A thesis update covering a process for deriving a gain-scheduled nonlinear controller for the motion control of an underwater vehicle, specifically an underwater buoyancy glider.

1. First Semester Thesis Update
Ashton Johnston
08/19/2022
Thesis Committee
Thesis Advisor: Dr. Neil Palumbo
Committee Member: Dr. Adam Watkins
Department Co-Chair: Dr. Cleon Davis

2. 2
Agenda
 Next Steps
o Increase Simulation Solve Rate
o Finalize Pitch Transfer Function
o Finalize Gain Tuning Procedure
o Assess Gain/Phase Margin
o Assess Time Response Characteristics
o Assess Robustness to Varying Plant
and Environment
o Compare Static vs. Gain-Scheduled
Controllers
o Thesis Document and Presentation
 Thesis Overview
 Schedule
 Progress
o Literature Review
o CAD Model
o CFD Simulations
o MATLAB Simulation and GUI
o Control Architecture
o Model Linearization
o Preliminary Gain-Scheduled Controller

3. 3
Thesis Overview
 Title: “A Gain-Scheduled Control Scheme for Improved Maneuverability and Power
Efficiency of Underwater Gliders”
 Goal: To develop a gain-scheduled controller that improves underwater glider
performance over a wider range of operating conditions.
 Motivation: The benefit of underwater gliders is their power efficiency; however, the
dynamics of an underwater glider are highly nonlinear, and they are under-actuated (i.e., they
don’t have direct control authority in all axes of motion), which makes motion control uniquely
challenging. In this thesis, it is proposed that a gain-scheduled motion controller will lead to a
vehicle that is more performant and robust to changing plant dynamics and environmental
disturbances, which will in turn minimize its deviation from the flight path and improve power
efficiency overall.

4. 4
 Originally conceived by Henry Stommel and Doug
Webb in 1989, underwater gliders are widely used
as ocean sensing platforms and have been used to
study the ocean interior for over 20 years.
 Their main benefits over other UUVs are their
power efficiency, long duration, and vertical
motion through the water column.
 They are trimmed to be neutrally buoyant in water
and produce forward motion by ingesting and
expelling water in order to dive and rise.
 Their main power usage comes from the pumping
needed to expel water in order to rise. If they
deviate from their path by too much, the pumping
energy needed to course correct can be
substantial, decreasing the vehicle’s efficiency.
 Therefore, minimizing path deviation, as well as
actuator motion to maintain the vehicle’s heading,
is paramount in maintaining high power efficiency.
Thesis Overview
Background

5. 5
 A simplified CAD model of a Slocum glider will be developed and used with CFD to derive the
nonlinear forces and moments acting on the vehicle over a wide range of operating conditions.
 A 6-DOF numerical model will be developed using the nonlinear forces and moments derived
from the CFD runs in order to generate a “truth” model of the vehicle.
 The nonlinear dynamics of the vehicle will be linearized in the pitch and yaw directions about
various operating points in order to derive linear representations of the vehicle in those axes for
controller synthesis.
 Linear control methods (e.g., PIDs) will be applied to the linear models and gains will be derived
at the various operating points.
 Those gains will then be “scheduled” based on the vehicle’s sensed operating point and applied
to the simulated vehicle in order to create a nonlinear controller.
 Finally, the stability and robustness of the gain-scheduled controller will be characterized and
compared to a static gain controller.
Thesis Overview
Approach

6. 6
Thesis Overview
Teledyne-Webb Slocum G3 Glider

7. 7
Schedule

8. 8
Literature Review
 A thorough literature review was done on modern underwater glider motion control
techniques
 80+ documents total
 60+ on glider
motion control
 Numbered and cataloged
for easy search and
referencing
NOTE: A sampling of the bibliography
can be found in the Appendix.

9. 9
CAD Model
 A simplified CAD model of a Slocum glider was developed in SOLIDWORKS
NOTE: Dimensional drawings can be found in the Appendix.

