculations in this text will regard the body-frame coordinate system origin to exist at the
vehicle's symmetric midpoint as explained in Section 2.4 and illustrated in Figure 2.3.
3.2
Vehicle Kinematics
The vector transformation from an inertial frame of reference to the body frame is as shown below. The
transformation is developed by using Euler angles (y,O,4), which describe the roll, pitch and yaw position
of the vehicle in inertial space. Roll, pitch and yaw are the rotations of the body about the x, y, and zaxes, respectively. The transformation R(y,O,4) was calculated by cascading the 3 separate angular
transformation matrices. The order of the rotations from the inertial frame of reference was: rotation
about the z-axis with the yaw angle 4, then rotation about the y-axis with the pitch angle 0, and finally
rotation about the x-axis with the roll angle y. The following coordinate transform relates a vector in
body-fixed coordinates with a vector in inertial or earth-fixed coordinates [14].
XB
cOCO
cOs 0
-Cqk0sV~S0cq0
cVgcq5+sy/sOsb0
sXBs#+[Cqs0C#
-syc+cVysOs#
-so]
sy/cO XI =R(y',O,q0)Xj
(3.1)
cyco]
The transformation matrix R(y, 0, 4) is orthonorinal, therefore its inverse is the same as its transpose.
Hence, the following relationship between body referenced vectors and inertial vectors is made.
X = R T(,
0, )XB
(3.2)
23
The motion of the body-referenced frame is described relative to an inertial frame of reference. The
general motion of the vehicle in six degrees of freedom is described by the following twelve states.
x
S=y
position of the body origin in inertial space
(3.3)
Euler angles with respect to inertial space
(3.4)
the body-referenced translation velocities
(3.5)
the body-referenced rotational velocities
(3.6)
Lz]
P
S=q
r
A vector can be used to represent the Euler angles that determine the roll, pitch and yaw position in
inertial space.
FV0-
S=0
(3.7)
For the case of infinitesimal Euler angles, the time rate of change of these Euler angles is equal to the
body-referenced rotation rate, @3, however, for larger Euler angles the physically determined rotation
rate, & , needs to be related to the time rate of change of the Euler angles as follows [14].
1
W = 0
0
0
-so
c
sqic
-sV
cVcO
P
Z=F-'(Z)Z=
(3.8)
q
r.
The derivatives of the body velocity, i;, and rotation rate, >, come from the equations in Section 3.3 for
the external forces and moments [X,Y,Z] and [K,M,N].
following relationships.
. = R (x,
The derivatives for
,#)_
.
and £ come from the
(3.9)
The vector & is the vector containing roll, pitch, and yaw rates: p, q, and r respectively. The inverse of
the transformation matrix F' (F) provides the following relationship [14].
24
E= 0
_0
cV/
sVC c
-s V
6= (E)
(3.10)
Vc
yIc_
Equations 3.9 and 3.10 provide a relationship between measurable quantities that are readily available,
and the variables that need defining. Note that F(E) is not defined for pitch angles at 0= ±900. This is
not a problem, as the vehicle motion does not ordinarily approach that singularity. If this situation were to
occur, then it would become necessary to model the vehicle motion through extreme pitch angles, and the
analysis could then resort to an alternate kinematics representation such as quatermions.
3.3
Vehicle Rigid-Body Dynamics
The locations of the vehicle centers of mass and volume (buoyancy) are defined in terms of the bodyfixed coordinate system as follows. The values for the center of mass and volume are shown in Table 2.2.
XCM
SB
YCM
CV
XB=
C
B
VJ
YCV
(3.11)
O
ZCV
ZCM
The vessel inertial dynamics were derived from the physics of the system and are written as follows in the
body-referenced frame. With xcM, ycM, and ZCM defining the location of the vessel's center of mass with
respect to the body-referenced coordinate frame origin. X, Y and Z are the external body forces applied
in the body-referenced directions of x, y and z respectively, as shown in equation 3.12-3.14 [14].
X =mz +qw-rv+4zCM -- yCM
Y =m[,>+ru- pw+xCMZ =m [+
pv -quy+
zCM
PyCM-x
CM p -(q2
C
(rzCM
+ PXCM
CM
+r2)XCM
2
qyp)r_(p
C
CM
2
+q2 )zCM
The angular dynamics were derived in a similar way. The body-referenced angular equations of motion
also include translation and rotational accelerations as well as velocities.
K = I
+ I,4 + Ix
+ (Iz -I,,)rq+I,(q 2 -r
+ m[ycM
M = I,
+ I,,4+ IJ
pq -Ipr
2)+J
(* + pv - qu) - zCM ( + ru - pw)]
+ (I - Izz)pr +Ix (r2 _ p 2 )+
+ m[zCM (zi + qw - rv) - XGM(v + pv
N
=IJp
(3.15)
+ Iz,4+Iz + (I,, -Ix)pq +I,,(p
2
_
q2)
-
- Iqp
(3.16)
- Ixqr
(3.17)
qu)]
+ Ipr
+ m[xCM (i + ru - pw) - YCM (zi + qw - rv)]
The equations (3.12-3.17) describe how the AUV will respond to external forces and moments acting in
each degree of freedom. These are nonlinear, coupled equations of motion.
25
3.4
Vehicle Mechanics
Given the body-referenced coordinate system, and assuming a symmetric homogeneous body, the angular
inertial properties can be described by a diagonal inertia tensor.
I=
IXX
0
0
0
I,,
0
-0
0
IZZ
(3.18)
Caribou's volume due to the short configuration's prescribed hull shape is 0.394m3 . This corresponds to
the mass of 394kg in fresh water. The wetted surface area, A,, is 3.378m2 . The frontal area, Af, is
1. 14m 2 . The moments of inertia were computed as follows, treating the vehicle as a homogeneous body
with the density of water. For the short configuration with parameters listed, the standard inertial
integration about the center of the body, /2, provided the following with the cross-terms nearly equal to
zero for the nearly symmetric vehicle.
Ixx =12 kg-m
2
(3.19)
2
IY, = IZZ =133 kg -m
For the extended hull AUV, the moments of inertia and added mass change. Adding the cylindrical term
increases the moment of inertia in the x-axis. Adding not only the cylindrical term, but also a term
corresponding to the parallel axis theorem for the extended original configuration increases the moment
of inertia in the y-axis and z-axis.
=
J-2pcrdx
(3.20)
2
p7r
I=
2
1
(3r2 +h2
(3.21)
x 2dx
LI
The result of the combined external forces and moments is described as follows:
X'
Y[
Z
B,bodylift +B,bodydrag
±FB,tailcone + FB,thrust +FB,hydrostatics
+
B,crossterms]
(3.22)
K
M
=
f B,bodylift
+MB,bodydrag +MB,tailcone
+
B,thrust
B,hydrostatics
B,crossterms]
TNd
These forces and moments were determined based on coefficients that are described in Chapter 4.
26
(3.23)
3.5
The Double-Gimbaled Duct Thruster
The tailcone of the Caribou AUV is a double-gimbaled duct thruster. The deflection angle of this duct is
a combination of the yaw rotation, known as the rudder angle, and the pitch rotation, known as the
elevator angle, due to the double-gimbaled arrangement. Rotational transformations were developed
to
relate the position of the duct to the body frame coordinate system. Table 3.1 notes the nomenclature
used in this development.
T_
Thrust
dE=8e
Commanded elevator angle in the body frame
Commanded rudder angle in the body frame
dR=8r
6
Desired elevator angle in the inertial frame
Desired rudder angle in the inertial frame
dEdesired= edesired
dRdesired=8rdesired
W
Roll angle in the body frame
Table 3.1: Variables used in duct thruster angle analysis
The tailcone is double gimbaled; therefore there are several transformations that need to be addressed
(Figure 3.1).
Z2
dR
z'
X(
ZB
dE
dE
Xi
dR
X2
YB
f
dE
y2
Figure 3.1: Body frame, elevator frame and rudder frame
The commanded elevator actuation is done relative to the hull, while the commanded rudder actuation
is
performed relative to the elevator position. Hence, the first rotation is made about the y-axis in
body
coordinates. This rotation is the commanded elevator angle of rotation. This new
coordinate system (1)
can be related to the body coordinate system (B) as follows:
x I = COS(Se)XB
-snie )B
yI =YB
zi = COS(5, )ZB + snt
(3-24
e )B
27
The second rotation is made about the z-axis in the (1) frame of reference. This rotation is the
commanded rudder angle of rotation. This new coordinate system (2) can be related to the coordinate
system in frame (1) as follows:
x2
= cos(, )x, + sin(3, )y
Y2 = cos(0, )yj - sin(5, )xl
(3.25)
Z 2 =Z
The thrust from the propeller is always in the negative x direction of the (2) frame and therefore may be
written as follows.
Frrust -- Tx 2
(3.26)
Combining the results of equations 3.24 and 3.25 with equation 3.26, the following relationship is
established in body coordinates.
=-TPx2
FTrust
-
-
-Tp cos(3r )xI- Tpsin(5r )y(
-Tp COS(Sr )[cos(5e )XB - sin(de )ZB
Tp sin(3r )[YB
= -T, [cos(5r) cos(5,)XB + sinQ5r)YB
Frust =-T,
cos(,r ) cos(e)]
sin(S)
-cos(,. ) sin(G)j
cos(5r ) sin(3 )ZB
FXB
(3.28)
=
FzB _
Equation 3.28 shows that the magnitude of the thrust force in the body frame directions depends upon
both the commanded elevator and rudder angle. From this equation, the imposed rudder deflection in the
body frame is tan-1 (-Fy/-Fx) and the imposed elevator deflection in the body frame is tan-1(Fz/-Fx).
Therefore from equation 3.28, the imposed rudder deflection in the body frame is tan-'[tan(6r)cos(6e)],
while the imposed elevator deflection in the body frame is 6e. Thus, the imposed elevator angle is
decoupled from the commanded rudder angle, but the imposed rudder angle in the body frame is
dependent upon both the commanded rudder and commanded elevator angles. This is because the
commanded rudder deflection is made relative to the elevator frame in the double-gimbaled duct thruster
arrangement.
However, if the commanded elevator angle is small, which it is set to be less than 15 degrees,
then the imposed rudder deflection and imposed elevator deflection are approximately the commanded
angles 6r and 6, respectively. The vehicle is modeled using this assumption, and the heading and
pitch/depth control are assumed to be done in a frame that is not affected by vehicle roll. This assumption
is made because the roll of the vehicle is generally small due to the large righting moment. This is how
Caribou's control system operated in the field for this work.
3.6
Duct Angle due to Vehicle Roll
The development in Section 3.5 does not take into account the roll of the vehicle. Generally this roll
angle is small; however, a more accurate control system may take this roll angle into account. When the
28
AUV rolls, a body referenced rudder deflection is not the same as an inertial referenced rudder deflection.
The same is true for the elevator angle. Therefore, in order to achieve a desired rudder or elevator
deflection in the inertial frame, a transformation that relates the desired inertial deflection with the
commanded deflection is needed. For this derivation we assume that the thrust, Tp, is constant and acts
along the x 2-axis. The vehicle rolls about the x-axis in body coordinates, shown in Figure 3.2. Again, a
transformation can be found that relates this rolling body position to the fixed inertial frame (I).
XB =
I
YB
Icos(y)Y
+ sin(y)Z,
(3.29)
ZB =cos()Z, - sin(V)Y
ZB
XB
Zi
X1I
Y1
Figure 3.2: Vehicle roll with respect to the body frame
Combining the results of 3.27 and 3.29, the following relationship is established that relates the roll angle
as well as the commanded rudder and elevator positions to the thrust vector in inertial space (I).
FT,.ust =
-T, [cos(r ) cos(3, )XB + slf(3r )
B-
cos(6,r ) Sif(35 )ZB]
=-T,{cos(Sr, )cos( 5 )XI + sin(,. )[cos(y)Y + sin(Vy)Z,]- cos(5, ) sin(t3, )[cos( or)Z, - sin(j')Y
-(cos(r)cos(Se))
]}
Fx
FI,,rust =T, -(sin(3r)cos(y))-(cos(3r)sin(S,)sin(y))
= Fr
-(sin()5r)sin(ig))+(cos(5r)sin(,)cos(t))j
LFz
(3.30)
Equation 3.30 shows that the thrust vector, in inertial space is determined by the commanded rudder,
commanded elevator and roll angles, as well as the thrust magnitude, Tp. From this equation, the imposed
rudder deflection in the inertial frame is tan-'(-Fy/-Fx) and the imposed elevator deflection in the inertial
frame is tan-'(Fz/-Fx). Therefore, in order to create a desired rudder deflection in the inertial frame, a
desired force along the inertial X and Y axes is needed as shown in Figure 3.3.
29
(rdesired = tan-'
U
X
dsre
d ii
~
(3.31)
dsie
I --. F
F
irisd
\Y
Figure 3.3: Desired rudder position in the inertial frame
Now using the results from 3.30 in equation 3.31 the following relationship is attained, which reduces to
equation 3.33. Notice that the thrust, Tp drops out from the equations, and thus has no bearing on the
results.
tan(rdesired)
sin(S,) cos(V) + cos(gr) sin(S) sin(yi)
FYdered
=
- F desied
)
cos(r) cos()
(
)tan(Sr )cos(ig)
rCOS()
+ tan(Se) sin(t)
tan(gr desired
-
(3.32)
(3.33)
COS(5e)
Likewise, in order to create a desired elevator deflection in the inertial frame, a desired force along the
inertial X and Z axes is needed as shown in Figure 3.4. Here equation 3.30 and 3.34 are combined to
form equation 3.35 which leads to equation 3.36.
desiredF=
tan(Se desired)
Fzdesied
de
_
dered
( e desired!)
a
30
tan
\
(3.34)
zdesired
- sin(5r)sin(V/) + cos(3,)sin(e)cos(V)
cos(r )cos(Se)
) sin(V) +
- tan(S3
cos(e
+ tan()s
t
) COS
)
(3.35)
(3.36)
z
d Edmired
-FxJ,. -
X
Y
Figure 3.4: Desired elevator position
Now the results of equation 3.33 and 3.36 are combined in matrix form to show the
following
relationship. The desired rudder and elevator angle are the deflection angles of the duct thruster
that are
desired in the inertial frame. These desired angles are achieved by setting the commanded
rudder and
elevator angles according to the relationship shown in equation 3.37, which depends also
on the roll
angle.
tan(Fedesired
a
-cos
tan(6,sied,)
-sin
_sin]V
cos
tan(e)
_ tan(r)/cos(8,)
(337)
Equation 3.37 is now arranged so that the commanded rudder and elevator angles are a function
of the roll
angle and the desired rudder and elevator inertial frame deflection angles.
tan(Se)
~cos Vf - sin
tan(Sr)/cos(,)j =
sin
cos y
V
tan(e desired
tan(gr desired
(3.38)
Equation 3.38 leads to equation 3.39 by inverting the roll transformation matrix.
tan(
Ltan(5,I/cos(e
cos y
--
sin V ~tan(e desired)
sinV
cosvt'
tan(grdesid
Finally, reduction from matrix form to equation form provides these results for the commanded
deflection
angles based on the desired deflection angles in the inertial frame and the roll angle
of the AUV.
Equation 3.40 represents the algorithm that could be used by the AUV's control system,
where the desired
rudder and elevator angles in the inertial frame are determined by the controller gains.
Then based on
these desired inertial angles and the vehicle's roll angle, the commanded elevator
and rudder angles are
determined. This would provide a more precise approach to the control of the AUV.
However, the roll
angle and the deflection angles are generally small, which show that under this assumption,
equation 3.40
becomes 3.41.
45e = tan [cos
tan(edesired
)+
sin y tan(r desired)]
4, = tan- {cos te - sin y tan(Sedesired )+ cosV/ tan(3rdesired
(3.40)
The one subtlety shown in equation 3.40 is that the desired rudder angle is related
to the commanded
elevator angle. As was noted earlier, the elevator moves relative to the hull, while
the rudder moves
relative to the elevator. The cosine term has been carried through the algebra and
shows up precisely as
31
expected in the commanded rudder equation. If the roll angle is assumed to be small, then equation 3.40
becomes 3.41, where the commanded elevator angle is same as the desired elevator angle in the inertial
frame and the commanded rudder angle depends on the commanded elevator angle and the desired rudder
angle in the inertial frame. This is the same as the derivation made in Section 3.5.
e
e desired
r= tan
[tan(rdesired )cos
5e
(3.41)
As was mentioned in Section 3.5, if the commanded elevator angle is small, as it is, then the commanded
elevator and rudder angle will be the same as the desired elevator and rudder angles in the inertial frame,
again assuming that the roll angle of the vehicle is small.
32
Chapter 4
Derivation of Nonlinear Coefficients
In this chapter, the coefficients defining the external forces and moments on the vehicle are defined. The
vehicle and fluid parameters necessary for calculating each coefficient are included in the section
describing the coefficient. This nonlinear model leads to the linearized initial model that is the first step
in the controller design process as shown in Figure 1.2.
4.1
Hydrostatics
The vehicle experiences hydrostatic forces and moments as a result of the combined effects of the vehicle
weight and buoyancy. The mass and weight of the vehicle is m and W, respectively, where W = mg . The
vehicle buoyancy is expressed as B = pVg, where p is the density of the surrounding fluid and V the
total volume of water displaced by the vehicle. The buoyancy force acts at the center of volume and the
weight act at the center of mass. These positions are expressed in terms of body-frame coordinates.
XCM
fkCM
=YM]
B$
__
CM
CV
X C YCVVI
(4.1)
B
ZCM _CV
Since the volumetric center and mass center do not coincide in the vehicle, there will be a righting
moment that is induced whenever the vehicle is rotated from its stable position. The position vector for
the vehicle's center of mass is X7c
defined relative to the body frame origin. The position vector for the
vehicle's volumetric center is X7v, again in body frame coordinates described from the body origin.
Using the previously defined rotational transformation matrix, R(, 0, ) from equation 3.1, the weight
and buoyancy forces can be transformed from the inertial frame to the body frame.
.i0
FB,,eight
= R(yf,0,#)
0
=
0
FBbyny = R(Vf, 0,#0) 0
-W
(4.2)
Bync
33
This transformation is used because the forces acting at the volumetric center and the center of mass
occur only in the z direction of the inertial frame. The induced moment was calculated as follows for the
opposing weight and buoyancy forces.
'I
B,weight
B
B,weight
B,buoyancy
BV
B,buoyancy
(4.3)
Note that the hydrostatics moment is stabilizing in pitch and roll, meaning that the hydrostatic moment
opposes deflections in those angular directions. The following, based on the above calculations, are
presented in coefficient form for consistency.
X hydrostafics
=(W
-
B)sinO
Yhydrostatics = (-W + B)sinqcosO
4.2
(-W + B)cos y cosO
Zhydrostafics
=
Khydrostatics
=(-Wycm +Bycv)cos jcos0+(WzcM - Bzcv)sin ycos
Mhydrostatics
=(WxCM - Bx c )cos
Nhydrostatics
=(-WxCM + Bx cv )sin
(
q cos 0 + (WzCM - Bzcy )sin0
y cosO + (-Wycm + Bycy )sin0
Hydrodynamic Damping
It is well known that the damping of an underwater vehicle moving at a high speed in six degrees of
freedom is coupled and highly non-linear. In order to simplify modeling the vehicle, we will make the
following assumptions:
*
We will neglect linear and angular coupled terms. We will assume that terms such as Y,, and
M,. are relatively small. Calculating these terms is beyond the scope of this work.
"
We will assume the vehicle is top-bottom (xy-plane) and port-starboard(xz-plane) symmetric. We
will ignore the vehicle asymmetry caused by the sonar transducer or any antenna. This allows us
to neglect such drag-induced moments as K
and Mulu1 .
*
We will neglect damping terms greater than second-order. This allows us to drop such higherorder terms as Y .
The principal components of hydrodynamic damping are skin friction due to boundary layers, which are
partially laminar and partially turbulent, and damping due to vortex shedding. Non-dimensional analysis
helps us predict the type of flow around the vehicle. Reynolds number represents the ratio of inertial to
viscous forces, and is given by the equation:
Re =-
Ul
V
(4.5)
where U is the vehicle operating speed, which is typically 1.5 m/s (-3 knots); I the characteristic length,
which for Caribou (without the middle extension) is 2.58 m; and v, the fluid kinematic viscosity, which
for seawater at 15'C, has a value of 1.190e-6 m2 /s.
34
This yields a Reynolds number of 1.3e6, which for a body with a smooth surface falls in the
transition zone between laminar and turbulent flow. However, the hull of Caribou is broken up by a
number of pockets, bulges and singularities, which more than likely trip the flow around the vehicle into
the turbulent regime. We can use this information to estimate the drag coefficient of the vehicle.
Note that viscous drag always opposes vehicle motion. In order to result in proper sign, it is
necessary in all equations for drag to consider v IV , as opposed to v 2 .
4.2.1
Axial Drag
Vehicle axial drag can be expressed by the following empirical relationship:
(4.6)
pAfCDuIU
DBody
The equation yields the following non-linear axial drag coefficient:
X
(4.7)
:
where p is the density of the surrounding fluid, Af is the vehicle frontal area (1.14m 2 ), and A, is the
rectangular planform area (1d). The axial drag coefficient of the vehicle was estimated from the Hydat
manual [15] as follows where the leading coefficient comes from Schoenherr's line [16].
CD
4.2.2
=0.0040 ' r1+60(d)
. Af
1
(4.8)
= 0.023
+0.0025-
d=
Crossflow Drag
The method used for estimating the hull drag is analogous to strip theory, the method used to calculate the
hull added mass: the total hull drag is approximated as the sum of the drags on the twodimensional
cylindrical vehicle cross-sections.
Slender body theory is a reasonably accurate method for calculating added mass, but for viscous
terms it can be off by as much as 100%. This method does, however, allow us to include all of the terms
in the equations of motion. In conducting the vehicle simulation, we will attempt to correct any errors in
the crossflow drag terms through comparison with physically correct behavior of the vehicle.
Crossflow drag coefficients are calculated as follows with Sin = 0.05m 2, the cross-sectional area of the
ring fin. Also, xfin is the distance from the body frame origin to the ring fin (xfin~ 1/2 in), and Cd - 1.1
from Hoerner for a cylinder [17]. Cdf is the coefficient for the ring fin's effect. Cdf = 0.1 + 0.7t, where t is
the fin's taper ratio. The model uses t = 2/3 for a ringfin. Equations 4.9-4.12 are:
Y
=Z
pC,
-
I
[
L/21
J2R(x)dx+CfSfi ]=-P[C(1.075)+C
S
L /2
M
=-NVIVI =
2
FC 2xR(x)dx -L /2
xfin CdfSfi
= p[C,(0.060) -
XfinCdf Sfin
35
Y,.j =-Zqlq
=
-
P[ C
Mqq =Nrrj= -_P
Cc
x
LL
22
x
R(x)dx -xfi |
I xjIR(x)dx
+x
=-- p[Cd,(0.054) - x, I
CdxS
dfSfi
I
X
4.2.3
fpnfiC(1.139)+x]
f
I CdfSfif]
=-L12
XfiI Cdf Sfi]
I
Rolling Drag
We will assume that without any control planes on the vehicle hull, the rolling drag is nearly negligible.
K
4.2.4
~10
(4.13)
Combining all damping effects
In vector form, the AUV damping forces and moments are as follows:
rK~pIpIP
X1 JJu u
Yvivvv l + ] ,rjrj
B,bodydrag
=
MB,bodydrag
ZWw w + Z qlql ql
4.3
M*WwIw|+ Mqjqjq\q
(4.14)
N[ vv + NrrrrJ
Added Mass
Added mass is a measure of the mass of the moving water when the vehicle accelerates. Ideal fluid forces
and moments can be expressed by the equations as follows. Due to body top-bottom and port-starboard
symmetry, the vehicle added mass matrix reduces to:
0
0
0
0
M2 2
0
0
0
M2 6
0
M33
0
M35
0
M4 4
A
A
M53
0
Mn5 5
0
0
0
0
M6 6 _
M11
0
0
0
J
0
0I
0
0
M62
(4.15)
which is equivalent to:
-X
0
0
-Y,
0
0
0o
-Z.
0
0
0
0
0
0
36
0
0
0
0-z. 0
0w
-Z.
- KP
0
-M,
0
-M4
0
0
0
0
-Y
0
0
0
-N
(4.16)
4.3.1
Axial Added Mass
To estimate axial added mass, we approximate the vehicle hull shape by an ellipsoid for which the major
axis is half the vehicle streamlined body length 1, and the minor axis half the vehicle diameter d. Blevins
gives the following analytical formula for the axial added mass of an ellipsoid [18],
X d = -m
= -a - p)c 2
3
2
(4.17)
where p is the density of the surrounding fluid (seawater), and cc is an empirical parameter measured by
Blevins and determined by the ratio of the vehicle length to diameter as shown in Table 4.1.
l/d
0.01
0.1
0.2
0.4
0.6
0.8
1.0
1.5
2.0
2.5
3.0
5.0
7.0
10.0
U
6.148
3.008
1.428
0.9078
0.6514
0.5000
0.3038
0.2100
0.1563
0.1220
0.05912
0.03585
0.02071
Table 4.1: Axial added mass parameter c [18]
For the short vehicle configuration, l/d = (2.511)/(0.53m) = 4.71. Therefore c = 0.068, and in,= 26.2.
For the long vehicle (1.05m extension) that was used in the field tests, l/d = 6.68 and cc = 0.040.
4.3.2
Crossflow Added Mass
Vehicle added mass was calculating using strip theory on both cylindrical and cruciform hull cross
sections. From Newman [19], the added mass per unit length of a single cylindrical slice is given as:
ma (x) = gcp R 2(x)
(4.18)
where p is the density if the surrounding fluid, and R(x) the hull radius as a function of axial position as
defined in equation 2.1. Integrating equation 4.18 over the length of the vehicle, we arrive at the
following equations for cross flow added mass:
37
nose
ma(x)dx
Y =-m 22 =tail
Z.
