Comprehensive System Identification of Ducted Fan UAVs

Comprehensive System Identification of
Ducted Fan UAVs
A Thesis
Presented to the Faculty of
California Polytechnic State University
San Luis Obispo
In Partial Fulfillment
of the Requirements for the Degree of
Master of Science in Aerospace Engineering
by:
Daniel N. Salluce
January 2004
© Copyright 2004
Daniel Salluce
ALL RIGHTS RESERVED
ii
APROVAL PAGE
TITLE:
Comprehensive System Identification of Ducted Fan UAVs
AUTHOR:
Daniel N. Salluce
DATE SUBMITTED:
January 2004
(SUBJECT TO CHANGE)
Dr. Daniel J. Biezad (AERO)
Advisor & Committee Chair
____________________________________
Dr. Mark Tischler (NASA/Army)
Committee Member
____________________________________
Dr. Jordi Puig-Suari (AERO)
Committee Member
____________________________________
Dr. Frank Owen (ME)
Committee Member
____________________________________
iii
ABSTRACT
The increase of military operations in urbanized terrain has changed the nature of
warfare and the battlefield itself. A need for a unique class of vehicles now exists. These
vehicles must be able to accurately maintain position in space, be robust in the event of
collisions, relay strategic situational awareness, and operate on an organic troop level in a
completely autonomous fashion. The operational demands of these vehicles mandate
accurate control systems and simulation testing. These needs stress the importance of
system identification and modeling throughout the design process. This research focuses
on the unique methods of identification and their application to a class of ducted fan,
rotorcraft, and unmanned autonomous air vehicles. This research shows that a variety of
identification techniques can be combined to comprehensively model this family of
vehicles and reveals the unique challenges involved. The result is a high fidelity model
available for the purposes of control system design and simulation.
iv
ACKNOWLEDGMENTS
The author would like to give special recognition to Dr. Daniel J. Biezad,
Department Chair at Cal Poly, San Luis Obispo, CA and Dr. Mark B. Tischler, U.S.
Army Aeroflightdynamics Directorate Moffett Field, CA. Without their support,
guidance, and organizational efforts this research would never have been possible. Also,
Dr. Colin Theodore, Jason Colbourne, and the whole of the Army/NASA Rotorcraft
Division at Moffett Field proved to be invaluable resources and facilitators in the
completion of this project.
v
TABLE OF CONTENTS
LIST OF TABLES........................................................................................................... viii
LIST OF FIGURES ........................................................................................................... ix
NOMENCLATURE ......................................................................................................... xii
CHAPTER 1 – Introduction and Motivation
1.1 Vehicles Examined ............................................................................................1
1.2 Scope..................................................................................................................8
CHAPTER 2 – Dynamic Model Identification Methods and Techniques
2.1 Identification Methods .................................................................................... 11
2.2 CIFER ............................................................................................................. 12
2.2.1 Flight Test Techniques..................................................................... 13
2.2.2 Bench Test Techniques .....................................................................14
2.3 Manufacturer Specifications ............................................................................14
2.4 Wind Tunnel Tests...........................................................................................15
CHAPTER 3 – Vehicle Identification
3.1 Areas of Identification .....................................................................................16
3.2 Bare-Airframe ID.............................................................................................17
3.2.1 Aerovironment/Honeywell OAV......................................................17
3.2.2 Allied Aerospace MAV ....................................................................35
3.2.3 Trek Aerospace Solotrek...................................................................46
3.2.4 Hiller Flying Platform.......................................................................48
3.2.5 Vehicle Scaling Laws and Comparisons...........................................52
3.3 Servo Actuator Identification...........................................................................56
3.4 Sensor Identification ........................................................................................94
3.4.1 Accelerometer Identification ............................................................95
3.4.2 Rate Gyro Identification ...................................................................96
3.4.3 GPS Receiver Identification .............................................................98
vi
3.4.4 Magnetometer Identification...........................................................101
3.4.5 Pressure Altimeter Identification ....................................................102
CHAPTER 4 – Flight Simulation
4.1 Simulated Sweeps ..........................................................................................104
4.2 Matlab Linear Model Determination .............................................................110
CHAPTER 5 – Conclusions.............................................................................................119
BIBLIOGRAPHY............................................................................................................120
APPENDIX A – OAV Proposal State Space Form .........................................................123
APPENDIX B – Frequency Response Bode Plots for all Actuator Cases ......................124
APPENDIX C – Actuator Generated TF Model Bode Plot Verification ........................135
APPENDIX D – Actuator Time Domain Verification of Final Models..........................157
vii
LIST OF TABLES
3.1 – OAV Measured Parameters during Flight Testing .......................................................18
3.2 – OAV Frequency Range of Good Coherence (rad/sec) .................................................19
3.3 – OAV Control Derivatives Extracted from Transfer Function Fits ...............................20
3.4 – OAV DERIVID Identified parameters and Certainties ................................................23
3.5 – OAV DERIVID Frequency Response Costs ................................................................23
3.6 – OAV Eigenvalues and Associated Eigenvectors of [F]................................................24
3.7 – MAV Physical Properties .............................................................................................35
3.8 – MAV Identified Stability Derivatives...........................................................................39
3.9 – MAV Identified Control Derivatives ............................................................................40
3.10 – Final Flight Test Identified MAV Derivatives............................................................42
3.11 – MAV Wind Tunnel Identified Derivatives and Flight Test Results ...........................44
3.12 – Pitching Moment Derivatives and Solotrek Fan Speed ..............................................47
3.13 – Pitching Moment Coefficient Summary .....................................................................53
3.14 – Pitching Moment with Blade Chord Summary...........................................................54
3.15 – Manufacturer Specifications for Servo Actuators Tested...........................................57
3.16 – Actuator Linkage Geometries .....................................................................................60
3.17 – Actuator Calibration Factors for Input and Output Channels to Degrees...................61
3.18 – Frequency Sweep Used for all Actuators....................................................................62
3.19 – Square Wave Parameters ............................................................................................63
3.20 – Actuator Bench Test Matrix........................................................................................65
3.21 – Actuator NAVFIT Frequency Ranges for CIFER Cases............................................67
3.22 – Actuator NAVFIT Results for all Cases .....................................................................68
3.23 – Actuator Nonlinear Characteristic Summary..............................................................74
4.1 – Linmod, Wind Tunnel, and Flight Test Results for i-Star 9”........................................114
viii
LIST OF FIGURES
Figure 1.1 – Land Warrior OAV Concept ..........................................................................1
Figure 1.2 – Hiller Helicopters Flying Platform – 1958 .....................................................3
Figure 1.3 – Aerovironment / Honeywell DARPA Phase I OAV – 2001 ..........................3
Figure 1.4 – Trek Aerospace Solotrek Ducted Fan – 2001.................................................4
Figure 1.5 – Allied Aerospace i-Star MAV 9” Vehicle – 2003..........................................4
Figure 1.6 – Detailed view of 9” MAV Design ..................................................................5
Figure 1.7 – MAV Stator and Vanes...................................................................................6
Figure 1.8 – Helicopter Body Axes System........................................................................7
Figure 1.9 – Helicopter Body Axes System Applied to the Ducted Fan ............................7
Figure 1.10 – Block Diagram of Basic DFCS Architecture ...............................................8
Figure 1.11 – Comprehensive Identification Schematic.....................................................9
Figure 2.1 – Sample Frequency Sweep Flight Test Command.........................................13
Figure 3.1 – Roll rate response frequency domain verification........................................26
Figure 3.2 – Pitch rate response frequency domain verification.......................................27
Figure 3.3 – Yaw response frequency domain verification ..............................................29
Figure 3.4 – Roll response time history verification.........................................................30
Figure 3.5 – Pitch response time history verification .......................................................31
Figure 3.6 – Yaw response time history verification ........................................................32
Figure 3.7 – Techsburg Wind Tunnel Setup for OAV......................................................33
Figure 3.8 – Techsburg OAV Pitching Moment to Airspeed ...........................................34
Figure 3.9 – On and Off Axis MAV Roll Frequency Responses .....................................36
Figure 3.10 – On and Off Axis MAV Pitch Frequency Responses ..................................37
Figure 3.11 – MAV Lateral Acceleration and Roll Rate Response to Roll Input ............40
Figure 3.12 – MAV Longitudinal Acceleration and Pitch Response ...............................41
Figure 3.13 – Pitching Moment Wind Tunnel Test Data for i-Star 9” .............................43
Figure 3.14 – Solotrek Wind Tunnel Test Results for Pitching Moment .........................46
Figure 3.15 – Hiller Flying Platform Pitching Moment Data ...........................................48
Figure 3.16 – Drag over a Flat Plate Perpendicular to Flow.............................................49
Figure 3.17 – Results of Removing Dummy Moment from Hiller Platform Test............50
ix
Figure 3.18 – Actuators Tested and Relative Sizes ..........................................................57
Figure 3.19 – Actuator Test Stand Apparatus...................................................................58
Figure 3.20 – Cirrus CS-10BB Mounted on Wooden Strip..............................................58
Figure 3.21 – Schematic Detailing Linkage Geometry.....................................................59
Figure 3.22 – Sample Chirp Input, Response, and Square Wave Time History...............64
Figure 3.23 – HS512MG Responses Illustrating Difference between 5V and 6V ...........70
Figure 3.24 – Sample Square Wave Response .................................................................72
Figure 3.25 – Linear Fit for Max Rate Determination......................................................73
Figure 3.26 – CS-10BB at 5V TH Illustrating Erratic Response at High Frequency .......75
Figure 3.27 – 94091 at 6V TH Illustrating Erratic Response at High Frequency.............75
Figure 3.28 – 94091 at 5V TH not Showing Erratic Response.........................................76
Figure 3.29 – DS8417 FR Illustrating Mismatch in Linear Model...................................77
Figure 3.30 – DS8417 TH Comparison to 1995 STI Findings .........................................78
Figure 3.31 – Magnitude Comparison for Linear & Nonlinear Model to Bench Test .....81
Figure 3.32 – Phase Comparison for Linear & Nonlinear Model to Bench Test .............82
Figure 3.33 – Error Function Fr and NAVFIT Transfer Function Fit ..............................83
Figure 3.34 – Rise Time Ratio Phase Lag Relationship ...................................................85
Figure 3.35 – Rise Time for Linear Model of DS8417 at 5V...........................................86
Figure 3.36 – Sweep Amplitude and Natural Frequency with Rate Limiting ..................87
Figure 3.37 – Simulink Actuator Blockset .......................................................................88
Figure 3.38 – Configurable Actuator Parameters .............................................................89
Figure 3.39 – 2nd Order Actuator Dynamics behind Mask ...............................................90
Figure 3.40 – DS8417 at 5V Time Domain Validation ....................................................91
Figure 3.41 – Accelerometer Model .................................................................................95
Figure 3.42 – Accelerometer Stationary Noise Model .....................................................96
Figure 3.43 – Rate Gyro Model ........................................................................................97
Figure 3.43 – Rate Gyro Response to Constant 15 deg/sec for 10 sec .............................98
Figure 3.44 – GPS Heading and Speed Model .................................................................99
Figure 3.45 – GPS Error and Discrete Signal Model......................................................100
Figure 3.46 – GPS Model Results...................................................................................101
Figure 3.47 – Magnetometer Model ...............................................................................102
x
Figure 3.48 – Magnetometer Depiction at 5 Gauss for 5 Seconds .................................102
Figure 3.49 – Pressure Altimeter Model.........................................................................103
Figure 3.50 – Pressure Altimeter at 15 feet for 5 seconds ..............................................103
Figure 4.1 – Simulink MAV Model................................................................................105
Figure 4.2 – Custom PC and COTS Simulation Environment .......................................106
Figure 4.3 – Simulink Sweep Generator GUI Built for Sweeps.....................................107
Figure 4.4 – Simulink GUI Generated Sweep ................................................................108
Figure 4.5 – MAV Flight Test Cross Coherence between Pitch and Roll controls ........109
Figure 4.6 – Cross Control Decoupling Block Diagram.................................................111
Figure 4.7 – LINMOD and Simulated Sweep Roll Frequency Response ......................115
Figure 4.8 – Effect of Removing Cross Control Coupling to Response.........................116
Figure 4.9 – Coupling Removed Illustrating linmod and Simulated Sweep Results......117
Figure 4.10 – Comparison of linmod and Flight Test Pitch Responses..........................118
xi
NOMENCLATURE
A
a1
b1
BW
c
C
CMPA
CMQA
CMRA
CR
F
G
H1
H2
I
j
L
M
N
p
pmixer
P
q
qmixer
r
R
rmixer
s
tˆR
v&
w
&
w
Y
x
X
Z
φ
θ
ϕ
ωn
ωˆ n
Lateral body acceleration
Vertical body velocity
Vertical body acceleration
Lateral Body Force
State Matrix
Longitudinal Body Force
Vertical Body Force
Roll attitude
Pitch attitude
Heading attitude
Natural Frequency
Normalized Natural Frequency
Propeller Rotational Velocity
Density
Propeller Coefficient
Time Constant
Damping Ratio
Phase Angle
t RNL
Area
First Fourier Coefficient
Second Fourier Coefficient
Bandwidth
Chord
Nondimensional Coefficient
Commanded Roll Rate
Commanded Pitch Rate
Commanded Yaw Rate
Cramer-Rao Bound
Plant Matrix
Control Matrix
Output Matrix Position
Output Matrix Rate
Inertia
Imaginary Variable
Rolling Moment
Pitching Moment
Yawing Moment
Roll body rate
Lateral mixer signal
Period
Pitch body rate
Longitudinal mixer signal
Yaw body rate
Radius
Pedal mixer signal (deg/sec)
Frequency Domain Variable
Linear : Nonlinear Rise Time
Rise Time Nonlinear
c, δ
CG
col
FS
Lat
lon
mixer
ped
prop
Command
Center of Gravity
Collective
Full Scale
Lateral
Longitudinal
Mixer
Pedal
Propeller
t RL
Rise Time Linear
rad
Radians
u
u
u&
v
Longitudinal body velocity
Input Control Matrix
Longitudinal body acceleration
Lateral body velocity
xx
yy
zz
dot
X-plane in the Direction of X
Y-plane in the Direction of Y
Z-plane in the Direction of Z
Time Derivative
Ω
ρ
σ
τ
ζ
∠
Subscripts
xii
CHAPTER 1 – INTRODUCTION AND MOTIVATION
1.1 Vehicles Examined
Interest and application of ring-wing type unmanned aerial vehicles (UAVs) has
increased within recent years. The military and commercial uses for a vehicle capable of
hovering and forward flight while remaining small and unmanned are countless. Military
operations on urbanized terrain (MOUT) have become an area of concern for the United
States military within recent years. An increased need for policing and securing
urbanized areas has become apparent with the conflicts in Iraq and Mogadishu. It is this
type of environment that dictates the especially challenging design of small-scale UAVs1.
Because of the nature of MOUT, precise station-keeping requirements and overall
increased risk of collision with obstacles are important. Add to that the need for small and
back-pack carried vehicles and it becomes apparent why the ducted fan design is
appealing. The Defense Advanced Research Projects Agency (DARPA) advanced
concept technology demonstrator (ACTD) projects yielded submissions which included
the Kestrel organic air vehicle (OAV) and i-Star micro air vehicle (MAV). Figure 1.1
shows the typical application of the OAV envisioned by the US Army.
Figure 1.1 – Land Warrior OAV Concept
-1-
Commercial interest has also been seen by companies and organizations looking
for stable camera and surveillance platforms. Bridge inspection, traffic monitoring, and
search and rescue in hostile environments all can benefit from use of a small unmanned
vehicle capable of hovering flight. A unique class of small rotorcraft UAVs (RUAVs)
incorporating all of the characteristics yields a small design with certain design
difficulties. These RUAVs possess the problem of making a small-scale vehicle
unmanned along with the inherently unstable nature of rotorcraft dynamics. The ducted
fan RUAV design fulfills the collision and troop handling safety requirements. However,
these ducted fans introduce a strong tendency to correct themselves in pitch and roll with
longitudinal and lateral velocity, respectively.
