Six-DoF Dynamic Modeling

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American Institute of Aeronautics and Astronautics1

Six-DoF Dynamic Modeling and Flight Testing of aUAV Helicopter

Subodh Bhandari and Richard Colgren

University of Kansas, Lawrence, Kansas 66045

Philipp Lederbogen

University of Stuttgart, D-70569 Stuttgart, Germany

and

Scott Kowalchuk

Virginia Polytechnic Institute and State University, Blacksburg, Virginia

This paper presents the continued research on UAV dynamic modeling at theAerospace Engineering Department of the University of Kansas using a ThunderTiger Raptor 50 V2 remote control (RC) helicopter. Previously, an uncoupledthree-degree-of-freedom (3-DoF) linear parameter varying (LPV) dynamic modelbased on stability and control derivatives was presented. The response obtainedfrom this model showed a high degree of correlation with the actual flight test data.The previous paper also highlighted the flight test instrumentation and datacollection procedures. As a result of the on-going research on the dynamic modelingof the UAV helicopter, a complete six-degree-of-freedom (6-DoF) LPV model hasbeen developed in the MathWorks MATLAB and Simulink environment. An

effort has been made to take into account the aerodynamic and inertial couplingbetween the longitudinal and lateral dynamics of the helicopter. Validation of themodel was accomplished by comparing the results of the simulation with flight testdata along with the data obtained from CIFER (Comprehensive Identification fromFrEquency Response) model. The 6-DoF model showed a good correlation betweenactual flight test data and CIFER.

_______________________________________Graduate Research Assistant, Department of Aerospace Engineering, 2129C Learned

Hall, and AIAA Student Member.

Associate Professor, Department of Aerospace Engineering, 2120D Learned Hall, andAIAA Associate Fellow.Graduate Student, University of Stuttgart Aerospace Engineering and Geodesy,Pfaffenwaldring 27, D-70569 Stuttgart, Germany, and AIAA Student Member.Graduate Research Assistant, Department of Aerospace and Ocean Engineering,Virginia Polytechnic Institute and State University, 215 Randolph Hall, Blacksburg, VA24061, and AIAA Student Member.

AIAA Modeling and Simulation Technologies Conference and Exhibit15 - 18 August 2005, San Francisco, California

AIAA 2005-6422

Copyright 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


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Nomenclature

a1s = longitudinal flapping co-efficientCH = main rotor H-forceFx, Fy, Fz = forces along X, Y, and Z axis

g = acceleration due to gravityIx = moment of inertia about X axisIxz = product of inertia about XZ planeIy = moment of inertia about Y axisIz = moment of inertia about Z axisL = aircraft rolling momentLu, Lq, Lv, Lp , Lr = rolling moment stability derivativesL 0, LA1, LB1, L 0T = rolling moment control derivativesM = aircraft pitching momentMu, Mw, Mq, Mv, Mp, M r = pitching moment stability derivativesM 0, MA1, MB1, M 0T = pitching moment control derivatives

N = aircraft yawing momentNw, Nv, Np , Nr = yawing moment stability derivativesN 0, , N 0T = yawing moment control derivativesp = roll rate in body fixed coordinate systemq = pitch rate in body fixed coordinate systemr = yaw rate in body fixed coordinate systemR = radius of the main rotoru, v, w = velocity component along X, Y, and Z axesXu, Xw, Xq, Xv, Xp, Xr = X force stability derivativesX 0, XA1, XB1 = X force control derivativesYu, Yq, Yv, Yp, Yr = Y force stability derivativesY

0, Y

A1, Y

B1, Y

0T= Y force control derivatives

Zw = Z force stability derivativesX 0 = Z force control derivatives = advance ratio

= main rotor blade angular velocityA1 = lateral cyclicB1 = longitudinal cyclic

o = main rotor collectiveoT = tail rotor collective

= pitch angle= roll angle

I. IntroductionThere has been a significant growth in the use of UAVs for a multitude of military

and civilian applications. This has led to substantial research on a variety of UAVs.Various organizations have been conducting research on both fixed and rotary wingUAVs over the past few years. The Department of Aerospace Engineering at theUniversity of Kansas (KU) is also conducting such research. Other universities that are


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involved in UAV research are the Georgia Tech, Carnegie Mellon University, and theMIT. Georgia Tech is conducting dynamics and control research on Aerial Roboticsusing an R-MAX UAV helicopter 1, while Carnegie Mellon is using its Yamaha R-50helicopter for research on parameter identification and control system design 2. MITresearchers, on the other hand, are using an X-Cell 60 UAV helicopter for autonomy

research and dynamic modeling of aerobatic maneuvers3, 4

.Intelligent Unoccupied Air Vehicles represent a major area of multidisciplinary,systems oriented research and development at KU. Current research is being conductedusing both fixed and rotary-wing UAVs. These vehicles are both purchased from themanufacturers and assembled as it is, are modified to meet the goals of our research, orare completely new vehicle designs developed by the Department of AerospaceEngineering. The research emphasis at KU is on designing, modeling, and flight-testingthese vehicles to develop accurate dynamic computer simulations. Another key researchtopic in this area to enable unpiloted vehicles as viable systems is the development of reliable autonomous control technologies implemented within embedded computersystems.