M33=M22=Y,
nose
M\ =-m
f
53
x ma(x)dx
tail
(4.19)
N= -iM62 =m 53 =-Mk
Yi =-M 2 6 =-m 62 =N,
Z4 =-M
35
-- iM 53 =M\
nose
M =-m
x 2 ma(x) dx
55 = -
tail
N. =-M
4.3.3
66
=-M
55
=Mq
Rolling Added Mass
As for the rolling added mass, we will assume that in absence of any fins, the rolling added mass term is
negligible.
KP = -44
~ 0
(4.20)
4.3.4 Added Mass Cross-terms
The remaining cross-terms result from added mass coupling, and are listed below:
Xwq = Ze
Xvr =- Y,
Xqq =Z
Yur = - X
wp
Zuq = XVp
u
Muwa =-(ZA
-Xa)
Nuva=-(XC- Y)
=- Z
Ypq =-
=Y
Z
Mvp =-Yi
NM =Za
X, =- Y
Z4
(4.21)
(4.22)
(4.23)
Yi
Mp=(K,-Nj)
Muq =Z
(4.24)
Np=(K,-Ma)
N_ =-Y.
(4.25)
The added mass cross-terms Muwa and Nuva are known as the Munk Moment [20] and relate to the pure
moment experienced by a body at an angle of attack in ideal, non-viscous flow. These effects are
included in Section 4.4.2.
4.3.5 Combining all Added Mass Cross-terms
X,,.vr + Xwqwq + Xrrrr+ X qq
F
ur±Y wp+Y ~qp
WP
B,crossterms =ur
Zuquq+
38
qp
ZvPvp + Zrrp
(4.26)
0
+ Mvp
M B,crossterms
Nurur + Npqpq + N wpj
4.4
1Muq+M,,pr
(4.27)
Body Lift and Moment
Vehicle body lift results from the vehicle moving through the water at an angle of attack, causing flow
separation and a subsequent drop in pressure along the aft, upper section of the vehicle hull. This pressure
drop is modeled as a point force applied at the center of pressure. As this center of pressure does not line
up with the origin of the vehicle-fixed coordinate system, this force also leads to a pitching moment about
the origin. Hoerner's estimate of body lift, which appeared reliable with a lack of specific experimental
measurements on the AUV, was used [21]. Hoerner's estimate includes all effects, including the Munk
moment effects as mentioned in Section 4.3.4.
4.4.1
Body Lift Force
The hydrodynamic lift is based on the body's angle of attack with respect to the flow. As defined below,
6w is the pitch angle of attack, and 8v is the yaw angle of attack. The variables u, v, and w are the AUV
velocity in the x, y, and z body directions, respectively.
35 = tan1
5,=tan1CIY~-
(4.28)
For this analysis, positive 8w implies that the nose is pitched down with respect to oncoming flow,
resulting in a positive pitch rotation. Positive 8v implies that the nose is rotated towards the port side with
respect to oncoming flow, resulting in a positive yaw rotation. These determinations were made in order
to remain consistent with control surface orientation in which positive control surface actuation means
rotation of the control surface in the positive pitch or yaw axes accordingly. Body lift is as follows:
(4.29)
LBody = - 1PAPCLu'
2
where A, is the rectangular planform area (1d). The coefficient for lift, CL, is described below. With I=
2.511m and d = 0.53m (I is the streamlined body length of AUV, and d is diameter), l/d = 4.71, d/1= 0.21.
The coefficient relationship below holds from d/l ~ 0.25 to 0.10. The following are in degree form [21]:
CLpitch
=0.0023.3w
CL
yaw
=0.0023 - 5v
(4.30)
When converted to a radian system the following relationships are developed.
CLpitch
=0.131 ow
CLyaw =0. 131.v
(4.31)
Therefore, the following coefficients for body lift are developed where A, is the rectangular planform
area, (dl), and for the short configuration A, = 1.34m2 .
39
1
u, =2
Zuw= --
-"pACL,yaw
1
2
pCL,pitch
(4.32)
4.4.2 Body Lift Moment
For bodies of revolution, these body lift forces generally do not intersect the desired reference frame
origin. Therefore, there is an implied moment associated with this offset force. Here this moment is
developed, which combines the Munk Moment and the Lifting Moment [21].
MBody =1pAplCM U2
(4.33)
2
From Hoerner, the moment coefficient for dl = 0.25, is 0.0022 which results in the following in radian
form.
0.126 -,5
CMpitch
CMyaw
=0.
126.3w
(4.34)
However, for d/l = 0.10, Cm= 0.0020 from Hoerner. Hence, for the smaller d/l ratio:
CMpitch
=
0.115
- (5V
CMyaw =0.115 -Sw
(4.35)
By interpolating between the values in equation 4.34 and 4.35 for the ratio d/l = 0.21 the following
relationship was established for the short vehicle configuration.
CMpitch
= 0.1 2 3 - 5v
CMyaw =0. 123
5W
(4.36)
Therefore, the following coefficients for body moments due to the body's hydrodynamics are:
N 1,,'
4.4.3
1
2 PAPCL,yaw
1
M uw =-AC
P p 'CL,pitch
(4.37)
Combining all Lifting Force and Moment Effects
In vector form, the AUV lift force and lift moments are as follows:
0
B,bodylift
MB,bodylift =Muw W
-uK]
Zuw_
4.5
0
(4.38)
_Nuv
Duct Thruster System
The Odyssey III duct thruster allows for the vector thrusting capability. Ring fin ducts have shown
significantly extended underwater flight performance in range and endurance [22]. The duct itself also
40
acts like a control surface and will be related to as a rudder and elevator although the dynamics are
different.
4.5.1
Duct Hydrodynamics
Figure 4.1: Caribou's duct fin elevator and rudder system
From the Principles of Naval Architecture the lift and drag of the duct, neglecting transient flow effects,
are given by [16]:
1
L= IfpAfU2CL (a)
D-pA
1
U 2 CD(a)
(4.39)
where,
CL (a)L=
C
aD
8a )+Dca
CD(a) =
Cd
+
C2
LARe
(4.40)
The minimum section drag coefficient, Cdo, is 0.010 while the standard Oswald efficiency factor,
e, is
0.90. The cross flow fin drag coefficient, CDc, is 0.81, while the aspect ratio and duct area are listed
as:
AR = diameter = 3.458
chord
Aeff = diam. x chord = 0.0498
(4.41)
McEwen and Streitlien [23] estimated aCL / aa for this ring wing shaped duct control surface on results
from [24], which referred to previous work done [25], and [26]. They found from Milewski's thesis [24]
that dCl/da = 3.4855 for AR=1.25, and that from the DSRV report [24] that 8 CL /aa = 5.1566 for AR =
4.3716. Between these two points they interpolated with a function of the form seen in van
Dykes
equation, and fit the ring wing results by allowing multiplicative parameters, r and s, in the definition
of
lift coefficient and aspect ratio as follows:
41
OC
aa )a=O
1+
2
-+
= 5.1
16
2
sAR
,z(sAR)2
(4.42)
log(1+re-91 'sAR)
where s and r are described as:
s = 4.305, 2)zr = 5.927
(4.43)
This development leads to a lifting force, developed using a linearized equation 4.40 for the lift
coefficient.
L
=
1
U 2 CL (a) = 131U 2 a
-pA
(4.44)
Hoerner offers an alternate approach to development of a lifting force equation. This involves an
alternative lift coefficient, aspect ratio, and effective area, where c = chord (0.12), d = diameter (0.40) and
i=0.80. [21]
4d
AR = -- = 4.4055
13.687
CL
(2 a=
)ao
Mf
A
1 +rA
2a-ir MzR
ef
ff=
1
dc = 0.0754
(4.45)
2
which leads to a lifting force of:
1
L =- pA U 2 CL (a)
2
=
143U 2a
(4.46)
ef
The lifting force relationship between equation 4.46 and 4.44 is quite similar, which bolsters our
confidence in the validity of the form developed by McEwen and Streitlien [23]. The velocity of the ring
fin duct in the body frame is a combination of the body translation velocity and rotational velocity:
vRo
_Bo
-B
Lw
(4.47)
qxyR}
v
- BoRo
[r]
LZR
The transformation matrix relating the ring frame of reference to the body frame is,
Cos SR
TR/B
where
42
6
E
sin5R
0
0 CosE
8R
cos
0
0
-sin
R
0
0
cosS
ir-S
COScE
is the elevator angle and 6 R is the rudder angle.
=siE
Rcos
i5R CsE
E
E
-
sinR
Cos
sE
R
cos
R sin
REl
sinCRsinS
CosS(E
E
r2
Ro
Figure 4.2: Duct coordinate frame
Using this transformation matrix, the duct velocity in the ring frame and the angle of attack (Figure 4.2)
are computed:
Ro x
= TRIB
RIBRo
i;VR
a=arcsin
Ro
(4.48)
IR
I
To form the transformation matrix that relates the ring frame to the forcing frame, the following vectors
are established:
I
R
2
=
" {2
x
=
T3 = T X2
(4.49)
0
Combining these vectors yields the transformation matrix for relating the ring frame to the forcing frame:
-D
TR I D
V[1
2
3
]
0
FLD
(4.50)
- LCombining the two transformation matrices and the forcing vector, the body referenced force from the
duct control surfaces is calculated:
B,tailcone
BIR
-
RILD FLD
4.51)
This force is created at the duct location, thus inducing a moment about the body frame origin. This
corresponding moment force is:
AlB,tailcone
_
BoRo
B'tailcone
(4.52)
43
4.5.2
Propeller Thrust
The thrust from the propeller can be directed at various orientations depending on elevator angle, 6 E, and
rudder angle, 8R, because the vehicle has a vectored duct thruster. Therefore, depending on the orientation
of the duct frame, as explained in Section 3.5, the applied force in the body-referenced frame is:
COS 3 E Cos 3R
(4.53)
sin R
B,thrust
-CosR
E
Esin
Since this force may not be coincident with the body axis, a moment is induced in the pitch and yaw
directions due to the thrust. The vector, FBT, is the vector from the origin of the body frame to the thrust
referenced frame origin, and xp < 0. Figure 4.3, shows the duct thruster propulsion system.
[P
Sl B,thrust
= (B
)i
(4.54)
Bthrust
0
Figure 4.3: Caribou's propulsion system
The propeller thrust can be described as follows, where Up is the speed of the water seen at the propeller,
n, is the propeller speed, D is the propeller diameter, and p is the water density [20].
T, =Krpn 2 D
KT =
U
1
-
2J
nD
(4.55)
s1
The thrust coefficient, KT is approximated as a linear function of the advance ratio, J. For Caribou,
and
so
that
the
p1
and
P2,
were
determined
P2 were modeled as 0.035 and 0.3, respectively. These coefficients,
model simulation would reach a steady state speed of 1.5m/s at 60% thrust, with a rise-time of 10
44
seconds, based on the modeled thrust and drag coefficients. The only controlled variable was, np, the
propeller speed which is modeled as the maximum speed multiplied by the desired percent thrust. The
maximum propeller speed, n,, is 500 rpm and the propeller diameter, D, is 0.375m.
np =p
The speed of the
fraction factor, w,
linear coefficients
linear coefficients
U = U(i - w)
%thrust
(4.56)
water seen at the propeller is usually less than the speed of the vehicle. The wake
is taken as 0.1, a typical value in ships and submarines. Table 4.2 and 4.3 list the nonfor the Caribou AUV in the short configuration and Table 4.4 and 4.5 list the nonfor Caribou with the 1.05m extension used in the field tests.
Parameter
Value
Units
Description
Xul ul
-13.8
kg / m
Axial Drag
X
-26.2
kg
Added Mass
Xwq
395
kg / rad
Added Mass Cross-term
2
Added Mass Cross-term
Xqq
32
kg-m / rad
XVr
-395
Xrr
32
kg / rad
kg-m / rad 2
Added Mass Cross-term
Added Mass Cross-term
-624
kg / m
Cross flow Drag
11r
2
-8
kg-m / rad
Ye
-395
kg
Added Mass
Yi
kg m / rad
Yur
32.4
-26
Added Mass
Added Mass Cross-term
Ywp
-395
kg / rad
kg / rad
Cross flow Drag
Added Mass Cross-term
2
Ypq
-32
kg-m / rad
Yuv
-90
kg / m
Body Lift Force
-624
kg / m
Cross flow Drag
Zqj qj
8
kg-m / rad 2
Cross flow Drag
zw
z.
-395
kg
Added Mass
-32.4
kg-m / rad
Added Mass
Zuq
26
kg / rad
Added Mass Cross-term
zvp
395
kg / rad
Added Mass Cross-term
Zrp
-32
kg-m / rad 2
Added Mass Cross-term
zuw
-90
kg / m
Body Lift Force
Added Mass Cross-term
Table 4.2: Short Caribou configuration non-linear force coefficients
45
Parameter
Value
Units
Kp~P
0.0
2
kg-m / rad
K
0.0
kg-m 2 / rad
Added Mass
MWI W1
16
kg
Cross flow Drag
M q qj
-229
kg-m2 / rad 2
Cross flow Drag
Me
M4
-32.4
kg-m
Added Mass
-127
2
kg-m / rad
Added Mass
Muq
32
kg m / rad
Added Mass Cross Term
MVP
32
kg-m / rad
Description
2
Mrp
-127
kg-m / rad
MUW
213
kg
NV
1 vj
-16
kg
2
2
Rolling Resistance
Added Mass Cross Term
2
Added Mass Cross Term
Body Lift Moment
Cross flow Drag
2
Nr1 rl
-229
kg-m / rad
Cross flow Drag
N
32.4
kg m
Added Mass
2
Nj
-127
kgm / rad
Added Mass
Nur
32
kg-m / rad
Added Mass Cross Term
NWP
32
kg-m / rad
2
Npq
127
kg-m / rad
Nuv
-213
kg
Added Mass Cross Term
2
Added Mass Cross Term
Body Lift Moment
Table 4.3: Short Caribou configuration non-linear moment coefficients
46
Parameter
Value
Units
Description
Xu ul
-15.0
kg/rm
Axial Drag
X
-26.2
kg
Added Mass
Xwq
632
kg / rad
Added Mass Cross-term
2
Added Mass Cross-term
Xqq
57
kg m / rad
Xvr
Xrr
-632
57
kg / rad
kg-m / rad 2
Added Mass Cross-term
Added Mass Cross-term
Yv VI
-944
kg / m
Cross flow Drag
2
Cross flow Drag
-34
kg m / rad
Y
Yi
-632
57.1
kg
kg-m / rad
Added Mass
Added Mass
Yur
-26
kg / rad
Added Mass Cross-term
YWP
-632
kg / rad
YrI
r
Ypq
-57
kg-m / rad
YUV
Z
-128
-944
kg / m
kg / m
Added Mass Cross-term
2
Added Mass Cross-term
Body Lift Force
Cross flow Drag
2
Zq1 qj
34
kg-m / rad
Ze
-632
kg
Added Mass
Zq
-57.1
kg-m / rad
Added Mass
Zuq
26
kg / rad
Added Mass Cross-term
Z
632
kg / rad
Zrp
-57
kg-m / rad
ZuW
-128
kg / m
Cross flow Drag
Added Mass Cross-term
2
Added Mass Cross-term
Body Lift Force
Table 4.4: Extended (1.05m) Caribou configuration non-linear force coefficients
47
Parameter
Value
Units
Description
0.0
kg-m2 / rad2
Rolling Resistance
Kb
0.0
2
kg-m / rad
Added Mass
MW
1 WI
33
kg
Cross flow Drag
K
2
M q qj
-1027
kg-m / rad
Me
-57.1
kg m
Added Mass
M4
-458
kg-m2 / rad
Added Mass
Muq
57
kg-m / rad
Added Mass Cross Term
MVP
57
kg-m / rad
Mrp
-458
2
kg-m / rad
M.
413
kg
Body Lift Moment
NVI VI
-33
kg
Cross flow Drag
Nr rl
-1027
kg-m2 / rad2
Cross flow Drag
N
57.1
kg-m
Added Mass
Nr
-458
2
kg-m / rad
Added Mass
Nur
57
kg-m / rad
Added Mass Cross Term
NWP
57
kg-m / rad
458
2
kg-m / rad
-413
kg
Npq
IV
2
Cross flow Drag
Added Mass Cross Term
2
Added Mass Cross Term
Added Mass Cross Term
2
Added Mass Cross Term
Body Lift Moment
Table 4.5: Extended (1.05m) Caribou configuration non-linear moment coefficients
48
Chapter 5
Linearized Model
In this chapter, the yaw plane linearized model is developed. The linearization includes the equations of
motion, as well as force and moment coefficients. The pitch plane linearized model is also developed
briefly since the derivation is straightforward and similar to the yaw plane development. This linearized
model are treated as the initial model that leads to the initial controller as shown in Figure 1.2.
5.1
Linearizing the Vehicle Equations of Motion
The set of equations that govern the motion of the vehicle were described in detail in Chapter 3, and the
governing dynamics were described in Chapter 4. Here we describe the linearization of the vehicle
kinematics, rigid-body dynamics, and mechanics. In the yaw plane model, there is only 3 degrees of
freedom: surge, sway, and yaw. Thus, heave, roll, and pitch are set to zero.
5.2
Vehicle Linearized Kinematics
The vector transformation from an inertial frame of reference to the body frame was shown in Chapter 3
(equations 3.1). The transformation was developed by using Euler angles (y, 0, 4), which describe the
roll, pitch and yaw position of the vehicle in inertial space. In the linearized yaw plane there is only one
angle of interest, . The roll and pitch angles are set to zero for the yaw plane model. The transformation
R(y, 0, ) was calculated by cascading the 3 separate angular transformation matrices. Here there is only
one transformation matrix. The following coordinate transform relates vectors between body-fixed and
inertial or earth-fixed coordinates.
XB
Fcos
-sin#0
o
cos#
i = RO
c,
(5.1)
The transformation matrix R4() is orthonormal, therefore its inverse is the same as its transpose. Hence,
the following relationship between body referenced vectors and inertial vectors was made.
X 1 = R|T(#)XB
(5.2)
49
The motion of the body-referenced frame is described relative to an inertial frame of reference. The
general motion of the vehicle in three degrees of freedom is described by the following six states.
=
=
=
=
[]
[#]
0
[i
r]
The derivatives of the body velocity i
position of the body origin in inertial space
(5.3)
Euler angle yaw with respect to inertial space
(5.4)
the body-referenced translation velocities
(5.5)
the body-referenced rotational velocity yaw rate
(5.6)
and rotation rate Co, come from the equations in Section 3.3 for
the external forces and moments [X,Y,Z] and [K,M,N].
following relationships.
The derivatives for X and t
.X = RO (#)i
come from the
(5.7)
Therefore using equation 5.1 in equation 5.7,
i= cos #u - sin #v
= sin Ou + cos v
#=
(5.8)
r
If the heading is kept around a constant value with only small perturbations, then that point can be
thought of as a zero reference operating angle, thus the angles will be small. Therefore, using Taylor
Series approximations, we can neglect quadratic and higher order terms so that cos(4) = 1, and sin(f) =
cos#O= 1-
sin # =
#2
+ higher order terms
2!2(5.9)
-
3!
+ higher order terms
Using linearization methods, we can define the body-referenced rates as follows:
u=U+iW
v =V
(5.10)
r =rF
where, U is the steady state value of u, and i is the perturbation about this steady state value. Likewise,
the steady state value of v and r is zero, but the perturbation is iY and i respectively. Therefore, using
5.10 and 5.9 in 5.8 the results are as follows.
50
x=U+W-#ir
f= #U
5.3
+vr
(5.11)
Vehicle Linearized Rigid-Body Dynamics
The locations of the vehicle center of mass and volume (buoyancy) are defined in terms of the body-fixed
coordinate system, for the yaw plane, as follows:
XCM
[CM
.YCM _
CV
CV
(5.12)
CV_
The vessel inertial dynamics are written in the body-referenced frame, with xCM, and ycM defining the
location of the vessel's center of mass with respect to the body-referenced coordinate frame origin. X,
and Y are the external body forces applied in the body-referenced directions of x, and y respectively. The
external forces in the Z direction are set to zero since this is a yaw plane model. Likewise, p, q and w are
zero. Therefore, the equations from Section 3.3 are simplified as follows (5.13, 5.14):
X = m
Y
-
v - yc
-r
2xCM
= m[1 + ru + rxCM - r 2yCM
In a similar way, the external moments K and M are set to zero, hence the following simplification.
N=I,,* + m[xCm(i)± ru) -yCm (t -rv)]
(5.15)
Now, by substituting earlier linearization results (5.10), and dropping higher order terms, as well as
assuming that ycm is equal to zero, the following linearized equations of motion are found.
X=mi
Y = ml$ +
N = IZZ
(5.16)
rU + rxCM
+m[ xCM(7
(5.17)
U)]
(5.18)
Notice that the equation of motion in the surge direction is decoupled from the sway and yaw motion.
From here on, in this chapter, u = W, v = iY, and r =
unless otherwise stated.
',
5.4
Vehicle Linearized Mechanics
The moment of inertia in the yaw plane remains the same. However, only the moment of inertia about the
z-axis, Izz, is considered because only rotation about that axis can occur. The results of the combined
external forces and moment in the linearized yaw plane is:
_Y_=[Bbodyift+B;bodydrag
± FB,tailcone + FB,thrust + FB,crossterms
(5.19)
51
IN] = [I
B,bodylift +
B,tailcone
B,bodydrag
B,thrust
B,crossterms
(5.20)
These forces and moments are determined based on coefficients that are described in Chapter 4. Notice
that the weight and buoyancy forces are neglected in this yaw plane model. Additionally, there is no
transformation needed to relate the vectored thrust in the inertial frame to the body frame due to the
vehicle roll, since the roll is set to zero. Hence, the desired inertial frame thrust vector is the same as the
commanded body frame thrust vector.
5.5
Yaw Plane Linearized Coefficients
5.5.1
Hydrostatics
The vehicle experiences hydrostatic forces and moments as a result of the combined effects of the vehicle
weight and buoyancy. The mass of the vehicle is m and the vehicle weight is W = mg. The vehicle
buoyancy is expressed as B = pVg , where p is the density of the surrounding fluid and V the total
volume displaced by the vehicle. The buoyancy force acts at the center of volume and the weight act at
the center of mass. These positions were expressed in terms of body-frame coordinates.
CM _ LYV
FCM
_YCM
_YCV
(5.21)
Since the volumetric center and mass center do not coincide in the vehicle, there will be a buoyancy
moment that is induced whenever the vehicle is rotated from its stable position. However, in the yaw
plane model, there are no forces in the z-direction, thus the weight and buoyancy forces are zero,
regardless of yaw orientation. The following relationships were developed in Chapter 4.
X hydrostatics =(W
Yhydrostatics
-
B)sin0
(5.22)
=(-W + B)sin V/cosO
Nhydrostatics =(-WxCM + Bxcv )sinq/cos0 + (-Wycm + Bycv)sinO
However, roll, xV, and pitch, 0, are set to zero, therefore:
hydrostatics
0
hydrostatics
0
N hydrostatics
0
5.5.2 Hydrodynamic Damping
Axial Drag
As developed in Chapter 4, the axial drag term was:
52
(5.23)
XU" =-
1
pAfCD
2 fD
(5.24)
In order to linearize this equation we take ulul as follows, and eliminate quadratic terms that are not
constants.
u u|=(U+WiUi+W)= (U 2 +2U +22)=(U
2
+2Us)
(5.25)
Now the axial drag term can be represented as a constant drag term and a linearized drag term as follows.
-
X
1
2
2
-PAf CD U 2 =x UUU
(5.26)
Xu = -pAfCDU= Xl, 2U
Crossflow Drag
The cross flow drag was developed in detail in Chapter 4. However, the coefficients are all nonlinear
terms that tend to zero when linearized. Therefore, the linear coefficients Yvd, Nvd, Yrd, and Nrd, from
Yvivi, Nvivi, Yrirl, and NrIrI, respectively, are set to zero, and any adjustments to these coefficients are made
through system identification later. In vector form, the AUV linearized damping forces and moments are
as follows:
FBbodydrag =[ I
IY
5.5.3
IU
'B,bodydrag =
dv + Y rI
[Ndv + Nrdr]
(5.27)
Added Mass
The added mass matrix is reduced to:
mil
0
L0
m
2
m62
0
~X6
01
0
m 26
0
(5.28)
=0
-
_0
m6
2
- N]
Axial Added Mass
The axial added mass remains unchanged. As was shown in Chapter 4:
Xf, =-m
4
1
=-a - pr
3
(l
dY2
I)2 2
(5.29)
Crossflow Added Mass
In a similar fashion as was shown in Chapter 4, the cross flow added mass coefficients are:
53
nose
m (x)dx
tail
nose
N =-m 6 2 =
xm
a
tail
Y =-M 2 6 =-m 6 2
(x)dx
(5.30)
=N'
nose
N =m 6 6 =
x
2
m
(x)dx
tail
Added Mass Cross-terms
The remaining cross-terms result from added mass coupling, and are listed below. However, all higher
order terms have been eliminated, as well as those associated with roll, pitch and heave rates.
Y, =-X
NU,=-Y
(5.31)
Now we linearize these coefficients, using the relationships from 5.10.
Ym
Nra =Y
= mUU
(5.32)
Nra =N~U
(5.33)
Representation of these added mass forces and moment cross-terms in vector form are as follows:
-.
FBcrossterms
0
Lyrar]
(5.34)
[Nrr]
(5.35)
AB,crossterms =
5.5.4 Body Lift and Moment
For the yaw plane model, the body lift and moment are as were developed in Chapter 4. However, here
we use the relationship from equation 5.10 as follows.