These ducted fan RUAVs have low inertias with most of the weight near the
center of the vehicle. Their small size and weight make for stringent volumetric and mass
restrictions. This leads to lower performance subsystems, especially sensors and
actuators. High degrees of cross coupling due to strong gyroscopic effects are created by
the fast spinning propellers. The unconventional designs that have little or no knowledge
base established make physics based modeling difficult2. Most RUAV types include the
ability for a wide range of scales to be produced. Because of the relative simplicity of
construction, bigger and smaller vehicles alike can be produced. Usually shorter design
cycles due to limited funding and demanding project requirements leave these vehicles in
need of accurate models early in the design cycle. Flight vehicles are available very early
in the design sequence and make for easier flight test based identification. These
characteristics combine to mandate accurate dynamic models. This research work will
focus on the comprehensive identification of these models.
-2-
The vehicles examined within the scope of this research are all very similar in
design in that they consist of mainly a ducted fan utilized for lift. The vehicles examined
are shown in Figures 1.2 – 1.5. Although the mission profiles for all of these vehicles
varies greatly, the two smaller scale surveillance vehicles, the Kestrel and the i-Star
MAV are most representative of future military operations on urbanized terrain (MOUT)
applications.
Figure 1.2 – Hiller Helicopters Flying Platform – 1958
Figure 1.3 – Aerovironment / Honeywell DARPA Phase I OAV – 2001
-3-
Figure 1.4 – Trek Aerospace Solotrek Ducted Fan – 2001
Figure 1.5 – Allied Aerospace i-Star MAV 9” Vehicle – 2003
Figure 1.2 depicts the Hiller flying platform. This vehicle underwent some testing
of the pitching moment characteristics of ducted fans back in 19583. For this purpose it
was included in the study. Figure 1.3 shows the Aerovironment/Honeywell teamed effort
technology demonstrator for DARPA. This vehicle was used for flight testing and
parametric modeling as well as for the identification of sensor packages. Figure 1.4
shows the Trek Aerospace Solotrek. This unique design underwent comprehensive wind
-4-
tunnel testing to study the characteristics of the ducted fan at varying propeller speeds.
Finally, Figure 1.5 shows the Allied Aerospace i-Star MAV vehicle. Pictured is the 9”
diameter vehicle. There is also a bigger cousin with a 29” diameter. Both of these
vehicles were used for actuator identification, flight testing, and simulation as part of
work for DARPA. Figure 1.6 shows a detailed view of the MAV.
Figure 1.6 – Detailed view of 9” MAV Design
The basic design of the ducted fan UAV incorporates a small COTS power plant
that is centered inside a duct. The flow of air in the duct is passed over stators for flow
straightening and over vanes which allow actuation to generate moments. Figure 1.7
shows the vanes and stators on the bottom of the 9” MAV design.
-5-
Duct
Lower
Center
Body
Stators
Vanes
Camera &
Proximity Sensor
Figure 1.7 – MAV Stator and Vanes
Great care is needed in specifying proper coordinate systems. It is not uncommon
to see these vehicles with their x-body axis out the nose, or main nacelle pointing up.
This causes issues because then the vehicle is at a 90° nose up orientation in hover. This
is a gimbal-lock orientation and is best avoided for standard Euler sequences. Figure 1.8
below illustrates the helicopter coordinate system used for this research and Figure 1.9
shows it applied to the ducted fan. Unless otherwise specified, all derivatives and
mention of moments are referred to in standard helicopter notation.
-6-
Figure 1.8 – Helicopter Body Axes System
YBody
XBody
ZBody
Figure 1.9 – Helicopter Body Axes System Applied to the Ducted Fan
All moments and forces are represented as positive in the directions shown with moments
being applied in accordance with the positive right-hand rule.
-7-
1.2 Scope
This research will focus on representing the entirety of the RUAV modeling.
Figure 1.10 shows a simplified block diagram depicting the operation of the vehicle.
Vehicle
Response
Digital Flight
Control
Servo-Actuators
Commanded
Inputs
Bare-Airframe
Dynamics
Sensors
Figure 1.10 – Block Diagram of Basic DFCS Architecture
It can be seen that simply modeling the bare airframe and its dynamics is not
enough to capture the whole nature of the vehicle. Due to the small size and limited
performance actuators and sensor packages, these areas heavily influence the nature of
flight. To accurately model the vehicle for flight control and simulation purposes, a more
expanded diagram would be required. Figure 1.11 represents the identification effort of
this research.
-8-
GPS
Rate Gyros
Accelerometers
InnerLoop
Closures
IMU
Sensors &
Telemetry
Control System
OuterLoop
Closures
Actuators
Pressure
Altimeter
Vehicle Dynamics
Rigid Body
Dynamics
SOURCES OF IDENTIFICATION
CIFER
Wind Tunnel or Other Empirical Data
Manufacturer and Bench Data
Unique Pitching
Moment
Characteristics
Figure 1.11 – Comprehensive Identification Schematic
Figure 1.11 shows that a number of techniques (described in Chapter 2) applied to a large
range of components are required to model the system. Each of these areas will be the
-9-
focus of this research. Various vehicles will be looked at in order to build up this compete
picture of the operation of these ring wing UAVs.
- 10 -
CHAPTER 2 – METHODS AND TECHNIQUES
2.1 Identification Methods
A combination of the characteristics of these small RUAVs makes system
identification an important and integral part of the design cycle. The need for a high
performing and robust control system is paramount to vehicle survivability and mission
performance. The design of the flight control system requires an accurate model across a
variety of operating conditions and input frequencies.
As previous work shows2, the use of Froude scaling the natural frequencies of
vehicles reveals the natural frequency would increase by the square root of a scale factor
measured in length. For example, making the vehicle 4 times smaller would increase the
natural frequency by 2. So, as vehicles become smaller, they require a higher bandwidth
control system. The need to operate at higher frequencies and in more of the available
flight envelope requires accurate models across large ranges of input frequencies. The use
of frequency domain techniques lends itself very nicely to accomplishing this modeling
challenge.
The NASA/Ames Research Center tool CIFER® (Comprehensive Identification
from Frequency Responses) is primarily used to identify low order equivalent systems
and parametric state-space models required across broad frequency ranges. This tool is
used extensively for the modeling of system dynamics in this effort.
The reliance on small scale, low performance components and sensors makes
characterizing the errors and inconsistencies of components important. Without exclusive
- 11 -
access to hardware inside of test vehicles, manufacturer data must be applied for error
and noise modeling. These tools and techniques combine to represent the comprehensive
identification of these vehicles.
2.2 CIFER
CIFER provides an environment and set of programs that perform the various
steps of the system identification process. Nonparametric modeling, in which no model
structure or order is assumed; in the form of frequency responses represented as Bode
plots are first extracted with CIFER. This then allows for the parametric modeling.
Transfer functions, low order equivalent (LOE) systems, or state-space models with
stability and control derivative representation3 are all used. The identification process can
be summarizes as4:
1. Nonparametric frequency response calculation from time history data
o Use of Chirp-Z Fast Fourier Transforms (FFT) and complex functions to generate
the frequency responses over multiple windows and samples
2. Multi-input frequency response conditioning
o Off axis control inputs’ contribution to on axis response is removed
3. Multi-window averaging of frequency responses
o Combination of different window sampling sizes
4. Parametric models fit to frequency responses
o Transfer function models fit to single input single output (SISO) systems
o State-space models fit to all controls and states for parameter extraction
5. Time domain verification of parametric models
When complete, this procedure yields accurate models to be applied for a variety
of tasks. CIFER does require flight test time histories in which the vehicle’s modes have
been excited by frequency rich inputs. It is not limited to vehicle dynamics either. This
tool can be used anywhere frequency domain analysis is needed. CIFER is a powerful
tool that incorporates all of the tools to needed to model in the frequency domain.
- 12 -
2.2.1 Flight Test Techniques
There are a number of techniques that need to be applied to ensure that the flight
test of the vehicle is useful and applicable to system identification. While outside the
scope of this research, it is sufficient to say that a combination of frequency rich
maneuvers as seen in Figure 2.1 and validation maneuvers like doublets are required. A
combination of sensing and telemetry equipment is needed to measure both the input
from the actuators and the vehicle response. Access to the IMU and servo signals is
required.
15
Rise
Time
Control Deflection (%)
10
Sine Frequency Sweep
Fall
Time
5
0
Zero
Duration
-5
Zero
Duration
-10
-15
0
15
30
Time (seconds)
Figure 2.1 – Sample Frequency Sweep Flight Test Command
- 13 -
45
2.2.2 Bench Test Techniques
Bench testing was used in cases where components were to be tested without
actually installing them on the vehicle or testing them while in flight. This method was
primarily applied to the testing of the servo actuators. The search for and classification of
actuators meeting the requirements of the vehicles made it impractical to install the
numerous actuators on the vehicle for testing. In this case, the actuators were tested while
hooked up to specific measuring equipment. Frequency domain analysis with CIFER was
applied to determine the dynamic characteristics of the components.
2.3 Manufacturer Specifications
The use of commercial off the shelf (COTS) devices and components for the
buildup of inertial measuring units (IMU) on the vehicles provides for manufacturer
specifications and ratings of component performance. This is important when direct
access of the components and hardware in the loop (HIL) bench testing is not available.
The identification of the rate gyros, accelerometers, magnetometers, GPS receiver, and
actuators all benefited from the provision of manufacturer identified errors and
performance specifications. In general, these specifications are slightly optimistic and
reflect the specific measuring procedure applied by the manufacturer. Averages are
usually presented by manufacturers while component-specific results are required in
some modeling cases. Due to time constraints and availability of hardware for testing,
- 14 -
manufacturer specifications are modeled and applied for the majority of telemetry and
measuring equipment aboard the vehicles.
2.4 Wind Tunnel Tests
Wind tunnel and other empirical data measured from the vehicles themselves play
an important role as well. As previously mentioned, these ducted fan RUAVs exhibit
unique corrective pitching moment characteristics due to large Mu and Lv derivatives.
Wind tunnel studies help to better characterize this. The need to accurately characterize
the behavior of the ducted fan in translational velocities has put emphasis on accurate
wind tunnel modeling. This type of physics-based modeling is used to draw some
conclusions regarding the nature of the strong pitching and rolling moment created when
the vehicle is in forward flight or in a cross-wind. It is also used to compare and correlate
the CIFER identified dynamics. In the case of the Solotrek vehicle, a wind tunnel was not
actually used. Similar techniques and methodology was applied to the vehicle although it
was suspended on top of a moving pickup truck. Regardless, wind tunnel tests and data
were used to validate and compare trends for most of the vehicles studied.
- 15 -
CHAPTER 3 – VEHICLE IDENTIFICATION
3.1 Areas of Identification
As mentioned in Chapter 2, the comprehensive identification of these vehicles
requires modeling and testing of the bare-airframe dynamics as well as all of the systems
and components onboard which directly affect the flight characteristics of the vehicle.
Figure 1.11 of Chapter 1 illustrates the areas of identification. The tools and techniques
outlined in Chapter 2 will be applied to the bare-airframe of the vehicles with conclusions
being drawn for scaling and correlation. COTS actuators will then be analyzed for there
dynamics and nonlinearities. Finally, all of the sensors and telemetry equipment used in
observation for the control system will be analyzed and modeled.
- 16 -
3.2 Bare-Airframe ID
The bare-airframe dynamics are perhaps the most unique aspect of these vehicles
and the way they fly. A small inertia with a large concentration of mass near the center of
the duct is inherent in the design. Combined with this, there is heavy coupling between
pitch and roll due to the gyroscopic effects of the fast spinning propeller. All of the
vehicles looked at utilize fixed pitch propellers. Figure 1.11 showed that the pitching
moment characteristics together with the whole of the bare-airframe rigid body dynamics
characterize the vehicle in uncontrolled flight.
3.2.1 Aerovironment/Honeywell OAV
The goal of the CIFER® system identification was to achieve an accurate MultiInput Multi-Output (MIMO) state-space model to support flight control development and
vehicle sizing for the DARPA Phase I test vehicle. The frequency range of interest was
0.1 –10 rad/sec. Frequency response analyses show that the important dynamic
characteristics in this frequency range are the rigid body dynamics.
Examination of the eigenvalues of the identified model reveals low frequency
unstable periodic modes in both the pitch and roll degrees of freedom. Excellent matches
between the model and flight data for the on-axis time responses confirm the accuracy of
the of the identified state-space dynamic model.
- 17 -
The CIFER identification is based on a set of flight test data gathered while flying
the prototype vehicle. The data was recorded at a nominal data rate of 23 Hz and included
vehicle rate and control mixer inputs. These are presented in Table 3.1.
Table 3.1 – OAV Measured Parameters during Flight Testing
Parameter
Measured Value
pmixer
CMPA
qmixer
CMQA
rmixer
CMRA
p
q
r
PP
QQ
RR
Frequency responses were generated with CIFER’s FRESPID tool from the test
data gathered from flying the proposal vehicle. Frequency ranges from ~0.35 – 20
(rad/sec) were used with four windows. The data was processed through MISOSA to
remove the effect of off-axis control inputs during the sweeps. COMPOSITE was used to
combine the four windows of data into a single response.
The frequency ranges used for the dynamic model identification were the ranges
when the coherence was good (values above 0.6). These frequency ranges are listed in
Table 3.2 and are used in the state space model identification in DERIVID. Examination
of the off-axis frequency responses indicates no significant cross-couplings between the
longitudinal and lateral degrees of freedom. These couplings are therefore not included in
the state space model. This is unique to this vehicle and differs from other vehicles tested.
It may be due to lack of excitation during flight test.
- 18 -
Table 3.2 – OAV Frequency Range of Good Coherence (rad/sec)
P
Q
R
CMPA
1-8
-
CMQA
1-8
-
CMRA
3-10
Because no significant cross-coupling between the longitudinal and lateral degrees of
freedom was observed, the state-space form would be modeled after the transfer
functions. The identified transfer functions appear as Equations 3.1-3.2.
18.68s(s + 0.0032)e−0.0477 s
=
pmixer (s + 2.0983)[−0.5761,1.7921]
(Equation 3.1)
21.07s2 e−0.0653 s
=
qmixer (s + 1.9496)[−0.7616,1.9349]
(Equation 3.2)
p
q
r
rmixer
=
20.81e−0.0718 s
s
(Equation 3.3)
The 3rd order denominator forms known as a “hovering cubics” (Equations 3.4 and
3.5) exemplify the dynamic modes for the longitudinal and lateral directions5. The control
derivatives for the state-space model were initially set as the free gain terms in the
numerators of the transfer functions. These values appear in Table 3.3.
∆ lateral − hover = s 3 + ( −Yv − LP ) s 2 + Yv LP s − gLv
(Equation 3.4)
∆ longitudinal − hover = s 3 + ( X u + M q ) s 2 + X u M q s − gM u
(Equation 3.5)
- 19 -
Table 3.3 – OAV Control Derivatives Extracted from Transfer Function Fits
Derivative
Lδ
Mδ
Nδ
Value
0.326
0.343
0.339
A state space form comprised of a set of four matrices (F, G, H1, and H2) known
as a quadruple was set up. This can be seen as Equations 3.6 – 3.13. The state vector ( x )
is presented as equation 3.8 (the subscript "rad" indicates that these quantities have the
units of rad and rad/sec). The three controls were pmixer, qmixer, and rmixer, as seen in
Equation 3.10 ( u ). The removal of cross-coupled terms yielded a final stability matrix
(F) to be fitted to the data (Equation 3.11). While the units of the states are in rad, rad/sec,
and ft/sec; the data is in deg/sec. A conversion factor of 57.3 (deg/rad) was multiplied
through the H1 matrix (Equation 3.13) and divided through the initial values of the
control derivatives (Table 3.3) in the G matrix (Equation 3.12). CIFER then tuned the
parameters in the F and G matrices to match the state space model’s frequency responses
to those for the flight test data.
x& = Fx + Gu
(Equation 3.6)
y = H1 x + H 2 x&
(Equation 3.7)
⎧ v ⎫
⎪p ⎪
⎪ rad ⎪
⎪ φrad ⎪
⎪
⎪
x =⎨ u ⎬
⎪q ⎪
⎪ rad ⎪
⎪ θ ⎪
⎪
⎪
⎩ rrad ⎭
(Equation 3.8)
- 20 -
⎧ p⎫
⎪ ⎪
y = ⎨q ⎬
⎪r ⎪
⎩ ⎭
(Equation 3.9)
⎧ pmixer ⎫
⎪
⎪
u = ⎨ qmixer ⎬
⎪r
⎪
⎩ mixer ⎭
(Equation 3.10)
⎡ Yv
⎢L
⎢ v
⎢0
⎢
F =⎢0
⎢0
⎢
⎢0
⎢0
⎣
0
LP
1
0
0
g
0
0
0
0
0
0
0
Xu
Mu
0
0
0
0
Mq
0
0
0
−g
0
0
0
0
0
0
0
1
0
0
0
⎡ Ypmixer
⎢L
⎢ p mixer
⎢ 0
⎢
G=⎢ 0
⎢
⎢ 0
⎢ 0
⎢
⎢⎣ 0
0
0
0
X qmixer
Mq
mixer
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
0⎥
N r ⎥⎦
0
0
0
0
0
0 ⎤
0 ⎥⎥
0 ⎥
⎥
0 ⎥
⎥
0 ⎥
0 ⎥
⎥
N r mixer ⎥⎦
0
0
0 ⎤
⎡0 57.3 0 0
⎢
H1 = ⎢ 0
0
0 0 57.3 0
0 ⎥⎥
⎢⎣0
0
0 0
0
0 57.3⎥⎦
(Equation 3.11)
(Equation 3.12)
(Equation 3.13)
It is worthwhile to note that many of the derivatives were set to zero in the
identification process. Because of the lack of acceleration data, the on-axis damping
parameters Xu, Yv, and Zw were unable to be determined in the model and were thus
removed from the CIFER model (fixed to a value of 0). A closer examination of the
- 21 -
transfer functions (Equations 3.1-3.3) will show that the longitudinal and lateral modes
are heavily reliant on the values of Lv and Mu, respectively. If these derivatives were the
only ones in the hovering cubic forms (Equations 3.4 and 3.5), the equations would
reduce to the degenerate forms seen in Equations 3.14 and 3.15. These forms contain one
real and one complex root for negative values of Lv and Mu. These roots describe the
dynamics of the system and show that Lv and Mu are the dominant terms required to
depict the three modes.