Figure 1: Yamaha R-MAX and Thunder Tiger Raptor 50 V2 Helicopters

This paper focuses on the dynamic modeling of the Thunder Tiger Raptor 50helicopter. The model being developed will eventually be used to model the dynamics of a Yamaha R-MAX helicopter (R-MAX). The two helicopters are as shown in the Figure1. Table 1 lists the mass and geometric characteristics of the two helicopters. Thedynamic modeling is based on stability and control derivatives derived from rigid bodyequations of motion 5, 6. The stability and control derivatives are functions of theaerodynamic, geometric and mass characteristics of the rotorcraft. Once validated


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through extensive flight-testing, such models can be used for the stability and controlanalysis of any other UAVs just by replacing the aerodynamic, geometric and masscharacteristics.

Comprehensive dynamic models, similar to the one discussed in this paper allowthe control system engineers to analyze the stability and control characteristics of a

specific UAV in-depth, which leads to reduced time and cost in the flight test phase.These models can also be used in the design of stability augmentation systems (SAS) andautomatic flight control systems with the goal of significantly reducing the overall cost of UAV design 7.

Table 1: Raptor and RMAX Helicopter SpecificationsParameter Raptor 50 V2 Yamaha RMAX

Fuselage Length 47.24 in. 142.8 in.Fuselage Width 5.51in.Main Rotor Diameter 52.95 in. 122.4 in.Tail Rotor Diameter 9.26 in 21.0 in.

GRmr 8.5:1 6.7:1GR tr 4.56:1 6.7:1Un-instrumented VehicleEmpty Weight 7.5 lbs. 140 lbs.Instrumented VehicleWeight+Fuel 10.5 lbs. 200 lbs.xcg 15.26 in. 29.0 in.ycg -0.47 in. 0 in.zcg 7.12 in. NAIxx 0.0769 lb ft s2 NA

Iyy 0.1973 lb ft s2 NAIzz 0.1912 lb ft s2 NAEngine Displacement 0.50 cu. in. 15 cu. In.Endurance 6 min. 60 min.Fuel cost $15 /gal. $2 /gal.Onboard Starting No YesWater Cooled No Yes

Currently, research is underway in the development of a coupled 6-DoFtheoretical model for the Raptor 50 V2 RC helicopter. This paper discusses the

background research leading to the development and validation of the 6-DoF dynamicsmodel. The instrumentation of the Raptor 50 along with the flight tests is also discussed.Finally, the simulation result of the 6-DoF model and the comparison of the result withflight data and CIFER model simulation is presented.

II. Previous Research on Dynamic ModelingThe Aerospace Engineering Department at The University of Kansas procured 2

Raptor 50 V2 RC helicopters for dynamics and control research. The helicopter was


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instrumented for flight data acquisition as discussed in the next section. A theoretical 3-DoF LPV model for longitudinal dynamics was developed. As shown in Figure 2, acomparison between the simulation and the flight data showed a good correlation forpitch rate and pitch angle 8, 9 . However, the comparison for forward and vertical velocitieswas not so satisfactory. The fact that the coupling between the longitudinal and lateral

dynamics was not considered in developing the 3-DoF model led to the conclusion thatthe disparity between the flight data and the simulation was the result of such neglectedcoupling effects.

Figure 2: 3-DoF Simulation vs. Flight Data for a Pitch Sweep in Hover

Unlike fixed wing aircraft, the helicopters possess significant coupling betweenlongitudinal and lateral dynamics. The coupling comes from different sources. One of the main sources is the flapping due to pitch and roll velocities 10. An offset in theflapping hinge also causes a coupling between the two dynamics. Helicopters like theRaptor 50 have a teetering rotor hinge and, therefore, no coupling due to the hinge offset.Reference 10 talks in detail about different sources of coupling between longitudinal andlateral dynamics of a helicopter.

Almost concurrently, a non-parametric model of the Raptor 50 was developedusing CIFER which is a system identification software tool developed by the U.S. Army,NASA, and Sterling Federal Systems 11. It is an interactive software for systemidentification and verification based on a comprehensive frequency-response approach.