10. 10
CFD Simulations
 CFD runs were performed at a variety of angles of attack (AOA) and sideslip
angles (SSA) using SOLIDWORKS 2021 Flow Simulation
NOTE: Full CFD setup and parameter list can be found in the Appendix.
Fluid Velocity (m/s)

11. 11
 The purpose of the runs was to identify the
nonlinear hydrodynamic force and moment
coefficients for the numerical model:
o Forces included Drag, Lift, and Sideforce
o Moments about the X, Y, and Z axes
 Typical angles of attack for gliders in steady
glides are less than 5 degrees; however,
larger angles are seen during inflections
 The AOA and SSA were set to various
combinations of the following angles:
o -30,-10,-5,-4,-3,-2,-1,0,1,2,3,4,5,10,30
 The CFD runs were repeated with the rudder
at 20 degrees in order to characterize the
effect of the rudder deflection on the
hydrodynamics of the vehicle
 The resulting forces and moments, with and
without the rudder deflection, were compiled
in Excel and transformed into the proper
coordinate frame and units for the model
 The Akima method was then used to
interpolate the coefficient values to 0.1 degree
precision and a lookup table was used to add
the coefficients to the model
 The effect of the rudder coefficients is
assumed to be linear with the angle of
deflection
 The addition of the nonlinear coefficients was
used to create a “truth” model of the glider
that better captured the nonlinear dynamics of
the real vehicle
CFD Simulations
Motivation

12. 12
CFD Simulations
Coefficient Curve Example

13. 13
CFD Simulations
Coefficient Curves
NOTE: Larger images of the CFD curves can be found in the Appendix.

14. 14
CFD Simulations
Coefficient Curve Discussion
 All of the force and moment coefficients are
nonlinear, and most of them vary widely in value,
with respect to angle of attack (AoA) and sideslip
angle (SSA).
 This indicates that averaging or curve-fitting the
coefficients will not replicate the true
hydrodynamics of the vehicle in simulation.
 This is particularly true for gliders because they
repeatedly change their glide direction during
transit, which effects the AoA and SSA.
 The rippling in the center of the curves is due to the
smaller interval steps used between ±5 degrees for
both AoA and SSA.
 The Akima interpolation method was chosen
because it tends to avoid the overshoot that occurs
with cubic-spline interpolation methods.

15. 15
CFD Simulations
Rudder Effects
 As expected, the addition of the rudder
results in significant changes to the side
force and yaw moment coefficients.
 Due to the high rudder design of the
Slocum glider, where the rudder is above
the axial center of the vehicle, it also
induces small changes in the roll and
pitch moment coefficients.
 This indicates that there is coupling
between the roll, pitch and yaw axes
when steering the vehicle.

16. 16
MATLAB Simulation and GUI
 A 6-DOF numerical model of a Slocum glider was developed in MATLAB that
incorporates CFD derived forces and moments
 A GUI was developed for quick and easy simulation setup, data plotting/logging, as
well as system characterization and analysis

17. 17
MATLAB Simulation and GUI
Simulation Architecture
A GUI is used for
setup and control of
up to 10 simulations,
as well as plotting
and logging
simulated data
Each simulation runs
until the user
defined Quest is
complete
The GNC object
manages the glider
actuators and
returns logs for
plotting
The Guidance object
transitions through
the glider state
machine
The Navigation
object estimates the
glider’s velocity and
position
The Control object
manages the
simulated actuators
and calculates the
control updates

18. 18
MATLAB Simulation and GUI
GUI Examples

19. 19
MATLAB Simulation and GUI
Simulation Flow Chart
Initialize actuators and GNC
Solve EoM* (1Hz)
Convert quaternions to
Euler angles
Run GNC update (1Hz)
Log data for plotting
Get dynamic coefficients
for next iteration
Check for end condition
Plot/log data
Update simulated sensors
Estimate velocity and
position
Check for
maneuver/mode
transition
Compute range and
bearing to waypoint
Update actuator positions
GNC Update
*EoM: Equations of Motion

20. 20
MATLAB Simulation and GUI
State Machine
…
Glide Down
Inflect Up
Glide Up
Inflect
Down
BE* is locked
Pitch and Heading
controllers are active
BE* is moved into position
Pitch is moved to initial position
Rudder is centered
Repeats until the waypoint
is reached
NOTE: Surfacing for comms/GPS
is not considered
*BE: Buoyancy Engine