∆lateral −hover = s 3 − gLv
(Equation 3.14)
∆longitudinal −hover = s 3 − gM u
(Equation 3.15)
CIFER allows for a measure of merit, or cost, of the final model fit to the
frequency responses. Lower costs are better fits. The final model had an excellent
average cost of 23.6. For the best possible fit, pure time delays were identified as
0.04205, 0.08730, and 0.07189 seconds for roll, pitch, and yaw responses, respectively.
The longitudinal delay was bigger in both the state space model and the transfer function
fits. However, the Cramer-Rao bound for the longitudinal delay was rather big (29%)
revealing that it was a correlated term in the minimization process. This may be due to
CIFER adjusting the value to make up for inconsistencies in the model or it is due to the
pitch sensor or flight control computer. All other Cramer-Rao bounds were acceptable,
(CR< 15%) indicating good reliability of the identified derivatives.
Table 3.4 contains the identified variables and their respective certainty during the
identification. A comparison with the control derivatives extracted from the transfer
functions (Table 3.3) reveals very close matches.
- 22 -
Table 3.4 – OAV DERIVID Identified Parameters and Certainties
Table 5 shows the cost functions for the transfer functions. They were all very acceptable.
Table 3.5 – OAV DERIVID Frequency Response Costs
The asymmetric design of the vehicle accounts for the difference in the values
between Lv and Mu. Figure 1.3 depicts the fact that the OAV design has nacelles or cargo
pods making it asymmetric. The ratio of the identified values (Lv : Mu = 0.7510) reflects
the relationship of the lateral and longitudinal inertias specified (Iyy : Ixx = 0.6312).
- 23 -
The final CIFER® identified state space dynamic model is presented in Appendix A.
The eigenvalues and their associated eigenvectors are given below in Table 3.6.
They have been normalized to the dominant mode. The eigenvectors are the
corresponding state values which identify the modes. The larger values indicate the states
which are dominant in the modes. A value of 1 in the eigenvector indicates which state is
the primary mode. From the eigenvectors and eigenvalues some interesting dynamics can
be noted.
Table 3.6 – OAV Eigenvalues and Associated Eigenvectors of [F]
Mode #
(Aperiodic Yaw Subsidence)
real
0.00E+00
Mode #2
(Lateral Low Frequency Periodic)
imaginary
real
0.00E+00
9.25E-01
Mode #3
(Aperiodic Roll Subsidence)
imaginary
Real
-/+1.60E+00
imaginary
-1.85E+00
0.00E+00
[zeta, omega]
[zeta, omega]
[zeta, omega]
[0.000E+00, 0.000E+00]
[-.500E+00, 0.185E+01]
[0.000E+00, 0.000E+00]
V
0.00E+00
0.00E+00
V
-8.20E-02
+/-1.42E-01
V
1.64E-01
0.00E+00
P
0.00E+00
0.00E+00
P
1.00E+00
-/+1.13E-08
P
1.00E+00
0.00E+00
PHI
0.00E+00
0.00E+00
PHI
2.70E-01
+/-4.68E-01
PHI
-5.40E-01
0.00E+00
U
0.00E+00
0.00E+00
U
0.00E+00
0.00E+00
U
0.00E+00
0.00E+00
0.00E+00
Q
0.00E+00
0.00E+00
Q
0.00E+00
0.00E+00
Q
0.00E+00
THETA
0.00E+00
0.00E+00
THETA
0.00E+00
0.00E+00
THETA
0.00E+00
0.00E+00
R
1.00E+00
0.00E+00
R
0.00E+00
0.00E+00
R
0.00E+00
0.00E+00
Mode #4
(Aperiodic Pitch Subsidence)
real
-2.04E+00
Mode #5
(Longitudinal Low Frequency Periodic)
imaginary
real
0.00E+00
1.02E+00
imaginary
-/+1.76E+00
[zeta, omega]
[zeta, omega]
[0.000E+00, 0.000E+00]
[-.500E+00, 0.204E+01]
V
0.00E+00
0.00E+00
V
0.00E+00
0.00E+00
P
0.00E+00
PHI
0.00E+00
0.00E+00
P
0.00E+00
0.00E+00
0.00E+00
PHI
0.00E+00
U
2.76E-01
0.00E+00
0.00E+00
U
-1.38E-01
-/+2.39E-01
Q
THETA
-3.55E-02
0.00E+00
Q
1.78E-02
-/+3.08E-02
1.00E+00
0.00E+00
THETA
1.00E+00
+/-2.21E-08
R
0.00E+00
0.00E+00
R
0.00E+00
0.00E+00
- 24 -
The identified state-space model yielded 7 eigenvalues. Two of these were
complex pairs, and three real. These 7 eigenvalues depict 5 modes. Mode #1 is the yaw
mode which was modeled with no yaw damping, thus the value of 1 for the yaw rate state
(r). Mode #2 is associated with the 2nd order periodic denominator term in the hovering
cubic because of the high values for the lateral velocity (v) and roll rate (p) states. This is
a low frequency unstable mode. Likewise, Mode #5 is from the 2nd order term in
longitudinal hovering cubic. This is seen by the larger eigenvectors for the states of
longitudinal velocity (u) and pitch rate (q). The remaining eigenvectors identify the 1st
order, aperiodic subsidence modes for roll (Mode #3) and pitch (Mode #4). These
eigenvalues are very close to the modes of the transfer function models (Equations 1-3).
The excellent agreement between the flight data and model can be seen in the
following frequency responses comparing the parametric state space model and the actual
flight test data.
- 25 -
Figure 3.1 – Roll rate response frequency domain verification
It can be seen in Figure 3.1 that the roll rate model fits very well in the regions of
good coherence. Only where there are dips in this signal to noise ratio does the model
start to yield poor results. These results were obtained without linear acceleration data.
- 26 -
Better sensors, at higher sampling rates together with linear acceleration data will yield
closer matches across broader frequency ranges.
Figure 3.2 – Pitch rate response frequency domain verification
- 27 -
The pitch rate response seen in Figure 3.2 illustrates the accuracy of the statespace model in regions of good coherence as well. The coherence is the ratio of output
power that is linearly related to input power. This means that high noise in this channel,
or wind gusts during the sweep can produce lower coherence. It can be seen that the
accuracy of the state-space model for the pitch rate deteriorates quickly at lower
frequencies.
- 28 -
Figure 3.3 – Yaw response frequency domain verification
The model revealed that there was no natural yaw damping for this vehicle. The
unstable hovering cubic is prevalent in the 1-3 (rad/sec) region. The fit was accurate at
higher frequencies before noise in the channel becomes a problem, as seen in Figure 3.3.
- 29 -
The identified models were compared with data taken by Aerovironment during
flight testing. It can be seen that the on-axis responses have an excellent match for all 3
controls. The quality of the match confirms that the identified model is accurate.
Figure 3.4 – Roll response time history verification
- 30 -
Figure 3.4 shows that even though the lateral dynamics were modeled without a
roll damping term, the control surface effectiveness term and Lv in the hovering cubic
accurately pick up the nature of the response.
Figure 3.5 – Pitch response time history verification
- 31 -
Likewise, Figure 3.5 above shows that the longitudinal degree of freedom is
captured and represented in the state-space model very accurately.
Figure 3.6 – Yaw response time history verification
Figure 3.6 shows the accuracy of the yaw degree of freedom. It stays accurate
regardless of being modeled as the simple integrator form with no yaw damping.
- 32 -
It can be seen that the Aerovironment Proposal prototype OAV was successfully
modeled with a state-space model. The identified model shows good agreement for both
the time and frequency responses. The identified system showed an unstable periodic
mode in the pitch and roll responses. Time delays were determined for all three channels.
The ratio of the lateral to longitudinal moment terms Lv and Mu reflect the ratio of the
inertias Iyy to Ixx. All of the modes dictated by the hovering cubic forms were identified,
but because of a lack of acceleration data the speed damping force derivatives could not
be accurately identified. The identified transfer function modes closely match the modes
of the identified state space dynamic model.
After flight test was completed for the purposes of identification, the OAV design
was further analyzed in the wind tunnel. The vehicle was put into the Virginia Tech
Stability Wind Tunnel by Techsburg, Inc. without the payload nacelles. A photograph of
the setup is shown as Figure 3.7.
Figure 3.7 - Techsburg Wind Tunnel Setup for OAV
- 33 -
Although part of a larger control surface and augmentation experiment, the
vehicle was tested in a baseline configuration similar to that seen in Figure 1.3. From the
tests, pitching moment information was extracted with varying wind speeds. Figure 3.8
shows the results of that test.
2
1.5
1
M (ft-lbf)
0.5
0
-50
-40
-30
-20
-10
0
10
20
30
40
-0.5
-1
-1.5
-2
u (fps)
Figure 3.8 – Techsburg OAV Pitching Moment to Airspeed
As Figure 3.8 shows, there is a unique pitching moment created when the vehicle
experiences some wind velocity across the duct. This is illustrated by the slope of the
tangent line depicted as a dotted line. In this case, the dimensional derivative about the
hover condition is 0.011. This is a corrective moment for velocities below some critical
velocity. A negative pitching moment is then created above this critical speed. In the case
of OAV as tested, this occurs at roughly 10 fps.
- 34 -
50
3.2.2 Allied Aerospace MAV
Flight test was performed on the MAV vehicle in a similar manner as was
described in the previous section for the OAV. Table 3.7 below shows the physical
properties for the vehicle as it was tested.
Table 3.7 – MAV Physical Properties
Physical Quantity
Mass (slugs)
C.G. (below duct lip - inches)
Propeller Speed (rad/sec)
Ixx (slug-ft^2)
Iyy (slug-ft^2)
Izz (slug-ft^2)
Iprop (slug-ft^2)
Value
0.233
2.25
1884.0
0.021
0.021
0.021
0.00012*
* value obtained from Allied Aerospace that contains the inertia of all of the rotating components.
Frequency responses for on and off-axis are presented as Figure 3.9. These include the
removal of off-axis control contributions by using the CIFER tool MISOSA.
- 35 -
30
MAGNITUDE(DB)
-10
-50
250
PHASE(DEG)
50
-150
1
COHERENCE
0.6
0.2
0.1
1
FREQUENCY (RAD/SEC)
10
100
F040P_COM_ABCDE_pcmd_pb - p/lat
F040P_COM_ABCDE_pcmd_qb - q/lat
F040P_COM_ABCDE_pcmd_rb - r/lat
Figure 3.9 – On and Off Axis MAV Roll Frequency Responses
Figure 3.9 shows the roll, pitch and yaw rate frequency responses to roll control.
Here there is good coherence for the on-axis responses, but no coherence in the off-axis
direction. The roll rate frequency response has a good coherence from 0.5 to 12 rad/sec
and this portion of the frequency response is used in the identification.
- 36 -
30
MAGNITUDE(DB)
-10
-50
250
PHASE(DEG)
50
-150
1
COHERENCE
0.6
0.2
0.1
1
FREQUENCY (RAD/SEC)
10
100
F040Q_COM_ABCDE_qcmd_qb - q/lon
F040Q_COM_ABCDE_qcmd_pb - p/lon
F040Q_COM_ABCDE_qcmd_rb - r/lon
Figure 3.10 – On and Off Axis MAV Pitch Frequency Responses
Figure 3.10 shows the pitch, roll and yaw rate frequency responses to pitch
control. As with the roll control responses, there is good coherence for the on-axis
response, but no coherence for the off-axis responses. This would indicate that there is
very little cross-coupling and the pitch and roll responses are essentially uncoupled. It is
uncertain why the gyroscopic coupling is not evident in the flight tests. A similar
- 37 -
approach was used for the accelerometer information. The parametric state space model
was setup as shown in Equation 3.16.
⎧ u& ⎫ ⎡ Xu 0 − g
⎪ q& ⎪ ⎢ Mu Mq 0
⎪ ⎪ ⎢
1
0
⎪⎪θ& ⎪⎪ ⎢ 0
⎨ ⎬=⎢
0
0
⎪ v& ⎪ ⎢ 0
⎪ p& ⎪ ⎢ 0
Lq 0
⎪ &⎪ ⎢
0
0
⎪⎩φ ⎪⎭ ⎣⎢ 0
0
0
0
Mp
0
Yv
Lv
0
0
0
Lp
1
0 ⎤ ⎧u ⎫ ⎡ 0
0 ⎥⎥ ⎪⎪ q ⎪⎪ ⎢⎢ 0
0 ⎥ ⎪⎪θ ⎪⎪ ⎢ 0
⎥⎨ ⎬+ ⎢
g ⎥ ⎪ v ⎪ ⎢ Ylat
0 ⎥ ⎪ p ⎪ ⎢ Llat
⎥⎪ ⎪ ⎢
0 ⎦⎥ ⎩⎪φ ⎭⎪ ⎣⎢ 0
Xlon ⎤
Mlon ⎥⎥
0 ⎥ ⎧ δ lat ⎫
⎬
⎥⎨
0 ⎥ ⎩δ lon ⎭
0 ⎥
⎥
0 ⎦⎥
(Equation 3.16)
The derivatives Mp and Lq result from the gyroscopic moments produced by the
rotating inertia of the propeller. This coupling is one of the unique aspects of the
vehicle’s dynamics. Taking into account the angular momentum of the spinning propeller
and dividing by the inertia of the total vehicle yields the moment produced by the
gyroscopic effects. This is shown as equations 3.17 and 3.18.
Lq =
I prop Ω
Mp =
(Equation 3.17)
I xx
I prop Ω
(Equation 3.18)
I yy
The values for Mp and Lq therefore can be used for the determination of propeller
inertia. This is possible because the rotational speed of the propeller remained mostly
constant and the inertia of the vehicle changed negligibly due to fuel burned. This is
useful because the inertia of the small propeller while spinning is hard to measure in any
type of simple experiment. A time delay was also added to the dynamics to account for
transport delays in the electronics.
- 38 -
A 0th/2nd order transfer function is included in the identification to take into
account the actuator dynamics. The form of this transfer function is as follows:
ωn
2
s + 2ζω n + ωn
2
TF =
2
The values of the damping and natural frequency of the actuator used were
obtained from bench tests of the actuator dynamics presented in section 3.3 for the
Airtronics 94091 servo actuator running at nominally 5 volts. The natural frequency for
this case is 28.2 rad/sec and the damping ratio is 0.52.
The DERIVID utility was used to identify the elements of the state-space model.
The stability derivative results are shown Table 3.8.
Table 3.8 – MAV Identified Stability Derivatives
Derivativ e
Xu
Mu
Mq
Mp
Yv
Lq
Lv
Lp
I pr op
Param Value
-0.1090
0.5014
0.000 +
0.000 +
-0.1090 *
0.000 +
-0.5014 *
0.000 +
0.000 +
COUP02
CR Bound C.R. (%)
0.04395
40.33
0.03412
6.805
......