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A number of flight test frequency sweeps were conducted in order to get frequencydomain data. The data were then used to identify single-input-single-output (SISO)transfer function models and multiple-input-single-output (MISO) state-space models.The models were simulated and the results compared with the flight data 12. Thecomparison showed a good correlation between the flight data and the CIFER model. The

improved quality of the match was attributed to the fact that the CIFER model, unlike the3-DoF dynamic model, took into account the coupling between the longitudinal andlateral dynamics of the helicopter.

III. Instrumentation of the Raptor 50 V2 and Flight Data ProcessingFigure 3 shows the instrumentation of the Raptor for the purpose of flight data

acquisition 13. Also shown is the stock helicopter that serves as a training helicopter. Thehelicopter was instrumented to obtain the pilot command inputs and the response of thehelicopter. These response measurements include the longitudinal, lateral, and verticalaccelerations along with pitch, roll, and yaw rates. The instruments used and theirpurpose are listed in Table 2. The flight test instrumentation package is capable of

simultaneously recording 12 analog channels and 4 digital channels while sampling from0.00027 to 512 Hz. The sampling frequency and number of channels being recordeddictates the data recording duration.

Table 2: Raptor 50 V2 Flight Article Instrumentation EquipmentInstrumentation Purpose

4 Position Transducers To measure the inputs from the pilot:Main Rotor Collective ServoPitch Cyclic ServoLateral Cyclic ServoTail Rotor Collective Servo.

Crossbow Dynamic MeasuringUnit (DMU) Model H6X To measure the 3 axis linear accelerations and the 3axis rotation rates.Crossbow 16 Channel DataAcquisition Unit Model ReadyDAQ AD2012

To collect and store flight data for post-processing.

Ultrasonic Sonar Sensor To measure the height above the ground from 0.5feet to 40 feet.

Power Supply (4 9-VoltBatteries)

To provide constant 18-volt power to thepreviously mentioned devices.

Governor To maintain constant main rotor rpm.


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Figure 3: Raptor 50 V2 Test Flight Vehicle

A number of flight tests were carried out to collect the flight data. The flight tests

accomplished were for both the hovering and forward flight conditions. Initially, theflight tests were done for the purpose of generating CIFER models and for the simulationof the 6-DoF mathematical model. Once the CIFER models were generated and thesimulation completed, further flight tests were carried out in to collect data for thevalidation of CIFER models and simulation results. Figure 4 shows an example of such aflight test for the hovering flight condition.


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Figure 4: Raptor 50 V2 in Hover

After the flight data is collected, it is necessary to filter the noise out and applythe proper conversion to interpret the data. The flight data is collected as analog signalsof 0-5 and 5 volts. The filtering of the data is done using a 2 nd order Butterworth filterwith a cutoff frequency of 6 Hz. All the post processing of the data has been done usingMATLAB and Simulink. Once the filtering is accomplished and conversion toengineering units is completed, the data is ready for use in the parameter identificationand dynamic model simulations. More information on post processing techniques can befound can be found in the References 8 and 12.

IV. 6-DoF Dynamic Model for RaptorThe dynamic model for the Raptor is derived using equations of motion for the

helicopter. The principles of Newtonian mechanics are used to derive the equations. Theequations relate the helicopter response to the forces and moments acting on thehelicopter. The equations used during model development process are the three force (X,Y, and Z) equations and the three moment (rolling, pitching, and yawing) equations asgiven below 14:

)( vr wqumF x += & (1))( w pur vmF y += & (2)

)( uqv pwmF z += & (3)q p I I I r qr I p I L xz y z xz x += )(&& (4)

)()( 22 r p I I I pr q I M xz y x y ++= & (5)r q I I I q pr I p I N xz x y z xz +++= )(&& (6)

The total forces and moments result from the contribution of the main rotor, tailrotor, horizontal tail, vertical tail, and fuselage. The equations given above are nonlinear.Small perturbation theory and Taylor series expansion can be used to linearize these


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equations. The linearized equations are much easier to use. Furthermore, the mainadvantage of the linearized equations is that they can be used in state-space modeling 15.The linearized equations, for example, can be written in the following form:

1100 B Ar pqwu X X X X r X p X q X w X u X um T M =+++++

&&

(7)

In the above equation, X u

X u = , X w X w = and so on are called the stability

derivatives and X X M

M 00

= , X A

X A1

1= et cetera are called the control

derivatives. The stability and control derivatives are functions of aircraft speed andaltitude similar to the aircraft lift, drag and moments. The formulae for these derivativesare similar to the formulae for lift and drag. For example, the formula for X u, the speed-damping derivative, is as follow

u

a

a

CH R A X s

s

bu = 11

2)( (8)