1
Yv =YU
N-N
=2PACLaU
2
1
NI = N.U = 2 pACLa xtU
54
(5.36)
(5.37)
5.5.5
Duct Thruster System
Duct Hydrodynamics
The coefficients for the lift and drag are now linearized from section 4.5.1 as follows:
CL (a)
C-a)
a = 5.la
CD W=Cdo =O0010
(5.38)
1
-- pAeffU 2 CD(a)
2ef
Xtailcone
00498
Aef
The velocity of the ring fin in the body frame is a combination of the body translation velocity and
rotational velocity. Here p, q, w, and zR are set to zero.
U+W
-Ro
_
Bo
B
0
U+W-yRr
xR
BoRo
(5.39)
+xRr
0
r
0
0
The transformation matrix relating the ring frame of reference to the body frame is simply in the yaw
plane since 6 Eis zero.
Cos
(R
sin tR0
5R
TRIB
CosR
0
0
(5.40)
0
1
u sin(di)
-(v-x r)cos(dk)
-
a
Figure 5.1: Ring coordinate frame in yaw plane
Using this transformation matrix, the ring velocity in the ring frame and the angle of attack (Figure 5.1)
are computed:
R
=RIB
x
= sin- 1
a = arcsin
-RoL
(5.41)
Ro
=TsRIR
usin t
V+Xr)cosS
2
+v + xr)
(5.42)
2
55
Now, if equation 5.42 is linearized about u, with (v + xRr) and aR assumed to be small, then equation 5.42
becomes:
a =3 R
(5.43)
xr
u
u
Using the relationships for lift and drag on the tailcone, the following forces were established for the three
distinct contributions to the angle of attack.
1
YRtailcone
(aCL
=
R
p
"AC
I P(a
NrRtailconeR
2
Y=
2g R
eff
>eff U
2
R R
Ba
(aC_
1
- pI
2
1
aa
LIAU
Nvt
(aC_
a
1
Aef UXR(5.45)
2 P(a
rt
L
2
a
AUXR
)
(5.46)
2
Py aa)
f
These three sets of control surface forces and moment in the body referenced frame were combined as
follows for this linearized yaw plane model:
FB,tailcone =
]
"'""(5.47)
oXtaiicone
MB,tailcone = [ANfRtailcone
+
Nvt + N,,
(54_
(5.48)
PropellerThrust
The thrust from the propeller can be directed variably depending on only rudder angle, 6R, in the yaw
frame. Therefore, depending on the orientation of the propeller frame, the thrust in the body-referenced
frame is:
Xthrust
Y5thrust
=T, cos
R
TP
R
T
= T sin
FKthrust1
B,thrust ~
s
L_ = th ru s t (5R
56
(5.49)
R
(5.50)
(5.51)
Since this force may not be coincident with the body axis, a moment is induced in the pitch and yaw
-T
directions due to the thrust. The vector rB , is the vector from the origin of the body frame to the thrust
referenced frame origin, and xp < 0.
(5.52)
F =
NORthrust3
R
(FB )XB,thrust
MB,thrust
P
snR
(5.53)
PTPSR
(5.54)
=[NRthrust5R
The propeller thrust can be described as follows, where Up is the speed of the water seen at the propeller,
nP is the propeller speed, D is the propeller diameter, and p is the water density, here equation 4.55 is
modified so that thrust operates linearly around a preset operating point, n.
U
T = KTpnp;YD
4
KT =,1 - /2J
J=
U
(5.55)
The propeller speed, n,, and water speed, U, seen at the propeller are the same as in Chapter 4.
n, = nu
%thrust
U
= U(1 - w)
(5.56)
Now, by combining the drag, lift, and tailcone forces and moments, only four coefficients remain in
equation 5.57 and 5.58.
Y=Y +Y +Y
t NY
N, YN d + v I
N, =Nd,+ N,,., + Nt
Y,.=Y,. +Y
r
rd
( 5 .5 7 )
+Y
r
r(5.58)
Nr =Nd+Nra+N58
The tailcone and thrust forces and moments are both dependent on the angle of the duct, thus they can be
combined as follows:
Y1R
&tailcone +
SRthrust
N 6±(5.59)
=
~~taico
N,5R = N(tilCone + NRthrust
Now, by combining all linearized coefficients and the equations of motion, equation 5.16, 5.17, and 5.18,
the surge, sway and yaw equations of motions are described.
m
= Xhydrostatics + X + XUu + Xid + Xtaiicone + Xthrust
m[1+rU +xcm ]=Yhydosttics + Yv + Yr +YY +
I
+ m[xcM (f) + rU)] = Nhydrostatics + N~v + Nr
YR RR
+ Nl:'+ N
(5.60)
+ N,1?
R
Equation 5.60 can be rearranged as follows:
57
(m
-
XU )
(m -Y
-
X
- Xtaicone
v+(mXCM - Yr
(mxCM - N,
V+(MU -Y )r =Yhydrostatics +R
-
- N,)
+(I
= Xhydrostatics + Xthrust
(5.61)
R
R
- Nv +(mxcmU - N, r = Nhydrostatics + NR
These equations of motion can be represented in matrix form as follows, where the third state is simply an
integrator.
(mxCM - Yr
(m-Y )
NJ
(mxCM
(I - NJ
0
Y
0)
0
(mU -Y)
0
0 i +
N,
(mxCMU - N,
1- -O
0
-1
V
Y
0 r
0
-
R
N
R
0
R
(5.62)
i + B = Ci
Finally, this matrix form can be rearranged into standard state space form as follows:
. = A + _Bi
(5.63)
For feedback systems, the vector y represents the state feedback dependent upon matrix C.
Cj
=
(5.64)
This system (5.63 and (5.64) can now be represented as a transfer function in the s-domain as follows:
GA UV ya(s) =
C(sL - A)- B
(5.65)
In order to determine the transfer function of the yaw plane system, matrix C is set as follows, to allow
the heading angle, p, to be passed back as a function of the rudder angle, 8R-
C = [0
0
1]
(5.66)
Equation 5.62 can be represented as follows to allow for a simplified development of the transfer
function.
Al
A2
0
A3
A4
0
0
0
1
9
~BI
+
B2
0
V
C
B3 B4
0
r
=C2
- 1 0-(P
-0-
0
5R
(5.67)
Using equation 5.63, 5.64 and 5.65, the following transfer function for the yaw plane system was
developed.
FA
(0_
R
-A4C1+
S 3 +S
2
4A
B 3 -A
2
A,A4 -
58
JA BAICG+ABAA(4A
A+
2
SLS A11A-A
As - A2I
2 A3 j A,,C
3
B2
A2 A3
+
3
4
_
A
_
B
qA
1 AC 2
3
-- AA 2 A
4 _A2A)
-4BBA
BA,
C]
4
I
+, FB
APB4 + A2B AB2- A4B2AB, - A2B4]ABj
_ ll
A A4 - A 2 A3 2
I
(5.68)
Equation 5.68 can be minimized by noticing that A 2 and A 3 are very small compared to rest of the
parameters. A 2 and A 3 are small because the vehicle's hull is nearly a symmetric ellipsoid. Therefore,
5.68 becomes:
BIC2-BC 2
AIA4
-(5.69)
SC2 +
___
-A4.
S3+S 2[
S
4
+
Al
3
'2
4
+[1
A, A4
A4
Substituting the values from 5.62 back into 5.69, results in equation 5.70. At s=oo, equation 5.70 becomes
the instantaneous initial value of the transfer function, and at s=0, equation 5.70 becomes the steady-state
value.
N
s (I - N.
(
45R
S3
-
Yv
S2
-YNR + NYR
+
]I,,
QzI
s=o,
(
-(5.70)
MCM U - N,.]
I(m - Y)
- N)
Y)
- -
- Y (mxCM U - N,. +(m U - Y,. N
(M - YV XIZZ - N
- N,
-- = 0
s=0,
5R
-L=
00
(5.71)
t5R
Equation 5.71 shows that any step response in the rudder does not have an immediate impact on the
heading angle. Also, for a rudder step response the steady state heading goes to infinity as time goes to
infinity which is to be expected since there is no restoring force in the yaw plane. By implementing a
differentiator, as shown in equation 5.72, equation 5.70 becomes equation 5.73, and 5.75.
r
r
S
S
- = P=
r
(5.72)
Therefore, for yaw rate as a function of rudder deflection angle the transfer function is:
N
r_
=
05R
s =),
s[I2
2
s +s++
r
R
-N,)
- Y,R + NY
()Iz
- N,)j]
N -
~~-v
-- = 0
(5R
(5.73)
(mCMU -N,.)]+-YV(mxCMU -N,.)+(MU -Y,.Nv
(m-Y
(
_,-N
(-Y
_(,-N
s = 0,
r
1R
=
-Y N 8 + NMY3
R
R
(5.74)
,v mCM U - N,. + (MU -Y,.)Nv
As shown in equation 5.74, for a constant rudder angle, the system reaches a steady turning rate. The yaw
acceleration as a function of deflection angle is shown in equation 5.75.
59
2
05R
s2
+S
1s
N
I [-Nj
S(m -Y,
U - N)
-CM
I
(m -Y,_)
s =0 ,
=
R
-_N,_
)
5R
zz- N)
Y
I, - N)
- YNOR +N
-Y(mxCMU - N, +(mU -Y)N
_
+_
(M -Y, XI.-
s = 0,
(,
NJ
-= 0
(5.76)
R
Notice that instantaneous yaw acceleration as a function of rudder deflection angle is simply the moment
forcing term, Nd, divided by the mass term, (Izz-Nradt). These results will be further discussed in Chapter
9, as related to the responses seen from field data.
60
5.6
Tabulated Linear Yaw Plane Coefficients
Setting steady state surge velocity to: U=1.5m/s
Parameter
Value
X =xuu U 2
-31
= X, 2U
X
Units
kg-m /
Description
S2
Constant Axial Drag
-41
kg / s
Axial Drag
X
-26.2
kg
Added Mass
Xthrust
Tp
kg-m / S2
Axial Thrust
Xtailcone
-0.6
kg-m /
S2
Xhydrostatics
0
kg-m/
S2
Hydrostatic Force
V
-75
kg / s
Cross flow Drag
= YrrCR
-0.5
kg-m / rad 2
Cross flow Drag
-395
32.4
kg
kg-m / rad
Added Mass
-39
kg-m / rad-s
Added Mass Cross-term
-135
kg / s
Body Lift Force
Yd = Y,
Yrd
Axial Drag from Tailcone
Ye
yi)
yr
ra =u,.U
0
hydrostatics
yt
kg-m/
S2
Added Mass
Hydrostatic Force
-196
kg / s
Tailcone Lift Force
247
kg-m / rad-s
Tailcone Lift Force
ySRtailcone
294
kg-m / S2 -rad
Tailcone Lift Force
yRthrust
Tp
kg-m / s 2 -rad
Thrust from Tailcone Angle
-406
kg / s
Lift Force from Translation
208
kg-m /rad-s
Lift Force from Rotation
YVt
Y, = Yd +
Yr = Yrd
y
-5
5 ay
£ Rtailcone
+ Y
ra
rt
s
6Rthrust
294+Tp
2
kg-m / s -rad
Lift Force from Tailcone Angle
Table 5.1: Short Caribou configuration linear force coefficients, U=1.5m/s
61
Setting steady state surge velocity to: U=1.5m/s
Parameter
Nvd =N
Nrd
Value
c
=Nr CR
Units
Description
-1.9
kgm /s
Cross flow Drag
-14
kgm
Cross flow Drag
2
/s
32.4
-127
kg-m
kg-m 2 / rad
Added Mass
Added Mass
48
kg-m2 / rad-s
Added Mass Cross Term
-320
kg-m/s
Body Lift Moment
Nhydrostatics
0
kg.m 2/s 2
Hydrostatic Moment
Nt
N,,
247
kg-m/s
Tailcone Lift Moment
-311
kg-m 2 / rad-s
NRftailcone
-371
2
kg-m / rad-S
N(&thrust
-1.26*Tp
kg-m 2 / rad-s2
N =Nvd + Nvi + Nv,
Nr =Nrd + Nra + N
-75
kg-m/s
Tailcone Lift Moment
Thrust Moment from Tailcone
Angle
Lift Moment from Translation
-277
kg-m2 / rad-s
Lift Moment from Rotation
-371-1. 26*Tp
kg'm 2 / rads 2
Lift Moment from Tailcone Angle
Ng
Nj
Nra =
Nur U
NVI = N'U
N8 R = N
3
Rtailcone +
N
4thrust
Tailcone Lift Moment
Table 5.2: Short Caribou configuration linear moment coefficients, U=1.5m/s
62
Setting steady state surge velocity to: U=1.5m/s
Parameter
Value
Units
Description
X=Xuju U
X= XJ 2U
-34
kg-m / s 2
Constant Axial Drag
-45
kg / s
Axial Drag
X
-26.2
kg
Added Mass
Xthrust
Tp
Xtailcone
-0.6
kg-m / s2
Axial Drag from Tailcone
Xhydrostatics
0
kg-m / S2
Hydrostatic Force
2
kg-m/
S2
Axial Thrust
vCv
-113
kg / s
Cross flow Drag
rjrjCR
-2.0
kg'm / rad2
Cross flow Drag
kg
kg-m / rad
Added Mass
Yi
-632
57.1
Yra = Y..Ur
-39
kg-m / rad-s
Added Mass Cross-term
YVI
-192
kg / s
Body Lift Force
0
kg-m / S2
Hydrostatic Force
Yvt
-196
kg / s
Tailcone Lift Force
Y,
349
kgm / rad s
Tailcone Lift Force
YRtailcone
294
kg-m / S2-rad
Tailcone Lift Force
Yvd = Y
rd =
Ye
=
YUU
hydrostatics
2
Added Mass
YRthrust
Tp
kg-m / S -rad
Thrust from Tailcone Angle
Yv = Yvd + Yv + Yvt
-501
kg / s
Lift Force from Translation
Yr = Yrd
308
kg-m / rad-s
Lift Force from Rotation
YR =Y
5
ra
Rtailcone +
rt
5Rthrust
294+T
kg-m/
2
s -rad
Lift Force from Tailcone Angle
Table 5.3: Extended (1.05m) Caribou configuration linear force coefficients, U=1.5m/s
63
Setting steady state surge velocity to: U=1.5m/s
Parameter
Value
Units
Description
Nvd= Nvl cv
-4.0
kg-m /s
Cross flow Drag
N,d =NrlrlCR
-62
kg-m2 /s
Cross flow Drag
Nr
57.1
Added Mass
Nj
-458
kg-m
kg m2 / rad
86
kg-m / rad-s
Added Mass Cross Term
N,, = N U
-620
kg-m/s
Body Lift Moment
Nhydrostatics
0
kg'm 2/s
Nt
349
kg-m/s
2
Hydrostatic Moment
Tailcone Lift Moment
2
N,,
-621
kg m / rad-s
NRtailcone
-524
kg-m2 / rad s 2
-1.78*Tp
N, = Nd + Nv, + Nv,
Nr =Nrd +Nra + N
-275
=
Added Mass
Nra = NurU
N(thrust
N5R
2
N5Rtalcone + N(Rthrust
22
kg-m2 /ra-s
2
kg-m/s
2
Tailcone Lift Moment
Tailcone Lift Moment
Thrust Moment from Tailcone
Angle
Lift Moment from Translation
-597
kg-m / rad-s
Lift Moment from Rotation
-524-1.78*T,
kg-m 2 / rad-s2
Lift Moment from Tailcone Angle
Table 5.4: Extended (1.05m) Caribou configuration linear moment coefficients, U=1.5m/s
64
Setting steady state surge velocity to: U=1.3m/s
Parameter
Value
Units
Description
-26
kg-m / S2
Constant Axial Drag
XU =Xuj 2U
-39
kg / s
Axial Drag
X
-26.2
kg
Added Mass
Xthrust
Tp
kg-m / S2
Axial Thrust
Xtaicone
-0.4
kg-m / S2
Axial Drag from Tailcone
Xhydrostatics
0
kg-m / S2
Hydrostatic Force
-113
kg /s
2
X = X1 l U
#
Yd =
V
Cross flow Drag
2
-2.0
kg-m / rad
y
Yi
-632
57.1
kg
kg-m / rad
Added Mass
Ya = Yu,U
-34
kg-m / rad-s
Added Mass Cross-term
-166
kg /s
Body Lift Force
Yhydrostatics
0
kg-m /
y,,
-170
kg / s
Tailcone Lift Force
Yt
302
kg-m / rad-s
Tailcone Lift Force
YRtailcone
221
kg-m / s-rad
Tailcone Lift Force
YRthruSt
T
kgm / s2 rad
Thrust from Tailcone Angle
,-r CR
Yd =
Y,1 =
YU
S2
Cross flow Drag
Added Mass
Hydrostatic Force
Yv
Y
+ Yvi +
Yt
-449
kg / s
Lift Force from Translation
Yr
rd
+ Yra + Y,,
266
kg-m / rad-s
Lift Force from Rotation
kg-m / s 2-rad
Lift Force from Tailcone Angle
Y5R
SRailcone
JRthrust
221+
Table 5.5: Extended (1.05m) Caribou configuration linear force coefficients, U=1.3m/s
65
Setting steady state surge velocity to: U=1.3m/s
Value
Units
Description
-4.0
kg-m / s
Cross flow Drag
Nrd =NrIICR
-62
kg-m2 / s
Cross flow Drag
Nr
Nj
57.1
-458
kg-m
kg-m2 / rad
Added Mass
Added Mass
Nra = NurU
75
kg-m2 / rad-s
Added Mass Cross Term
-537
kg-m/s
Body Lift Moment
Parameter
NVI = NU
Nhydrostatics
0
kg-m /s
Hydrostatic Moment
Nt
302
kg-m/s
Tailcone Lift Moment
N,,
-538
2
kgm
2
2
/ rad-s
Tailcone Lift Moment
2
2
N(&talcone
-393
kg-m / rad'S
Tailcone Lift Moment
NLRthrust
-1 .78*Tp
kgm 2 / rad'S2
Nv =Nvd+ N,1 + Nv,
Nr =Nrd + Nra + Nrt
-239
kg-m/s
Thrust Moment from Tailcone
Angle
Lift Moment from Translation
-525
kg-m2 / rad-s
Lift Moment from Rotation
N3 R = NRetailcone + NiRthrust
-393-1 .78*Tp
kgm 2 / rad'S2
Lift Moment from Tailcone Angle
Table 5.6: Extended (1.05m) Caribou configuration linear moment coefficients, U=1.3m/s
66
5.7
Pitch Plane Linearization
In the pitch plane model, there are only 3 degrees of freedom: surge, heave, and pitch. Thus, sway, roll,
and yaw are set to zero. The model derivation shown here is very brief since the same straightforward
approach is used here as was used for the yaw plane derivation.
XB
-. cos6 -sin0 -lsI
=RO(0)X,
[sin9 cos6]
f
= R| (O)XB
(5.77)
(5.78)
The motion of the body-referenced frame is described relative to an inertial frame of reference. The
general motion of the vehicle in three degrees of freedom is described by the following six states.
[]
=
position of the body origin in inertial space
(5.79)
Euler angle yaw with respect to inertial space
(5.80)
the body-referenced translation velocities
(5.81)
the body-referenced rotational velocity pitch rate
(5.82)
Z
F0=
[]
c = [q]
The equations of motion from Section 3.3 are linearized as follows:
X = M(
Z =m
M = I,
+ qzCM)
(5.83)
(5.84)
- qU + xCM
+ m[zCM i
-
XCM (ii' - qU)]
(5.85)
The locations of the vehicle centers of mass and volume (buoyancy) are defined in terms of the bodyfixed coordinate system as follows:
CM _XCM
CV
[V(5.86)
zCVi
LZCM _
Since the volumetric center and mass center do not coincide in the vehicle, there will be a buoyancy
moment that is induced whenever the vehicle is rotated from its stable position. By linearizing equation
4.4 and setting roll, y, and yaw, p, to zero the hydrostatic equations become:
Xhydrostatics =(W - B)sinO ~ (W - B)O
Zhydrostatics =(-W + B)cos0 ~ (-W + B)
Mhydrostatics = (WxCM -
(5.87)
Bxcv)cos 0 + (Wzcm - Bzcv)sin0
~ (WxCM - Bxcy) + (Wzcm - Bzcv)O
67
The linearized cross flow drag in the pitch plane was developed in the same manner as the cross flow drag
in the yaw plane in Section 5.5.2. Therefore, the linear coefficients Zwd, Mwd, Zqd, and Mqd, from ZWJWi,
Miww, Zqgqi, and Mqq, respectively, are set to zero, and any adjustments are made through system
identification later.
The added mass matrix is reduced to:
Mil
0
0
0
M3
0
m53
3
]
-- Xl
0
0
0
Z
- Z4
0
-MfV
-
=
M5
(5.88)
The remaining added mass cross-terms result from added mass coupling, and are listed below. However,
all higher order terms have been eliminated, as well as those associated with roll, pitch and heave rates.
Muq =Z4
Zuq = X,
(5.89)
Now we linearize these coefficients, using the pitch model relationships similar to those shown in
equation 5.10 for the linearized yaw plane model.
Zqa = ZuqU
(5.90)
Mqa = MuqU
(5.91)
The body lift and moment developed in Chapter 4 are linearized here.
Z 1 = ZUU =
MW
1 = MUWU =
pACLaU
(5.92)
PA, CL a XlU
(5.93)
2
1
2
The coefficients for the duct fin lift and drag are the same for the pitch plane as they were for the yaw
plane. The velocity of the ring fin in the body frame is a combination of the body translation velocity and
rotational velocity. Here p, r, v, and yR are set to zero. The angle of attack in the pitch plane is computed
in the same way as the angle was computed in the yaw plane. Figure 5.2 shows this relationship for the
pitch plane.
Ro
a = arcsin
R
_usin3E+(-xRq)COs3E
sin
(5.94)
Now, if equation 5.94 is linearized about u, with (w - xR q) and 6E assumed to be small, then equation 5.94
becomes:
a = tg + U
68
U
(5.95)
(w-xRq)cos(dE)
W-XRq
dE
dE
c
usin(dE)
Figure 5.2: Duct coordinate frame in pitch plane
Using the relationships for lift and drag on the tailcone, the following forces were established for the three
distinct contributions to the angle of attack
SEtailcone
E
2
Oa
UP(
2gE
5AefU
L
(5.96)
M2Etailcon
1
Z,,=
fI
U
t 2Pyaa)
= 2, p
Z
LAC
2
M
U
D)eff
E
eff
8L )e-1
E R
(8CL
A
Z,=-P
AeUxR
R
a Ae~R
a
p(IOL
M,=
2
Ba
(5.97)
2Af
~
x
The thrust from the propeller can be directed variably depending on only elevator angle, 8E, in the pitch
frame. Therefore, depending on the orientation of the propeller frame, the thrust in the body-referenced
frame is:
Xthst =TP
-Tp
6Ethrust E
MEthrustSE
cosSE
(TBT)xB'thrust
P
(5.98)
inp E
p psin E
E-T
PTP8E
(5.99)
(5.100)
Now, by combining the drag, lift, and tailcone forces and moments, only four coefficients remain.
Z
=W Zd
+ ZW + Z,
Zq = Zd + Zwt + Zq ,1
MW = Mw + MWI + M,,
Mq = Mqd + Mqa + M5
The tailcone and thrust forces and moments are both dependent on the angle of the duct, thus they can be
combined as follows:
ZE
SEtailcone + ZEthrust
M5E
= M gEtailcone + MgEthrust
(5.102)
69
Now, by combining all linearized coefficients and the equations of motion, equation 5.83, 5.84, and 5.85,
the surge, heave and pitch equations of motions are described.
(mX X
(m-Z)
-(mxCM
- B)+ Xth(
- Xi.cone = (W
- X
-Xu
-(mxcM
+Z,)4-Zvw-(mU+Z)q =(-W+B)+ZS3E E(5.103)
+ M)iv+(I, - M 4 q - Mw+(mxcMU - M, )q = (WxCM -BxCV)
+(WCM
- Bzcv)O
+ ME 3 E
These equations of motion can be represented in matrix form as follows (5.104), assuming that W-B, and
xcM=xcv, and zcv=O. This is completely parallel to the yaw plane case, except that here the separation of
the center of mass and center of volume create a restoring moment in the pitch plane.
(M-Z
(MXCM
-(MXCM
)
+
00
(I,, - Mq
0
0
1-
0
-ZW
-(mU+Z,)
+ -M
0
wl
(mxCMU-M,)
-WzcM
q
-1
0
0
0
L
E
E
E
0
Ai + _Bi = Ci
Finally, this matrix form can be rearranged into standard state space form as follows:
. = Ai + Bi
(5.105)
For feedback systems, the vector y represents the state feedback dependent upon matrix C.
(5.106)
= C-i
This system (5.105 and 5.106) can now be represented as a transfer function in the s-domain as follows:
GA UV
pitch(s)
= C(sI - A)- B
(5.107)
In order to determine the transfer function of the yaw plane system, matrix C is set as follows, to allow
the pitch angle, 0, to be passed back as a function of the elevator angle, 6 E-
C = [0
0
1]
(5.108)
Equation 5.104 can be represented as follows to allow for a simplified development of the transfer
function.
Al
A2
0
A
A4
0
_0
0
1- &
W*
+
BI
B2
0
W
B
B4
B
q
0
- 1
0 -LO
C,
=C2
gE
(5.109)
-0
Using equation 5.105, 5.106 and 5.107, the following transfer function for the pitch plane system was
developed and approximated as was done for the yaw plane system by noticing that A 2 and A3 are very
small compared to rest of the parameters, due to the vehicle's hull matching closely with that of a
symmetric ellipsoid.
70
BIC 2 + -B 3 C1
L- A 4- I
A1A 4
B++S
B 4 -B 2 B+ AB
A1 A4
LA
1 A4
rC2
_
(E
S 3 +S2
+
IB
(5.110)
5
LA1A4I
Substituting the values from 5.104 back into 5.110, results in equation 5.111. At s=oo, equation 5.111
becomes the instantaneous initial value of the transfer function, and at s=O, equation 5.111 becomes the
steady-state value.