......
......
......
......
......
......
......
......
......
......
......
......
......
Insens.(%)
10.92
2.729
......
......
......
......
......
......
......
+ Eliminated during model structure determination
y Fixed value in model
* Fixed derivativ e tied to a free derivativ e
Yv = 1.000E+00* X u ( COUP02 )
L v =-1.000E+00* M u ( COUP02 )
The value of the rotating inertia (Iprop) was insensitive in the identification and
was dropped from the list of active elements. This is because there was no good
coherence in the off-axis roll and pitch rate responses, which result for the gyroscopic
- 39 -
effects from the rotating inertia. Ultimately this made for the coupling derivatives in the
model to become zero as well.
The control derivatives were identified as shown in Table 3.9.
Table 3.9 - MAV Identified Control Derivatives
Derivativ e
X lon
M lon
Yl at
L lat
øl at
øl on
COUP02
CR Bound C.R. (%)
0.01692
5.955
0.01103
4.705
0.01876
7.519
0.01056
5.902
......
......
4.599E-03
6.796
Param Value
-0.2841
-0.2343
0.2495
-0.1789
0.06767 *
0.06767
Insens.(%)
2.058
2.149
2.544
2.614
......
3.272
* Fixed derivativ e tied to a free derivativ e
ølat = 1.000E+00* øl on ( COUP02 )
Figure 3.11 shows the identified model’s roll and lateral acceleration responses for the
roll sweep.
40
p/lat
Magnitude(DB)
20
0
0
-20
-20
-40
-40
-60
150
100
Phase (Deg)
Phase (Deg)
50
100
50
0
0
-50
-50
-100
-100
-150
-150
-200
1
ay/lat
Magnitude(DB)
20
1
Coherence
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.1
1
10
Frequency (Rad/Sec)
Coherence
0.2
100 0.1
1
Frequency (Rad/Sec)
Flight results
COUP02 - Identification Results
Figure 3.11 – MAV Lateral Acceleration and Roll Rate Response to Roll Input
- 40 -
10
Figure 3.12 shows the same for the longitudinal acceleration and pitch rate response to
pitch input.
40
q/lon
Magnitude(DB)
20
20
0
0
-20
-20
-40
-40
-60
150
-100
Phase (Deg)
100
-150
50
-200
0
-250
-50
-300
-100
-150
-350
-400
1
1
Coherence
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.1
1
10
Frequency (Rad/Sec)
ax/lon
Magnitude(DB)
Phase (Deg)
Coherence
0.2
100 0.1
1
Frequency (Rad/Sec)
10
Flight results
COUP02 - Identification Results
Figure 3.12 – MAV Longitudinal Acceleration and Pitch Rate Response to Pitch Input
The combination of Figure 3.11 and Figure 3.12 show that the identified model
agrees with the flight test data. There are some inconsistencies, but overall the costs of
the fits were low and the model agrees with flight test results. The final identified
parameters are outlined in Table 3.10.
- 41 -
Table 3.10 – Final Flight Test Identified MAV Derivatives
Derivativ e
Xu
Mu
Mq
Mp
Yv
Lq
Lv
Lp
I pr op
X l on
M l on
Ylat
L lat
ølat
ølon
Param Value
-0.1090
0.5014
0.000 +
0.000 +
-0.1090 *
0.000 +
-0.5014 *
0.000 +
0.000 +
-0.2841
-0.2343
0.2495
-0.1789
0.06767 *
0.06767
+ Eliminated during model structure determination
y Fixed value in model
* Fixed derivativ e tied to a free derivativ e
M p = 8.971E+04* I pop ( PIT21 )
L q =-8.971E+04* I pop ( PIT21 )
Yv = 1.000E+00* X u
L v =-1.000E+00* M u
ølat = 1.000E+00* øl on
The identification of the MAV vehicle benefited from also having wind tunnel
tests performed by Allied Aerospace. These tests were completed to build up a nonlinear,
test data based, table-lookup bare airframe and control simulation. MAV is a family of
vehicles. Both the larger 29” vehicle and smaller 9” vehicle were put into the wind tunnel
with the fans spinning at various speeds while the attitude and wind velocity was varied.
This was done to determine moment and force values with angle of attack and beta as
well as lateral, longitudinal, and vertical velocities.
There were issues with the 9” wind tunnel results. To illustrate the wind tunnel
method for the MAV (which is similar to the wind tunnel tests performed for OAV by
- 42 -
Techsburg) the pitching moment response to gusts was analyzed. Figure 3.13 shows a
summary of the data collected for the pitching moment.
i-Star-9 Pitching Moment Characteristics
Pitching Moment (ft-lb)
0.4
0.2
0
-0.2
0
20
40
60
80
100
120
-0.4
-0.6
-0.8
-1
Shroud Velocity (fps)
Figure 3.13 – Pitching Moment Wind Tunnel Test Data for i-Star 9”
Figure 3.13 shows that a linearization was completed for the first 30 knots and is
shown. The slope of this line represents the dimensional derivative Mu. What is curious
here, and will be discussed in further detail in the next sections, is the nature of the
pitching moment response to increases in speed. As the vehicle experiences a cross wind
in hover, it will pitch in the positive direction. This represents a corrective moment.
However if the gust is strong enough, it will actually experience a negative moment.
The method illustrated above was repeated for all of the major flight derivatives
to obtain the values portrayed in Table 3.11. Table 3.11 compares both 9” and 29”
vehicles as well as the 9” flight test results where appropriate.
- 43 -
140
Table 3.11 – MAV Wind Tunnel Identified Derivatives and Flight Test Results
I-Star Vehicle
Derivative
9”
29”
Wind Tunnel
Flight Test
- 0.476
- 0.344
-0.1090
- 0.476
- 0.344
-0.1090
(Fixed to Xu)
(Fixed to Xu)
- 0.349
- 0.212
n/a
- 0.046
0.004
-0.5014
(Fixed to –Mu)
(Fixed to –Mu)
(Fixed to –Mu)
Lp
0
0
0
Mu
0.046
0.003
0.5014
Mq
0
0
0
Mp
n/a
n/a
0
Lq
n/a
n/a
0
Nw
- 0.056
- 0.006
n/a
Nr
0
n/a
n/a
X lon
- 0.190
- 0.157
-0.2841
Ylat
0.156
0.123
n/a
Z col
- 0.012
- 0.264/100
n/a
Llat
- 0.218
- 0.418
n/a
M lon
- 0.387
- 0.548
-0.2343
N ped
0.669
0.555
n/a
N col
-0.005
- 0.057/100
n/a
Xu
Yv
Zw
Lv
(Fixed to Xu)
- 44 -
Table 3.11 shows that all of the dimensional derivatives for the 29” vehicle are
larger than the 9” values. This is to be expected because the larger vehicle should
experience larger forces and moments to go with its increased mass and inertias. It also
shows that the flight test and wind tunnel results are all of the same sign and fairly close.
The only exception is that of the difficult derivative Mu. Wind tunnel testing revealed a
much smaller value for this critical derivative (0.003) than the flight test (0.5014).
- 45 -
3.2.3 Trek Aerospace Solotrek
Although nothing like the other vehicle’s examined, the Trek Aerospace (now
Trek Entertainment, Inc.) Solotrek does possess ducted fan technologies which are
common to the MAV and OAV. One of the Solotrek’s ducted fans (Figure 1.4) was
inserted into the NASA Ames 7’ x 10’ wind tunnel at Moffett Field for aerodynamic
testing. Forces and moments were recorded with various wind tunnel and fan speeds
while the ducted fan was mounted at 90° to the flow.
The pitching moment was recorded with varying forward speeds and propeller
RPM. The results of that test are shown in Figure 3.14. This data could be used for
determination of dimensional pitching moment derivatives.
200
180
Pitching Moment (ft-lbs)
160
1800 rpm
2200 rpm
2600 rpm
3000 rpm
140
120
100
80
60
40
20
0
0
20
40
60
80
100
Wind Tunnel Speed (fps)
Figure 3.14 – Solotrek Wind Tunnel Test Results for Pitching Moment
- 46 -
120
Figure 3.14 shows how increasing the fan speed increases the pitching moment.
By fitting lines to the data for 0 to 20 knots, a linear representation of the pitching
moment derivative is obtained for this low speed condition. This is shown in Figure 3.14
as dashed lines. The slopes of these lines are the dimensional derivatives. They are
summarized in Table 3.12. Figure 3.14 also shows that some critical velocity may exist
when the derivative will actually swing to negative. This is seen in the 1800 RPM case to
be around 70 fps.
Table 3.12 – Pitching Moment Derivatives and Solotrek Fan Speed
Pitching Moment Derivative Mu
Fan Speed
(rpm)
⎛ ft-lb ⎞
⎜
⎟
⎜ ft ⎟
⎝ sec ⎠
1,800
2,200
2,600
3,000
1.034
1.376
1.933
2.589
This wind tunnel testing was the extent of identification work completed for the Solotrek
vehicle.
- 47 -
3.2.4 Hiller Flying Platform
The Hiller Flying Platform along with a dummy mannequin was attached to the
top of a truck and possessed equipment to measure moments and forces as it was driven
at Moffett Field in 1958. The results of the tests by Sacks3 are the basis for the pitching
moment identification.
The primary data of concern is that of the pitching moment directly measured
with increasing truck speed. The results of those runs are presented in Figure 3.15.
450
400
Pitching Moment (ft-lbs)
350
300
250
200
150
100
50
0
0
20
40
60
80
Speed (fps)
Figure 3.15 – Hiller Flying Platform Pitching Moment Data
The truck test was performed with the fan running at the speed required to keep
the vehicle in hover. However, it also contained a dummy 6 foot tall, 175 lb man.
Because this comparison is primarily focused on the pitching moment characteristics of
- 48 -
the duct, the effects of the man need to be removed from the above moments. This is
done by approximating the man as a flat plate (6’ x 2’). While crude, this investigation is
merely to establish a trend with the pitching moment characteristics of ducted fan
vehicles.
The relationship for the drag on a flat plate for Re > 1000 is presented as Figure 3.16.
Figure 3.16 – Drag over a Flat Plate Perpendicular to Flow
With the approximation in size of the man, a drag coefficient of CD = 1.1 is found
from Figure 3.15. It follows that the drag of the man will vary with velocity as in
Equation 3.19.
D plate =
1 2
ρ v ACD
2
- 49 -
(Equation 3.19)
It is known that the dummy was placed directly on top of the platform, so it is
assumed that the drag will have a moment arm of 3 feet above the platform, or half the
height of the plate used to approximate the drag. This allows the determination of
moment produced with airspeed due to the dummy. This is calculated and then subtracted
from the actual data in Figure 3.15 to produce Figure 3.17.
450
Hiller Test Results
400
Approximate Dummy Moment
Approximate Duct Pitching
Moment
Pitching Moment (ft-lbs)
350
Linear Fit for 20 knts
300
250
200
150
100
50
0
0
20
40
60
80
Speed (fps)
Figure 3.17 – Results of Removing Dummy Moment from Hiller Platform Test
It can be seen that the moment from the dummy is increasing with truck speed.
Removing the effect of the dummy produces the green line. This is then used to fit a line
to determine the average slope from 0 to 20 knots (33.8 fps). This slope of this dashed
line is the dimensional pitching moment derivative, Mu.
- 50 -
M u PLATFORM = 5.11
ft-lb
ft
sec
This dimensional derivative is naturally much larger than the other values looked
at for the other vehicles. This makes sense because this is a much larger vehicle. It is a
positive number for hover. However, it will go negative if the wind velocity reaches some
critical speed. In this case, that velocity is 55 feet per second. This follows the trend of
the other vehicles.
- 51 -
3.2.5 Vehicle Scaling Laws and Comparisons
It becomes apparent that the ducted fans looked at all share some basic
characteristics in one way or another. One of the main advantages of the RUAV designs
mentioned in Chapter 1 is that these vehicles can hover. Hovering flight leaves these
vehicles highly susceptible to wind in station-keeping applications. Of particular interest
is the derivative Mu. This derivative characterizes the vehicle very well in hovering flight
(as seen with OAV flight test: Equation 3.15) in the hovering cubic. To understand the
nature of the vehicles and fully characterize and identify their flight, some time is needed
to understand the pitching moment characteristics.
In order to compare the pitching moment characteristics of the four vehicles, Mu
must be nondimensionalized to take into account the size of the vehicles, the propeller
effects, and the ducts themselves. To do this, the nondimensional pitching moment
definition for rotorcraft is applied:
M ~ pitching moment
CM =
M
ρ ~ density
Ω ~ blade rotation speed (rad/sec)
ρ A ( ΩR ) R
2
R ~ duct radius
A ~ duct area
This method primarily accounts for duct size with the radius terms, and fan speed Ω.
Because the condition we are most interested in is low speed around hover, we
look at the derivative about zero to 20 knots airspeed for the vehicles. In other words, the
slope of a line fit to the pitching moment vs. airspeed data is calculated for only the low
speed condition. This value is then nondimensionalized with the above method. It is
- 52 -
apparent that the size of the duct is the driving factor in the aerodynamic pitching
moment. In fact, this nondimensionalization by the third power of the radius follows what
was observed for ducted fans by Sacks3.
This approximation of the way the pitching moment varies with duct size is used
to compare the three vehicles. The geometries of the vehicles are used here to determine
the dimensional and nondimensional parameters for comparison (Table 3.13). In the case
of the Solotrek fan, the four different fan speeds are presented.
Table 3.13 – Pitching Moment Coefficient Summary
Vehicle
Flying Platform
Wind Tunnel
OAV
Flight Test
1,800 RPM
2,200 RPM
Solotrek
2,600 RPM
3,000 RPM
Wind Tunnel
i-Star 9”
Flight Test
i-Star 29”
Pitching Moment Derivative Mu
⎛ ft-lb ⎞
⎜
⎟
⎜ ft ⎟
⎝ sec ⎠
5.11
0.011
0.00643
1.034
1.376
1.933
2.589
0.00323
0.5014
0.11652
Nondimensional
CMu
7.95 x 10-5
1.09 x 10-5
6.52 x 10-5
3.21 x 10-5
2.86 x 10-5
2.87 x 10-5
2.90 x 10-5
1.30 x 10-6
2.01 x 10-4
1.14 x 10-6
It is evident from Table 3.13 that the values are within the same order of
magnitude and show positive speed stability for most of the vehicles and methods. Wind
tunnel values seem to differ from the other values. The largest values are seen with the
flight test for MAV and wind tunnel results for OAV. The values for the different fan
speed for the Solotrek duct are all closely related, demonstrating that the same method is
nondimensionalizing well for vehicles of varying prop speeds.
- 53 -
Table 3.13 reveals that this method may not be accounting for the entirety of
dominant characteristics for ducted fan vehicles. This is seen in the way the Solotrek
differs from the other smaller chord vehicles. To account for more specific geometries, a
method which better characterizes the propellers was also investigated. This
nondimensionalization uses the chord and radius of the rotating propellers to
nondimensionalize the pitching moment:
CM =
M ~ pitching moment
M
ρ ~ density
ρσ A ( ΩR ) R
2
Ω ~ blade rotation speed (rad/sec)
R ~ duct radius
bc
σ=
πR
A ~ duct area
b ~ # of blades
c ~ mean blade chord
Table 3.14 represents the results of this method.
Table 3.14 – Pitching Moment with Blade Chord Summary
Vehicle
Flying Platform
Wind Tunnel
OAV
Flight Test
1,800 RPM
2,200 RPM
Solotrek
2,600 RPM
3,000 RPM
Wind Tunnel
i-Star 9”
Flight Test
i-Star 29”
Pitching Moment Derivative Mu
⎛ ft-lb ⎞
⎜
⎟
⎜ ft ⎟
⎝ sec ⎠
5.11
0.011
0.00643
1.034
1.376
1.933
2.589
0.00323
0.5014
0.11652
Nondimensional
CMu
4.48 x 10-4
1.03 x 10-4
6.15 x 10-4
2.90 x 10-4
2.58 x 10-4
2.60 x 10-4
2.61 x 10-4
2.45 x 10-5
3.80 x 10-3
5.20 x 10-5
This method yields values similar to the previous methods in Table 3.13. The
numbers here are more closely related and show that the nondimensionalization is an
- 54 -
adequate way to characterize the different pitching moment characteristics for these
vehicles. It is can be seen that the derivatives for the i-Star class of vehicles differ
considerably from the other ducted fans analyzed. In the case of the wind tunnel results
for these two vehicles, the 9” value (2.45 x 10-5) and the 29” value (5.20 x 10-5) are of the
same order of magnitude, but an order lower than all of the other vehicles. This suggests
that there may be something unique about the i-Star design, or that there was something
unexplainable happening with the wind tunnel tests of the vehicles. Flight test revealed
that the 9” vehicle actually had a very large value for Mu (3.80 x 10-3). This is an order
larger than the other vehicles, and a full two orders greater than the wind tunnel results
for the same vehicle. This could be due to the fact that Mu was found to be so dominant in
the identification.