These derivatives represent the stability or controllability of the aircraft and form thebasis for state-space representation of equations of motion given by:

u B x A x +=& (9)u D xC y += (10)

where x is the state (u, w, q, , v, p, r, ) vector, A is the system matrix consisting of stability derivatives, B is the control matrix consisting of control derivatives, C is theoutput matrix (which is usually an identity matrix), D is the matrix representing couplingbetween input and output (which for aircraft applications usually consists of zeros), u isthe input vector ( ),,,

0110 T M B A , and y is the output vector. The state-space equations

can be used for the simulation of aircraft motion following a pilot input using MATLABand Simulink. The state-space equations have a further advantage in that the equationscan be used to simulate not only the single input single output (SISO) motion, but alsothe multiple input multiple output (MIMO) motion of the aircraft.

The 6-DoF model developed for the Raptor 50 employs the state-spacerepresentation of the equations of motion. As mentioned earlier, some excellent progresshas been made in the development of the dynamic model for the hovering flightcondition. The development of the model for the forward flight is underway. There is aslight difference between the models for hovering flight and forward flight due to theeffect of coupling between longitudinal and lateral dynamics of the helicopter. Rotor

flapping is the main source of this coupling. Reference 10 discusses in detail about rotorflapping and the coupling between longitudinal and lateral dynamics of the helicopter.But, the effect of the rotor wakes in the induced flow and consequently in the couplingbetween longitudinal and lateral dynamics has not been accounted for in Reference 10,resulting in an off-axis response in the wrong direction. This has been a topic of researchover the last few years. As a result of this research, it has been possible to accuratelymodel the coupling between the longitudinal and lateral dynamics of the helicopter,resulting in the correct off-axis response 16, 17, and 18 .


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The present hover model accounts for the coupling between longitudinal andlateral dynamics of the helicopter, and the effect of the rotor wake. The model also takesinto account changing flight conditions due to pilot input. This means that the aircraftstates, A and B matrices, and outputs are constantly changing with time. In other words,parameters are varying with changing flight conditions over the time leading to a LPV

model, unlike the linear parameter invariant (LPI) models in which the parameters areheld fixed. The state-space equations for the LPV model are given by:)()()()()( t ut Bt xt At x +=& (11)

)()()( t u Dt xC t y += (12)For a coupled, 6-DoF dynamic model, A and B matrices also consist of the

derivatives that represent the coupling between the longitudinal and lateral dynamics of the helicopter. In hover, the longitudinal and lateral dynamics are weakly coupled andthe coupling between directional dynamics and the other two dynamics are almostnegligible. However, the model being discussed considers the coupling between all thethree dynamic axes of the helicopter. The A and B matrices for the 6-DoF coupled modelare given by the equations (13) and (14) 19. In these equations, the terms such as X v, MA1,

Yu, Lr et cetera represent the coupling effect between the various dynamics. Though thereare a number of derivatives that come from the coupling between the various dynamics,very few of them have a pronounced effect on the helicopter dynamics. The others havea negligible effect on the dynamics. It has been found that the exclusion of thosederivatives from the model does not affect the results of the simulation. For example, thederivative X r, a change in the X-force due to a change in the yaw rate, has a negligibleeffect on the dynamics of the helicopter and can be excluded from the model.

The coupled 6-DoF model in state-space form thus obtained is used for thesimulation of the helicopter motion using Simulink. The Simulink model is as shown inthe Figure 5. It can be seen that the output of the simulation is being fed back into themodel in order to take into account the changing flight conditions after any perturbation

from the reference flight condition. At each time step of the simulation, the modelcalculates new A and B matrices depending on the flight condition at that time step. Thisis the essence of an LPV model. The benefit of an LPV model is that any changes in theflight velocities and attitude angles are considered while calculating the stability andcontrol derivatives that constitute the A and B matrices. Thus, the model will be able tomore accurately predict the behavior of the helicopter after the perturbation.