0
_
S2U
+E
_(M - Z,)
s =oo,
- M+)S +
SE
-(5 Z,E
s (IYY--_MO
(M -Z
F- Z+(mxc
31 U -
M)
5E
-0
s = 0,
MJ
(mU - Zq)M" - (m - Z,)Wzc1 +
(m
I: - Mq)
o
+ M"Z E
' XIYY-
I,
o
=
-
J (m-
M4)
--ZM
(5E
8
E
+±M,Z
8
E
Z ,Wzc
M
512
(5.112)
ZWWz
This shows that any step response in the elevator does not have an immediate impact on the pitch angle.
However, for an elevator step response the steady state pitch angle settles to a value shown in equation
5.112, as time goes to infinity, which is to be expected since the righting moment is the restoring force in
the pitch plane. By implementing a differentiator, as shown in the yaw plane derivation, 5.112 becomes
5.113 and 5.114.
s =
9
= 0
s=0,
q - 0
(5.113)
As shown in equation 5.113, for a constant elevator angle, the pitch rate does not reach a steady state and
initially is zero.
.M
s=
=
s=0, 9
- 0
(5.114)
S
,yy - MF
5E
As shown in equation 5.114, notice that the instantaneous pitch acceleration as a function of elevator
deflection angle is simply the moment forcing term, Mde, divided by the mass term, (Iyy-Mqdot). These
results will be further discussed in Chapter 9, as related to the responses seen from field data.
5.8
Depth Model
The AUV velocity along the inertial z-axis depends on the vehicle speed, U, the pitch angle, 0, as well as
the side slip velocity in the body referenced heave direction. Depth is the negative of inertial position, z.
- z =U sin(O) -wcos(O)
U
- Z ~ S
(5.115)
Assuming side slip, w, is small and the pitch angle is relatively small, then inertial velocity in the z
direction is simply U*0. This is a major assumption that dramatically affects the depth controller design,
because the pitch angle can reach up to 30 degrees in field operations.
71
5.9
Tabulated Linear Pitch Plane Coefficients
Setting steady state surge velocity to: U=1.5m/s
Parameter
Value
Units
Description
x=X uu U 2
-31
kg-m / s 2
Constant Axial Drag
XU = Xjj 2U
-41
kg / s
Axial Drag
X
-26.2
kg
Added Mass
Xthrust
T,
kg-m /
S2
Axial Thrust
Xtaiicone
-0.6
kg-m /
S2
Axial Drag from Tailcone
2
Hydrostatic Force
Xhydrostatics
(W-B)O
kg m /s
Zwd = Z~j~jc,
-75
kg / s
Zqd = zqiqlcQ
0.4
kg-m /rad
Z- Z
-395
kg
Added Mass
Z4Z
-32.4
kg-m / rad
Added Mass
Zqa = Zu U
39
kg-m / rad-s
Added Mass Cross-term
ZW1 = ZUU
-135
kg / s
Body Lift Force
Zhydrostatics
(-W+B)
kg-m / S2
Hydrostatic Force
-196
kg / s
Tailcone Lift Force
-247
kg-m / rad-s
Tailcone Lift Force
-294
kg-m / S2 -rad
Tailcone Lift Force
kg-m / s 2 -rad
Thrust from Tailcone Angle
zwt
z.t
Zq,
z JEtaiCONe
Z (Ethrust
z
Cross flow Drag
2
Cross flow Drag
Z, = Zwd + ZW1 + Z,,
-406
kg / s
Lift Force from Translation
zq = Zqd + Zqa + Z
-208
kg-m / rad-s
Lift Force from Rotation
-294-Tn
kgm/S 2 -rad
Lift Force from Tailcone Angle
SE
=
ZEtailcon
Ethrust
Table 5.7: Short Caribou configuration linear force coefficients, U=1.5m/s
72
Setting steady state surge velocity to: U=1.5m/s
Parameter
Value
Units
Description
Mwd = MWH C
1.9
kg-m /s
Cross flow Drag
Mqd =
2
-14
kg-m / s
Cross flow Drag
-32.4
kg-m
Added Mass
M4
-127
2
kg-m / rad
Added Mass
Mqa =Mqu U
48
kg-m2 / rad-s
Added Mass Cross Term
MW1 = MU
320
kg-m/s
Body Lift Moment
(WXCM-BXCV) +
kg-m 2/s 2
Hydrostatic Moment
Tailcone Lift Moment
qlqlCQ
Mhydrostatics
(Wzcm-Bzcv)O
M
-247
kg-m/s
Mqt
-311
kg m2 / rad-s
-371
MSEtailcone
-1.26TpT
M5Ethrust
2
2
Tailcone Lift Moment
kg-m / rad-s
Tailcone Lift Moment
kg-m 2 / rad-S2
Thrust Moment from Tailcone
Angle
MW = Mwd + M W + M,,
75
kg-m/s
Lift Moment from Translation
Mq= Mqd +Mqa + Mqt
-277
kg-m2 / rad-s
Lift Moment from Rotation
-371-1. 26*Tp
kg-m 2 / rad-s2
Lift Moment from Tailcone Angle
M '5E
Etailcone + M
EthruSt
Table 5.8: Short Caribou configuration linear moment coefficients, U=1.5m/s
73
Setting steady state surge velocity to: U=1.5m/s
Parameter
Value
Units
Description
2
-34
kg-m /s 2
Constant Axial Drag
X = Xul 2U
-45
kg / s
Axial Drag
Xi
-26.2
kg
Added Mass
Xthrust
TP
X
X.,U
kgm /S
2
Axial Thrust
Xtailcone
-0.6
kg-m / S2
Axial Drag from Tailcone
Xhydrostatics
(W-B)O
kg-m /s 2
Hydrostatic Force
-113
kg / s
Cross flow Drag
2
Zqd = ZqlqcQ
2.0
Z-Z
-632
kg
Added Mass
Z4Z
-57.1
kg-m / rad
Added Mass
39
kg-m / rad-s
Added Mass Cross-term
-192
kg / s
Body Lift Force
(-W+B)
kg-m /s 2
Hydrostatic Force
ZWt
-196
kg / s
Tailcone Lift Force
Zqt
-349
kg-m / rad s
Tailcone Lift Force
Zqa = ZuqU
Z,, = zU
Zhydrostatics
-294
ZtEtaicone
kg-m /rad
2
kg-m / S -rad
2
Z5Ethrust
Cross flow Drag
Tailcone Lift Force
kg'm / s -rad
Thrust from Tailcone Angle
Z4 = Zwd + ZWI + Zw,
-501
kg / s
Lift Force from Translation
Zq = Zqd + Zqa + Zqt
-308
kg-m / rad-s
Lift Force from Rotation
-294-T,
kg-m / s 2 -rad
Lift Force from Tailcone Angle
zE
Ftailcone +
Z5Ethrust
Table 5.9: Extended (1.05m) Caribou configuration linear force coefficients, U=1.5m/s
74
Setting steady state surge velocity to: U=1.5m/s
Parameter
Value
Units
Description
4.0
kg-m /s
Cross flow Drag
Mqd =M qlqc Q
-62
kg-m 2 / s
Cross flow Drag
Mq
-57.1
kg-m
Added Mass
Mg
-458
kg-m2 / rad
Added Mass
Mqa =Mqu U
86
kg-m2 / rad-s
Added Mass Cross Term
MW
1 = M U
620
kg-m/s
Body Lift Moment
(WxCM-Bxcv) +
(WZCM-Bzcv)O
kg-m 2/S 2
Hydrostatic Moment
M.,
-349
kg-m/s
Tailcone Lift Moment
Mqt
-621
kg-m 2 / rad-s
Tailcone Lift Moment
MdEtailcone
-524
kg-m2 / rad-s
2
Tailcone Lift Moment
MSEthrust
-1.78*Tp
kg'm2 / rad s 2
275
kg-m/s
Thrust Moment from Tailcone
Angle
Lift Moment from Translation
-597
kgm 2 / rad-s
Lift Moment from Rotation
-524-1.78*T,
kg-m2 / rad-s2
Lift Moment from Tailcone Angle
Mwd = MWI
C
Mhydrostatics
MW =Mwd + M 1 + M.,
Mq Mqd + Mqa +Mqt
M-E
A
Etailcone
+ mEthr4,t
Table 5.10: Extended (1.05m) Caribou configuration linear moment coefficients, U=1.5m/s
75
Setting steady state surge velocity to: U=1.3m/s
Value
Parameter
Units
Description
2
-26
X, =Xuj 2U
-39
kg / s
Axial Drag
X
-26.2
kg
Added Mass
T,
kg-m /s 2
Axial Thrust
X = X,,u
Xthrust
'tailcone
Xhydrostatics
S2
Constant Axial Drag
-0.4
kg m /
S2
Axial Drag from Tailcone
(W-B)O
kg m /
S2
Hydrostatic Force
-113
Zwd = ZwiwC w
kg-m /
kg / s
Cross flow Drag
Zqd =ZqqlCQ
2.0
kgm / rad
Cross flow Drag
zw
-632
kg
Added Mass
Z4
-57.1
kg-m/ rad
Added Mass
Zqa = Zuq
34
kg-m / rad-s
Added Mass Cross-term
ZW1 = Z.U
-166
kg / s
Body Lift Force
2
Zhydrostatics
(-W+B)
zwt
-170
kg / s
Tailcone Lift Force
Zqt
-302
kg-m / rads
Tailcone Lift Force
Zs5taicone
-221
kg-m / S -rad
Tailcone Lift Force
Zdthrust
kg-m /
S2
2
Hydrostatic Force
-T,
kg-m / s rad
Thrust from Tailcone Angle
Zw = Zwd + ZWI + Z,
-449
kg / s
Lift Force from Translation
Zq = Zqd + Zqa + Zqt
-266
kgm / rad s
Lift Force from Rotation
ZE
Etailcone +
Ethrust
-22 1-Tn
2
kg-m/ s -rad
Lift Force from Tailcone Angle
Table 5.11: Extended (1.05m) Caribou configuration linear force coefficients, U=1.3m/s
76
Setting steady state surge velocity to: U=1.3m/s
Parameter
Mwd = MWIWI
c
Mqd =M qqc Q
Description
4.0
kg-m / s
Cross flow Drag
2
kg-m / s
Cross flow Drag
-57.1
kg-m
Added Mass
-458
kg-m2 / rad
Added Mass
75
kg m 2 / rad s
Added Mass Cross Term
537
kg-m/s
Body Lift Moment
2
kgm /S
M wt
(WxCM-Bxcv) +
(Wzcm-Bzcv)O
-302
Mqt
8
Mhydrostatics
Etailcone
EtalCOne
Hydrostatic Moment
Tailcone Lift Moment
-538
kg-m2 / rad-s
Tailcone Lift Moment
-393
kg-m 2 / rad-s2
Tailcone Lift Moment
kgm 2 / rad
2
Thrust Moment from Tailcone
Angle
239
kg-m/s
Lift Moment from Translation
+ Mqt
-525
kg-m2 / rad-s
Lift Moment from Rotation
+ MgEthrust
-393-1.78*Tp
kg-m 2 / rad-s2
Lift Moment from Tailcone Angle
MW= Mwd + MW
1 + MWt
Mq= Mqd +
2
kg-m/s
-1.78*T
M6Ethrust
M'E
3
Units
-62
M4
M~
1=
M
Value
qa
Table 5.12: Extended (1.05m) Caribou configuration linear moment coefficients, U=1.3m/s
77
78
Chapter 6
Vehicle Simulation
In Chapter 6, the equations of motion are combined with the forcing components to complete the
derivation of the overall equations of motion. The Matlab simulation is discussed, and the overall
nonlinear and linear approach to the simulation is explained. This simulation is used to adjust the
dynamic model in the controller design process as outlined in Figure 1.2.
6.1
Complete Nonlinear Equations of Motion
Combining the equations for the vehicle kinematics (Chapter 3) with the rigid-body dynamics (Chapter
4), we arrive at the combined nonlinear equations of motion for the Caribou AUV in six degrees of
freedom.
Translation along the x-axis (Surge):
mX[ + qw - rv +
XHS
u
u
U
q
CM p -(q
.qC
+ Xqqqq
Xq
+ Xvr
2
+ r 2()XCM
+ X,,rr + Xthrust
tailcone
Translation along the y-axis (Sway):
m[- + ru - pw + xC
+Yv
VI
IV
v1rr
Irl+ Y,
YHS
+
-
PzCm + (rzCM + px~C
-
+ Y wp
Ypq
+Y
I
+ Yur+Yw
ur
2r +
qYvY
Yuv
.'c C6.2
~
+Ythrust +
(6.2)
tailcone
Translation along the z-axis (Heave):
m[+ pv - qu + pyCM
ZHS +Z W1
w1|+ Z qq|+
xCM +(PXCM +qyC)r (P2 +q )zcm
(6.3)
Z p'v+ Z44++Zuquq+ Z,,vp+ Z,,rp + Zuw+Zthrust + Ztaicone
Rotation about the x-axis (Roll):
79
I,
+ IY,4 + I_ + (I,, - IY )rq +I,(q2 - r 2) + I,,pq - I,,pr +
~,p P K~P(6.4)
m[yc
yCvM
pv - qu - zCM ( + ru pw)] = KHS~ ±K
+ KPPI1 pp+ Kfip
Rotation about the y-axis (Pitch):
Ip + I,,4 + Iyf + (I - I,,)pr +Ix(r2 _ p 2) + Iqr - Iyqp
+ m[zcM ( + qw - r - xcm (v+ pv - qu)] = MHs +M 1w w|w+
M
q
||
(6.5)
+ Mj* + M/I + MvpvP + Mrp+Muquq+MUMuw + Mthrust + Mtaicone
Rotation about the z-axis (yaw):
ip + Iz,4 + Ij
+ Mmc
+ (I, - I)pq +I, (p 2
-
q 2) + Iypr - Ixqr
+ m - pw - ycm (5 + qw - rv)] = NHs + N
+ Nv9 + N
vIv|+ N rIr r|
(6.6)
+ N wp + Npqpq + Nrur + Nuv + Nthrust +Ntaicone
These six equations can be transformed into state space notation for simulation purposes. It is convenient
to separate the acceleration terms from the other terms in the vehicle equations of motion. The mass
acceleration terms can be set equal to the remaining forcing terms.
(m - X,)u +mzCM4 - MyCMi =X
(6.7)
=>
(m -Y)>+(mxCM
- Y )* -mpzCM
(m - Zj)V -(mxCM
+ Z)4 +mPyCM =
(IU - K)p+ I.,
+ Ix*-
mkyM m
IYX P +(IYY - M)4 + I, -(mxCM
Iz, p + Iz,4 + (Izz - N, ) + (mxCM-
Y
(6.8)
3z
(6.9)
zCM =
K
(6.10)
(6.11)
+ M,)Vv+ mtzCM =ZM
N, )1 - mycmt0=
N
(6.12)
Now the remaining forcing terms are represented as follows:
JX
= XHS + X
+(Xrr
u1u
± (Xwq
m(qy
+ mxCM)r 2
SY = Y'S +Y viv1 V+Y
CM
P
+ m)vr + (Xqq + mxCM
+ X,,
+
2
Xta(icone
r rl +(Y ,. -m)ur +(Y, + m)wp+(Yq - mxcm)pq(
VI
VIrII(6.14)
- mrqzm + m(r 2 +P
80
- m)wq + (X,
2
)YCM
+YUV +Ythust + Ytailcone
lZ=Z
1
s +Z"w- W|+Z
mrqyCM
-
+ m(P
2
+ m)uq +(Z, - m)vp +(Zp - mxcM)rp
qlq|q+(Z,,q
(6.15)
+ q 2 )zCM + Zu"uw + Zthrust + Ztaicone
+ I, pr
K=KHs +K pip p|+(I,,- Izz)rq +I,(r2 - q 2 ) - I ,,pq
5
m[yCM (pv - qu) -
-
ZCM
M=MH + Mi1W|+
+ (Mrp - I
I
I. 5
(p
2
-
r
(N, + mxcM)wp
(6.18)
-I,)pq + (Nu, - mxCM)ur + Nuv + Nhrs,
+ I
+I, (q 2
+ Ntaiicone
(6.17)
+ Mthrust
+ Izqp + m[zCM(rv - qw)]
2)--I1xqr
N=Ns + NivIv|+ Ni rjr|+
+ (Npq
6.2
q|I q|+(Mvp + mxcm )vp
M,
+ Izz )rp+ (Muq - mxCm uq + M uW
+
± Mtacone
(6.16)
(ru - pw)]
_ p)
-
I, pr + I, qr + m[yCM (qw
-
rv)]
Nonlinear Simulation States and Matrices
Now, these equations (6.7-6.18) are written in matrix form as follows.
0
0
0
mzCM
-myCM
LX
0
m-Y
0
- mzCM
0
MxCM -Y
>Y
0
0
m-X
0
-mzCM
0
-
MyCM
mxCM
m-Z.
myCM
MyCM
IXX - K.
-mxCM
-Mv
-m-1 CM -Z4
yx
0
-N
I
x -M4
(6.19)
0
IZ
I,
Iz, - N,
IZY
p
q
Y-M
*
_-IN-_
Y_ K
Equation 6.19 can be rearranged now as the inverted matrix C times the forcing matrix F.
m-X
0
0
0
V
0
m-Y
P
0
0
p
0
q4
mZCM
- mzCM
0
*_
- mycM
mCM
-
m-Z,
N,
MyCM
mxCM -Mv
0
-1
0
mzCM
-mZCM
myCM
IXX
-
K
Izx
I,y
I
MyCM
>zX
mxCM -
0
-MCM
-
-Z4
0
Z
IxZ
J K
I
I M
Izz - N
(6.20)
= C-F
-IN-_
The state that will be used during the simulation is the combination of the states described in equations
3.3, 3.4, 3.5, and 3.6. Here they are combined as follows to form a vector z.
81
F 3x]
z T =[x
y
9 #
y
z
u
v
w p q
±
r]
(6.21)
=-
LB 6 1
The derivative of z with respect to time is 2 . This vector, ±, is composed of three separate vectors,
k,
, and B. These vectors are defined as follows from equations 3.9, 3.10, and 6.20, in order to
account for the angular rotation.
u
.i = RT (V/,,0,#0)
V
P
S=F-(V/,O,#) q
(6.22)
r
B = C-'F
6.3
Linear Simulation States and Matrices
In a similar fashion, the simulation states and matrices were derived for the simulation. From the yaw and
pitch plane linearized coefficients the following states and matrices were developed and implemented into
simulations. From equation 5.60, the motion in the body frame x-direction is modeled as follows for both
the yaw plane and the pitch plane simulation models.
(m - X, d - Xi - Xu
6.3.1
(6.23)
- Xtaiacone = Xhydrostatics + Xthrust
Yaw Plane
From the kinematics and dynamics explained in Chapter 5, the matrix representation of the yaw plane
dynamics, sway and yaw, was developed in equation 5.62 and repeated here in equation 6.24.
(m -Y )
(0mxCMN)
(mxCM - y)
(Iz - N)
0
0
0
- Y,
- + -N
1-O
(mU -Y )
0
V
U
-1
0
r =
(mxCM
L 0
A. +B = Ci
-
Nr)
Oq
Y5R
.R
R
(6.24)
0
This matrix form leads to a standard state space form which was developed in Chapter 5 and is
implemented as the linear simulation model for the yaw plane. In this form, J is a vector representing
sway rate, yaw rate, and yaw position, and ii represents input to the system, which is the rudder angle in
the yaw plane model. The last channel is just an integrator to capture the heading angle, (p, from the yaw
rate, r.
[V1
X=
Ai + Bii
r
L(Pi
82
ii = (R
(6.19)
From equation 5.8, the relationship between body frame velocities and inertial frame velocities is shown
in state form.
cos#
-sin#
0 u
=sin#
cos#0
0 v
0
ij rj
x
0
(6.26)
The transfer function for the yaw plane model, as derived in Chapter 5, is reprinted below, along with the
values for this transfer function from the initial linearized model from Chapter 5, for the yaw plane.
][ -]YvNR + N,
SF(NgR
(10
-NJ
S(I
5R
-
YF(xUN,
.(M - YJ
(m-Y XIz-N.
s3
(6.27)
) +S-Y(mxc U - N,. +(m U - Y,.N,
CMU+S
(M - Y,X(IZ - NJ
(ZZNJ
p _ -0.6944s -0.4325
(R
Y]
+1.471s
2
initial yaw model
+0.2891s
The poles for this transfer function are: 0, -1.2369 and -0.2337, while the zeros are at -0.6229 and minus
infinity. The pole at the origin is due to the lack of any restoring force in the yaw plane. Therefore, the
gain of this function is infinity as any set rudder angle will cause the AUV to continually go in circles, as
was noted in equation 5.71. Also, the negative eigenvalues of the system show that the model is open
loop stable.
6.3.1
Pitch Plane
From the kinematics and dynamics explained in Section 5.7, the matrix representation of the pitch plane
dynamics, heave and pitch, was developed in equation 5.104 and repeated here in equation 6.28.
(m--Z,)
-(MxCM
0
+M)
-(mxM+ Z,)
(I,, -M)
0
0~0
0
0
-Z]
4 + - M,
L 0
i
-(mU +Zq)
(mxU - M )
-1
-
0
W
Wzcm
q
0
10
Z-E5
=
'5E
E
(5E
0
+ Bi = Ci
This matrix form leads to a standard state space form which is implemented as the linear simulation
model for the pitch plane, where 3c, is a vector representing heave rate, pitch rate, and pitch position, and
where ii represents input to the system, which is the elevator angle in the pitch plane model. The last
channel is just an integrator to capture the heading angle, 0, from the yaw rate, q.
x= Ai + Bil
=q
ii= 5E
(6.29)
-0
83
The relationship between body velocities and inertial velocities is shown in state form.
~cosO
S
[i=[-sinO
sinO
01F u
cosO
0 w
0
1__q_
N_
_ 0
(6.30)
The transfer function for the pitch plane model, as derived in Chapter 5, is reprinted below (equation
6.31), along with the values for this transfer function from the initial linearized model from Chapter 5.
V M 1-ZVI
3
&
(E S3
_
s
Z"
2
(MX'U -
(I,,
(M+(Zw)
=
0
M
+
F-
(,,
- M
(mxMU
- M
-(
Z,
0.6944s - 0.43 25
s +1.471s 2 +0.4288s + 0.06328
-
3
5E
z'
m -- Z, XI,, - Mj
(m -
- MJ
+M
- ZJAM.
- qIMj
-
(m -
Z)WzcM
+
1
Z Wz C
(m -
Z, XI,, - MW
initial pitch model
The poles for this transfer function are: -1.1442, -0.1632±0.1693i, while the zeros are at -0.6229 and
minus infinity. The gain for this transfer function is -6.83. The difference between the yaw and pitch
plane transfer function is simply that the pitch plane also has the righting moment. This component
creates the oscillatory behavior in the poles, as well as a gain that corresponds to the steady state pitch
angle in which the righting moment equates with the moment created by the elevator deflection. Also, the
negative eigenvalues of this system show that the model is open loop stable. From Section 5.8, the
vehicle depth as a function of elevator angle is shown below for the open loop system. The vehicle depth
is the negative of the inertial position along the z-axis.
U
-z
-
= --
0
E-
-1.0416s -0.6488
s4
initial d epth mod
el
The poles and zeros for this depth transfer function are the same as those for the pitch transfer function,
with the addition of a pole at the origin for the integrator.
6.4
Computer Simulation
As described earlier in Chapter 1, the simulation was implemented using Matlab code. The model code
works by calculating the forces and the moments on the vehicle as a function of vehicle velocities and
attitude for each time step. These forces determine the vehicle body-fixed accelerations and earth-relative
rates of change. These accelerations are then used to approximate the new vehicle velocities, which
become the inputs for the next modeling time step. All of this is computed internally using a prescribed
ODE function in Matlab.
The vehicle model requires two inputs:
*
84
Initial conditions, or the starting vehicle state vector z(t=0), as well as the initial and final times.
*
Control inputs, or the vehicle thrust, and rudder and elevator angles, either given as a predetermined vector, when comparing the model output with field data, or calculated at each time
step, in the case of control system design.
Using these inputs, the simulation integrates over the range prescribed, producing a time stamped vector z
as a final output. The simulation package has been constructed in a way to allow a full non-linear
approach, as well as linear yaw plane or pitch plane simulations. In addition, a minimization technique
was implemented to allow the model to search for appropriate coefficients when matching the model with
field data during system identification. The approach uses a Nelder-Mead simplex (direct search)
method. More of this will be described in the Chapter 9.
85
86
Chapter 7
Tailcone Testing and Modeling
The actuation system of the Odyssey III AUV provided additional challenges in control and modeling of
the system. Based on field data retrieved from previous survey missions, the software and drivers used to
control and operate Caribou included significant delay times in posting data for other clients as well as
unnecessary filtering of position data. The tailcone actuation system was rigorously tested and improved
by R. Damus, an engineer at MIT Sea Grant's AUV Lab, to help eliminate drift, stiction, and excessive
delays in actuation. Bench tests of the tailcone were also completed in order to develop a low-order
model that encompassed the actuator dynamics as well as any computational delays in the system. This
tailcone model is included in the initial dynamic model that is shown as a first step in controller design in
Figure 1.2.