21. 21
MATLAB Simulation and GUI
Equations of Motion
Newton’s 2nd Law
Coefficients Calculated from CFD
NOTE: Full derivation will be in the final document.
Combined
Reorganized
The rudder induced forces and moments are added to the
coefficients in the simulation prior to solving.
Example: 𝐷 = 𝐷𝑣𝑒ℎ𝑖𝑐𝑙𝑒(𝛼, 𝛽) + 𝐷𝑟𝑢𝑑𝑑𝑒𝑟(𝛼, 𝛽)

22. 22
 The 2 main control goals for underwater
gliders are:
o Pitch angle
o Heading angle
 The pitch angle set point is usually chosen to
balance the vehicle’s speed over ground and
glide efficiency
 The heading angle is used to steer the
vehicle between waypoints in order to reach
areas of interest
 The dynamics of underwater gliders is highly
nonlinear due to the balancing of stabilizing
forces (buoyancy and gravity) and
hydrodynamic forces (lift and drag)
 Additionally, gliders are under-actuated and
are sensitive to mis-trim and environmental
changes, making robust control challenging
 The Slocum glider uses a “pitch vernier” (a
moving ballast weight) to control the pitch
angle of the vehicle during steady glides
 The shallow water version of the Slocum glider
uses a rudder to steer, while the deep-water
version uses a rolling ballast weight to cause
banked turns in order to steer
 For the purposes of this thesis, only the
rudder-controlled version of the Slocum glider
is considered
Control Architecture

23. 23
PID
Actuator
Dynamics
Pitch
Dynamics
Pitch
Command
Pitch
Angle
Disturbances
+
-
+
+
Control Architecture
Pitch Control Loop
Sensor
Dynamics
Measured
Pitch
Pitch
Error
Commanded
Battery
Position Max
Min

24. 24
PID
Actuator
Dynamics
Yaw
Dynamics
Heading
Command
Heading
Angle
Disturbances
+
-
+
+
Control Architecture
Heading Control Loop
Sensor
Dynamics
Measured
Heading
Heading
Error
Commanded
Rudder
Angle Max
Min

25. 25
KP
E(s)
+
Control Architecture
PID w/ Anti-Windup
KI
KD
s
U(s)
Ka
+
+
+
-
-
+ 1
𝑠
 Anti-windup logic is used to limit the
growth of the PID integrator
 In cases where the actuator is either
rate limited, or reaches it minimum or
maximum position, the integrator can
continue to grow
 This can lead to significant lag in
control due to the integrator needing
to “unwind” before the commanded
position can move away from the limit
 The anti-windup logic shown here is
used to cap the integrator in case of
actuator limits being reached
 NOTE: The anti-windup logic is
ignored for the purposes of gain
tuning

26. 26
Model Linearization
 Due to the nonlinear nature of the
vehicle dynamics, linear control
theory is ill-suited for controller
synthesis and gain tuning
 In order to use PID controllers,
linearized models of the vehicle’s
pitch and yaw dynamics must be
developed
 The linearized models must then be
assessed at various operating points
in order to tune the PID gains

27. 27
Model Linearization
Longitudinal Model
 Leonard and Graver derived a longitudinal
(i.e., front to back) model of a general
underwater glider’s dynamics

28. 28
Model Linearization
Linearized Longitudinal Model
 Graver then linearized the model about operating points
defined by the desired glide path angle and speed,
resulting in the following state space representation
NOTE: The A matrix coefficient definitions can be found in the Appendix.

29. 29
Model Linearization for Control Synthesis
Current MATLAB Implementation
 This linearized model has been coded in MATLAB
and checked against values that Graver
presented in his publications.
 It was then solved at all combinations of the
following parameters:
o Glide Path Angle (deg): -45, -40, -35, -30, -25,
-20, -15, -10, 10, 15, 20, 25, 30, 35, 40, 45
o Glider Speed (m/s): 0.1, 0.2, 0.3, 0.4, 0.5
 The individual linearized solutions were then
used to generate a discrete time transfer
function from the control force on the axial
battery position to glider pitch angle.
 A built in PID gain tuning function was then
used to select gains that would stabilize the
system and produce a reasonable step response
at the operating point.
 The Akima method was then used to interpolate
the gain values to 0.01 m/s and 0.1 degree
precision, and a lookup table was generated that
is used in sim to select gains based on the
estimated state of the vehicle.
 As the glider speed and orientation change in sim,
new gains are selected from the table and used in
the pitch control loop shown previously.
 NOTE: This is an interim process that was used to
test and validate the gain-scheduling procedure.
It is NOT the final methodology that will be used
for tuning controller gains. The transfer function
used for this demonstrative controller is not in the
correct form, and the gain selection methodology
will be based on natural frequency and damping
coefficient selection.