To briefly summarize and conclude, all four of the ducted fan vehicles exhibit
likeness in pitching moment characteristics. The only anomaly seen is with the i-Star
vehicle which shows relatively higher and lower CMu values in comparison to the other
vehicles and the method of identification.
- 55 -
3.3 Servo Actuator Identification
The goal of the actuator test program was to measure a set of data that was used to
identify models of the actuator dynamic response characteristics. These actuator models
include linear transfer functions of the input/output relationships as well as non-linear
actuator properties such as actuator rate and position limits.
The identification was performed using the CIFER. Linear 0th/2nd order transfer
functions capturing the actuator dynamics were identified. Testing allowed for the
determination of the maximum angular rates and positions using linear curve-fitting of
the square wave responses. An explanation of the construction of the actuator block
diagrams built is also included. The actuators are a critical part of the flight control
system and it is important to have accurate models of the dynamics and limits of the
actuators themselves. Individual blocks were created for each actuator corresponding to
each of the tested 5 volt and 6 volt conditions. This section also includes a time domain
validation of the actuator models.
The goal of bench testing the control surface actuators was to collect a set of
bench test data that will be used to identify the actuator dynamics. This test data was also
used to determine the position and rate limits of the actuators. The significance of other
non-linear actuator properties, such as hysteresis and stiction, are also evaluated from the
bench test data.
The bench testing was carried out in accordance with CIFER flight test techniques
wherever possible. Five separate actuators from four manufacturers were tested. The
- 56 -
actuators varied in size, weight, cost, and performance. The manufacturers’ specifications
are presented in Table 3.15. Figure 3.18 shows the relative sizes of the actuators tested.
Table 3.15 – Manufacturer Specifications for Servo Actuators Tested
MODEL NUMBER
WEIGHT
(oz)
TORQUE
(oz/in@ 4.8V)
RATE
(deg/sec)
L
(in)
W
(in)
D
(in)
JR PROPO DS8417
2.03
0.80
0.80
0.32
0.19
82.0
42.0
53.0
18.0
7.0
600.0
352.9
285.7
500.0
1000.0
0.73
0.39
0.50
0.44
0.37
1.52
1.33
1.12
0.91
0.90
1.32
1.18
1.17
0.87
0.61
HITEC HS-512MG
JR PROPO DS368
AIRTRONICS 94091
CIRRUS CS-10BB
Figure 3.18 – Actuators Tested and Relative Sizes
The test apparatus was comprised of a rigid aluminum base stand with allowances
for the actuators to fit inside without moving. For the smaller actuators, small wooden
strips were used to ensure rigid mounting. The actuator horns were connected to horns on
potentiometers using clevises. The potentiometers offered little to no load resistance. The
mechanical apparatus can be seen in Figure 3.19.
- 57 -
Figure 3.19 – Actuator Test Stand Apparatus
A close up of the small Cirrus CS-10BB servo mounted on the test fixture in the wooden
strip is presented as Figure 3.20.
Figure 3.20 – Cirrus CS-10BB Mounted on Wooden Strip
- 58 -
It is noticeable from the figure that the servo horn and the potentiometer horn are
not the same length. This means that the deflection of the potentiometer horn will not be
the same as the deflection of the servo horn. All attempts were made to keep these
lengths the same.
Measurements of all the actuators and the various geometries accounting for the
aforementioned differences were taken with precision calipers and recorded as seen in the
schematic in Figure 3.21.
Figure 3.21 – Schematic Detailing Linkage Geometry
It is apparent that because the ‘center-center’ distance is different from the ‘hornhorn’ measurement, the servo deflection will not be 90° when the potentiometer is at 90°.
The geometries for all of the actuators are presented in Table 3.16.
- 59 -
Table 3.16 – Actuator Linkage Geometries
HORN
SERVO
DS8417
JR94091
DS368
HS12MG
CS-10BB
SERVO
INPUT
POT
SERVO
w/
POT @
90°
VOLT
HORNHORN
(in)
SERVO
HORN
(in)
POT
HORN
(in)
CENTERCENTER
5
3.482
0.994
0.975
6
3.482
0.994
5
3.688
0.757
6
3.688
0.757
5
3.527
0.495
0.468
3.539
-45°
60°
-48.610
32.539
-50
50
2102
4086
88.611°
6
3.527
0.495
0.468
3.539
-45°
60°
-48.610
32.539
-50
50
2102
4085
88.611°
5
3.51
0.509
0.469
3.544
-55°
50°
-43.471
39.408
-50
50
1880
3870
86.170°
6
3.51
0.509
0.469
3.544
-55°
50°
-43.471
39.408
-40
40
1987
3670
86.170°
5
3.67
0.504
0.468
3.652
-45°
60°
-43.814
39.785
-50
50
1930
3963
92.077°
6
3.67
0.504
0.468
3.652
-45°
60°
-43.814
39.785
-50
50
1935
3969
92.077°
MIN
MAX
MIN
(deg)
MAX
(deg)
MIN
(103)
MAX
(103)
MIN
MAX
3.460
-40°
60°
-40.741
34.174
-50
50
1810
3728
91.268°
0.975
3.460
-40°
60°
-40.741
34.174
-50
50
1809
3727
91.268°
0.669
3.719
-60°
68°
-46.419
30.644
-40
40
1851
3895
87.653°
0.669
3.719
-60°
68°
-46.419
30.644
-40
40
1854
3882
87.653°
(in)
The most non-linear case was observed for the HS12MG where problems with the
horns also resulted in binding and interference at larger deflections. For this reason, the
maximum commanded deflection was limited to 80% of the maximum actuator
deflection when testing this actuator.
The potentiometer apparatus was located next to Allied Aerospace’s HIL
simulation test stand. This utilized the ADC and DAC capabilities of the vehicle
hardware to feed the actuators the Pulse Width Modulation (PWM) from the stimulus
files prepared in accordance with CIFER flight test techniques.
The two primary measurements required for the CIFER identification were the
sweep commanded into the actuator and the potentiometer reading as a result of the
actuator moving. Because of the nature of the recording equipment, calibration factors
were required to convert the input and output signals to degrees. These calibration factors
were determined using the geometries shown in Table 3.16 and are presented in Table
3.17.
- 60 -
Table 3.17 – Actuator Calibration Factors for Input and Output Channels to Degrees
CALIBRATION FACTOR
SERVO
VOLTAGE
DS8417
JR94091
DS368
HS12MG
CS-10BB
IN Channel
OUT Channel
(degrees/unit input)
(servo deg/POT units)
5
0.000749
0.0391
6
0.000749
0.0391
5
0.000963
0.0377
6
0.000963
0.0380
5
0.000811
0.041
6
0.000811
0.0409
5
0.000829
0.0416
6
0.001036
0.0492
5
0.000836
0.0411
6
0.000836
0.0411
The hardware fed signals from -50,000 to 50,000 to the servos and recorded
potentiometer deflection from roughly 1500 to 4500. The calibration factors in Table 3.17
relate these to degrees of command and deflection of the servo. They are a result of the
geometries and readings for each actuator-voltage combination tested.
Data was recorded at 50 Hz and there was no filtering of the input and output
channels. An unidentified glitch was observed in the output signal and showed itself as a
signal spike at roughly every 5 samples (0.1 sec). This was evaluated and it was
determined to be minor in identifying the dynamics. With that exception, there was very
little noise in the signals.
Frequency sweep actuator commands were used to generate test data from which
frequency responses of control surface response due to actuator command could be
identified. From these frequency responses, transfer functions of the actuator dynamics
were extracted. The non-linear effects, such as rate and position limits were identified by
using a square-wave command.
- 61 -
The time histories of the actuator command signals were computer generated
using the frequency sweep code that was described for the flight test frequency sweep
maneuvers. The inputs to this code specify the various parameters of the frequency
sweep. These parameters are shown in Table 3.18 for the sweeps used in the tests.
Table 3.18 – Frequency Sweep Used for all Actuators
Description:
Control axis
Total duration of sine sweep
Duration of zero signal
Time for signal fade-in
Time for signal fade-out
Signal sample rate
Minimum frequency of sweep
Maximum frequency of sweep
Filter cut-off frequency
Amplitude of control input
Maximum allowable amplitude
Noise random flag
Units:
sec
sec
sec
sec
Hz
Hz
Hz
Hz
%
%
-
Value:
1
30
2
3
1
50
0.1
10.0
-1
10, 50, (80),100
100
-1
The signal amplitudes used to drive the actuators during the frequency sweep tests
were 10, 50 and 100% of the maximum pulse width amplitude and was generated with
computer code. In the case of some of the smaller actuators (DS368 & HS512MG), the
100% input was brought down to 80% because of clevis interference at higher
deflections. White noise is not required in the command signals for actuator testing. A
cut-off filter could be included to ensure that the frequency content of the command
signal does not go beyond a maximum frequency. This is not required for bench testing
and no filter cut-off frequency was set, indicating that the signal should not be filtered.
- 62 -
Figure 2.1 shows an example frequency sweep time history generated with the computer
code.
A 100% square wave was used to drive the actuators to their position limits. A
50% square wave was also used to determine rates for smaller peak to peak deflections.
The parameters for the square wave are shown in Table 3.19.
Table 3.19 – Square Wave Parameters
Description:
Total duration of wave
Signal sample rate
Amplitude of positive step
Positive step hold time
Amplitude of negative step
Negative step hold time
Units:
Value:
sec
Hz
% max
sec
%
sec
~30
50
50, 100
0.5
50, 100
0.5
The amplitude of the actuator signal is the percentage of the maximum pulse width
amplitude that drives the actuators in each direction. As an example, the chirp input,
response time history, and square wave used for the DS8417 is presented in Figure 3.22.
- 63 -
Chirp Input
50
40
30
Deflection (deg)
20
10
0
0
5
10
15
20
25
30
20
25
30
-10
-20
-30
-40
-50
Time (sec)
Potentiometer Response
50
40
30
Deflection (deg)
20
10
0
0
5
10
15
-10
-20
-30
-40
-50
Time (sec)
60
Deflection (deg)
40
20
0
0
1
2
3
4
5
6
7
8
9
10
-20
-40
-60
Tim e (sec)
Figure 3.22 – Sample Chirp Input, Response, and Square Wave Time History
- 64 -
The test matrix is provided as Table 3.20. It outlines the recorded file name, model
number, and conditions of the actuator tested. The CIFER case name for the frequency
sweep cases is also listed if identification was completed.
Table 3.20 – Actuator Bench Test Matrix
CIFER
NAME
DS8417_1
DS8417_2
DS8417_3
DS8417_4
DS8417_5
DS8417_6
DS8417_7
HS512MG1
HS512MG2
HS512MG3
HS512MG4
HS512MG5
HS512MG6
DS368_1
DS368_2
DS368_3
DS368_4
DS368_5
DS368_6
94091_1
94091_2
94091_3
94091_4
94091_5
94091_6
CS10BB_1
CS10BB_2
CS10BB_3
CS10BB_4
CS10BB_5
CS10BB_6
TEXT FILE NAME
DS8417_TEST_RUN.TXT
DS8417_100_5.TXT
DS8417_100_5_2.TXT
DS8417_50_5.TXT
DS8417_10_5.TXT
DS8417_100_6.TXT
DS8417_50_6.TXT
DS8417_10_6.TXT
DS8417_100_5_square.TXT
DS8417_50_5_square.TXT
DS8417_10_5_square.TXT
DS8417_100_6_square.TXT
HS-512MG_100_5.TXT
HS-512MG_50_5.TXT
HS-512MG_10_5.TXT
HS-512MG_100_5_square.TXT
HS-512MG_100_6.TXT
HS-512MG_50_6.TXT
HS-512MG_10_6.TXT
HS-512MG_100_6_square.TXT
HS-512MG_80_6_square.TXT
DS368_100_5.TXT
DS368_50_5.TXT
DS368_10_5.TXT
DS368_100_5_square.TXT
DS368_100_6.TXT
DS368_50_6.TXT
DS368_10_6.TXT
DS368_100_6_square.TXT
94091_80_5.TXT
94091_50_5.TXT
94091_10_5.TXT
94091_80_5_square.TXT
94091_80_6.TXT
94091_50_6.TXT
94091_10_6.TXT
94091_80_6_square.TXT
CS-10BB_100_5.TXT
CS-10BB_50_5.TXT
CS-10BB_10_5.TXT
CS-10BB_100_5_square.TXT
CS-10BB_100_6.TXT
CS-10BB_50_6.TXT
CS-10BB_10_6.TXT
CS-10BB_100_6_square.TXT
MODEL
NUMBER
VOLTAGE
AMPLITUDE
(% max)
SAMPLES
RECORD
TIME
(sec)
JR PROPO DS8417
JR PROPO DS8417
JR PROPO DS8417
JR PROPO DS8417
JR PROPO DS8417
JR PROPO DS8417
JR PROPO DS8417
JR PROPO DS8417
JR PROPO DS8417
JR PROPO DS8417
JR PROPO DS8417
JR PROPO DS8417
HITEC HS-512MG
HITEC HS-512MG
HITEC HS-512MG
HITEC HS-512MG
HITEC HS-512MG
HITEC HS-512MG
HITEC HS-512MG
HITEC HS-512MG
HITEC HS-512MG
JR PROPO DS368
JR PROPO DS368
JR PROPO DS368
JR PROPO DS368
JR PROPO DS368
JR PROPO DS368
JR PROPO DS368
JR PROPO DS368
AIRTRONICS 94091
AIRTRONICS 94091
AIRTRONICS 94091
AIRTRONICS 94091
AIRTRONICS 94091
AIRTRONICS 94091
AIRTRONICS 94091
AIRTRONICS 94091
CIRRUS CS-10BB
CIRRUS CS-10BB
CIRRUS CS-10BB
CIRRUS CS-10BB
CIRRUS CS-10BB
CIRRUS CS-10BB
CIRRUS CS-10BB
CIRRUS CS-10BB
5
5
5
5
5
6
6
6
5
5
5
6
5
5
5
5
6
6
6
6
6
5
5
5
5
6
6
6
6
5
5
5
5
6
6
6
6
5
5
5
5
6
6
6
6
100
100
100
50
10
100
50
10
100
50
10
100
100
50
10
100
100
50
10
100
80
100
50
10
100
100
50
10
100
80
50
10
80
80
50
10
80
100
50
10
100
100
50
10
100
n/a
1478
1464
1461
1452
1466
1464
n/a
1067
1392
1123
1380
1462
1482
1496
1125
1464
1466
1450
1404
917
1459
1471
1462
1253
1458
1462
1474
1241
1467
1458
1467
1417
1463
1475
1440
1519
1466
1469
1475
1198
1463
1476
1467
1145
n/a
29.56
29.28
29.22
29.04
29.32
29.28
n/a
21.34
27.84
22.46
27.60
29.24
29.64
29.92
22.50
29.28
29.32
29.00
28.08
18.34
29.18
29.42
29.24
25.06
29.16
29.24
29.48
24.82
29.34
29.16
29.34
28.34
29.26
29.50
28.80
30.38
29.32
29.38
29.50
23.96
29.26
29.52
29.34
22.90
- 65 -
Three of CIFER’s subprograms were utilized to perform the identification.
FRESPID (frequency response identification) was used to generate multiple responses at
different window lengths for each condition. COMPOSITE (multi-window averaging)
was used to average the results of the FRESPID cases into one response. NAVFIT
(transfer function fitting) was used to identify the 0th/2nd order transfer function of the
actuator dynamics from the COMPOSITE results. These linear models are required for
the optimization of the control system using CONDUIT. A strong effect of the nonlinear
characteristics on the responses was observed. Correlation to previous studies on
nonlinear actuators is provided which explains some of the inaccuracies in the linear
model.