7.1
Experimental Setup
In the closed-loop controlled system, the error from heading and depth is sent through the controller to the
system plant. The plant consists of the actuator system and the vehicle's hydrodynamics. The actuator
system itself had its own dynamics, in addition to a delay (Figure 7.1). The desired tailcone position,
rudder and elevator angle, were passed from the controller to the actuation system. In an ideal system, the
actuation system would have a transfer function of one. However, Caribou's actuation system is rate
limited and deflection angle limited. In the field, Caribou's deflection angle limit is usually set to ± 15
degrees for both the rudder and elevator angle. The rate limit was set at 15 degree/sec, which is
dependent upon the actuator capabilities. These limitations, in addition to computational delays, result in
a transfer function much different than the ideal case of one. Therefore, the actual tailcone position is
measurably different than the desired tailcone position. In order to fully develop a model of the vehicle,
as well as understand control strategies more fully, this actuation system needed to be modeled.
ta htonc
Desired
+
Controller
tailconc
Delay
DActator
o
N
Iydrodamics
Figure 7.1: Closed loop system
87
In order to develop a representative transfer function of the vehicle's actuation system, experimental
testing was required. To develop an understanding of the transfer function for the actuator, the system
response was studied over a range of frequencies and amplitudes. The experimental setup, Figure 8.2,
involved mounting the vehicle's Crossbow INS (inertial navigation system) system to the tailcone. These
tailcone INS values were then fed back into the MOOS operating system of the AUV as a client, strictly
for logging purposes only. In order to simulate the vehicle's motion, a fake INS heading and pitch were
sent into the MOOS operating system from an additional outside source. This arrangement allowed the
entire system to run on one platform, while receiving sensory data from an additional source. In order to
capture the time delay and actuator dynamics of the system, the controller proportional value was set to
one and the other values were set to zero. This allowed the controller to simply represent a transfer
function of one. The AUV operated then as if it were at sea, receiving fake sensory information and
acting upon that information by actuating the rudder and elevator. The AUV received changes in the INS
heading and pitch (as a fake signal), and commanded the tailcone based on the controller proportional
gain of one. The desired rudder and elevator positions were logged, in addition to the actual rudder and
elevator position provided by the tailcone INS system. The fake heading and pitch sensory information
was logged as well, all with the same timestamp.
INS
](e'
ciig
<11b1
from HlTernate source
INS tailno
Ie
Position
Ha
Figure 7.2: Experimental setup
7.2
Experimental Results
The heading/rudder tests was completed independently of the pitch/elevator tests. By passing fake
sinusoidal heading and pitch data, with varying amplitudes and frequencies, to the AUV, a typical Bode
plot was established based on the actual tailcone positions and desired heading and pitch angles. (Figure
7.3).
As seen in Figure 7.3, the cutoff frequency ranged from 1.5 to 4.5 rad/sec (0.24 Hz to 0.72 Hz)
for the various heading/rudder tests. The stepper motors that control the rudder and elevator are rate
limited actuators. Therefore, at larger peak to peak amplitudes, the cutoff frequency is less than for
smaller peak to peak amplitudes commands. Caribou's tailcone is limited to travel l5degrees/sec.
Therefore, for peak to peak amplitudes of 30 degrees, the highest frequency that can be passed without
attenuation is 0.25 Hz (1.57rad/sec) because the time required, by the rate limited motors, to complete one
cycle is 4 seconds. However, for a peak to peak amplitude of 10 degrees, the time required to complete
one cycle is 1.33 seconds, which corresponds to a possible cutoff frequency of 0.75 Hz (4.71radl/sec).
7.2.1
Heading - Rudder Results
The fake INS heading data consisted of three sets of data with various amplitudes (30, 20 and 10 degrees
peak to peak). The frequency of the data varied from a period of 40 seconds (0.025 Hz, 0.16 rad/sec) to a
period of 0.5 seconds (2 Hz, 12.57 rad/sec). The three sets of data were plotted on a Bode Diagram.
Using these sets of data, a rudder actuator model was developed as detailed in the following section.
88
Bode Diagram
-5 -
-
-10 -0
15 -;
-4
-20-
model TF
0 30kpk
-25 - + 2 pk-pk
10 pk-pk
-30
-
-
100
10
102
0
-100
-200 -
-
0
model TF
30 pk-pk
+
20 pk-pk
-300-
-
.
10 pk-pk
-400
107
10
10U
102
Frequency (rad/sec)
Figure 7.3: Experiment results for rudder with a 1 " order model
7.2.2 Pitch - Elevator Results
The fake INS pitch data consisted of three sets of data with various amplitudes (30, 20 and 10 degrees
peak to peak). The frequency of the data varied from a period of 40 seconds (0.025 Hz, 0.157rad/sec) to a
period of 0.5 seconds (2 Hz, 12.566rad/sec). The three sets of data were plotted on a Bode Diagram
which consisted of magnitude and phase plots for the varying frequency. The cutoff frequency ranged
from 1 to 3 rad/sec (0.16 Hz to 0.48 Hz) for the various pitch/elevator tests, as shown in Figure 7.4.
Using these sets of data, an elevator actuator model was developed.
89
Bode Diagram
-5 -
0 -10-15 -20-
model TF
30 pk-pk
20 pk-pk
10 pk-pk
-25-30
-
-
-
10 0
10
102
0
-100
D
-200 -
a-
-
-300-
model TF
30 pk-pk
20 pk-pk
10 pk-pk
-
-400
10
10U
10
10
Frequency (rad/sec)
Figure 7.4: Experiment results for elevator with a I" order model
7.3
Tailcone Actuator Model
By using this data for the various trials, a dynamic model could be established that was representative of
the AUV's tailcone actuation system consisting of separate rudder and elevator models.
7.3.1
Rudder Model
As a first approximation, the heading/rudder data was modeled as a 1 s' order system (Figure 7.3).
However, this model proved to not have enough roll off in magnitude and not enough phase shift at
higher frequencies. Since the roll off of the data appeared to be roughly 40dB/decade, which is consistent
with 2 "d order systems, a 2 "d order model, corresponding to equation 7.1 was tuned to fit the data (Figure
7.5).
G 2 nd order rudder (S
Is'2
-+-+I
10
10.
90
(7.1)
Bode Diagram
0
-10
-0-±
~-20--30-40,
0
+
model TF
30 pk-pk
20 pk-pk
10 pk-pk
--10
10
102
0
-100 -
-300-
-.....
- model TF
0 30 pk-pk
+ 20 pk-pk
10 pk-pk
-
--
-
-400
10-
10
Frequency (rad/sec)
1
102
Figure 7.5: Experiment results for rudder with the 2 "d order model
This second order model fits the magnitude plot quite nicely for the data corresponding to the 20 degree
peak-to-peak test, however, in order to model the system about a controlled heading, the rudder actuation
model should encompass how the rudder responds at lower amplitudes. Thus, the nd order model was
2
modified to approximate the 10 degree peak-to-peak test data. Additionally, the phase plot shows that
there is much more phase lag at higher frequencies, than shown by a 2 "d order model alone. Thus, in
order to maintain this second order response with respect to magnitude, but add additional phase lag, a
time delay was added to the system. Ogato describes a function which represents an ideal time delay as
shown below in equation 7.2 [27].
G(s) = e-sT
(7.2)
The magnitude of this function can be shown to always equal one, thus not affecting the first order model
shown previously.
=|G(jo)|=e=G(s)
=|cosoiT - j sin oTI =1
(7.3)
The phase of this function can be shown to drop off linearly with increased frequency, depending on the
time constant T.
ZG(s) = -ofT
(7.4)
91
The time delay function can be represented in an alternate way. This allows easier implementation for
modeling and simulation purposes. For modeling purposed, this modeled delay was truncated and
modified as follows to approximate the time delay as shown in equation 7.5.
-+
T-Ts(Ts)
1
e-sT
2
2
8
Ts
2
(Ts)2
3
(Ts)
+ ...
S)
48
8
e-sT
(Ts)
48
9
Ts
(TS)2
2
9
2
7
Ts
7
25
(Ts)
(7.5)
25
This time delay model was then implemented, in addition to the modified second order model, and plotted
along with the data (Figure 7.6).
Bode Diagram
0
-10
-20
Cn
-30
+30
model TF
pk-pk
20 pk-pk
10
-40
pk-pk
10
10
101
100
0
-
*-
-100
A-
C)
0
0
(n
C-
-200
-300
model TF
-)
30 pk-pk20 pk-pk
10 pk-pk
-400
101
100
Figure 7.6: Experiment results for rudder with the 2 "d order model and time delay
102
This second order model, along with the time delay, closely models the lower rudder amplitudes. We
model at the lower amplitudes because the control system spends most of its time there. An overall
rudder actuator model was then developed based on this second order model and the time delay model.
The time constant T was set to 0.3, which resulted in an observed time delay of 0.3 seconds.
dea (
Gudderacuao,(s)=-G2ndorde,(s)Gtimes)
s2 3s
-+-+I
25
92
TS
(Ts)
2 7
9 Ts
-+-+
.2
7
(Ts),
9
10
252
25_
(7.6)
.00014S4
dder actuator
0.0036s32 - 0.0429s +0.2222
+0.00279s
+0.02535s
2
+0.10950s +0.22222
(7.7)
This rudder model transfer function was then used in simulation and control and has poles, from the
second order model, at -3.7500±3.3072i, and has a DC gain of 1.0. The zeros from the time delay are
equal and opposite of the poles from the time delay so that there is no attenuation due to the time delay in
the modeling. These poles are at -5.9524±5.1281i, while the zeros are at 5.9524±5.1281i. This rudder
dynamic model has poles and zeros as shown in Figure 7.7.
x
0
4
x
x
x
Real AxIs
Figure 7.7: Pole-Zero plot for the rudder dynamic model
The rudder system shows a bandwidth of 4.5 rad/sec (0.72 Hz), at -3dB as shown in Figure 7.6. This
shows that the rudder system has fairly low performance, and should be incorporated into the overall
vehicle model, so that the control system can be designed around this performance.
7.3.2
Elevator Model
As was done for the rudder model, the pitch/elevator data was modeled as a 1 st order system as a first
approximation (Figure 7.4). However, like the rudder system, this model proved not to have enough roll
off in magnitude and not enough phase shift at higher frequencies. Since the roll off of the data appeared
to be roughly 40dB/decade again, a 2 "d order model was tuned to fit the data (Figure 7.8).
G2nsordereieato(s)= [1s2 16s
_10
(7.8)
10]
93
Bode Diagram
Q
0
-10 -.
+
CO
-20-
--
-30 -
-40 1-
model TF
30 pk-pk
20 pk-pk
10 pk-pk
..-
.
........
. ...
10 2
10010
0
-100-
-200-300 -
model TF
30 pk-pk
-
20 pk-pk
10 pk-pk
-400
100
10-
102
101
Frequency (rad/sec)
Figure 7.8: Experiment results for elevator with the 2 "d order model
The second order model fits the magnitude plot quite nicely for the data corresponding to the 20 degree
peak-to-peak test, however, in order to model the system about a controlled depth, the elevator actuator
model should encompass how the elevator responds at lower amplitudes. Thus, the 2 "d order model was
modified to approximate the 10 degree peak-to-peak test data. Additionally, the phase plot shows that
there is much more phase lag at higher frequencies. Thus, in order to maintain this second order response
with respect to magnitude, but add additional phase lag, a time delay was added to the system. The same
procedure was followed for the elevator as was done for the rudder. This time delay model was then
implemented, in addition to the modified second order model, and plotted along with the data (Figure
7.9). An overall elevator actuator model was then developed based on the second order model and the
time delay model. The time constant T was set to 0.3, which resulted in an observed time delay of 0.3
seconds.
Geievator
elvtractuator(s) = G 2 nd
Geievtor
actuator
orde,(s)G,,
orer
merwdela,
0.00014s4
=
9
Ts+ (Ts)2
T
2
4s
+S1(s
9
100
10
_ _2
9
20
I-- )
(7.9)
20
0.0045s2 - 0.0333s + 0.2222
+0.00279s 3 +0.02450s 2 +0.12222s +0.22222
This elevator model transfer function, was then used in simulations and control and has poles, from the
second order model, at -10.0 and -3.33, and has a DC gain of 1.0. The zeros from the time delay are equal
and opposite of the poles from the time delay so that there is no attenuation due to the time delay in the
94
modeling. These poles are at -3.7037+5.9720i, while the zeros are at 3.7037±5.9720i. This tailcone
dynamic model has poles and zeros as shown in Figure 7.10.
Bode Diagram
00
G
10 -0
-+-
0
Ca
+
0-
-2
-30-
model TF
0
30 pk-pk
+
20 pk-pk
10 pk-pk
-40-
10 0
n
101
1 02
101
102
III
-100 C)
U)
V
U)
U,
-200 -
0~
-300
0
+
model TF
30 pk-pk
20 pk-pk
10 pk-pk
-4VV
100
10
Figure 7.9: Experiment results for elevator with the 2 "d order model and time delay
0
x
x
x
E!
-4
x
-10
-8
-G
-4
0
-2
0
2
4
Real Axis
Figure 7.10: Pole-Zero plot for the elevator dynamic model
95
The elevator system shows a bandwidth of 3.0 rad/sec (0.48 Hz), at -3dB as shown in Figure 7.9. This
elevator system response is substantially slower than the rudder system response, and should also be
incorporated into the pith plane models, so that the control system can be designed around this
performance.
96
Chapter 8
Initial Controller Design
In order to develop a control system for the Odyssey III class AUV, Caribou, modeling of the system as a
whole was considered. This system model consisted of the tailcone dynamics as well as the vehicle
dynamics. The pitch plane and yaw plane are controlled independently, since the roll of the vehicle is
usually less than 5 degrees, and thus the individual planes can be treated as decoupled states. Chapter 8
explains the design of the initial control system, controller Al, based on the initial model, model A. This
initial controller design is the second step in the controller process as shown in Figure 1.2.
8.1
Heading Controller
The heading controller controls on the negative feedback of the actual heading of the vehicle as is shown
in Figure 8.1.
Dcsrid
I
ading
+
+-
or
I Icading
Controller
rudder.\
*.~d(1
i
o
Rudder Actuator
Dynamics
actual
position
AUV Yaw Plane
1
Statc
I lydrodynamics
hcading
Figure 8.1: Heading control diagram
In addition to the vehicle hydrodynamics, the rudder actuator dynamics are modeled to provide a
complete model of the system. The heading controller is a proportional, derivative, integral controller.
The yaw plane portion of the system is modeled as explained in Chapter 5 for the dynamics and Chapter 7
for the rudder actuation system.
Grudder actuator(s)
9
Ts+ (Ts)2
2
7
25
s
3s
9-+ Ts (Ts)
-- + -+
- +
-25
16
-_2
7
25
G
G
Vw(s)
= C(sI - A)-B
(8.1)
_
The heading controller transfer function is shown in equation 8.2.
97
Gheading controller(s)=
8.1.1
K, + K
sK
S
d
d sp=h l
(8.2)
Heading Controller Design without Tailcone Dynamics
A controller was first designed without the tailcone dynamics. The heading controlled open loop transfer
function for this system was:
Gyawplane (s)= [d
K
'j
+
s
sL - A~ -1
]
(8.3)
Figure 8.2 shows the Root Locus for this system for various gains of feedback, along with the position of
the locus for the gains chosen, as denoted by the black asterisks on the plots. The right plot in Figure 8.2
is a closer look at the left plot around zero. Due to constraints in Root Locus modeling in Matlab, the plot
on the right does not appear to represent a typical smooth locus path, when in fact; the path is actually
rounded and smoother than is displayed in Figure 8.2. Because there is no restoring moment in the
heading system, there exists an open loop pole at the origin. The closed loop system draws this pole away
from the right hand plane of the Root Locus plot as shown in the right plot of Figure 8.2. The bandwidth
of this initial heading controller and initial dynamic model, without the rudder system model, is 0.6942
rad/sec, which is slow.
100.6
-
0.4-
0.2
-05
P4
Z3
Z
Z2
P3
02-
C
I
-0.25
44-05
-0.4-
-1.8
-.
-
-0.62
-1.6
-1.4
-1.2
-1
-0.8
Real Ais
-0!6
-0.4
-0.2
-2!5
-2
-1.5
Real Aes
-1
-0.5
0
xle
Figure 8.2: Root Locus for heading system without tailcone dynamics
8.1.2
Heading Controller Design with Tailcone Dynamics
The rudder system is modeled as a second order system with a time delay as developed in the previous
chapter. The closed loop bandwidth is shown to be near 4.5rad/sec (0.72 Hz) for the modeled rudder
system. This bandwidth is low, therefore, the rudder dynamics could not be neglected and are included as
follows for the heading controlled open loop transfer function.
98
9
K
Gd,,()
s2+K
-
s+K.
p
i'
s
s
- -+
25
j -2 - TS7
1
3s
-
10
9
+
-+ --
_ _2
(Ts)2
+
+
7
1
jS)2
252 [C(sI- A) _B]
(8.4)
ST)2
25 _
The root locus for this system, which includes the tailcone rudder model, is shown for various gains of
feedback, along with the position of the locus for the gains chosen, as denoted by the black asterisks in
Figure 8.3. The bandwidth of this initial heading controller and initial dynamic model, including the
rudder system model, is 0.0015 rad/sec, which is substantially slower than the bandwidth for the model
without the tailcone. Therefore, including the model of the tailcone is needed for proper design.
0.5
P7!
0.4
Z4
0.3
P5
0.2
0.1
P4.
0
Z3
P4:
Z2
P3
-0.1
-2
-0.2
P6
-4
-0.3
Z5
P8
-0.4
-6
-0.5
-8
-10
-8
-6
-4
-2
0
Real Axds
2
4
6
8
-1.2
-1
-0.8
-0.6
Real A~ds
-0.4
-0.2
0
X 10,
P1, P2
Z0p
01--
-0.5 k
-1
-3
-2.5
-2
-1.5
Real
-1
ss
-0.5
0
Xlon
Figure 8.3: Root Locus plot for the heading system with tailcone dynamics
The top right plot, and the bottom plot of Figure 8.3 show a closer look at the top left Root Locus plot.
By adding the rudder model, the Root Locus has changed from Figure 8.2 to Figure 8.3. In comparing the
top right plot in Figure 8.3 to the left plot in Figure 8.2, the poles have moved slightly further away from
the right hand plane. The major difference is the addition of four extra poles that model the rudder
system. These rudder dynamic poles, in the top left plot of Figure 8.3, help show that the system can go
unstable when high gains are used. The initial gains were chosen to minimize the oscillatory behavior, as
99
well as show quick response. The Root Locus in Figure 8.3 shows the best controller that could be
devised which balances response time and robustness, based on the initial dynamic model.
8.2
Pitch Controller
The pitch controller controls on the negative feedback of the actual pitch of the vehicle as is shown in
Figure 8.4. The difference between the heading system and pitch system is that there is a righting
moment in the pitch system from the separation of the center of buoyancy and the center of mass.
Desired
Pitch
delired
+
Pitch
rror
actual
clevaItor
Controller
Al V Pitch
1levator
Elevator Actuator
P
AUN
Pitch Plane
1
position
I
Stat
Ilydrodynamic s
Dynamics
atctuol pitch
Figure 8.4: Pitch control diagram
In addition to the vehicle hydrodynamics, the elevator actuator dynamics are modeled to provide a
complete model of the system. The pitch controller is a proportional, derivative, integral controller. The
pitch plane portion of the system is modeled as explained in Chapter 5 for the dynamics and Chapter 7 for
the elevator actuation system.
9
Gelevator actuator
(S)
_-100
Ts
(Ts)2-
9
202
1
3s 4s
2
9
TS
(Ts
10
L_2
9
20
GAUV
pitch (s)
=
sI-
- B
(8.5)
_
The pitch controller transfer function is shown in equation 8.6.
Gpitch controller
(s)
= K, + Kds +
K.
i-
FKds2±+K~s±+KI1
S
(8.6)
S
8.1.1 Pitch Controller Design without Tailcone Dynamics
A controller was first designed without the tailcone dynamics included. The pitch controlled open loop
transfer function for this system was:
Gpitch plane(S) =
+
+K
s
100
C(s-
A) t
f]
(8.7)
The root locus for this system is shown for various gains of feedback, along with the position of the locus
for the gains chosen, as denoted by the black asterisks in Figure 8.5. The bandwidth of this initial pitch
controller and initial dynamic model, without the elevator system model, is 1.3236 rad/sec, which is slow.
1
0.8F
0.6k
0.4k
P1I
gn 0.2W
Z2 ZI
P3
0i
CMI
P2
-0.4
-0.6-0.8
-1
.
-2
-2 5
-1.5
-1
0
-0.5
Real Axis
Figure 8.5: Root Locus for pitch system without tailcone dynamics
The Root Locus plot in Figure 8.5 differs from the plot in Figure 8.2 only because in the pitch system the
separation of the center of mass and the center of volume add a restoring righting moment. Since there
exists this restoring moment, there are no open loop poles at the origin as there are in the heading system.
8.1.2
Pitch Controller Design with Tailcone Dynamics
The elevator system is modeled as a second order system with a time delay in the previous chapter. The
closed loop bandwidth is shown to near 3rad/sec (0.48 Hz) for the modeled elevator system, which is low.
Therefore, the elevator dynamics could not be neglected and are included as follows for the pitch
controlled open loop transfer function.
Gpitchplane(S)
[
Kd
Kds
2+Ks+K
s±KsK
3s
4s
--
+ - 1
100
10
9
Ts
2
9
9
-_2
-2
(Ts) 2
20
S
+
9
-C(sI-A)-
]
(8.8)
)s
2
20 _
The root locus for this system, which includes the tailcone model, is shown for various gains of feedback,
along with the position of the locus for the gains chosen, as denoted by the black asterisks in Figure 8.6.
The bandwidth of this initial pitch controller and initial dynamic model, including the elevator system
model, is 2.5733 rad/sec, which is substantially faster than the bandwidth for the model without the
tailcone. Therefore, including the model of the tailcone is needed for proper design.
101
-
6
Z3
P52
4 -1
2
0.5
P7
P4
P3
P4
P3P3
E
Z;
E
P2
-0.5
-2
-4 --
P6
-10-~
-14
-12
g
0
8
-hp
~
-6 -4
0
2
Z4
-.
-3
4 have4
Real AXIS
-2.5
-2
-1.5
-1
-05
Real AidIs
0
0.5
1
1.5
Figure 8.6: Root Locus plot for the pitch system with tailcone dynamics
The right plot of Figure 8.6 shows a closer look at the left Root Locus plot. By adding the elevator
model, the Root Locus has changed from Figure 8.5 to Figure 8.6. In comparing the right plot in Figure
8.6 to Figure 8.5, the poles have moved slightly further away from the right hand plane. The major
difference is the addition of four extra poles that model the elevator system. These elevator dynamic
poles, in the left plot of Figure 8.6, help show that the system can go unstable when high gains are used.
The initial gains were chosen to minimize the oscillatory behavior, as well as show quick response. The
Root Locus in Figure 8.6 shows the best controller that could be devised, based on the initial dynamic
model. One difference between this pitch system and the heading system is that elevator dynamics in the
pitch system are slightly different than the rudder dynamics in the heading system. The main difference
in the system dynamics is that the pitch system has a restoring righting moment due to the separation of
the center of mass and the center of buoyancy. This major difference, as well as the gains chosen to meet
the design goals, creates the major differences between the Root Locus models of the heading and pitch
systems.
8.3
Depth Controller
The depth controller controls on the negative feedback of the actual depth of the vehicle as is shown in
Figure 8.7. This outer depth loop contains an inner pitch loop described in Section 8.2.
)
...
cpih
i.d
+
rror
Depth
Controller
dcsired
pitch
position
actual
desircd
clev.lor
1rrr
+
-
Pitch
Controller
p
clevtior
Elevator Actuator
Dynamics
IPitin
AUV Pitch Plane
Iydrodynanics
actual pitch
alti
dkpth
Figure 8.7: Depth and pitch control diagram
The depth controller is a proportional, derivative, integral controller. This depth controller's output acts
on the closed loop pitch control system. However, the pitch control system feeds back the actual pitch
position which is needed for the pitch loop, but the depth loops needs feedback of actual depth. Root
Locus is designed for single input single output systems (SISO). In this thesis we want to do SISO
control only, therefore, in order to perform Root Locus analysis on this depth controlled system, pitch
needs to be converted into depth. The AUV velocity along the inertial z-axis depends on the vehicle
102
speed, U, the pitch angle, 0, as well as the side slip velocity in the body referenced heave direction. The
vehicle depth is the negative of the position along the inertial z-axis.
-
z
= U sin(9) - w cos(9)
(8.9)
Assuming side slip, w, is small and the pitch angle is relatively small, then inertial velocity in the
z direction is simply U*O. This is a major assumption that dramatically affects the depth controller
design, because the pitch angle can reach up to 30 degrees in field operations. At these angles the side
slip begins to make an impact. However, under normal operating conditions, the AUV stays in nearly
level flight with smaller pitch angles, and thus this depth controller design approach seems appropriate,
while keeping a conservative depth control system design in mind.
P5
P!
6
Z4
1.5
4
P5
2
0.5-
P3
P9
0
0-
P4
-2
-0.5-
-4
-1
P8
-6
Z5
-1.5
P6
-12
-10
-8
-6
-2
-4
0
2
4
-2.5
Real Axis
X
-2
-1.5
-1
-0.5
Real Ayis
0
0.5
1
1.5
10
8-6
2
P, P2
ZI
-2--
-4
-6
-18
-16
-14
-12
-10
8Axs
Real A
R
4
es
2
w
102
iy,
Figure 8.8: Root Locus plot for the depth-pitch system with tailcone dynamics
Therefore, by integrating U*0, the position in the inertial z-axis is approximated. Hence, the
open loop transfer function for the depth controlled system is as follows, with desired depth in, and actual
depth out.
Gep,,
(s) -
Kd ,depths
2 +K
pdepthS+Kdepth
dephph+
Gpitch plane (s)
U(8.10)
Gplane
(S ) _ s._
103
This transfer function is different than the open loop transfer function developed in Section 5.8 that
showed a linear relationship between pitch angle and depth rate. Here the open loop depth transfer
function used the nested closed loop pitch loop. Software on Caribou is set up in this configuration, and
therefore, our modeling is done in analogous fashion.
The root locus for this system, which includes the tailcone elevator model, is shown for various
gains of feedback, along with the position of the locus for the gains chosen, as denoted by the black
asterisks Figure 8.8. The bandwidth of this initial depth controller and initial dynamic model, including
the tailcone dynamics, is 0.2056 rad/sec, which is slow.