30. 30
Model Linearization for Control Synthesis
Steering Model
 Due to the glider needing to dive and rise
through the water column in order to
generate forward motion, it is impossible
to completely decouple the longitudinal
and lateral dynamics when developing a
steering model.
 In fact, the turning rate of a glider directly
correlates with the pitch angle and speed
of the vehicle.
 Additionally, due to the high rudder on
the Slocum glider, any deflection of the
rudder also induces a pitch and roll
moment on the body, further complicating
the dynamical model.
 For these reasons, the “truth” simulation of
the Slocum glider, which includes the CFD
derived forces and moments of the vehicle
at various operating points, was used to
derive the yaw rate model.
 The 1st-order Nomoto approximation is a
commonly used model for ship and UUV
heading control, where the vehicle
dynamics can be described by a gain and
time constant like so:
𝑟
𝜕𝑅
𝑠 =
𝐾
1 + 𝑇𝑠
𝜓
𝜕𝑅
𝑠 =
𝐾
𝑠(1 + 𝑇𝑠)

31. 31
Model Linearization for Control Synthesis
Steering Model Derivation
 In order to calculate the steady-state gain of the
1st-order models, the rudder and depth rate of
the simulated vehicle were commanded to
specific values and the steady-state yaw rate of
the vehicle was averaged over a 500 second
period.
 This was done at appropriate combinations of
the following parameters:
o Pitch Angle (deg): -35, -30, -25, -20, -15, -10, 10,
15, 20, 25, 30, 35
o Depth Rate (m/s): -0.35, -0.25, -0.15, 0.15, 0.25,
0.35
 A negative depth rate corresponds to an upward
glide, so only positive pitch angles were used
with negative depth rates. And similarly, only
negative pitch angles were used with positive
depth rates.
 To find the time constant, the depth rate of the
simulated vehicle was commanded to a value, and
upon reaching steady-state, the rudder was
deflected to a specific angle and the rise time of
the response was measured.
 A spot check of various combinations of depth
rates and pitch angles gave an average time
constant of 58 seconds.
 The steady-state yaw rates were then normalized
by dividing them by the rudder angle deflection
and averaged along the pitch axis in order to
derive the steady-state gain values for the
transfer function.
 A similar procedure as the one used to tune the
pitch PID gains was then used to tune the PID
gains for the heading controller.

32. 32
Model Linearization for Control Synthesis
Steady-State Gain Table Example
Steady State Gains
Glider Up
Steady State Gains
Glider Down
𝜓
𝜕𝑅
𝑠 =
𝐾
𝑠(1 + 58𝑠)
Normalized Yaw Rates
Glider Down
Normalized Yaw Rates
Glider Down
Normalized Yaw Rates
Glider Up
Normalized Yaw Rates
Glider Up
Unstable Glide Condition
Not Considered
Unstable Glide Condition
Not Considered
Not Considered
Not Considered

33. 33
Preliminary Gain-Scheduled Controller
Pitch Gains
 The pitch compensator gains are selected based on the vehicle’s glide angle and total velocity through the water
 To do this, an AoA lookup table is generated during the linearization procedure and used to estimate the glide
angle based on the measured pitch angle where: Glide angle = Pitch – AoA
 The uniform shape of the gain curves is due to MATLAB’s built-in gain tuning function

34. 34
Preliminary Gain-Scheduled Controller
Glide Up Heading Gains
 The heading compensator gains are selected based on the vehicle’s pitch angle and depth rate
 The uniform shape of the gain curves is due to MATLAB’s built-in gain tuning function

35. 35
Preliminary Gain-Scheduled Controller
Glide Down Heading Gains
 The heading compensator gains are selected based on the vehicle’s pitch angle and depth rate
 The uniform shape of the gain curves is due to MATLAB’s built-in gain tuning function