Following the test matrix yielded 5 actuators with 2 different voltages and 3
different sweep magnitudes. These 30 cases were processed in CIFER and frequency
responses were generated within FRESPID. A single sweep was used for each of the
conditions. Five frequency responses were generated for each case based on window size
for the FFT routine within CIFER. 5, 10, 15, 20, and 25 second windows were used.
These responses were averaged into one response for each case using COMPOSITE. The
COMPOSITE response is the response used for the transfer function fitting.
The responses were analyzed for regions of best coherence in order to ensure
fidelity of the responses to be used for linear model fitting within NAVFIT. Plots for each
of the FRESPID generated frequency responses for each case are presented at the end of
this memo in Appendix B.
Table 3.21 shows the responses used for identification and the frequency ranges
where NAVFIT was used to fit a transfer function. The case names for each of the
- 66 -
frequency response curves shown in Appendix C can be referenced to the case names in
Table 3.21.
Table 3.21 – Actuator NAVFIT Frequency Ranges for CIFER Cases
NAVFIT FREQ RANGE
(rad/sec)
(rad/sec)
MIN
MAX
CIFER NAME
MODEL NUMBER
VOLTAGE
AMPLITUDE
(% max)
DS8417_1
DS8417_2
DS8417_3
DS8417_4
DS8417_5
DS8417_6
DS8417_7
JR PROPO DS8417
JR PROPO DS8417
JR PROPO DS8417
JR PROPO DS8417
JR PROPO DS8417
JR PROPO DS8417
JR PROPO DS8417
5
5
5
5
6
6
6
100
100
50
10
100
50
10
1
1
1
1
1
-
35
35
45
35
35
-
HS512MG1
HS512MG2
HS512MG3
HS512MG4
HS512MG5
HS512MG6
HITEC HS-512MG
HITEC HS-512MG
HITEC HS-512MG
HITEC HS-512MG
HITEC HS-512MG
HITEC HS-512MG
5
5
5
6
6
6
100
50
10
100
50
10
1
1
25
35
1
1
-
35
35
-
DS368_1
DS368_2
DS368_3
DS368_4
DS368_5
DS368_6
JR PROPO DS368
JR PROPO DS368
JR PROPO DS368
JR PROPO DS368
JR PROPO DS368
JR PROPO DS368
5
5
5
6
6
6
100
50
10
100
50
10
1
1
1
1
-
25
30
25
30
-
94091_1
94091_2
94091_3
94091_4
94091_5
94091_6
AIRTRONICS 94091
AIRTRONICS 94091
AIRTRONICS 94091
AIRTRONICS 94091
AIRTRONICS 94091
AIRTRONICS 94091
5
5
5
6
6
6
100
50
10
100
50
10
1
1
1
1
-
35
35
35
35
-
CS10BB_1
CS10BB_2
CS10BB_3
CS10BB_4
CS10BB_5
CS10BB_6
CIRRUS CS-10BB
CIRRUS CS-10BB
CIRRUS CS-10BB
CIRRUS CS-10BB
CIRRUS CS-10BB
CIRRUS CS-10BB
5
5
5
6
6
6
100
50
10
100
50
10
1
1
1
1
-
35
35
35
35
-
It became apparent after generating responses for the 10% max deflection cases
that the signals were not adequate for system identification work. Although the coherence
was good, the responses did not resemble 0th/2nd order forms of 0-dB gain at low
- 67 -
frequency and a break at -40 dB per decade at the natural frequency. Because the 0th/2nd
forms were not valid, these responses were rejected from system identification results.
Table 3.22 includes the complete NAVFIT results for natural frequency and damping
ratio for each case. The NAVFIT cost function result for each case is also presented.
Table 3.22 – Actuator NAVFIT Results for all Cases
CIFER NAME
MODEL NUMBER
VOLTAGE
AMPLITUDE
(% max)
ζ
ωn
(rad/sec)
τ
(sec)
COST
DS8417_1
JR PROPO DS8417
5
100
0.4986
20.4054
0.0110
37.804
DS8417_2
JR PROPO DS8417
5
100
0.5074
20.3759
0.0067
33.020
17.862
DS8417_3
JR PROPO DS8417
5
50
0.5166
33.0836
0.0055
DS8417_4
JR PROPO DS8417
5
10
-
-
-
-
DS8417_5
JR PROPO DS8417
6
100
0.5034
22.5944
0.0054
26.364
DS8417_6
JR PROPO DS8417
6
50
0.6556
50.9824
0.0155
1.324
DS8417_7
JR PROPO DS8417
6
10
-
-
-
-
HS512MG1
HITEC HS-512MG
5
100
0.5472
13.9439
0.0079
26.653
HS512MG2
HITEC HS-512MG
5
50
0.5606
21.9789
0.0121
11.352
HS512MG3
HITEC HS-512MG
5
10
HS512MG4
HITEC HS-512MG
6
100
0.5352
16.4602
0.0132
42.696
HS512MG5
HITEC HS-512MG
6
50
0.5243
22.9373
0.0127
12.813
HS512MG6
HITEC HS-512MG
6
10
-
-
-
-
DS368_1
JR PROPO DS368
5
100
0.5920
11.2955
0.009
65.993
DS368_2
JR PROPO DS368
5
50
0.5136
16.4615
0.0042
32.826
DS368_3
JR PROPO DS368
5
10
-
-
-
-
DS368_4
JR PROPO DS368
6
100
0.5168
12.1254
0.006
59.259
DS368_5
JR PROPO DS368
6
50
0.5039
18.0762
0.0117
31.008
DS368_6
JR PROPO DS368
6
10
-
-
-
-
94091_1
AIRTRONICS 94091
5
100
0.5446
17.429
0.0054
27.490
94091_2
AIRTRONICS 94091
5
50
0.5108
21.3608
0.0048
9.593
94091_3
AIRTRONICS 94091
5
10
-
-
-
-
94091_4
AIRTRONICS 94091
6
100
0.5302
18.8425
0.010
19.964
94091_5
AIRTRONICS 94091
6
50
0.5489
23.4087
0.0073
13.922
94091_6
AIRTRONICS 94091
6
10
-
-
-
-
CS10BB_1
CIRRUS CS-10BB
5
100
0.5345
18.3889
0.0036
19.862
6.294
CS10BB_2
CIRRUS CS-10BB
5
50
0.5019
26.2309
0.0045
CS10BB_3
CIRRUS CS-10BB
5
10
-
-
-
-
CS10BB_4
CIRRUS CS-10BB
6
100
0.5273
21.0582
0.0077
16.154
CS10BB_5
CIRRUS CS-10BB
6
50
0.5192
29.0844
0.0069
5.641
CS10BB_6
CIRRUS CS-10BB
6
10
-
-
-
-
- 68 -
Table 3.22 shows that for the first actuator tested, the same 100% sweep at 5V
was applied. The NAVFIT results for these same sweeps show nearly identical results.
This was done to ensure repeatability and consistency of the test. The frequency
responses and the transfer function fits are presented by CIFER name (referenced in
Table 3.22) in Appendix C.
As Table 3.22 shows, in general, all the actuators running the sweep to only 50%
instead of the full 100% yielded a noticeably higher natural frequency and higher
damping ratio. This is because the smaller deflections allow the actuator to reach higher
frequencies before the rate limit is reached. This is evident in the frequency responses for
all the actuators as illustrated for the HS512MG in Figure 3.23.
- 69 -
6 Volts
5 Volts
Figure 3.23 – HS512MG Responses Illustrating Difference between 5V and 6V
- 70 -
The time delays all seemed to be around 0.005 – 0.010 seconds. These numbers
cannot be taken as the pure transport delay because some of the delay is being absorbed
in the second order form that NAVFIT determines after iterating for the best fit.
Running the actuators with more power (6V) yields slightly higher damping ratios and
higher natural frequencies for all of the actuators.
The costs for each fit seem to be very reasonable and show that the second order
model is quite valid for the responses exhibited by all of the actuators. The only response
that may have been the subject of error is the 50% deflection sweep at 6V on the DS8417
which shows a really low cost and a noticeably higher natural frequency than the rest of
the responses. This is due to the fact that for the frequency range analyzed, the magnitude
did not break sharply. This left a relatively flat response for which a second order form
was easily fitted.
More constrictive frequency ranges were chosen to study the effects this range
had on the fit presented by NAVFIT. It was determined that the frequency range had little
effect on the transfer function fit unless it went below the break frequency. As the
frequency responses show, all of the responses demonstrate clean breaks at their natural
frequencies and 0-dB gain at low frequencies with the exception of the 50% deflection on
the DS8417 at 6V.
The primary tool used for the determination of the nonlinear properties of the
actuators was the square wave shown in Figure 3.22. The square wave commanded a near
instantaneous change from maximum to minimum deflection. Using the geometric
calibration factors in Table 3.17, the maximum actuator deflection was calculated for the
- 71 -
0.5 seconds that the actuator was at the maximum position. This is where it was receiving
a PWM length of 1.0 ms (negative max) to 2.0 ms (positive max).
A linear curve fit was used between the test points where the response to the
change in deflection was constant. This meant that although the first change from -100%
to 100% occurred at a given time, for all the actuators there was still a transient response
due to the dynamics of the actuator that were ignored. Most fits actually started at up to
0.1 second after the commanded change. Figures 3.24 and 3.25 illustrate this for the JR
PROPO DS8417 for full 100% deflection at 6V. The response data in Figure 3.24 shows
the measurement spike every fifth data point that was described earlier. The presence of
this spike does not have a significant effect on the identification results.
60
Response
40
Command
20
0
0
0.2
0.4
0.6
0.8
1
-20
-40
-60
Figure 3.24 – Sample Square Wave Response
- 72 -
1.2
4000
3500
3000
2500
y = -13429x + 9284.1
R2 = 0.9958
2000
1500
1000
500
0
0.4
0.42
0.44
0.46
0.48
0.5
0.52
0.54
0.56
0.58
Figure 3.25 – Linear Fit for Max Rate Determination
The square wave commanded a maximum and minimum deflection. This is
shown in Figure 3.24 as 50 and -50 degrees, respectively because the actual limits were
not known during testing.
Figure 3.24 clearly shows that the servo was saturated. It also shows the transient
response. The figure shows how the maximum positions can be read from the plot. It is
asymmetric because the servo horn was not able to be positioned at exactly the 0°
location due to the teeth on the gear. To correct for this, the position limits were fixed to
be symmetric about zero degrees.
Figure 3.25 shows how nicely a linear curve fit could be accomplished. By using
the slope from that line and applying the calibration factor from Table 3.17, the
maximum rate in degrees/second was found. This was repeated for each of the actuators
to yield the final nonlinear characteristics for the actuators as shown in Table 3.23.
- 73 -
Table 3.23 – Actuator Nonlinear Characteristic Summary
SERVO
DS8417
JR94091
DS368
HS12MG
CS10BB
VOLTS
5
6
5
6
5
6
5
6
5
6
POSITION
(deg)
+/- 37.4571
+/- 38.5315
+/- 40.5744
+/- 41.4394
+/- 41.7992
RATE
(deg/sec)
MIN
MAX
-437.2
437.1
-524.5
534.9
-402.3
422.0
-435.2
467.8
-219.8
220.5
-264.6
265.7
-328.1
317.6
-376.5
404.7
-474.7
442.1
-567.2
536.4
It is apparent from Table 3.23 that there are different rates for different directions
on the actuators. The test stand was mounted horizontally, so gravity is was not the cause.
The DS368 proved to have the best symmetry in its rates where the smaller and lighter
CS-10BB showed to be more asymmetric. Many factors can contribute to this
asymmetry. Because there is a motor with an armature inside, the brushes on the motor
may be conditioned to one direction.
It should be noted that the square wave used for the first test was repeated at 50%
maximum deflection. There were not enough data points at which the actuator rate was
saturated to fit a valid linear curve at this deflection. For this reason, all results used a full
100% deflection command in the square wave to ensure saturation of the rate.
The sampling rate of 50 Hz and nature of the square wave did not reveal any
identifiable stiction or hysteresis. Although they undoubtedly exist, the methods
employed here did not reveal any substantial findings. More accurate potentiometers,
- 74 -
higher data rates, and tighter tolerances on the test equipment may have revealed these
nonlinearities.
It should be mentioned that observing the smaller actuators like the CS-10BB and
94091 revealed that at very high frequencies the actuator demonstrated output not directly
correlated to the input. This sporadic output is visible in the time responses shown in
Figures 3.26 and 3.27.
50
40
Deflection (deg)
30
20
10
0
0
5
10
15
20
25
30
-10
-20
-30
-40
-50
Time (sec)
Figure 3.26 – CS-10BB at 5V Time History Illustrating Erratic Response at High
Frequency
40
30
Deflection (deg)
20
10
0
0
5
10
15
20
25
-10
-20
-30
-40
-50
Time (sec)
Figure 3.27 – 94091 at 6V Time History Illustrating Erratic Response at High Frequency
- 75 -
30
50
40
Deflection (deg)
30
20
10
0
-10
0
5
10
15
20
25
-20
-30
-40
-50
Time (sec)
Figure 3.28 – 94091 at 5V Time History not Showing Erratic Response
Interestingly, the 94091 at 5V did not display this asymmetric response to the
extent that the 6V case did (Figure 3.28). The coherence for these actuators in this
frequency range still remains relatively high, indicating that the output is correlated with
the input. What the time histories reveal though is that the oscillations do not occur about
0°. These smaller actuators have issues tracking the input symmetrically at high
frequencies. The nature of the sporadic response was observed in all of the actuators to
some extent, but not more so than in the 94091 at 6V and CS-10BB at 5V and 6V. The
errors in tracking the input signal at high frequencies associated with these small
actuators must be a consideration when selecting an actuator for high bandwidth
applications.
It is known that the nonlinear characteristics of the actuators, especially rate
limiting, will have an effect on the accuracy of the linear transfer function models. It was
observed that although the magnitude fits were accurate for some of the NAVFIT results,
the match of the linear second order system on the phase curve did not fully characterize
the response. This was investigated further in an attempt to add fidelity to the model. The
DS8417 showed the worst correlation between the linear model response and the
- 76 -
30
response and the response obtained from test data. Figure 3.29 shows the phase of the
DS8417 at 5V with a 100% sweep.
Figure 3.29 – DS8417 Frequency Response Illustrating Mismatch in Linear Model
As Figure 14 illustrates, the phase is not fully characterized by the second order
fit at frequencies beyond 10 rad/sec. The mismatch shows itself as more time delay roll
off at higher frequencies.
Previous work completed by STI during investigation of PIOs due to nonlinear
vehicle characteristics5 determined that the mismatch in phase lag was due to the rate
- 77 -
limit of the actuators. A comparison of the time histories observed by STI and those of
the DS8417 at 5V is presented as Figure 3.30.
Chirp Input
Chirp Input
Actuator Response
50
40
30
Deflection (deg)
20
10
0
20
20.5
21
21.5
22
-10
-20
-30
-40
-50
Time (sec)
Figure 3.30 – DS8417 Time History Comparison to 1995 STI Findings
- 78 -
The time histories show that the output from the actuator is clearly rate saturated.
Work presented by STI shows that a linear describing function could be generated to fit
the data in the frequency domain for a given frequency range, but not for all frequency
ranges. For a more accurate match over broader frequency ranges, an exact sinusoidal
describing function is required. To compute this function, the Fourier integrals are first
computed for the input and output fundamentals as shown in the following equations.
According to Klyde, McCruer, and Myers, these integrals are computed for f(t) being
either the input or output periodic forcing function. For our case these are both sinusoids
with period P, so the input describing function’s a1 term is always zero. Using these to
characterize the magnitude and phase of the describing functions yields the following
relationships5.
In these equations, δc is the actuator deflection commanded and δ is the actual
output after rate saturation. Having these open frequency domain representations allows
us to characterize the response in order to explain the discrepancies in the phase plots.
According to Klyde et al, this difference can be characterized with an error function by
- 79 -
finding these integrals and comparing to the frequency responses generated by the bench
test data.
That method was effectively applied for the DS8417 actuator by applying a rate
limiter on the identified models within Simulink and using FRESPID to then generate a
frequency response. The responses for the bench test, NAVFIT linear model response,
and the NAVFIT model with rate limit in Simulink are shown in Figure 3.31.