If the pitch loop was very fast compared to the depth loop, the transfer function shown in
equation 8.10, would just be the Controller*(U/s), because the pitch loop would be approximately one.
However, for consistency, the entire system model was included in the depth controller Root Locus
design shown in Figure 8.8. The top right and bottom plots are simply closer looks at the top left Root
Locus plot in Figure 8.8. In comparing the pitch design, Figure 8.6, to the depth design, Figure 8.8, the
main difference is the addition of several poles near the origin along the real axis. The remainder of the
closed loop poles stayed near the position of the pitch design, while the locus of the closed loop poles
changed significantly. Like the heading and pitch loops, the initial depth controller gains were chosen to
minimize the oscillatory behavior, as well as show quick response, however, the depth gains were
designed conservatively to allow the AUV realistic results. The Root Locus in Figure 8.8 shows the best
controller that could be devised, based on the initial dynamic model.
Table 8.1 shows the controller
gains for this initial design, which is referred to as controller Al in the proceeding chapters.
Heading
Controller
Kp
Al
0.65
Bandwidth
Kd
Ki
1.5
0.001
Pitch to Elevator
Ki
Kp
Kd
Ki
0.01
0.8
1.9
0
limit
Depth to Pitch
K
Kp
Kd
Ki
0
0.13
0
0.001
limit
Ki
limit
0.004
0.0015 rad/sec
2.5733 rad/sec
0.2056 rad/sec
Table 8.1: Initial controller gains based on the initial model
The pitch to elevator controller, known as the pitch loop, did not have integral control, because steady
state error is only of interest in depth and heading. The depth to pitch controller, known as the depth
loop, did not have derivative control, because the conservatively designed proportional control allows for
little overshoot in field response. These gains were the values initially used in the field tests to control
Caribou to a desired depth and heading, which allowed for the open loop tests to commence, as explained
in the following chapter.
104
angle between 0 and 15 degrees. Note that only positive elevator angles were used to avoid hitting the sea
floor since the tests took place in shallow water.
Using the results from these various rudder and elevator step response tests, a more accurate
model was developed, and consequently the controller was redesigned to improve the maneuverability of
Caribou underwater as outlined in Figure 1.2.
9.2
Results from the Field
During a typical System Identification mission, the AUV was controlled to the specified depth and
heading. At this point the first step actuation would occur, in either rudder or elevator, but not both. The
actuation step would last 10 seconds, and then the vehicle would be controlled again, in both planes, to
the original controlled depth and heading. After the AUV was under control again, the next step response
would happen. During a typical rudder step response mission, the vehicle would under go a series of four
steps. In a typical elevator step response mission, the vehicle would under go only a series of two steps,
since the time needed to regain the controlled depth and heading during an elevator step mission was
much longer than during the rudder step response missions.
Figure 9.1, shows a typical rudder step response mission for commanded rudder steps of -10, 10, 15, and 15 degrees. Notice that the yaw response shows that the commanded and actual rudder responses
are not the same. The actual rudder response has an offset of -6 degrees compared to the commanded
rudder position. Figure 9.3 shows a run in which the rudder was actuated +5 degrees for 30 seconds,
brought back under control for 60 seconds, and then actuated to +7 degrees for 30 seconds. The response
of Caribou with these two commands is nearly equal and opposite, which bolsters the assumption of the
negative 6 degree offset.
50 -
--
I
-
40 -30
3010'1
Thrust(%)
Rudder(deg)
Elevator(deg)
Depth*1 O(m)
-..
..
--
--
20 -10
-
50 -
.....
.
....-
-.
.-.
0
10
-1t
0
-5 0 -1
00
-
-1 5 0 -
-. -- -
-.
.. .
.
....
..
.. .. ..
. . ... .
..
-...-.-.
-...
. ... . ...
-. - -.
yawRate
pitchRate-
10---
roliRate
0'w
-5 --1 0 -
..
--..
300
350
400
450
......
-..
........... . .
500
550
time (s)
Figure 9.1: Rudder step response mission for commanded angles of -10, 10, -15, and 15 degrees
106
Chapter 9
System Identification
In Chapter 9, the system identification process is explained. In order to validate and improve the dynamic
model of the AUV, various field data was gathered. The model responses were compared to the
responses from the field, for the same series of actuator inputs, namely thrust and rudder and elevator
deflection angles. The dynamic model was then improved so that rates from the simulation model more
closely represented rates seen in the field. Open-loop stability of the system is addressed as well as
turning rate, and turning radius. The system identification and model adjustment process leads to an
improved model as outlined in Figure 1.2.
9.1
System Identification Process
The first step in control system design is to develop a reasonably accurate model of the system that you
are trying to control. The second step is then to use this model to develop a control system that meets the
determined criteria for adequate control for that particular system. For Caribou, the nonlinear model is
shown in Chapter 4, while the decoupled linear model in the yaw plane and the pitch plane is shown in
Chapter 5, with the tailcone model developed in Chapter 7. The initial control system was developed
based on the linear yaw and pitch models, Chapter 8. The goal of the control system design is to control
the AUV within 10cm of the desired depth, as well as within 2 degrees of the desired heading.
Generally, an AUV such as Caribou will be tuned heuristically in the field, because in most cases
the vehicle can be roughly tuned to control depth and heading after a short amount of time in the field.
However, this rough control generally is refined by adjustments made to the controller during much
further testing in the field. In addition, when the payload changes, the controller needs retuning. While
this heuristic approach has worked satisfactory in the past, a more streamlined approach has many
advantages over the numerous days of trial and error work in the field. The goal of this work was to
develop an initial controller based on textbook models of the AUV. Using this controller, we could
control the AUV to a desired depth and heading, accurately enough, so that we could turn the control
system off and perform designed open-loop maneuvers. These open-loop maneuvers consisted of simple
step responses of various angles for both rudder and elevator during separate missions. The thrust was
kept constant during these step response tests, and the control system in the opposite plane was left on.
Therefore, during a rudder step response mission, Caribou dove to a prescribed depth, maintained a rough
heading and depth and then the heading controller was turned off while the rudder angle was set to a
specific angle between 0 and ±15 degrees. Likewise in an elevator step response mission, at the
prescribed depth the depth and pitch controllers were turned off while the elevator was set to a specific
105
In Figure 9.1, some periods of saturation are seen, as the controller tries to recover from the step. In this
figure, the rudder angles of -10, -15, and 15 degrees show good clean responses. Figure 9.2, shows a
typical elevator step response mission for commanded elevator steps of 3 and 5 degrees. The response to
the 3 degree step is not as clean as the response for the 5 degree step. An elevator bias was not detected
in the data or in the laboratory.
60 -
-...
-.......
__Thrust(%)
40-
-
---
Rudder(deg)
Elevator(deg)
Depth*1O(m)
20-
v
0
p
- -
40 20-00-0
S -2 0 -
-.
S -4 0 -
-. -.
--- -.
-6 0 -
.. ..
20
.. - ~~.
..
- -.
-.
.
.
-. -.
....... ......
....
..
..
-...
..-..
.- ..-. ... -- .-..
. ..... .
..... ..........
--
-8 0 -15
10 -
-..
. ..
-.. ..-
. ..
-.
..
-..
.-...
..
--.
.. -..
...
. -...
...h..-.
-.. ..
-
- -
--
-..
.-.
-.
.-...
-...
--.-.
-.-.
-- yawRate
pitchRate
-
time (s)
Figure 9.2: Elevator step response mission for commanded angles of 3, and 5 degrees
9.3
Model Adjustments
9.3.1 Model Adjustment Procedure
After the open-loop response data was gathered in the field, the dynamic model parameters were adjusted
to better fit the field data responses. Figure 9.4 shows the procedure used to adjust the model parameters.
The entire adjustment process was completed using Matlab software. The simulation block consists of a
3-degrees of freedom (DOF) yaw plane and pitch plane, and a combined 6-DOF simulator as well. The
process of parameter adjustment begins with picking which portions of data will be modeled. Thus, the
recorded field data is screened for the step responses that provide consistent rates and no apparent
unrepeatable system disturbances. For example, in simulating a yaw plane run, seven of the available
step response runs were used out of a possible 14 runs. These seven runs were for rudder angles smaller
than 10 degrees, and all showed yaw rate transients with minimal pitch and depth response. Figure 9.3
shows a typical suitable yaw plane run for steps of 5' and 70 with rudder bias of -6'. Likewise, in
choosing elevator step response runs for the simulation, runs with minimal heading change, and smooth
pitch rate transients were chosen. Figure 9.5 shows a typical suitable pitch plane run for a 100 step. A
107
decision must also be made on which parameters will be adjusted in the simulation and which parameters
will be held constant, as well as the number of iterations the simulation will be allowed to step through.
The factor here that was considered was that a balance between a small number and large number of
iterations was needed so that the simulation would not over minimize the error and provide impractical
model parameters, and at the same time, converge long enough on a viable solution.
-
- ---
5 0 ----40 -
-
-
...
30 -
- . - -. .
-
20 --
---..-
10 -
-
-
-.
.
.
7
-
_
Elevator(deg)
---
-.-.
Thrust(% )
Rudder(deg)
Depth*10(m)
-
0-
-
-50 -
0
-10
50 -~
0
~ ~~~~
3..4..45
3.
20
o..........t....
.....
A
V
-
Nm
-10V_
nryawRate
pitchRate-
ropmRate
250
400
350
300
450
time (s)
Figure 9.3: Typical usable rudder step response mission
Field response
trajactory
Nominal
parametersI
Adjusted
Parameters
Simulation --w RMVS rate error
I
Nelder-Mead modifies parameters
to minimize RMS rate error
Figure 9.4: Model adjustment simulation process
108
Thrust(%)
Rudder(deg)
Elevator(deg)
Depth*10(m)
60 4020
\
0
0
4)
-.. . ....
.
-..
-50
-1 0 0
. -. -...
.. .....
...t.
-..
....... .........
p..
!
260
time (s)
yawRate
pitchRate
-rollRate
-
240
220
200
-
20
-
280
300
320
Figure 9.5: Typical usable elevator step response mission
Given these initial input parameters, response runs, and number of iterations, the simulation would
proceed iteratively searching for a solution that minimized the difference in the yaw rate of the field data
with the yaw rate of the simulated response. In order to maximize the effectiveness of this process, only
3-DOF simulations were used. Hence, in the pitch plane simulation, the pitch rates were compared and
the difference was minimized. The rates of all of the included runs were used simultaneously to establish
an average error each time through the simulation. This allowed various step responses with various
deflection angles to all be encompassed together with one simulation. In order to determine which
coefficients needed to be adjusted, the transfer function of the AUV was calculated for the yaw plane in
equation 5.70, and for the pitch plane in equation 5.111.
Y
Z, M,
NY
N,.YsNsRzY,
N
m U xcM
Zq Mq ZSE MSE ',, Zw Mqm W UxcM ZCM
(9.1)
(9.2)
These parameters have a direct influence on the transfer function of the system and the response of the
linear model simulation. However, not all of these parameters need to be adjusted. The mass, m, weight,
W, speed, U, and center of mass along the x-axis xcM are quantities that are accurately known. In
addition, the parameters Izz and Nrdot are always coupled as (Izz-Nrdot) and thus constitute only one
parameter. Likewise, in the pitch plane (Iyy-Mqdot) is coupled. One further reduction in parameter
uncertainty comes from the instantaneous acceleration rate per deflection angle. From equation 5.76, and
5.114, the instantaneous acceleration rates per deflection angle are as follows. These relationships are
used in Section 9.3.2 to reduce the number of uncertain parameters.
109
(
-=
9.3.2
J
N
1
SM8E
-~
R
(5E
yy
(9.3)
MJ-
Acceleration Analysis Direct from Data
Using the data from the field tests, for various deflection angles, the initial acceleration rate was
measured, Table 9.1. This acceleration rate was then non-dimensionalized by dividing by the deflection
angle. The average initial acceleration rate was then determined to be -0.51 sec-2 in the yaw plane.
Rudder
Deflection
Yaw
Acceleration
Acceleration/
Deflection
Angle (deg)
(deg/sec 2 )
Angle (sec-2 )
-15
-15
-15
6.5
9.5
9.5
-0.43
-0.63
-0.63
-11
-9
5.0
4.0
-0.45
-0.44
9
-4.5
-0.50
4
4
-3
-1
-1
-1
1
1
-2.0
-2.0
1.4
0.5
0.6
0.7
0.8
0.5
-0.50
-0.50
-0.47
-0.50
-0.60
-0.70
-0.80
-0.50
Average
-0.51
Table 9.1: Yaw initial acceleration rates
Using the relationship from equation 9.3 and this acceleration rate,
as shown.
N
R
(IZZ -NJ=
(Izz-Nrdot)
can be directly related to Ndr
(9.4)
8
-0.51
Likewise, the initial pitch acceleration rate was determined to be -0.46sec-2 as can be seen from the data in
Table 9.2.
Elevator
Pitch
Acceleration/
Deflection
Acceleration
Deflection
Angle (deg)
(deg/sec2 )
Angle (sec 2 )
5
10
15
L:_
-2.5
-4.4
-6.5
-0.50
-0.44
-0.43
Average
-0.46
Table 9.2: Pitch initial acceleration rates
_
Using the relationship from equation 9.3 and this acceleration rate,
Mde as shown.
110
(Iyy-Mqdot)
can be directly related to
I
- M
q
SM8
6E
- 0.46
Therefore, the parameters to be adjusted in the yaw plane are: Y,
pitch plane they are: Z
9.3.3
M
Zq Mq
Z'E
M5E
(9.5)
=
Z
N,
Y, N,
Y,
N8 R
Y, and in the
z CM
Model Adjustments Made
The original dynamic model, model A which is explained through Chapter 4 and 5, was used to determine
the initial controller, controller Al that was developed in Chapter 8. This initial model was adjusted
based on the first day of system identification tests. This first adjusted model was model B, as shown in
Table 9.3.
Model A->B
Initial
Model A
Adjusted
Model B
Nrdot
-570
-390
428
-858
354
-644
-632
469
-458
58%
1.5m/s
-596
-408
438
-841
361
-576
-632
469
-458
Zw
Mw
Zq
Mq
Zde
Mde
Zwdot
-570
390
-428
-858
-354
-644
-632
Thrust
U
Yv
Nv
Yr
Nr
Ydr
Ndr
Yvdot
Izz
zCM
lyy
Mqdot
Model C1 3
Percent
Change
Initial
Parameters
Adjusted
-4%
-4%
-2%
2%
-2%
12%
0%
0%
0%
-449
-239
266
-525
251
-447
-632
469
-458
58%
1.3m/s
-445
-159
331
-520
264
-513
-616
548
-458
-416
422
-455
-984
-384
-548
-632
37%
-8%
-6%
-13%
-8%
18%
0%
-449
239
-266
-525
-251
-447
-632
-0.021
-0.021
0%
469
-458
469
-458
0%
0%
Extra olated from C1 3
Percent
Change
Model
C1_0
Model
C1_5
Model
C2_0
-1%
-33%
24%
-1%
5%
15%
-3%
17%
0%
40%
1.0m/s
-368
-123
254
-414
211
-407
-616
548
-458
58%
1.5m/s
-496
-183
382
-590
301
-584
-616
548
-458
80%
2.0m/s
-624
-242
510
-767
391
-760
-616
548
-458
-450
157
-343
-599
-270
-514
-652
0.2%
-34%
29%
14%
8%
15%
3%
-372
122
-263
-475
-215
-408
-652
-502
181
-396
-682
-307
-585
-652
-631
239
-529
-888
-399
-762
-652
-0.021
-0.015
-29%
-0.015
-0.015
469
-458
659
-458
41%
0%
659
-458
659
-458
-0.015
659
-458
Table 9.3: Original and adjusted models A, B, and C
The second day of system identification tests allowed for improvements in testing, such as
controlling the alternate plane, and longer step durations. Using this second set of data, model C was
created. The adjustment and simulation process that created model C from the field data of the second
tests, was modified from the initial adjustment process that created model B. Improvements included the
simulation of all files done simultaneously, elimination of poor quality runs, and conducting the
simulations with an AUV speed of U=1.3m/s instead of U=1.5m/s as was done with model B. Also, the
yaw plane model adjustments were completed first, and these improved coefficients were then used as the
initial parameters for the pitch plane simulation, which allowed a closer model/data match. In addition,
111
the initial parameters were redeveloped that better approximated the AUV at the speed U=1.3m/s, as
shown in Table 9.3 under the Nominal Parameters column in Model Cl_3.
Yaw Plane Model
.
s3
A
- 0.694s - 0.433
+1.471s 2 +0.289s
0
B
- 0.662s - 0.414
s 3 +1.454s 2 +0.322s
0
C1 0
- 0.405s - 0.140
s 3 +0.764s 2 +0.102s
0
-1.2369
-1.1818
-0.5923
-0.2337
-0.2724
-0.1716
-0.6229
-0.6648
-0.3465
3.3780x10 5
-1.4964
3.8199x10 5
-1.2843
1.2729x10 5
-1.3795
C1 3
0.5 10s -0.216
C1 5
-0.58 Is- 0.275
C2 0
- 0.756s - 0.454
Poles
Zeros
-Zers
Stability C
Steady State Gain (r/dr)
Yaw Plane Model
s3
+0.948s
2
+0.149s
s3 +1.07s
2
+0.186s
s 3 +1.377s
2
+0.295s
0
-0.7486
0
-0.8523
0
-1.1124
-0.1996
-0.2178
-0.2649
Zeros
-0.4231
-0.4739
-0.6009
Stability C
Steady State Gain (r/dr)
1.8713x10 5
-1.4420
2.3247 x10 5
-1.4822
3.6903x10 5
-1.5409
Poles
Table 9.4: Yaw plane model information
Pitch Plane
Model
Transfer
Function
A
B
C1_0
- 0.694s -0.433
s3 +1.471s2 + 0.429s + 0.063
- 0.592s - 0.323
s3 + 1.468S2 + 0.378s + 0.047
-0.124
s +0.767s + 0.1 90s + 0.024
-1.1442
-1.1813
-0.4724
Poles
-0.1632+0.1693i
-0.1433+0.1381i
-0.1472+0.1711i
-0.1632-0.1693i
-0.1472-0.1711i
-0.3406
2
Zeros
-0.6229
-0.1433-0.1381i
-0.5555
Steady State
Gain (0/de)
-6.83
-7.02
-5.17
Pitch Plane
C1_3
C1_5
C2_0
Transfer
Function
- 0.460s -0.191
s 3+0.954S2 + 0.243s + 0.029
- 0.524s - 0.244
s3 +1.079s 2 + 0.283s + 0.032
- 0.682s - 0.403
s3 +1.390S2 + 0.403s + 0.041
Poles
-0.6497
-0.1522+0.147 1i
-0.1522-0.1471 i
Zeros
-0.4156
-0.7646
-0.1573+0.1332i
-0.1573-0.1332i
-0.4660
-1.0399
-0.1750+0.0929i
-0.1750-0.0929i
-0.5902
-00
-00
-00
-6.57
-7.52
-9.87
Zers-0
Model
Steady State
Gain (0/de)
112
-0.365s
3
,
III
Table 9.5: Pitch plane model information
The simulations were completed with the slower speed for several reasons. First, during the step
responses, the speed of the AUV dropped quickly to around 1.3m/s instead of the steady state speed of
around 1.5m/s at 58% thrust. Second, the speed of the AUV, as recorded on the Doppler Velocity Log,
fluctuated from 1.2m/s to 1.5 m/s during most of the step response mission which was most likely due to
the slight pitching and rolling of the vehicle, as well as loss in speed. While we were primarily interested
in the initial response dynamics, the simulation included the first 5 to 10 seconds of the step response,
where the speed of the AUV was predominately 1.3m/s.
After generating model C at 1.3m/s, the results were extended to speeds of U=1.5m/s, shown in
Table 9.3, as well as speeds of 1.0m/s and 2.0m/s as shown in Table 9.3 for thrust levels of 40% and 80%
respectively. Notice that Izz and Iyy were adjusted based on the relationships shown in equations 9.4 and
9.5. Parameters Nrdot and Mqdot were held constant. The transfer functions, as well as steady state values,
open loop poles and zeros, and stability factor are shown in Tables 9.4 and 9.5 for the yaw plane and pitch
plane models, respectively.
While the initially developed coefficients were the same for both the pitch and yaw plane models,
model A, the adjusted models B and C, do not maintain this strict symmetry as is seen in Table 9.3. Most
of the coefficients are similar between the two planes, within 5%-10%. These larger differences are likely
do to some lack of sufficient data in the pitch plane, and perhaps may also be due, in part, to some error in
the adjusted model. The one major difference is that between Iz and Iyy, which may be due to the
ballasting of the particular payload used during the system identification tests.
After developing model C, at U=1.3m/s, the new set of parameters was individually varied by 10% to see
the resulting change on the averaged Rate Error Squared. These changes are displayed in Table 9.6.
Notice that the most sensitive coefficients have a direct effect on the yaw or pitch rate of the system.
When Nr was varied by 10%, the error changed by 32%, but when Mq was varied by 10%, the error
changed by 13%. Similarly, when Ndr was varied by 10% the error changed by 21%, but when Mde was
varied by 10%, the error changed by only 3%. These large differences, between the two planes, are not
what were expected. However, these differences could be attributed to a lack of sufficient data in the
pitch plane, in which only four runs were used in the simulation, while seven were used in the yaw plane
simulation.
Rate
Model C
Yv+10%
Nv+10%
Yr+10%
Nr+10%
Ydr+10%
Ndr+10%
Yvdot+10%
Error
Squared
0.5152
0.5219
0.5347
0.5190
0.6802
0.5186
0.6214
0.5163
Model C
Zw+10%
Mw+10%
Zq+10%
Mq+10%
Zde+10%
Mde+10%
Zwdot+10%
zCM+10%
Table 9.6: Rate error
Percent
Difference
Difference
-0.0067
-0.0195
-0.0038
-0.165
-0.0034
-0.1062
-0.0011
-1.3%
-3.8%
-0.7%
-32.0%
-0.7%
-20.6%
-0.2%
0.3611
0.3871
-0.026
-7.2%
0.3338
0.0273
7.6%
0.3736
-0.0125
-3.5%
0.4064
-0.0453
-12.5%
0.3557
0.0054
1.5%
0.3489
0.0122
3.4%
0.3603
0.0008
0.2%
0.4529
-0.0918
-25.4%
squared sensitivity to parameter changes
113
Using several more pitch plane runs could make the error deviation more sensitive to changes in
the parameters. The most important thing to point out is that these step response tests may not be enough
to accurately model these parameters. This simulation adjustment process simply reduces the error
between the field response and the simulation response, by adjusted the listed parameters, therefore, more
elaborate system identification tests may provide more expectant results, for model parameters.
9.4
Stability and Verification of the Improved Model
9.4.1
Vehicle Stability
The adjusted dynamic model of the AUV shows that Caribou is open-loop stable in both the pitch and
yaw planes. It has been generally thought that the vehicle was in fact open-loop unstable. However,
several indicators point that the adjusted model is in fact accurate in showing open-loop stability. The
first indicator is the stability factor.
C =Y,(N, -mxcU) + N,(mU -Y
(9.6)
If the stability factor C is positive then the open-loop stability of the model in the yaw plane is shown to
be true [20]. The value of C is shown in Table 9.7 and it shows a positive open-loop stable system. The
eigenvalues of the open-loop model are shown to be negative in Table 9.7 which also indicates open-loop
stability. Note that the zero eigenvalue in the yaw plane is due to the fact that there is no restoring force
as there is in the pitch plane.
Pitch Plane
Speed U
Model C1 3
1.3m/s
Model C1 5
1.5m/s
Eigenvalues
-0.6497
-0.1522+0.1471i
-0.1522-0.1471i
-0.7646
-0.1573+0.1332i
-0.1573-0.1332i
Open-loop zeros
0/dE (steady state)
-0.4156
-6.57
-0.4660
-7.52
Yaw Plane
Eigenvalues
Model C1 3
0
Model C1 5
0
-0.7486
-0.8523
-0.1996
-0.2178
Open-loop zeros
Stability Factor C
-0.4231
1.871x10 5
-0.4739
2.324x10 5
r/dR (sec 1) (steady state)
-1.44
-1.48
Table 9.7: Stability indicators
Finally, looking back at Table 9.1, for small rudder deflection angles, only a modest initial yaw
acceleration occurs (- 0.5 deg/sec for Ideg rudder deflection) compared to the large initial yaw
acceleration for larger deflection angles (~9.5 deg/sec for 15deg rudder deflection). This indicates that
the rudder angle drives the response because the yaw acceleration doesn't appear to be the similar for
smaller perturbations as it is for the larger perturbations. Likewise, the pitch plane displays the same
behavior, as seen in Table 9.2. From equation 5.74, the steady-state turning rate of the AUV per rudder
deflection angle is:
r
(R
114
YN' +N Y(
Y
xc U - N, )+(mU-Y(M
)N
(9.
Table 9.8 shows the calculated rates and turning radii computed from model Cl_3. From equation 5.112,
the steady-state pitch angle of the AUV per elevator deflection angle is:
O
-Z W
M'E +MWZ5E
E
ZWWzcM
(9.8)
Table 9.7 shows the steady state pitch angle per deflection angle for model C _3 and Cl_5.
Rudder Angle
r (rad/sec)
r (deg/sec)
R = U/r (m)
U (m/s)
5deg =0.087rad
I0deg =0.175rad
l5deg =0.262rad
5deg 0.087rad
10deg 0.175rad
l5deg =0.262rad
-0.125
-0.251
-0.376
-0.129
-0.258
-0.386
-7.163
-14.325
-21.488
-7.392
-14.783
-22.175
10.4
5.2
3.5
11.6
5.8
3.9
1.3
1.3
1.3
1.5
1.5
1.5
Table 9.8: Turning rates and radii
9.4.2
Model Verification in the Pitch Plane
During the system identification testing period, an attempt at heuristic tuning was tried with Caribou. The
result is shown in Figure 9.6. The attempt only provided a less stable controller. However, we used this
run to determine frequency characteristics of the AUV in the pitch plane. From this data, a natural
frequency period of 12.5 seconds was discerned. The magnitude and phase of the transfer function are
shown below in Table 9.9 for the data run and the A, B and Cl_3 model. Model B and Cl_3 do a much
better job of modeling the pitch plane transfer functions.