36. 36
Preliminary Gain-Scheduled Controller
Pitch Response Comparison
Pitch SP
Scheduled
Gains
Static
Gains
Pitch
Command
Scheduled
Gains
Static
Gains
Static
Gains
Scheduled
Gains
Static
Gains
Scheduled
Gains
Pitch Response Pitch Error
Pitch Integrator Pitch Command
The pitch response time was decreased, but substantial ringing was added. More work is being
done to derive the proper transfer function for the glider’s pitch response.
Time (sec) Time (sec)
Time (sec)
Time (sec)

37. 37
Preliminary Gain-Scheduled Controller
Heading Response Comparison
Pitch SP
Scheduled
Gains
Static
Gains
Static
Gains
Scheduled
Gains
Pitch
Command
Scheduled
Gains
Static
Gains
Static
Gains
Scheduled
Gains
Heading Response Heading Error
Heading Integrator Heading Command
The scheduled gains drastically improved the heading response of the vehicle. Further work is
being done to tune the gains and characterize the stability and robustness of the controller.
Time (sec) Time (sec)
Time (sec)
Time (sec)

38. 38
Summary
 Underwater gliders are an important tool for understanding the interior of our planet’s
oceans and their use is expected to increase in the coming decades.
 A variety of control schemes have been theoretically developed for these vehicles, but
most still use simple static gain PID controllers.
 Due to the nonlinear nature of the vehicle’s dynamics, a gain-scheduled control scheme
was proposed in order to improve the performance and power efficiency of underwater
gliders.
 A simplified CAD model of a Slocum glider was developed and CFD was used to determine
the nonlinear hydrodynamic coefficients that govern the vehicle’s motion.
 A 6-DOF simulation was developed in MATLAB that incorporates these coefficients and acts
as a “truth” simulation of the vehicle for controller testing and analysis.
 Preliminary gain-scheduled controllers were developed and applied to the 6-DOF
simulation, showing improvement over static gain controllers.

39. 39
Next Steps
 Increase Simulation Solve Rate
 Finalize Pitch Transfer Function
 Finalize Gain Tuning Procedure
o Natural frequency and damping ratio
selection
o Gain/Phase margin tuning
 Assess Gain/Phase Margin
o Linearized model
o Nonlinear simulation using:
• Added delay/gain block after
compensator outputs
 Assess Time Response Characteristics
o Rise/Settling Time Comparisons
 Assess Robustness to Varying Plant and
Environment
o Actuator rates/drag and lift
coefficients/density/currents
 Compare Static vs. Gain-Scheduled Controllers
o Compare stability/frequency/time metrics
o Path deviation
o Actuator travel
 Thesis Document and Presentation

40. © The Johns Hopkins University 2021, All Rights Reserved.