- 80 -
Figure 3.31 – Magnitude Comparison for Linear & Nonlinear Model to Bench Test
Figure 3.31 shows that as expected, the addition of the rate limiter in the model
causes the response to break sooner; this yields a lower natural frequency. The addition
of the rate limit increases the magnitude accuracy of the model over the linear NAVFIT
result. However, as Figure 3.32 shows for the phase of the same three responses, the
- 81 -
addition of the rate limit actually causes a dip in the response (10 ~ 25 rad/sec) instead of
matching the bench test data better. This is most likely due to the fact that other
nonlinearities exist and become more influential at higher frequencies. More accurate test
equipment and a higher sampling rate would be required to identify these.
Figure 3.32 – Phase Comparison for Linear & Nonlinear Model to Bench Test
- 82 -
Performing the frequency response arithmetic within CIFER allowed the
frequency response quotient of the rate saturated response to the identified linear
NAVFIT model to be generated. NAVFIT was then used to try to characterize the error
with a linear transfer function. This resulted in the responses shown in Figure 3.33 for the
error function.
Figure 3.33 – Error Function Frequency Response and NAVFIT Transfer Function Fit
- 83 -
This response shows that the error function has a maximum phase lag of 32
degrees at approximately 14 rad/sec. However, the maximum error is an important
parameter because it is directly related to the ratio of linear and nonlinear rise times ( tˆR )5.
This lag cannot be characterized with a pure time delay, as shown by the NAVFIT result.
However, the magnitude response of the error is almost entirely at zero, indicating that
the inclusion of the rate limit in the model accurately models the magnitude as was seen
previously in Figure 3.31. The loss of phase fidelity starting at around 12 rad/sec is a
relatively high frequency for control system design and shows that the linear model with
the rate limiter would be fairly accurate for simulation purposes.
- 84 -
Results from STI utilizing the exact describing function yielded Figure 3.34. The
frequency is normalized by the actuator bandwidth ( ωn ) to represent the ratio ωˆ n . This
generates the family of curves relating the difference in phase to the ratio of linear rise
time to nonlinear rise time ( tˆR =
t RL
t RNL
).
Figure 3.34 – Rise Time Ratio Phase Lag Relationship
For the maximum phase error of 32 degrees seen in Figure 3.33, Figure 3.34
predicts a rise time ratio of tˆR = 0.17 at a normalized frequency of ωˆ n = 0.6. Looking at
the step response of the linear NAVFIT results without rate limiting, we see the rise time
to be t RL = 0.08 sec, as shown in Figure 3.35.
- 85 -
Figure 3.35 – Rise Time for Linear Model of DS8417 at 5V
Determining the rise time from the nonlinear, rate-limited model was
accomplished by analyzing the square wave time responses and found to be t RNL = 0.192
sec. Comparing this rise time to the linear rise time reveals a ratio of tˆR = 0.38. Although
not exactly the predicted 0.17, the only nonlinearity that was included in this model was
the rate limiting.
As mentioned previously from Figure 3.34, the predicted maximum difference in
phase lag would be expected at a normalized frequency of ωˆ n = 0.6. The bandwidth of the
DS8417 is approximately ωn = 20 rad/sec (Table 8). The error function in Figure 3.33
shows the maximum additional lag to occur at 14 rad/sec. This corresponds to a
normalized frequency of ωˆ n = 0.7. This is very close to the predicted frequency where the
additional lag is most apparent and is consistent with the STI trend.
- 86 -
The fact that the rate saturated during the sweep was readily noticeable in the fact
that all the natural frequencies and damping ratios were higher for the 50% sweeps than
the 100% ones. Plotting this trend as in Figure 3.36 shows that as expected, the natural
frequency drops with increased sweep amplitude. This trend is also evident in Figure 16
where the addition of the rate limit effectively causes the response to break sooner and
illustrates how much an effect the rate limit has on the response.
1.2
1
0.8
0.6
0.4
0.2
0
0%
20%
40%
60%
80%
100%
Amplitude of Sweep (% of max deflection)
Figure 3.36 – Sweep Amplitude and Natural Frequency with Rate Limiting
The result of the comparison to the STI data is that the general trends of the data
are correct. The addition of the rate limit in the model effectively corrects the magnitude
of the response. According to the error function in Figure 3.33, the model should lose
fidelity in the phase of the response around 14 rad/sec where the error is at a maximum.
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With the nature of the linear and nonlinear characteristics of the actuators
determined modeling and validation of the actuators was performed. The modeling was
done in a way which could be used for control system optimization and simulation. The
model is built within Simulink and includes the linear 0th/2nd transfer function form and
the identified nonlinear characteristics of rate and position limits. The validation of the
models is accomplished in the time domain by feeding the models the same chirp input
used in the test and comparing the responses to bench test responses.
With the actuator models identified, Simulink block diagrams were created to be
used in the inner loop block diagrams for MAV control system optimization and
simulation. The blockset can be seen in Figure 3.37.
Figure 3.37 - Simulink Actuator Blockset
Each block is configurable when double clicked, but reflects the CIFER identified
results for each voltage based on the results presented in Table 3.22 for the dynamics and
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Table 3.23 for the maximum rates and positions. The mean average of the 50% and 100%
sweep deflections were incorporated into the blocks because they were quite similar.
The block is left configurable to allow specification of the exact characteristics for the
condition and max deflections being used for the application, as seen in Figure 3.38.
Figure 3.38 – Configurable Actuator Parameters
The physical characteristics of the actuator from the manufacturer are also
presented in the header of the block parameters dialogue. A Matlab (HTML-based) help
file is also accessible through the parameters dialogue.
- 89 -
The blocks are all masks with the same underlying block diagram as shown in Figure
3.39.
Figure 3.39 - 2nd Order Actuator Dynamics behind Mask
It can be seen that a first order Pade approximation of the time delay is used.
Because no observable hysteresis was recorded, all of the blocks have values of zero for
this parameter, but it can still be specified within the parameters dialogue.
- 90 -
Verification of the identified models was accomplished by using the same sweep
input fed into the actuators during bench testing. A typical result is shown for the DS8417
in Figure 3.40 with all actuator model validations appearing at the end of this memo
within Appendix D.
Figure 3.40 – DS8417 at 5V Time Domain Validation
Figure 3.40 shows that the response has been captured in the model which
includes the rate and position limits. The non-linearties not accounted for and the
asymmetric response, begin to show as a loss of fidelity beyond approximately 13 ~ 16
- 91 -
rad/sec. The total deflection of the actuator and phase are not fully modeled at these
higher frequencies, as seen when zooming in on the response in Figure 3.40. From the
error function presented previously in Figure 3.33, we see that the maximum difference
in phase shows itself at 14 rad/sec (2.2 Hz). This corresponds to what is seen here in the
time domain. Any accurate modeling beyond 5 Hz would require more accurate test and
data acquisition equipment, in addition to more complex nonlinear, open-form models.
The goal of the actuator test program was to measure a set of data that was used to
identify models of the actuator dynamic response characteristics. These actuator models
include linear transfer functions of the input/output relationships as well as non-linear
actuator properties such as actuator rate and position limits.
The responses of the actuators were modeled by using CIFER to generate
frequency responses and then fit 0th/2nd order transfer functions. The position and rate
limits of the actuators were determined by analyzing the response to the square wave
input. It was found that the phase characteristics for some of the actuators were not fully
captured with the linear models. Comparing to known theory revealed the extent to which
the maximum rate of the actuator affects the response. The inclusion of the rate limit in
the model significantly improved the accuracy of the magnitude but some differences are
still seen at higher frequency due to nonlinear effects that are not included.
The identified actuator dynamics and nonlinear rate and position limits were used
to construct a set of Simulink actuator blocks. These blocks are customizable and include
the manufacturer specifications. A time domain validation of the models showed them to
be accurate up to the highest frequency range of interest for flight control work. When
comparing the manufacturer listed rate limit specifications (3.15) with those obtained
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from testing (3.22), it was found that the true actuator rate limits were lower than those
quoted. All of the actuators demonstrated increased bandwidth, damping ratios, and rate
limits when powered at 6V instead of 5V. The smallest and fastest actuators have issues
tracking the input at high frequencies. The CS-10BB at 5V and 6V and the 94091 at 6V
exhibited these characteristics. Based on bandwidth, maximum rate, weight, and size the
Airtronics 94091 is the best performing when run at 5V. It is one of fastest actuators
tested while remaining the 2nd lightest. Its performance is comparable to the much larger
and heavier JR DS8417 while being much smaller.
The manufacturers’ specified maximum torques of the actuators tested varied
considerably. This is an important factor because the application will drive the amount of
torque required. All bench tests were conducted with the actuators unloaded and no
conclusions could be made about the effect of load on the actuator response.
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3.4 Sensor Identification
The identification of the sensors and their respective errors is an area that requires
some attention. Because these vehicles are unmanned they usually utilize their control
systems in a conservative manner. Expanding the envelope of operation would be
beneficial to the overall performance and mission success. However, the small size of the
vehicles leaves them susceptible to low performance sensors. Knowing the limitation of
the components and the effects they have on the control systems is important.
All of the vehicles utilize inner loop controllers to stabilize the airframe. This is
usually comprised of proportional, rate, and integral feedback. This PID controller is
usually adequate to control the vehicle nicely in hover and forward flight. In some cases,
the need for cross feed in pitch and roll or pitch and yaw was deemed necessary due to
high coupling and large propeller inertias. In flight test however, this proved
unwarranted. The reliance on the highest performing, small-packaged, rate gyro is high.
Magnetometers are used for heading determination. The accelerometers are needed for
determination of lateral and longitudinal speed as well as vertical speed. This is
complimented with a pressure altimeter. Ultimately, machine or synthetic vision, laser
ranging equipment, and other advanced telemetry would be needed for accurate position
and landing requirements. GPS with selective availability (SA) off working nominally at
1 Hz was used for outer loop position control. All of these areas need to be modeled to
have a working model of the entire system (1.11).
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3.4.1 Accelerometer Identification
Modeling of typical accelerometers was done with the representative Crossbow
CXL04LP3. This is the accelerometer present on the Honeywell OAV. The
accelerometers were modeled with white noise and random bias. Figure 3.41 shows how
this was done. According to the manufacturer, the modules could report up 0.2 g of max
bias. This would be erratic and slowly switching between positive and negative. A
random number is filtered to ensure subtle changes between positive and negative.
Hysteresis was also identified to be no more than 0.1 g. The noise coming into the system
was identified as 10 mg RMS.
Figure 3.41 – Accelerometer Model
Figure 3.42 shows the noise and nonlinear effects the model has while the sensor is
stationary over a period of 10 minutes.
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Figure 3.42 – Accelerometer Stationary Noise Model
3.4.2 Rate Gyro Identification
Identification of the rate gyros was performed on the Inertial Science RRS75.
This was also part of the OAV sensor package. The piezoelectric rate gyros (3.43) were
modeled in a similar fashion as the accelerometers. The parameters are different; they are
based on Inertial Science specifications. The description from Inertial Science specified
the noise as a function of the bandwidth at which the gyros were run. The expression
was:
Noise =
deg
sec
BW
0.01
It can be seen that as the bandwidth increases, the RMS of noise will as well.
- 96 -
Figure 3.43 shows that gyros were modeled with the noise specified from the
manufacturer as well as Hysteresis and a slow drift modeled as a sine wave of low
frequency.
Figure 3.43 – Rate Gyro Model
The hysteresis was identified as a 0.1 wide dead zone, and the max bias specified
was 0.02 deg/sec. Figure 3.43 shows the model’s response to a constant 15 deg/sec input.
- 97 -
Figure 3.43 – Rate Gyro Response to Constant 15 deg/sec for 10 sec
3.4.3 GPS Receiver Identification
To model the GPS error and characteristics, a lot of tie was spent studying the
nature of the test data provided for the µ−BLOX GPS-MS1E receiver used on the
Honeywell OAV. The GPS manufacturer supplied detailed metrics as well as actual test
data to verify the accuracy of the model. Figure 3.44 shows how the manufacturer’s
specifications were implemented in Simulink. The actual positions north and east in feet
are biased by a low frequency random number that sweeps the position about the origin
to a max error of 10 feet. The random numbers are set to a variance to closely meet the 5
meter Circular Error Probability (CEP 50%) specification provided by µ−BLOX which
quantifies the error by predicting that at least 50% of the GPS’s readings will lie within a
- 98 -
5 meter circle centered about the true position. The modeling was completed for the case
of Selective Availability (SA) off.
The module was running at a 1 Hz sampling rate. This was modeled with a zero
order hold. The speed calculation was modeled by applying a unit delay and taking the
difference of the positions and dividing by the sample time. Figure 3.44 appears as the
main green block in Figure 3.45, which shows how the speeds were combined and the
heading calculated from the north and east positions.
Figure 3.44 – GPS Heading and Speed Model
- 99 -
Figure 3.45 – GPS Error and Discrete Signal Model
- 100 -
Figure 3.46 shows the modeled fluctuation of position over a 2 hour period assuming
the sensor is stationary at (0,0).
Figure 3.46 – GPS Model Results
3.4.4 Magnetometer Identification
Identification of the magnetometers used for heading determination was performed on
the Honeywell HMC 2003 used on the OAV. The magnetometers were modeled with a
max noise of 0.001 gauss, and a small Hysteresis 0.002 gauss wide. The only other
specification modeled was the 40 microgauss resolution specified by Honeywell. Figure
3.47 shows the model, while Figure 3.48 depicts a 5 sec reading at 5 gauss.
- 101 -
Figure 3.47 – Magnetometer Model
Figure 3.48 – Magnetometer Depiction at 5 Gauss for 5 Seconds
3.4.4 Pressure Altimeter Identification
Identification was performed on the Motorola MPX 4115A based on manufacturer
specifications. Motorola specified a max noise error of 0.03 inches of Hg. This was
scaled to an approximate linear relationship in the standard troposphere relating pressure
to altitude. Figure 3.49 depicts the final model.
- 102 -
Figure 3.49 – Pressure Altimeter Model
Figure 3.50 shows the model’s response to constant 15 foot reading for 5 seconds.
Figure 3.50 – Pressure Altimeter at 15 feet for 5 seconds
- 103 -
CHAPTER 4 – Flight Simulation
The wealth of identification information and models were applied to a full nonlinear
simulation. This model was used to extract a linear state-space model about hover as well
as investigate certain flying qualities. Automated sweeps were fed through the model in
an attempt to simulate flight test sweeps which were unavailable and evaluate the effects
of the nonlinear effects. The model used was that of the Allied Aerospace MAV.
Although it was found to be the most troublesome in correlating the Mu derivatives with
the other vehicles, it was the timeliest and possessed the most information from wind
tunnel testing, sensors, actuators, and flight control laws. This vehicle was also in a Phase
I DARPA ACTD program at the time of writing.
4.1 Simulated Frequency Sweeps
An industry supplied Simulink model was used to feed frequency sweeps in of
varying parameters in order to create time history responses for use in CIFER. Figure 4.1
shows the top level Simulink model used.
- 104 -
Figure 4.1 – Simulink MAV Model
Figure 4.1 shows the special code written to handle the unique task of real-time
simulation on a PC running COTS equipment. Special code was written to throttle
Matlab’s Simulink to run in near real-time. This is seen as the TimeKeeper subsystem
block. Although no guarantee of frame sizes and determinism is made within the timer
code, it nevertheless works quite well. Code written to handle joystick input from the
Logitech Strike Force 3D USB Joystick is also required. Output for such things as
graphics and sound are provided by special software utilizing a 100 Base-T network
shares the computing load. Together, these subsystems combine to create a unique and
powerful simulation environment shown in Figure 4.2.
- 105 -
Figure 4.2 – Custom PC and COTS Simulation Environment
While outside the scope of this research, it suffices to say that the environment
allows for some unique monitoring and evaluation of the overall simulation. Other
subsystems were built up to handle the flow of state variables and the creation and
formatting of CIFER specific time history text files.
Special code was also written to handle the sweep of the vehicle. As it would
become apparent, and mentioned in the proper methods to frequency domain
identification, the nature of the sweep used to generate responses is extremely important.
For this reason, the changing of parameters in a timely manner is valuable. This was
accomplished with special code and a graphical user interface (GUI) which handles the
specification of parameters. This sweep GUI is depicted in Figure 4.3.
- 106 -
Figure 4.3 – Simulink Sweep Generator GUI Built for Sweeps
Using the GUI and code in Figure 4.3, the sweep of Figure 4.4 was used to
simulate a sweep through the actual vehicle with all of its included sensors and nonlinear
actuators.