Field Data
Transfer Functions
Elevator to
Pitch to
Elevator to
(Figure 9.7)
Pitch
Depth
Depth
Magnitude
1.33 rad/rad
2.59 m/rad
3.44 m/rad
Phase
-3.5 deg
-111 deg
-114.5 deg
Model A
Magnitude
1.71 rad/rad
2.98 m/rad
5.10 m/rad
Phase
34.78 deg
-90 deg
-55.2 deg
Magnitude
1.32 rad/rad
2.58 m/rad
3.42 m/rad
Phase
33.0 deg
-90 deg
-57.0 deg
Magnitude
1.39 rad/rad
2.59 m/rad
3.60 m/rad
Phase
28.8 deg
-90 deg
-61.2 deg
Model B
Model C1 3
Table 9.9: Pitch plane transfer functions
These results show that the adjusted model transfer functions approximate the actual responses that were
seen in the field. Differences in phase may be due to recording time delays, and unmodeled dynamics.
115
50
40
30
20
10
0
---- Thrust(%)
-.
- . -. .Rudder(deg)
.
Elevator(deg)
-.-.
-
Depth*10(m)
-10
40
..... I .......
.......
....... __
_
__
_
_
[.
..
. . ..
. .
20
0
-20
~~..........
..
. .........
......
............
.........
......
. ....... ......
.............- .....
-40
pitch
-
-60
--
-
15
*
a
*
-.
10
-
.
..
. .
.
.
.
.
.
.
..
1
___yawRate
.pitchRate
rol-Rate
-
5
0
-5
. ............
-
-
.....
.
~~~~~....
...
. .
.
.
.
.
-10
300
.
.
..
.
--...
320
360
340
380
400
....
....
.....
.......
420
-....
-...
440
460
480
time (s)
Figure 9.6: Closed-loop straight run with heuristically tuned controller
9.5
Model Comparisons
The initial set of coefficient values used in modeling Caribou, is referred to as model A. Model A was
used to develop the initial controller, controller Al. After the first day of system identification tests, this
model was adjusted to better fit the data from the step tests. This improved model is model B. Model B
was used to improve controller Al, and this improved controller, controller B, was then used on the
second day of tests during closed-loop maneuvers. The improved controller provided a much better
controlled AUV than controller Al did, as is explained in Chapter 10 and 11. After the second day of
system identification tests the step test data was used to create another improved model, model C as was
explained in Section 9.3.1. Figures 9.7-9.10 show the model approximations to the recorded data. Table
9.10 lists the rate error for each model as compared to the field data rate. Notice that model B is more
accurate than model A, and model C appears to most closely model the AUV dynamics.
Average Rate ErrorSquared (deg/sec)
Mission
19 39
19 39
Angle
100 rudder
150 rudder
Model A
0.62
0.37
Model B
0.56
0.22
Model C_ 3
0.32
0.11
20 35
50 elevator
0.94
0.38
0.36
22 08
150 elevator
7.96
4.10
3.71
Table 9.10: Model improvements
116
Figure 9.7 and 9.8 show step responses for rudder angles of 10' and 15*. The raw data from the field is
shown, as well as the responses for the various models. Notice that model B and model C are both better
approximations to the actual response than model A. Therefore, heading controller designs based on
model B or C would be thought to have better results when tried in the field. This is actually the case, as
is explained in Chapter 10 and 11.
0
-.. -. -. -.
-1
-2
- ....
-3
-........-. ... .
.....
Data
ModelA
odelB
ModeIC1.3
-M
-.
-.
-..
....
CA
-4
. .
-.
.. - ..
.....-..
.. .
..
-.
. .....-..
....
.....
. .. .. ..-...-.
-........
-..
...
-..
......................--..
...
-5
376
377
378
380
379
381
383
382
384
time (s)
Figure 9.7: Yaw model improvements, 100 rudder angle
Figure 9.9 and 9.10 show step responses for elevator angles of 50 and 150. Again, the raw data from the
field is shown, as well as the responses for the various models. In the pitch plane, model B and model C
are both better approximations to the actual response than model A. Therefore, pitch and depth controller
designs based on model B or C would most likely have better results when tried in the field. Chapter 10
and 11 explain the improvements seen in the field.
Data
-
ModeLA
. .
-4
-
C.,
0
ModelB
ModelC1.3
-
-6
c,
-8
............
..
....
...................
-.
.....
...... ............... ..
..-.
....-.
-101
457
458
459
460
461
462
463
464
time (s)
Figure 9.8: Yaw model improvements, 150 rudder angle
Notice, that in these figures the adjusted models, model B and C, do not perfectly model the field
data. One of main reasons this does not happen, is that the simulation adjustment process take into
account numerous runs with various deflection angles. The model is adjusted to approximate all of these
runs simultaneously and therefore, more exact results are difficult to attain. Due to slight changes in the
alternate plane as well as environmental effects, the actual rates for each step response are not textbook
data sets. However, these several figures show that measurable improvements in modeling are made by
simply using step response tests. To develop a very precise model, several other system identification
tests would need to be used in the field to capture the various dynamic effects. However, for
improvements in control design, this simple approach works well.
117
I
...........
............... .......... ...
.....
......... ...... ............ .......
. ..........
. ...
Data
ModeLA
ModelB
ModeIC1.3
....... .......... .......................
-2 ..................
....
........... .........
- ......
........
.......
......
...
.. .........
.........
....
.........
-4 . ....
...
......... ..
......
......
........
........
-3 .......... ........ ..
.....
.....
........ ......... . ....
-304
303
302
301
30E
307
306
305
t7
time (s)
Figure 9.9: Pitch model improvements, 50 elevator angle
........
0
........
..........
........... ......... ...
..............
.......
..
. ...........
-2
-4
............ .............
.....
.........
.......
Da t a
Mode[A
...........
.............
M od e lB
ModeIC1.3
...................
............
........................... ............... ...
...
.........
-6
.....
.......
..
......
...
-10 . ...
..
....
..
...
....
. ..
..
..
..
....
..
....
...
...
....
.....
. ..
. ....
.....
.
. ..
..
..
..
....
...
..
...
...
..
...
..
....
.
....
..
.....
...
..
..
.
..
..
..
.....
-8 ...
.....
..
.....
..
..
...
.
..
....
.....
..
..
....
...
..
...
....
.......
.
.................
-12
334
335
336
337
-
338
339
time (s)
340
341
342
Figure 9. 10: Pitch model improvements, 15' elevator angle
118
..............
............
343
344
Chapter 10
Controller Redesign
The controller redesign is explained in detail in Chapter 10. The closed loop poles of model A, B and C
with the various controller gains are shown in Table 10.2. Also, these controller gains are evaluated at
several other thrust levels to determine if stability remains at other speeds. These results are compared
with tests completed in the field. Appendix B shows all of the Root Locus plots for the various models
and controller systems. The controller redesign is the final part of the control design process outlined in
Figure 1.2.
10.1
Root Locus of Models and Controllers
10.1.1 Controllers
Controller Al was developed prior to the system identification tests as explained in Chapter 8. Controller
A2 was an attempt to tune the AUV heuristically in the field after running Caribou with the original
controller, Al. This attempt resulted in a less controlled vehicle than with controller Al, as seen in
Figure 9.6. After the first day of tests, model B was developed as an improvement to model A, as
explained in Chapter 9. Using model B, an improved controller was developed and used during the
second day of tests. Table 10.1 lists the various controller gains used during the field tests.
Heading
Controller
A_
Al
A2
B
_
Kp
Kd
Ki
_.65
_.5_
_._
0.65
0.65
0.55
1.5
1.5
1.35
0.001
0.001
0.01
Pitch to Elevator
limit
_._
0.01
0.01
0.1
Kp
_.8i
0.8
1.3
0.55
Kd
_.9
Ki
_
1.9
0
1.9 10
1.2
0
Depth to Pitch
limit
Ki
Kp
K
0
0
0
0.13
0.09
0.12
0
0
0
Ki
Ki
limit
0.001
0.001
0.004
0.004
0.004
0.05
.
Table 10.1: Controller gains used in the field
The heading controller controls on the error in heading angle and has output of desired rudder.
This controller is aggressive and works to eliminate steady state error as well. The depth to pitch
controller controls on the error in vehicle depth, and has output of desired pitch. This outer depth loop is
only as quick as the AUV is capable of reaching desired depths and pitch angles. The depth loop is
slower than most loops, and thus does not have a derivative gain. The pitch to elevator controller controls
on the error in the vehicle's pitch. This loop has output of desired elevator.
119
Model
Controller
Heading Poles
-5.2235
A
Al
Bandwidth (rad/sec)
B
A
Bandwidth (rad/sec)
B
B
Bandwidth (rad/sec)
CI_5
Al
Bandwidth (rad/sec)
CI_5
A2
C1_5
B
Bandwidth (rad/sec)
-0.8257 1.8075i
-0.4480 ± 0.1984i
0.0015
2.5733
-5.2415 6.3269i
-7.37843.
-1.1043 1.73330
-0.3937 0.20271
-0.0015
-11.2392
-4.1095 ± 6.6275i
-0.9928 ± 1.7040i
-0.3823 ± 0.2137i
1.8994
-5.2658 ± 6.2422i
-7.1316
-1.2322 ± 1.6775i
-0.3559 0.1762i
-0.0196
1.3804
-5.213
-5.2413 12534
2.2141
-10.8638
-4.0079 6.3945i
-1.3313 ± 1.3031i
-0.3331 0.2520i
-1.0730 1.5458i
-0.3568 0.1897i
-0.0015
-4.0583 +6.5558i
-0.8941 + 1.47261
-0.4056 ± 0.1637i
0.0015
B
Bandwidth (rad/sec)
Depth Poles
-11.3711
-4.1207 ± 6.72521
-0.7766 t 1.69551
-0.4387 60.2747
-0.1603
-0.0081
1.1977
6.234i-11.0888
-11.1038
2.1066
6.234i-11.0724
-11.0827
0.2056
-11.2252
-4.0959 ± 6.6247i
-0.9631 + 1.6042i
-0.3585 ±0.2787i
-0.1404
-0.0082
0.0069
-10.8538
-3.9984 ± 6.3936i
-1.3264 ± 1.2082i
-0.2861 0.3015i
-0.0664 ± 0.0099i
0.1991
-4.0439 ± 6.553 i
-0.8277 ± 1.3390i
-0.4072 ±0.2356
-0.1654
-0.0081
0.2187
-5.2413
-5.213
12534
-1.0730
1.5458i
-4.0381 +6.5530i
-4.0283 . 6.5510i
-0.6464 + 1.2232i
0.1897i
-0.7254
± 1.31981
-0.6353 4 0.18921
-0.0015
-0.6051 ± 0.14301
0.0015
2.1178
-0.1152
-0.0123
0.1477
-5.2666 ± 6.1712i
-10.7623
-3.955 .76.34671
-1.2057 1.4678i
-0.3186 + 0.1775i
-0.0195
-3.9652 + 6.3476i
-1.1786 ± 0.96191
-0.3851 ± 0.2200i
0.0161
-5.2232
C2_0
-11.3885
-4.1377 ± 6.7295i
-7.6623
-1.0086± 1.8438i
-0.3737 ± 0.2152i
-0.0015
-0.3568
Bandwidth (rad/sec)
6.4265i
Pitch Poles
-5.232
6.40451
-0.9938 1.8113i
-0.3673 0.1645i
-0.0195
2.1718
-1.1520 +0.7876i
-0.3483 +0.30221
-0.1074
-0.0496
1.3150
0.2373
1
6.445i-10.942
-10.9588
-4.0136 ± 6.4554i
-1.1147 + 1.1888i
-4.0295 + 6.4576i
-0.3536 0.3216
-1.1466 1.36011
-0.1833
-0.4098 ± 0.21671
. 1-0.0415
1.5899
0.3558
Table 10.2: Closed loop poles for model A, model B and model C
120
The pitch loop does not have an integrator gain, because the depth loop has one and the ultimate goal is to
control about a depth and not a pitch, therefore, elimination of steady state error in depth is more
important than elimination of steady state error in pitch. The pitch loop is an inner loop surrounded by
the depth loop which is the outer loop.
10.1.2 Improved Model B and Initial Controller Al
After model B was developed, the Root Locus plots for the system were developed to see where
controller Al needed improvement. The Al heading controller appears to be pretty well designed, except
for the slower poles locations around -1.10+1.73i as shown in Table 10.2. These poles look like they
need to be slightly more damped. Looking at model B, the Al pitch controller design's major drawback
is the location of the poles at -0.99±1.70i. It appears that if these poles could be damped more, then that
pitch loop would be much more stable. The design flaw in the depth loop is that there are two poles
around -0.96+1.60i that are not damped enough. The other highly undamped poles in the loops are
probably fast enough to not affect the system after a short transient time, and that is why focus is on the
slower poles near the origin.
10.1.3 Improved Model B and Improved Controller B
After model B was developed, controller Al was modified, and an improved controller, controller B, was
used for the second day of system identification testing. The improvements in the closed loop pole
placement are shown in Table 10.2. The under damped poles of the various Al control loops were
dampened. The heading poles of interest were moved to -1.23+1.68i, which dampened them. From field
results, the heading system seemed to work well with controller A, and was actually slightly improved
with controller B, as is explained in Chapter 11. Controller B was an improvement for the pitch loop.
The slower under damped poles were increasingly damped using controller B. These closed loop poles
show a pitch loop that is appropriately damped for a quick, but stabilizing, response, with poles of interest
moved to -1.33+1.30i. The depth control system now appears to have enough damping which enables a
quick, but stabilizing response, with poles of interest now at -1.33+1.21i.
10.1.4 Improved Model Cl_5 and Initial Controller Al
The significant amount of data collected during the second day of system identification tests, along with
improvements made in the simulation and model adjustment portions of the design, allowed for a third
and most accurate modeling of the AUV, model C. This most accurate model is model C at 1.5m/s. The
closed loop poles of model C at 1.5m/s with the initial controller Al, show similar results to model B with
controller A1 when the results of listed in Table 10.2 are compared. These closed loop poles for model C
are shown in Table 10.2. Like was seen with model B, this controller handles the heading loops well.
Only slight improvements seem to be needed in adding damping to the poles at -1.07+1.55i. The pitch
loop seems to need improvements in damping to the poles at -0.89 1.47i, as was also noted for the model
B. Likewise, the depth loop appears to need an increase of damping to draw a couple of the slower poles
closer to the real axis, namely poles at -0.83 1.34i.
10.1.5 Improved Model Cl_5 and Heuristically Tuned Controller A2
An attempt was made at heuristically tuning controller Al, however, this new controller, A2, proved to be
even less effective. The heading controller for Al and A2 are the same, and thus the closed loop poles are
the same. For this model and controller, notice that slower poles, in the pitch and depth loops, are even
less damped than with controller Al. This leads to a very unstable system that has uncontrollable
121
oscillations as seen in Figure 9.6. The pitch loop has poorly damped poles at -0.73± 1.32i. The depth loop
also appears to be less stable with controller A2 when compared to the system as controlled by Al. This
depth loop has several slow, highly under damped poles near the origin, -0.65±1.22i, that lead to this
oscillatory response. This combined with the instability of the pitch loop, leads to an oscillatory behavior
that remains uncontrollable as shown in Figure 9.6.
10.1.6 Improved Model C1_5 and Improved Controller B
The most accurate model of the AUV, model C, and the improved controller, B, provide closed loop poles
much improved from the original design. Here the closed loop poles of interest in the yaw plane are 1.21±1.47i, which are greatly improved from the original design. In the pitch loop, the poles of interest
were damped significantly to -1.1 8±0.96i, while the poles in the depth loop became increasingly damped
to -1.15±0.79i.
These significant improvements in controller design show that the system identification process
used here, not only works well, but produces results in a simple forward design process. Figures 10.110.3 show the Root Locus plots for this most accurate model of the AUV, C, and the most accurate
controller, B.
20
6
15
P7 J
-,-7
Z4
4
10
P5
I.Z41
5
2
P4-
0
-5
-
-
-
E
P8
Z5
-2
-10
P6
-4
-15
P8
Z5
-6
-20
-30
-20
-10
0
Real AMs
10
20
-8
-10
-6
-4
-2
0
Real A~s
2
4
6
8
0.03-
0.2 9
0.02
0.1
-
0.01 -
Z3
0
PI, P2
Z2
P3
- ----------- ----ZI
P1, P2
ZI
-0.01
-0.2
-0.02
-0.2'-
-0.03
-0.6
-0.5
-0.4
-0.3
Real
-0.2
-0.1
0
0.1
as
-0.07
-0.06
-0.05
-0.04
-003
RealdB
-0.02
-0.01
Figure 10.1: Root Locus plot for the heading system, model Cl15 and controller B
122
0
0.01
Figure 10.1 shows the typical Root Locus plot for the yaw plane model and heading controller. Each
change in controller or model changed the loci of the closed loop poles in the Root Locus plot, as can be
seen in Appendix D for all models and all controllers. The closed loop poles are shown in Table 10.2.
The upper right plot of Figure 10.1 is a magnification of the upper left plot. The bottom right plot in
Figure 10.1 is a magnification of the bottom left plot, which is a magnification of the upper right plot.
Figure 10.2 shows the pitch system, which appears to be a well designed controller. The slower poles are
adequately damped, and the response due to the faster poles that are under damped looks to die out
quickly.
20
Z3
15
P5
4
10
P5
5
Z3
2
P7
0-
P4
E
-5
Z4
P6
-2
-10
-4
-15
P6
-6
-
-20
-30
-25
-20
-15
-10
-5
0
Real Axis
5
10
15
20
-12
-10
-8
-6
-4
Real Axis
-Z4
0
-2
2
4
0.3
0.2
1P
0.5
PI
0.1
-P3
Z2 Z I
-
-
-0.5 - -
-
- -
Z2
0
P2
E
ZI
-0.1
--
72
-0.2
-2
-1.5
-1
-0.5
Real ALss
0
0.5
-0.6
-0.5
-0.4
-0.3
-02
Real Anes
-0.1
0
0.1
Figure 10.2: Root Locus plot for the pitch system, model Cl15 and controller B
The upper right plot of Figure 10.2 is a magnification of the upper left plot. The bottom right plot in
Figure 10.2 is a magnification of the bottom left plot, which is a magnification of the upper right plot.
The faster poles, shown furthest from the imaginary axis of the Root Locus plot shown in the upper plots
of Figure 10.2, have underdamped poles. However, the transient response of the AUV due to these poles
dies out quickly, while the slower poles that are closer to the imaginary axis as shown in the bottom plots
of Figure 10.2 produce responses that do not die out quickly. Therefore, the damping ratio, and response
speed of these poles is of the most interest in the controller design. In comparing the results from the
initial controllers, Al and A2, to this improved controller, B, these slower closed loop poles of the
123
improved controller are damped more, and generally have a faster response time. This is true in the yaw
plane, Figure 10.1, as well as this pitch system shown in Figure 10.2. The depth system displays this
same improvement, as shown by the plots in Figure 10.3.
Finally, Figure 10.3 shows the Root Locus plots for the depth loop. Here again, controller B seems to
provide the system with a quick response, and adequate damping.
P
7
1.5
10
1
Z4
P7
0.5
4
P5
2
0
P9
Z3 P3
Z4 4
P4
a 0 0-
P6
-4
P8
-6
Z5
-1
-1
----6
-10C
-1.5
-20
0
Real Axis
5
-1.5
-2
-2.5
-1
-0.5
Real Axis
1
0.5
0
0.4 -
0.3 -
0.06-
P3
0.2-
0.040.1
0.02-
Z
Z3_Z2
0
P1, P2
V
-0.1
PI, P2
F
-0.02
-0.2
P4
-0.04
-0.3 -
-0.06
-0.4
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
Real Axis
-0.1
0
0.1
0.2
0.3
-0,14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
Real Axis
Figure 10.3: Root Locus plot for the depth system, model Cl_5 and controller B
As can be seen in Figure 10.1-10.3, controller B does an excellent job of controlling the system that is
modeled as model C, which is the most accurate model developed thus far. Based on the improvements
made in the simulation and the data taken in the field, Model Cl_5 most accurately models the vehicle,
while controller B is the best designed control system. This most accurate controller, B, is compared with
the initial controller, Al in the closed loop tests shown in Chapter 12. These comparisons show the vast
improvements made in the controller gains, through system identification work.
124
10.2 Controller Gains at Different Thrust Levels
10.2.1 Change of Thrust Field Test
One of the final tests completed was a closed-loop run that incorporated a several different thrust levels.
Controller B was used, and the mission was a simple straight line run at a heading of -80* and 4m depth.
Figure 10.4 shows this run in which the initial thrust was 40%. This thrust was increased to 60% and then
onto 80%. As you can see from the figure, thrust at 40% did not provide enough force to cause Caribou
to break free of the surface effects and dive to the desired depth. The 60% thrust portion of the mission
performed very well, as was expected because 60% thrust provides a speed of around 1.5m/s, which the
control system, controller B, was designed around. Finally, the 80% thrust portion proved to be unstable.
Figures in Appendix D show the Root Locus plots for this system and help explain the instability.
80
-
60
-
-
40 -
Thrust(%)
Rudder(deg)
-.-.-.-
40.... -.....
-
20-
--....-..-....... .
Elevator(deg)
D epth*10O(m )
_
-
-
0
20 -
..
.....
---..
.....
...
.......
0-2200 ---
yawt
...
....
-.-.
-----
..
. ....
...
--..
.........
-...
j.
I
pitchRate
-
___-roflRate
- --
-10 -
250
350
300
400
450
time (s)
Figure 10.4: Field test at 40%, 60% and 80% thrust with controller B
10.2.2 Improved Model C2_0 and Improved Controller B
The accurate model of the vehicle at 1 .3m/s, model Cl_3, was used to develop the model at 1 .5m/s,
model Cl_5. This same extrapolation was used to develop a model of the vehicle at 2.Om/s and 80%
thrust, model C2_0. The closed loop poles from this model, C2_0, and controller B, are shown in Table
10.2. For this heading system the slower poles at -0.9911.81i are underdiamped. Also, in the pitch loop,
underdamped poles exist at -1.15+ 1.36i. In the depth loop, closed loop poles at -1.1211.19i show that the
system may display uncontrollable characteristics as these poles are also underdamped. These closed
125
loop poles show that the system was underdamped in all three loops for model C at 2.0m/s with the
improved controller, B, that was used for the run displayed in Figure 10.4. The vehicle did roll and pitch
significantly during the 80% thrust period, which may be due to all three loops being underdamped. This
rolling and pitching motion removes the decoupled nature of the pitch and yaw plane. Therefore, it can
be assumed that the combination of the underdamped poles of all three control loops, led to the
uncontrollable nature of the vehicle at speeds higher than the control system design speed, which was
1.5m/s.
126
Chapter 11
In Chapter 11, the closed-loop test results are shown and explained. For many of the tests, controller Al
was used as well as controller B. This allows a direct comparison of the initial designed controller and
the redesigned controller for maneuvers under closed-loop control.
Closed-Loop Controller Comparisons
11.1
Tailcone Problems
Prior to completing the system identification tests, the tailcone was tested extensively in the laboratory, as
explained in Chapter 7. The tailcone's timing and response were greatly enhanced through software
modifications made by R. Damus, an engineer at MIT Sea Grant. However, due to the nature of the
stepper motors used in Caribou's tailcone, and the algorithm implemented to control these motors, there
were still persistent problems with the tailcone's response behavior.
The stepper motors tend to lose their absolute position count, and therefore, software checks this
position on a regular basis to correct the drift. However, due to the way the tailcone was implemented,
the actual zero rudder position is offset by -6 degrees. Therefore, this check causes the rudder to swing
negative at an interval of every 10 seconds, which is the checking frequency. Hence, heading control
about steady state is hard to achieve with great accuracy as is seen in the closed-loop control missions in
the proceeding figures. It appears that almost every 10 seconds the rudder reacts to this negative swing.
Due to reasons not yet fully understood, at a higher command frequency or at a constant deflection angle,
this negative swing does not occur. Thus, the step responses seemed to return valid data. The elevator
does not experience these effects.
Figure 11.1 shows two sets of plots. The upper plot shows the desired rudder position from the
command, as well as the actual rudder position, as measured by an accelerometer in the same fashion as
in Chapter 7. These two sets of rudder command are then fed into the vehicle simulation that utilizes
model Cl_5. The second plot shows the model simulation yaw rates associated with these two very
different rudder paths. This tailcone problem makes system identification, control and modeling
extremely difficult. However, steps were made in this work regardless. This problem is apparent in the
closed loop runs shown in Section 11.2-11.4. Notice that the heading oscillates nearly every 10 seconds,
as the rudder swing occurs and then is corrected. The pitch and roll of the vehicle are probably coupled to
this occurrence, as well as the depth, but to a lesser degree. Therefore, using an AUV with a better
tailcone system, the closed loop response with this improved controller can be expected to be even better
than the results shown in the proceeding sections.
127
0-
0--
1o
-Actual
rudder
-_Desiredrudder
-0Sim.with actual rudder
Sim. with desired rudder
2560
2580
2600
2620
2640
2660
2680
time (s)
Figure 11. 1: Desired rudder and actual rudder simulation responses
To demonstrate further that controller B was controlling heading better than is observed from data plots, a
straight run mission was simulated for a constant heading of -80*. Figure 11.2 shows this simulated run
that includes the rudder problems seen in Figure 11. 1. Within the simulation, the rudder was swung
negative by 10', every ten seconds, to simulate the results seen in the laboratory, as shown in Figure 11. 1.