41. 41
Appendix

42. 42
Literature Review
 [1] Hiroshi Akima. A new method of interpolation and smooth curve fitting based on local
procedures. Journal of the ACM, 1970.
 [2] Pradeep Bhatta. Nonlinear stability and control of gliding vehicles. PhD thesis, Princeton
University, 2006.
 [3] D Cowling. Full range autopilot design for an unmanned underwater vehicle. IFAS
Proceedings Volumes, 1996.
 [4] Ali Hussain et al. Underwater glider modelling and analysis for net buoyancy, depth and
pitch angle control. Ocean Engineering, 2011.
 [5] Cotroneo et al. Glider and satellite high resolution monitoring of amesoscale eddy in the
algerian basin: Effects on the mixed layer depth and biochemistry. Journal of Marine Systems,
2015.
 [6] D. Mercado et al. Aerial-underwater systems, a new paradigm in unmanned vehicles.
Journal of Intelligent and Robotic Systems, 2017.
 [7] Darshana Makavita et al. Fuzzy gain scheduling based optimally tuned pid controllers for an
unmanned underwater vehicle. International Journal of Conceptions on Electronics and
Communication Engineering, 2014.
 [8] Eriksen et al. Seaglider: A long-range autonomous underwater vehicle for oceanographic
research. IEEE Journal of Oceanic Engineering, 2001.
 [9] Isa et al. A hybrid-driven underwater glider model, hydrodynamics estimation, and an
analysis of the motion control. Ocean Engineering, 2014.
 [10] Joshua Graver et al. Underwater glider model parameter identification. Symposium on
Unmanned Untethered Submersible Technology, 2003.
 [11] Li et al. Vertical motion control of an underwater glider with a command filtered adaptive
algorithm. Journal of Marine Science and Engineering, 2022.
 [12] Liu et al. Using petrel ii glider to analyze underwater noise spectrogram in the south china
sea. Acoustics Australia, 2018.
 [13] Mahmoudian et al. Dynamics and control of underwater gliders ii: Motion planning and
control. Technical report, Virginia Center for Autonomous Systems, 2010.
 [14] Noh et al. Depth and pitch control of usm underwater glider: performance comparison pid
vs. lqr. Indian Journal of Geo-Marine Sciences, 2011.
 [15] Sang et al. Heading tracking control with an adaptive hybrid control for under actuated
underwater glider. ISA Transactions, 2018.
 [16] Sherman et al. The autonomous underwater glider “spray”. IEEE Journal of Oceanic
Engineering, 2001.
 [17] Tchilian et al. Optimal control of an underwater glider vehicle. Dynamics and
Vibroacoustics of Machines, 2017.
 [18] Wagawa et al. Observations of oceanic fronts and water-mass properties in the central
japan sea: Repeated surveys from an underwater glider. Journal of Marine Systems, 2019.
 [19] Wang et al. Dynamic modeling and motion simulation for a winged hybrid-driven
underwater glider. China Ocean Engineering, 2011.
 [20] Wang et al. A backseat control architecture for a slocum glider. Journal of Marine Science
and Engineering, 2021.
 [21] Wang et al. Vertical profile diving and floating motion control of the underwater glider
based on fuzzy adaptive ladrc algorithm. Journal of Marine Science and Engineering, 2021.
 [22] Webb et al. Slocum: An underwater glider propelled by environmental energy. IEEE
Journal of Oceanic Engineering, 2001.
 [23] You Liu et al. Steering control for underwater gliders. Frontiers of Information Technology
and Electronic Engineering, 2017.
 [24] Yu et al. Development and experiments of the sea-wing underwater glider. China Ocean
Engineering, 2011.
 [25] Ziaeefard et al. Effective turning motion control of internally actuated autonomous
underwater vehicles. Journal of Intelligent and Robotic Systems, 2017.
 [26] Joshua Graver. Underwater Gliders: Dynamics, Control and Design. PhD thesis, Princeton
University, 2005.
 [27] Isa and Arshad. Buoyancy-driven underwater glider modelling and analysis of motion
control. Indian Journal of Marine Sciences, 2012.
 [28] Isa and Arshad. Neural networks control of hybrid-driven underwater glider. Oceans –
Yeosu, 2012.
 [29] Isa and Arshad. An analysis of homeostatic motion control system for a hybriddriven
underwater glider. International Conference on Advanced Intelligent Mechatronics, 2013.

43. 43
Literature Review
 [30] Isa and Arshad. Modeling and motion control of a hybrid-driven underwater glider. Indian
Journal of Geo-Marine Sciences, 2013.
 [31] Isa and Arshad. Development of a hybrid-driven autonomous underwater glider with a
biologically inspired motion control system. Asian Control Conference, 2015.
 [32] Leonard and Graver. Model-based feedback control of autonomous underwater gliders.
IEEE Journal of Oceanic Engineering, 2001.
 [33] Nina Mahmoudian and Craig Woolsey. Underwater glider motion control. IEEE Conference
on Decision and Control, 2008.
 [34] Fletcher Paddison. The talos control system. Johns Hopkins APL Technical Digest, 1982.
 [35] Jan Petrich and Daniel Stilwell. Robust control for an autonomous underwater vehicle that
suppresses pitch and yaw coupling. Ocean Engineering, 2010.
 [36] Daniel Rudnick. Ocean research enabled by underwater gliders. Annual Review of Marine
Science, 2016.
 [37] Rugh and Shamma. Research on gain scheduling. Automatica, 1999.
 [38] De Souza and Maruyama. Intelligent uuvs: Some issues on rov dynamic positioning. IEEE
Transactions on Aerospace and Electronic Systems, 2007.
 [39] Gregory Stewart. A pragmatic approach to robust gain scheduling. IFAC Symposium on
Robust Control Design, 2012.
 [40] Henry Stommel. The slocum mission. Oceanography, 1989.
 [41] Yang and Ma. Sliding mode tracking control of an autonomous underwater glider.
International Conference on Computer Application and System Modeling, 2010.
 [42] Feitian Zhange and Xiaobo Tan. Passivity-based stabilization of underwater gliders with a
control surface. Journal of Dynamics Systems, Measurement, and Control, 2015.
 [43] Mingxi Zhou. The approach of improving the roll control of a slocum autonomous
underwater glider. Master’s thesis, Memorial University of Newfoundland, 2012.
NOTE: Additional documents will be referenced in the final document.