- 107 -
Figure 4.4 – Simulink GUI Generated Sweep
Of note from Figure 4.4 is that the sweep does not have a fade in and fade out
time associated with it as was seen in Chapter 2, Figure 2.1. This is due primarily to the
fact that for a 300 second sweep, the amount of energy going in to the system in the low
frequency region needs to be high. In a piloted sweep, there is usually plenty of lower
frequency data due to doublets and natural oscillation by the pilot. The parameters for
this sweep can be seen as entered in the GUI in Figure 4.3.
From the start, sweeping the vehicle proved to be problematic within Simulink.
The simulation environment is isolated and protected from naturally occurring
oscillations and energy other than that of the sweep entered. Also, by the nature of the
simulation, all coupling is hard-wired directly into the simulation. This means that the
addition of noise to break up off-axis coupling will still show high degrees of correlation
to on-axis inputs.
The RUAV class of vehicles analyzed all use spinning propellers inside a duct for
lift. With small vehicle inertias and very high speed propellers, gyroscopic coupling
occurs between pitch and roll. The angular momentum of the spinning propeller will
- 108 -
cause a pitching moment to be exerted on the vehicle when its angular momentum vector
is moved in roll. The reverse is true if moved in pitch; a rolling moment is produced. This
effect is apparent in the stability derivatives Mp and Lq. Due to sign conventions in
standard helicopter coordinate systems, Mp will be a positive value and Lq will be
negative. It is these gyroscopic effects that make simulating a sweep through the vehicle
difficult. They directly correlate the roll and pitch controls and make it difficult for
CIFER, or any identification tool to determine which input is creating which output. This
was seen when the MAV vehicle was flight tested. The actual flight test data revealed
correlation and cross-control coherence between the roll and pitch commands. This is
shown in Figure 4.5.
1
COHERENCE
0.6
0.2
0.1
1
FREQUENCY (RAD/SEC)
10
100
Cross Coherence between Pitch and Roll
Figure 4.5 – MAV Flight Test Cross Coherence between Pitch and Roll controls
It is readily evident that there is a large amount of coherence at some gyroscopic mode
between 2 ~ 7 rad/sec.
- 109 -
4.1 Matlab Linear Model Determination
Assuming that there are no other sources of coupling in pitch and roll besides
gyroscopic effects, the coupling could be calculated from what is known about the
angular momentum of the propeller and would be the key to modeling and sweeping the
simulation. Equations 3.17 and 3.18 from the MAV bare airframe identification are
repeated here.
Mp =
I prop Ω
I yy
(Equations 3.17 and 3.18)
Lq =
I prop Ω
I xx
A look at these equations shows that there would be a linear relationship between
the amount of moment received in pitch or roll due to the cross control’s generated
response. In fact, as Figure 4.6 shows, the dynamics in pitch and roll can be separated
entirely.
- 110 -
Figure 4.6 – Cross Control Decoupling Block Diagram
Figure 6 shows that by applying Equations 3.17 and 3.18, equivalent control
inputs are generated from the off-axis responses. For a pitch command, a pitch response
and a roll response are generated. Because the gyroscopic nature is known, it can then be
applied to come up with an equivalent roll command input. The similar approach is used
with the roll command. To illustrate how this is possible, we look at the linearization
results from Matlab.
- 111 -
Linearization of a nonlinear Simulink model is accomplished by the following steps:
1.
Identify inputs, outputs, and states.
2.
Invoke the trim function to bring all controls to yield desired states.
3.
Run the linmod function to generate quadruple matrices.
4.
Adjust linmod minimum step size and tolerance as needed.
With the model trimmed, linmod was used to generate the model setup (based on states
occurring as integrators in Simulink) presented in Equations 4.1 – 4.8.
x& = Fx + Gu
(Equation 4.1)
y = H 0 x + H1 x&
(Equation 4.2)
⎧ p⎫
⎪q ⎪
⎪ ⎪
⎪r ⎪
⎪ ⎪
⎪u ⎪
⎪ ⎪
x = y = ⎨v ⎬
⎪ w⎪
⎪ ⎪
⎪φ ⎪
⎪ ⎪
⎪θ ⎪
⎪⎩ψ ⎪⎭
(Equation 4.3)
⎧ δ lat ⎫
⎪δ ⎪
⎪ lon ⎪
u=⎨
⎬
⎪ δ col ⎪
⎪⎩δ ped ⎪⎭
(Equation 4.4)
- 112 -
⎡ Lp
⎢M
⎢ p
⎢ Np
⎢
⎢Xp
F = ⎢ Yp
⎢
⎢ Zp
⎢ 1
⎢
⎢ 0
⎢ 0
⎣
Lq
Mq
Nq
Xq
Yq
Zq
0
1
0
⎡ Llat
⎢M
⎢ lat
⎢ N lat
⎢
⎢ X lat
G = ⎢ Ylat
⎢
⎢ Z lat
⎢ 0
⎢
⎢ 0
⎢ 0
⎣
Llon
M lon
N lon
X lon
Ylon
Z lon
0
0
0
Lr
Mr
Nr
Xr
Yr
Zr
0
0
1
Lu
Mu
Nu
Xu
Yu
Zu
0
0
0
Lcol
M col
N col
X col
Ycol
Z col
0
0
0
Lv
Mv
Nv
Xv
Yv
Zv
0
0
0
Lw
Mw
Nw
Xw
Yw
Zw
0
0
0
0
0
0
0
g
0
0
0
0
0
0
0
−g
0
0
0
0
0
0⎤
0 ⎥⎥
0⎥
⎥
0⎥
0⎥
⎥
0⎥
0⎥
⎥
0⎥
0 ⎥⎦
Lped ⎤
M ped ⎥⎥
N ped ⎥
⎥
X ped ⎥
Yped ⎥
⎥
Z ped ⎥
0 ⎥
⎥
0 ⎥
0 ⎥⎦
(Equation 4.5)
(Equation 4.6)
H0 = I
(Equation 4.7)
H1 = 0
(Equation 4.8)
With this setup, the derivatives were calculated and Table 3.11 is repeated here as
Table 4.1 and expanded upon with the results from linmod.
- 113 -
Table 4.1 – Linmod, Wind Tunnel, and Flight Test Results for i-Star 9”
I-Star Vehicle
Derivative
9”
LINMOD
Wind Tunnel
Flight Test
- 0.344
-0.1090
- 0.344
-0.1090
(Fixed to Xu)
(Fixed to Xu)
- 0.212
n/a
0.004
-0.5014
(Fixed to –Mu)
(Fixed to –Mu)
Xu
-0.4003
Yv
-0.4003
Zw
-0.1737
Lv
-0.2373
Lp
0
0
0
Mu
0.2373
0.003
0.5014
Mq
0
0
0
Mp
2.6261
n/a
n/a
Lq
-2.6261
n/a
n/a
Nw
-0.004
- 0.006
n/a
Nr
0
n/a
n/a
X lon
-0.1554
- 0.157
-0.2841
Ylat
0.1233
0.123
n/a
Z col
-0.0027
- 0.00264
n/a
Llat
-0.412
- 0.418
n/a
M lon
-0.8361
- 0.548
-0.2343
N ped
1.1416
0.555
n/a
N col
0.0004
- 0.00057
n/a
- 114 -
It can be seen right away that the results from linmod agree very well with the
wind tunnel results. This is to be expected because the simulation is based on a table
lookup scheme directly based on tables from the wind tunnel data. This shows that
linmod is working and the vehicle is trimmed in a hover state.
Returning now to the simulated sweeps, we can overlay the frequency response
for the simulated sweep with the results of linmod. This is done in Figure 4.7.
Figure 4.7 – LINMOD and Simulated Sweep Roll Frequency Response
Figure 4.7 illustrates how the simulated sweep breaks down due to the crosscoupling in pitch and roll. Once MISOSA is used in an attempt to remove the
- 115 -
contributions of pitch input on the roll response, the result is a loss of coherence about the
gyroscopic mode (2~4 rad/sec) and a stable phase characteristic- which is know to be
untrue. Comparing the linmod results to the FRESIPD case where all the off-axis
contribution is intact reveals a better match, but misses the nature of the response and is
tainted by the fact that a good amount of energy was put into the system from the pitch
coupling.
From Figure 6 it is evident that we can model and validate the system the without
the pitch and roll coupling by treating each response as uncoupled. We can then
superimpose the coupling as linear feedback into the off-axis control. The nature of the
coupling is known already so we can avoid the breakdown in coherence. With the
coupling removed, Figure 4.8 shows the dramatic change in cross-control coupling.
Figure 4.8 – Effect of Removing Cross Control Coupling to Response
- 116 -
Figure 4.8 shows that the coherence drops dramatically when the inertial coupling
is removed. This means that once the coupling is removed from the model by removing
the propeller inertia, the coupling all but disappears. This proves that the coupling
diagram in Figure 4.6 would be a valid approach for correction and Figure 4.9 illustrates
how well the results of sweeping the model and the results of linmod agree.
Figure 4.9 – Coupling Removed Illustrating linmod and Simulated Sweep Results
Figure 4.9 shows that the results of linmod and simulated sweep match up very
well and that this method of treating the coupling as an external, linear, effect works from
a modeling point of view. A similar approach was used for the pitch response to be
compared with the actual flight test data. The results of the linmod model, the parametric
- 117 -
state space model determined with CIFER, and the actual frequency response from flight
test data is shown in Figure 4.10.
Figure 4.10 – Comparison of linmod and Flight Test Pitch Responses
Figure 10 shows excellent agreement between the flight test results and the linear
model determination with linmod within Simulink on the wind tunnel data-based model.
Although there are some differences in the phase and magnitude of the response from
linmod, these results are deemed fairly good considering the use of limited flight test
data. It is interesting to note that all the models reveal a lack of fidelity at about 2 ~ 3
rad/sec. This is gyroscopic coupling mode.
- 118 -
CHAPTER 5 – CONCLUSIONS
The need for accurate simulation models of small scale, ducted-fan, unmanned air
vehicles has lead to the development of techniques unique to this class of vehicles. Taken
as a whole, this research activity shows that by combining existing industry tools with
new techniques a fairly high fidelity model can be constructed. This model
comprehensively contains sensor and high fidelity actuator models along with nonlinear
bare airframe models.
Models and trends were developed by analyzing a number of different vehicles
spanning almost 50 years. All the vehicles showed that the ducted fan is vulnerable to a
high degree of pitching and translation at slow speeds due to a strong effect of the lateral
and longitudinal moment derivatives, Mu and Lv. This class of vehicles also shows that
the coupling of roll and pitch due to the spinning ducted fan proves troublesome during
identification. This is avoided by identifying the linear coupling and then removing it
from the correlated responses. The use of flight test results, simulation analysis, and wind
tunnel data all may be required to ensure proper modeling techniques.
Sensor performance is seen to be less than desirable due to the small packaging
and weight of the available components. In the area of actuation, the maximum rate of the
servos was seen to have profound effects on high bandwidth performance. This is
important to consider because almost all MOUT exercises require some sort of higher
bandwidth maneuvering.
Overall, this research has shed some light on some of the unique tasks and
procedures for the system identification of ducted fan unmanned air vehicles.
- 119 -
BIBLIOGRAPHY
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Environment and Its Application to a Helicopter,” IEEE Control System
Magazine, Vol. 19, No. 4, pg 22-33, August 1999.
12. Tischler, M.B. and M.G. Cauffman, “Frequency-Response Method for Rotorcraft
System Identification: Flight Application to BO-105 Coupled Rotor/Fuselage
Dynamics.” Journal of the American Helicopter Society, 1992. 37/3: p. 3-17.
- 121 -
Manufacturer References:
Crossbow CXL04LP3
http://www.xbow.com/pdf/Accelerometer/LP/LP%20Accel.pdf
Crossbow Technology, Inc.
41 Daggett Drive
San Jose, CA 95134-2109
Phone:
(408) 965-3300
Fax:
(408) 324-4840
Email:
info@xbow.com
JR Components 8700G Super Servos
Saturation identified from World Class Models:
http://www.worldclassmodels.com/cgi-bin/agora/agora.cgi?product=servos
µ−BLOX GPS-MS1E
http://www.u-blox.ch/gps/gps-ms1e/ubloxgps performance.pdf
Zuercherstrasse 68
P/O Box 78
8800 Thalwil
Switzerland
Email:
info@u-blox.com
Phone (UK):
+44 (0) 1622 618628
Inertial Science RRS75
RRS75.pdf
Peter Moon
Inertial Science, Inc.
(805) 499-3191, (805) 498-4882 Fax
http://www.inertialscience.com
pjmoon@inertialscience.com
Honeywell HMC 2003
http://www.ssec.honeywell.com/magnetic/datasheets/hmc2003.pdf
Motorola MPX 4115A
http://e-www.motorola.com/webapp/sps/prod_cat/prod_summary.jsp?code=MPX4115&catId=M98716
- 122 -
Appendix A
OAV Proposal Vehicle
Identified State-Space Quadruple and Form
x& = Fx + Gu
y = H1 x + H 2 x&
⎧v ⎫
⎪ p⎪
⎪ ⎪
⎪φ ⎪
⎪ ⎪
x = ⎨u ⎬
⎪q ⎪
⎪ ⎪
⎪θ ⎪
⎪ ⎪
⎩r ⎭
⎡ 0
⎢ −0.197
⎢
⎢ 0
⎢
F =⎢ 0
⎢ 0
⎢
⎢ 0
⎢ 0
⎣
⎧ pmixer ⎫
⎪
⎪
u = ⎨ qmixer ⎬
⎪
⎪
⎩ rmixer ⎭
⎧ p⎫
⎪ ⎪
y = ⎨q ⎬
⎪r ⎪
⎩ ⎭
0 32.17
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0⎤
0 ⎥⎥
0
0⎥
⎥
−32.17 0 ⎥
0
0⎥
⎥
0
0⎥
0
0 ⎥⎦
0
0
.2623 0
0
0
0 ⎤
⎡0 57.3 0 0
⎢
H1 = ⎢ 0
0
0 0 57.3 0
0 ⎥⎥
⎢⎣0
0
0 0
0
0 57.3⎥⎦
- 123 -
0
0 ⎤
⎡ 0
⎢.2958
0
0 ⎥⎥
⎢
⎢ 0
0
0 ⎥
⎢
⎥
G=⎢ 0
0
0 ⎥
⎢ 0
.3013
0 ⎥
⎢
⎥
0
0 ⎥
⎢ 0
⎢ 0
0
.3629 ⎥⎦
⎣
H2 = 0
Appendix B
Frequency Response Bode Plots for all Actuator Cases
- 124 -
DS8417 – 5V
- 125 -
DS8417 – 6V
- 126 -
HS512MG – 5V
- 127 -
HS512MG – 6V
- 128 -
DS368 – 5V
- 129 -
DS368 – 6V
- 130 -
94091 – 5V
- 131 -
94091 – 6V
- 132 -
CS-10BB – 5V
- 133 -
CS-10BB – 6V
- 134 -
Appendix C
Actuator Generated Transfer Function Models
Bode Plot Verification
- 135 -
DS8417 – 100% - 5V
- 136 -
DS8417 – 100% - 5V
- 137 -
DS8417 – 50% - 5V
- 138 -
DS8417 – 100% - 6V
- 139 -
DS8417 – 50% - 6V
- 140 -
HS512MG – 100% - 5V
- 141 -
HS512MG – 50% - 5V
- 142 -
HS512MG – 100% - 6V
- 143 -
HS512MG – 50% - 6V
- 144 -
DS368 – 100% - 5V
- 145 -
DS368 – 50% - 5V
- 146 -
DS368 – 100% - 6V
- 147 -
DS368 – 50% - 6V
- 148 -
94091 – 80% - 5V
- 149 -
94091 – 50% - 5V
- 150 -
94091 – 80% - 6V
- 151 -
94091 – 50% - 6V
- 152 -
CS-10BB – 100% - 5V
- 153 -
CS-10BB – 50% - 5V
- 154 -
CS-10BB – 100% - 6V
- 155 -
CS-10BB – 50% - 6V
- 156 -
Appendix D
Actuator Time Domain Verification of Final Models
- 157 -
Time Domain Verification
DS8417 5V
DS8417 6V
- 158 -
94091 5V
94091 6V
- 159 -
CS-10BB 5V
CS-10BB 6V
- 160 -
DS368 5V
DS368 6V
- 161 -
HS-512MG 5V
HS-512MG 6V
- 162 -