After the first rudder correction, the simulator allowed steady state to nearly be reached before beginning
corrections every ten seconds as seen in the lab and in the field. This provides a frame of reference for
how the simulation and vehicle would correct under field disturbances. As seen in Figure 11.2, these
rudder correction spikes cause the heading to change by some 3'. These results are analogous to the
results seen in the field for a controlled constant heading, as seen in Figures 11.3 and 11.4. These
simulation results, helped convince us that the heading controller developed from system identification
tests is in fact as nearly tuned as possible, considering the spiking rudder correction of the tailcone
system.
128
2
_ R dder Corrections
-
-10
-78
-
-
--
---
-- desire~d
- - actual
-
-70
8-
-\1 -\
--
-\
\--j-.
..
.aris..s
C
C...r...er
-77
Siuainsragtrnwthrde
ueorect1.2:
-1Fig-.
4
oreto
rbe
CO)
20
40
60
time (s)
80
100
Figure 11.2: Simulation straight run with rudder correction problem
11.2 Controller Comparisons
The initially designed controller, Al, and the redesigned controller, B, were used separately for the same
missions to verify the improvements made through system identification tests, redeveloped models and
redesigned controllers. Figure 11.3 and 11.4 in Section 11.2.1 display two straight runs at a controlled
heading and a controlled depth. The mission displayed in Figure 11.3 was made using controller Al,
while Figure 11.4 was made using controller B in the field. Notice that the heading control is very similar
for each run, as expected from the gains used, while the pitch and depth control seem to be much better
with the improved controller, B.
In Figure 11.5 and 11.6, two missions that consisted of heading changes at a controlled depth are
displayed. Notice that the transients died out rather quickly for the system that used controller B, while
the system with controller Al only seemed to remain oscillatory. The depth change mission with constant
heading is shown in Figure 11.7 and 11.8. Again, the system using controller B responds much better to
the commands, than the system that uses controller Al. Table 11.1 shows the comparisons between
controller B and controller Al.
Controller B
Controller Al
2.00
±2.50
Heading Error
2.50
+100
Pitch Error
10cm
50cm
Depth Error I
Table 11.1: Closed-loop improvements
129
11.2.1 Controlled Straight Run
-
60
40 -
-..
..
-...
-. -
-
Thrust(%
-Rudder(deg)
20-
Elevator(deg)
j-
~
0*~I
~
0-
I
*1O(rn)
,~Dept
0
-
4)-50
-~~yaw
-pitch
-roll
-150 --I
5-
!
!
yawRate
pitchRate
rollRate
-
........................
0
W
-5
440
420
400
380
460
480
time (s)
540
520
500
560
Figure 11.3: Controlled straight run at 4m depth and -80* heading, controller Al
6 0 t-
..
. -1 1 1 1
40
+* *--.+* ++
1 .1 .
1
. I.
-
--
-
Thrust(%)
-
Rudder(deg)
Elevator(deg)
--- Depth*10(m)
-
..~
..
~~~
...
20
-.
--..
. ......
- *-
] ....-......I . . I1
-.
0
0
U)
4)
0)
4)
0
-
.
-
50
-10 0 -- -------- '
0
4)
U)
4)
4)
0)
4)
*0
-
-.
--- -.....
50
'V,
yawRate
pitchRate
rollRate
a
A
Al
-
w
-5160
180
200
220
240
time (s)
260
280
300
320
Figure 11.4: Controlled straight run at 3m depth and -80* heading, controller B
130
11.2.2 Controlled Heading Change Run
50
Thrust(%)
-
40
30
20 10
0
-10 -
Rudder(deg)
Elevator(deg)
-.-.--.
Depth*10(m)
-
-.
4020
-20
-
-.
-0
-40 -
....
..
.
-
--..
.... ..
-..
..
-
yawRate
pitchRate
roliRate
10
5 -. . . .
......7.
...
. ....
--
-
-20-
10-
...
- -.
-.
-
..
4 ... ..
..
- -...
-.
- -.
-..--..
.
-...
l
-
. -.
20
-5-
-10
350
300
250
200
150
time (s)
Figure 11.5: Controlled heading change from -800 to -600 to -800 at 4m depth controller Al
-
-.- --.
---
Thrust(% )
Rudder(deg)
Elevator(deg)
50
40
30
20
10
Depth*10(m)
0
-10
20
0
a.
-20 - . ......
roll
-40
.
.
.....
..
..
pitch
.
.
.
.
.
-60
-80
yowat
-pitchRate
rollRateq
10
5 -
0
-1 0
...-.
-
300
-...
-
400
350
-.
-.-.-.
..
450
50(
time (s)
Figure 11.6: Controlled heading change from -80' to -60* to -800 at 4m depth, controller B
131
11.2.3 Controlled Depth Change Run
60
1111
A .
'
I
vV
40
V
-
-
..
.
-1
0
-
Depth*10(m)
-
20 -
I
Thrust(%)
-- Rudder(deg)
Elevator(deg)
__
-
(h
0
0
0
0
*0
-yaw
pitch
-1
roll
yawRate
j
1
C.,
0
irolRate
5
(1£
0
0
0
-.
0
-5
....
i
10
......................................................
150
200
250
300
time (s)
350
400
450
Figure 11.7: Controlled depth change from 4m to 5m to 4m at -80* heading, controller Al
60
-_-
40
-
Thrust(%)
Rudder(deg)
Elevator(deg)
Depth*10(m)
20
0
5 1) ,
-. ....
1
.............
I
f
0
U,
0
0
0)
0
*0
-50
-100 10
C.)
4)
C,,
U,
0
0
5
-
.. .. . . .
. .. . . .. . ....
.
. . . ..
. . . .. . . . ..
-
r..
. ..
yawRate
pitchRate
E
Sr
0
0)
0
0
-5
100
150
200
250
time (s)
300
Figure 11.8: Controlled depth change from 4m to 5m to 4m at -80' heading, controller B
132
11.3 Large Coupled Heading and Depth Changes under Control
A very aggressive mission was a coupled depth and heading change maneuver. The individual heading
and depth changes discussed in Section 11.2 were modest tests, while this coupled test is quite aggressive.
This test was completed only with the improved controller, B. The vehicle was commanded to 4m depth
at -80' heading for 60 seconds and then to 7m depth at -40* heading for a second set of 60 seconds. The
third set of 60 seconds commanded the AUV to Im depth at -100* heading. Finally, the last 30 seconds
of the mission was a simple pitch up maneuver. As can see be seen in Figure 11.9, Caribou performed
these maneuvers remarkably well while using the improved controller, B.
80 - 60-
--
- --
-- -
-Thrust(%)
- Rudder(deg)
--.--.-..
-Elevator(deg)
40Depth*10(m)
20-
0 -
..
- .....
-yaw
-100
1
-
y w
te
10rollRate
15 -100 -
- -
--
--
_
-pitch
-..
--..
-----
. ..... . ...
-.......-..
- -Y
-w-a-r- i-- --
- -a-
200
-e
250
300
350
400
time (s)
Figure 11.9: Controlled coupled heading and depth change, controller B
133
11.4 Shallow Im Depth Mission
The final closed-loop maneuver was a shallow, lm depth mission. The thought was that this may be very
difficult considering that surface effects at shallow depths may cause the vehicle to surface. Figure 11.10
displays a im depth run in which controller B was utilized. Caribou was programmed to stay at the
surface for the first 30 seconds, in order to build up speed. This is mainly the reason why there is such a
large overshoot at the initial dive. The remainder of the mission was a success. The vehicle was
programmed to pitch up with 30 seconds left in the mission, which is at time = 280 seconds in Figure
11.10. These results suggest that maneuvering under the Arctic ice may be possible at very shallow
depths, although that case represents a hard boundary as opposed to a free surface.
50-
--.Thrust(%)
-
40 -Elevator(deg)
Rudder(deg)
0(m)
30 -Depth*1
040-
..
-.
---..
.....
........
.....
.. ... ....
-
-80
5 -
-roll
-
yawRate
pitchRate
roliRate
0
y
-5 -
-
--160
180
200
220
240
time (s)
260
280
300
320
Figure 11.10: Controlled shallow run at 1m depth and -80' heading, controller B
134
yaw
pitc-
340
Chapter 12
Conclusions and Future Work
A dynamic model of the Odyssey III Class Autonomous Underwater Vehicle, Caribou, was initially
developed in order to develop a control system for this AUV.
The dynamic model included
hydrodynamic and hydrostatic effects, as well as inertial and added mass characteristics. The tailcone
system, which consists of an adjustable vector duct thruster was tested and modeled in a laboratory
setting. Using this combined nonlinear model, a linearized model was developed about the operating
point. Using this linear model, an initial control system was designed for the heading, pitch and depth
control loops.
A series of system identification open-loop step response tests were completed in the yaw and
pitch planes independently, using the initial controller to reach correct operating conditions. Using these
results, and the simulation model, the linearized coefficients from the initial model were adjusted so that
the simulation response closely matched the response seen in the field data for the same step input in
rudder or elevator commands. Using this improved model, the control gains were redesigned.
Results from closed-loop tests showed that the redesigned controller was superior to the initially
designed controller. These results show that a systematic approach to controller design, that is based on
first principles does not only work, but produces results that are hard to attain through heuristic tuning
alone. This work shows that by performing a few hours worth of tests, and then running the simulation
process with the recorded data, the proper control gains for an AUV can be discerned in only a very short
amount of time, while conventional methods can lead to days, if not weeks, of tuning in the field. This
work shows that time and money can be saved by using a systematic approach to control, especially when
vehicle configurations are often changed frequently
The simple system identification step response tests may not have excited all of the interesting
dynamics of the AUV, and thus, several other different tests could be utilized in the future in order to
develop a better list of dynamic coefficients. These tests may include runs in which sinusoidal rudder or
elevator commands are used in open-loop tests. Other tests that would be useful are low speed
maneuvering and acceleration tests, as well as working through all of the tests at various levels of thrust.
Several improvements need to be made to Caribou. First, a better tailcone system should be
implemented that does not operate using the stepper motor system currently being utilized, since the
current setup can lead to large errors in actual position vs. commanded position. Ideally this new system
would allow for duct position feedback that can be recorded as a mission variable. This improvement
would allow for a much greater ease in model simulation, as the actual duct position would be known and
not inferred from the commanded position. One other improvement that could be made in software is to
change the yaw and pitch rates used by the MOOS system. Currently, the software calculates the rates
based on position data received from the sensor. An improvement would be to use the rates available
directly from the accelerometer as to avoid calculation errors and time delays.
135
136
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Appendix A
Parameter Definitions
Right-hand body coordinates system:
x-axis pointing towards the bow
y-axis pointing portside
z-axis pointing upwards
Coordinate system origin is axisymmetrically located at the body midpoint 1/2.
Series 58, Model 4154 Gertler polynomial [5]
length = 84in (2.13m)
maximum diameter = 21in (0.53m)
hull shaped described by y 2 = aix +a 2 x 2 + a 3x 3 + a4 x 4 + a5 x5 + a6x 6
a, =
a2 =
a3 =
a4 =
a5 =
a6 =
1.0
2.149653
-17.773496
36.716580
-33.511285
11.418548
The hull is extended by adding cylindrical midsections at the point of maximum diameter (x=0.4)
Roll
Pitch
Yaw
Roll rate
Pitch rate
Yaw rate
Surge
Sway
Heave
p
q
r
u
v
w
X
Y
Z
K
M
N
body force applied in the body-referenced x-direction (N)
body force applied in the body-referenced y-direction (N)
body force applied in the body-referenced z-direction (N)
body moment applied in the body-referenced z-direction (N m)
body moment applied in the body-referenced z-direction (N m)
body moment applied in the body-referenced z-direction (N m)
external
external
external
external
external
external
V/
0
#
Euler angle (radians)
Euler angle (radians)
Euler angle (radians)
rad/sec
rad/sec
rad/sec
m/s
m/s
m/s
inertial-referenced
inertial-referenced
inertial-referenced
body-referenced
body-referenced
body-referenced
body-referenced
body-referenced
body-referenced
Ringfin
6R
rudder angle (radians)
(5E
elevator angle (radians)
Propeller
T
Propeller thrust (N)
U,
Water speed seen at the propeller (m/s)
np
D
Propeller speed (rad/sec)
Propeller diameter = 0.3m
139
140
Appendix B
Root Locus Plots for Various Models and
Controllers
Initial Model A and Initial Controller Al
B. 1
I
8
0.5-
0.40.30.2-
0.1
-.
-
0-
E
-0.1 -2
(I
4.2 -
-4
-0.3-0.4-
-6
-0.5
.
I
-10
-8
-6
-4
.
I
2
0
-2
Real Axis
4
6
-0.6
Real Axis
-0.8
-1
-1.2
8
-0.4
-0.4
-0.2
-012
0
0
1-
0.5
2
-
0
-0.51--
-1
-3
-2.5
-2
-1.5
Reais
-1
A
-0.5
dl
0
Figure B.1: Root Locus plot for the heading system, model A and controller Al1
141
6
1.5
4
2
0.5
0
0
E
-2
-0.5
-
-4--1
-6-(
-14
4
-12
-10
-8
-6
-4
Real Axis
-2
0
2
-.
4
-3
-2.5
-2
-1.5
-1
Real Axis
-0.5
0
05
5
1
Figure B.2: Root Locus plot for the heading system, model A and controller Al
6
1\1'
1.5-
4-
2-
0.5
0-
0
-2-
-0.5
-4
-1
-6
-1 5
-12
-10
-8
-6
-4
Real Axis
-2
0
2
4
-16
-14
-12
-10
-2.5
-2
-1.5
-1
-0.5
Real Axis
0
0.5
x 10
86-
4-
0
E
-2 -
-4-
-6
-18
-8
Real Axis
-6
-4
-2
0
2
x 103
Figure B.3: Root Locus plot for the heading system, model A and controller Al
142
1
1.5
B.2
Improved Model B and Initial Controller Al
I
6
20
15
10
2
5
.8
'V-
E
-5
-2 --
-10
-4
-15
-20
-6
I
-20
-30
10
0
-10
-10
20
-8
- . . . . I .
-6
-4
-2
0
Real Axis
-3
-2.5
-2
Real Axds
2
4
6
-0.5
0
1.sL
x 10
0.2
0.1
0.5
0
0
E
-0.5
-0.1
-1
-0.2
-1.5
-0.J 1
-0.7
111
-0.6
-
.1
-0 .5
.1 .
1
-0.4
-
I
-0.3
Real Axis
I 1 .
-0.2
1
-0.1
1
1-4
0
-3.5
-3Real
Axis
-1.5
Figure B.4: Root Locus plot for the heading system, model B and controller Al
143
20
6
15
4
10
2
5
w0-
0
-- 5-
-2
-10
-4
-15
-6
-20
--30
-20
0
-10
10
-8 L
20
I
-12
-10
-8
-6
Real Axis
-4
-2
Real Axis
0
2
4
6
2
0.3
1.5
0.2
-o
0.1
0.5
0
0
-0.5
-0.1
0-
-1 -0.2
-1.5 -0.3
-2 -
-0.4 -4
-3
-2
-1
Real Axis
0
1
2
-08
-0.7
-0.6
-0.5
-0.4
-0.3
Real Axs
-0.2
Figure B.5: Root Locus plot for the pitch system, model B and controller Al
144
-0.1
0
0.1
2
10
1.5
~-
-
~
O
0.5
0
'
-0.5
E
------- --------------1
-5
-1.5
-10
-2
-2.5
-15
-20
-10
-5
Real Axis
-3
10
5
0
-2
2
1
0
-1
Real Axis
0.80.01
0.6-
-
0.40.005 0.2
-
n
-
0-
2
-0.2
-0.005
-0.4
-
-0.6
-0.01
-0.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
Real Axos
-0.4
-0.2
0
0.2
0.4
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
Real AAus
Figure B.6: Root Locus plot for the depth system, model B and controller Al
145
B.3
Improved Model B and Improved Controller B
25 -
820
6-
15
10
4-
5
2
1'
0
-- 5
-2
-10
04
-15
-4
-20 k
-6
-25L
-30
-20
-10
0
10
Real Axis
-8
20
-10
-8
-6
-4
-2
0
2
Real Axis
4
8
6
0.3
0.03 0.2
0.02 0.1
0.01 0
U.
8
-0.1
-0.01-0.2
-0.02-
-0.3-
-0.03
-0.7
-0.6
-0.5
-0.4
-0.3
Real Axis
-0.2
-0.1
0
01
-006
-0.05
-0.04
-0.03
-002
Real Axds
-0.01
0
Figure B.7: Root Locus plot for the heading system, model B and controller B
146
0.01
0.02
20
6
15
4
10
2
5
0
.
- --.-
-5
-2
-10
-4
-15
-6
-20
-10
-12
20
10
0
-10
-20
-30
-8
-4
Real Axis
-6
Real Axis
-2
0
4
2
0.51.5
0.4
0.3
1
0.20.5
0.1
4-
-
0-
E
.9 -0.1 -0.5
-0.2
-1
-0.3-0.4--
-1.5
K,
-3
-2.5
-2
-1.5
-1
-0.5
Real Axis
.
0
.
0.5
S-0.5
1
-0.8
-0.6
-0.4
-0.2
0.2
Real Axis
Figure B.8: Root Locus plot for the pitch system, model B and controller B
147
2-
10
1.5-
5
05
0
W
0
h
-5
7
/
0
1. -
/
-10
4
5
-1
--
/
-20
-15
-10
-5
0
5
-3
Real Axis
-2.5
-2
-1.5
-1
-0.5
Real Axis
0
0.5
1
1.5
0.06
0.3
0.04
0.2
0.02
0.1
0
0
-C
6
-0.1
-0.02
-0.2 -
-0.04
-0.3-
-0.06
-0.7
-0.6
-0.5
-0.4
-0.3
Real Axis
-02
-0.1
0
0.1
-0.14
-0.12
-0.1
-0.08
-0.06
Real Axis
-0.04
-0.02
Figure B.9: Root Locus plot for the depth system, model B and controller B
148
0
0.02
Improved Model Cl_5 and Initial Controller Al
B.4
I
20
61
15
10
4
5-
2
..
--- -.-.-.-.-.-.
--.
m
0
0
-5
-- 2
-107
-4
-15
-6
-20 -25
-20
-5
-10
-15
0
Real Axis
5
10
15
-10
25
20
-8
-6
-4
-2
0
Real Axis
2
4
6
8
x 10,
0.3
0.2
0.1
0.5
00
E
-0.5
-0.1
-1
-0.2
-1.5
-0.3
IIIIk_
-0.7
-0.6
_I_------__
-0.5
-0.4
_
_---.
-0.2
-0.3
Real Axis
-0.1
0
_I_.
0.1
1
------
-4
-2.5
-3
-2.5
-2
-1.5
Real Axis
-1
-0.5
0
0.5
x 10"
Figure B.10: Root Locus plot for the heading system, model Cl_5 and controller A1
149
20
15-
6
10-
4
5
-
2
0
*0
E
2
--5
0
-2
-10 -4
-15
-6
-20
-30
-25
-20
-15
-10
-5
Real Axis
0
5
10
15
20
-12
-10
-8
-6
-4
-2
Real Axis
0
2
4
0.3
1.5
0.2
1
-
0.5
-N
0.1 -
0
6
0
0-
-0.5
-0.1
-
-1
-0.2-1.5
-3
-2 5
-2
-1.5
-1
-0.5
Real Axis
0
0.5
1
1.5
-0.3 -0.6
-0.5
-04
-0.3
-0.2
-0.1
Real Axs
Figure B. 11: Root Locus plot for the pitch system, model Cl_5 and controller Al
150
0
0.1
6
2
1.5
1
0.5
-15
10--
.-
0
0
E
-0.5
-1
-1.5
-2
-15
-10
-5
Real Axis
0
-3
5
-2.5
-2
-1.5
-1
-0.5
Real Axis
0
0.5
1
1.5
2
0.01
0.4
0.008
0.3
0.006
0.2
0.004
0.1
0.002-
0
0-
E -0.1
-0.002
-0.2
-0.004-
-0.3
-0.006-
-0.4
-0.008
-0.5
-0.01
-1
-0.8
-0.6
-0.4
Real Axis
-0.2
0
-
-0.02
-0.015
-0.01
Real Axis
-0.005
0
Figure B.12: Root Locus plot for the depth system, model Cl_5 and controller Al
151
B.5
Improved Model C 1_5 and Heuristically Tuned Controller A2
20-
6
15 4
102
5
0
0
E-
E'
-2
-10
-4-15
-6-
-20
-30
-20
-10
0
10
-12
20
-10
-8
-6
Real Axis
1.5
f
-4
Real Axis
-2
0
2
4
0.5
0.4-
I
0.3
0.2-
0.5
0.1
0-
0
-
-
E -0.1
-0.5
-0.2-0.3-
-1
-0.4 -0.5-
-1.5
-2.5
-2
-1.5
-1
-0 5
Real Axis
0
0.5
1
1.5
-0.8
-0.6
-0.4
-02
Real Axis
0
Figure B. 13: Root Locus plot for the pitch system, nodel C1_5 and controller A2
152
0.2
10
1.5
8
6
4
0.5 F
2
0
0
E -2
-0.5 1-
-4
-6
-1
-8
-1.5
-10
-15
-10
0
-5
Real Axds
5
-2.5
-1.5
-2
0
-0.5
Real Ads
-1
0.5
1
1.5
2
0.5-
0.02
0.4-
0.015
0.3
- - - - -
-
-
-
-
03
0001
0.1-
0.005
-
0:0
0
E
-0.005 -
-0.2-
-
-0.01
-0.3-
-0.015-0.4-0.02-0.5
-1
-0.8
-0.6
-0.4
Real Axis
-0.2
0
-0.05
-0.04
-0.03
-0.02
Real Axs
-0.01
0
Figure B.14: Root Locus plot for the depth system, model Cl_5 and controller A2
153
B.6
Improved Model C1_5 and Improved Controller B
20
6
15
-I-X
4
10
5
2
4.-
0~
0
E
-5
0
S
-2
-10-15-
-4
-20
-6
-25--30
-20
-10
0
Real Axis
10
20
-10
-8
-6
-4
-2
0
Real Axis
2
4
6
8
-0.02
-0.01
0
001
0.30.03-
0.20.0201 0.01
-
E~
/
-0.1
-0.01
-
-0.02-
-0.2
-0.03-0.3-0.6
-0.5
-0.4
-0.3
Real Axis
-0.2
-0.1
0
0.1
-007
-006
-0.05
-0.04
-003
Real Axis
Figure B .15: Root Locus plot for the heading system, model Cl_5 and controller B
154
20
6
15
4
/
/
10
L
5
2
------.
0
-5
-2
-10
-4
-15
-6
-20
-30
-25
-20
-15
-10
-5
Real Axis
0
5
10
15
I
20
-12
I
-10
0.3
-8
-6
4
Real Axis
-2
0
2
4
-
1
0.2F
0.5
0.1
E
-
0-0.1-
-0.5
-0.2-
-2
-1.5
-1
-0.5
Real Axis
I
0
I
0.5
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
Real Axis
0
0.1
Figure B. 16: Root Locus plot for the pitch system, model Cl_5 and controller B
155
1.5
10
8 6
4
0.5
2 0 -
ZI0 -
*
E
-2
-0.5-
-4
6
-6 -
*
-1
-8 -10
-1.5
5
0
-5
-10
-15
-20
-2
-2.5
-1.5
-1
1
0.5
0
-0.5
Real Axis
Real Axis
0.4 -
0.06
0.3 0.2
0.04 -
F
0.1
0.02 -
0
-0.02
-0.2
-0.04
-0.3
-0.06
-0.4
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
Real Axis
-0.1
0
0.1
0.2
0.3
I
-0.14
-0.12
I
-01
-008
-0.06
-004
-0.02
Real Axis
Figure B. 17: Root Locus plot for the depth system, model C1_5 and controller B
156
0
0.02
B.7
Improved Model C2_0 and Improved Controller B
20
6
15
4
10
5
2
0
.5
2
-5
-2
-10
-a-
-4
-15
-6
-20
-25
-20
-15
-10
-5
0
Real Axis
5
10
15
20
25
-
-6
-4
-2
0
Real Axis
-0.04
-0.03
Real Axis
2
6
4
8
0.3
0.030.2
0.020.1
0.01 -
/
0
5d
.5
01
-0.01
-0.1
-0.02 -0.2
-0.03
-0.3
-0.7
-0.6
-0.5
-0.4
-0.3
Real Axis
-0.2
-0.1
0
-0.07
-0.06
-0.05
-0.02
-0.01
0
0.01
Figure B. 18: Root Locus plot for the heading system, model C2_0 and controller B
157
20
x
15
10
5
2
0
-5
_2~
-10
-15
-6-
-20L
\
-25
-30
-15
-20
-10
0
-5
5
15
10
20
-12
-10
-8
-6
Real Axis
0
-2
-4
Real Axis
2
4
0.3
1.50.2
1--
0.5
0.1
-
0
0
-0.5
-0.1
-1
-0.2
-1.5--
-0
-3
-2.5
-2
-1.5
-1
-0.5
Real Axis
0
0.5
1
15
-0.6
-0.5
-0.4
-0.3
Real Axis
-0.2
-0 1
Figure B. 19: Root Locus plot for the pitch system, model C2_0 and controller B
158
0
10
2
8
1.5
6
4
0.5
2
I)
0
0
-2
-0.5
-4
-1
-6-
-1.5
-8-2
-10-3
10
5
0
-5
Real Axis
-10
15
-2
-1
Real Axis
2
1
0
0.4
0.04
0.3
0.03
0.2-
0.02
0.1
-
0.01
0--.
-
0
0
-0.01
-0.1
-0.02 -
-0.2
-0.03
-
-0.3 F
-0.04 -0.4
__
-__
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
Real Axis
-0.2
-0 1
_
0
_-0.05
0.1
0.2
-0.12
-0.1
-0.08
-0.06
Real Axis
-0.04
-0.02
0
Figure B.20: Root Locus plot for the depth system, model C2_0 and controller B
159