44. 44
CAD Model
General Dimensions
NOTE: All dimensions are in inches

45. 45
CAD Model
Wing/Tail Dimensions
NOTE: All dimensions are in inches

46. 46
CAD Model
Tail/Rudder Dimensions
NOTE: All dimensions are in inches

47. 47
 A local mesh was applied to all surfaces
of the CAD model, excluding the minor
elements on the rear face of the tail.
 Local mesh settings:
o Level of Refining Fluid Cells: 3 out of 9
o Level of Refining Cells at Fluid/Solid Boundary: 3
out of 9
o Characteristic Number of Cells Across Channel: 14
o Maximum Channel Refinement Level: 1 out of 9
o Small Solid Feature Refinement: 1 out of 9
o Maximum Height of Slots to Close: 3.9cm
 Global mesh settings:
o Type: Automatic
o Level of Initial Mesh: 6 out of 7
o Ratio Factor: 1
 Fluid and Thermal Characteristics:
o Fluid Type: Water
o Flow Type: Laminar Only
o Cavitation: None
o Wall Thermal Condition: Adiabatic
o Roughness: 50 micrometers
o Pressure: 14.7 lbf/in2
o Temperature: 20.05 ℃
 Velocity Parameters:
o Defined by: Aerodynamic Angles
o Velocity: -0.5 m/s (0.97 knots)
o Longitudinal Plane: YZ
o Longitudinal Axis: Z
CFD Simulations
Setup

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CFD Simulations
Computation
 Computation Domain:
o Type: 3D Simulation
o X Distance: ±2.13 meters
o Y Distance: +2.02/-1.75 meters
o Z Distance: +1.85/-4.35 meters
 Calculation Control Options:
o Stop Criteria: 500 Iterations and Refinement Finished
o Global/Local Refinement Levels: 7 out of 7
o Approximate Maximum Cells: 2,000,000
o Refinement Strategy: At iterations 80 and 160
o Relaxation Interval: 100
 PC Specs:
o Processor: Water Cooled Intel i7-10700 (2.9GHz)
o GPU: NVIDIA GeForce GTX 1650
o RAM: 32GB DDR4 3600MHz
o Hard Drive: 1TB SSD NVMe m.2
 Typical run time: 1-3 hours per permutation

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CFD Simulations
Drag Coefficient Curves

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CFD Simulations
Lift Coefficient Curves

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CFD Simulations
Sideforce Coefficient Curves

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CFD Simulations
Roll Moment Coefficient Curves

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CFD Simulations
Pitch Moment Coefficient Curves

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CFD Simulations
Yaw Moment Coefficient Curves

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MATLAB Simulation and GUI
Quaternions
Quaternions parameterize orientation using four
parameters and one constraint. This avoids the gimbal
lock singularities that occur with Euler angles.
Euler’s theorem of rotation states that any rigid body
rotation may be parameterized by specifying an axis of
rotation and a rotation angle about that axis.
Let 𝒄 = 𝑐1, 𝑐2, 𝑐3
𝑇
be the unit vector along the axis of
rotation and let δ be the rotation angle. If we define
the quaternion vector as:
where q is subject to the constraint:
Then the corresponding rotation matrix may be written as:
The quaternion parameters may be written in terms of the
XYZ Euler angles ψ, θ, Φ as:

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Model Linearization
Linearized Longitudinal Model