=

r pvw

r pvqu

r pvqu

r pvqwu

w

r pvqwu

N N N N

L L L L L

Y gY Y gY Y

M M M M M M

gg Z

X X X g X X X

A

0000tancos0100tansin00

000coscossinsin0

00000cos00

000cossin00sincos00

0cos

(13)


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=

T

T

T

T

N N

L L L L

Y Y Y Y

M M M M

Z

X X X

B

A B

A B

A B

A B

00

0110

0110

0110

0

110

000000

0000

0000

(14)

states

B1

To Workspace6

position

To Workspace5

B

To Workspace4

A

To Workspace3

A1

To Workspace2

control_inputs

To Workspace1

state_space

To Workspace

Velocity Input Position

Position

input_collective

FromWorkspace4

input_tail

FromWorkspace3

input_lateral_cyclic

From

Workspace2

input_long_cyclic

FromWorkspace1

Flight Data Comparison

inputs

inputs1

inputs2

states

states2

Dynamic Equations

m

State Space

CommandInput

AB

A1B1

Calculate next A & BMatrix

v

p

phi

r

u

w

q

theta

Figure 5: 6-DoF Simulink Model

V. Raptor 50 V2 Simulation Results vs. Flight Data and CIFERA number of flight tests were carried out to collect the helicopters response to

remote pilot input. The primary forms of the pilot input for these tests were either puresinusoidal or frequency sweeps. This data was generated for system identification usingCIFER. A frequency sweep is a type of sinusoidal excitation of the helicopter, the sweep


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going from low frequencies with small amplitudes to high frequencies with higheramplitudes within the frequency range of interest. In some of the flights, a doublet inputwas also used.

The remote pilot input extracted from the flight data was used to stimulate the 6-DoF dynamic model of the Raptor. The same data was used to extract the transfer

function (TF) model for the single input single output case and the state-space model forthe multiple input multiple output case using CIFER. The TF models obtained were bothin pole zero form and simple gain and time delay form. For the purpose of thisdiscussion, simple gain and delay models were used. The results of the 6-DoF simulationwere then compared with flight data and later with the response obtained from the CIFERmodel. The comparisons are shown in the following figures.

47 48 49 50 51 52 53 540

20

40

u ( f t / s

)

47 48 49 50 51 52 53 54-10

-5

0

w ( f t / s

)

47 48 49 50 51 52 53 54-1

0

1

q ( r a

d / s )

47 48 49 50 51 52 53 54-1

0

1

time (s)

t h e t a ( r a

d )

47 48 49 50 51 52 53 54

0

20

40

v ( f t / s

)

47 48 49 50 51 52 53 54-0.5

0

0.5

p ( r a

d / s )

47 48 49 50 51 52 53 54-0.5

0

0.5

p h i ( r a

d )

47 48 49 50 51 52 53 54-0.5

0

0.5

time (s)

r ( r a

d / s e c

)

SimulationFlight Data

Figure 6: 6-DoF Simulation vs. Flight Data for a Pitch Sweep

Figures 6 and 7 show the comparison between the flight data and the simulationresults for the pitch and roll sweeps respectively. The simulations shown are for amoderately short period of time. When the simulations are run for a longer period, theresponses are found to show an increasing error. A research effort is currently underwayto mitigate this problem. It can be seen from these figures that the correlation between the


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simulation results and the flight data is very good, with some exceptions for off-axisresponses. Though there has been improvement in the off-axis responses after theincorporation of rotor wake effects, it is felt that more work has to be done in this area.When compared to the 3-DoF longitudinal model, it is noticed that the velocities andangular rates are in better agreement with the flight data. It is noteworthy to mention

here that stabilizer bar dynamics have not been included within the model. The stabilizerbar has a pronounced effect on controllability, and on the pitch and roll damping of thehelicopter 20. Further improvement in the simulation response will be seen once the bardynamics are included.

288 289 290 291 292 293 294 295-50

0

50

u ( f t / s

)

288 289 290 291 292 293 294 295

-20

0

20

w ( f t / s

)

288 289 290 291 292 293 294 295-0.5

0

0.5

q ( r a

d / s )

288 289 290 291 292 293 294 295-0.5

0

0.5

time (s)

t h e t a ( r a

d )

288 289 290 291 292 293 294 2950

20

40

v ( f t / s

)

288 289 290 291 292 293 294 295-1

0

1

p ( r a

d / s )

288 289 290 291 292 293 294 295-0.5

0

0.5

p h i ( r a

d )

288 289 290 291 292 293 294 295-1

0

1

time (s)

r ( r a

d / s e c

)

SimulationFlight Data

Figure 7: 6-DoF Simulation vs. Flight Data for a Roll Sweep

In Figure 8, a comparison between the flight data, the CIFER model response, andthe 6-DoF model response is shown for pitch rate. Similarly, Figure 9 shows acomparison for roll rate. It is obvious from these figures that there is a high degree of correlation between the flight data and the CIFER model response, while the 6-DoFmodel response is seen to follow the trend closely. However, the CIFER models obtainedso far are SISO models for the pitch, roll, and yaw rates. Some MIMO models fromCIFER have also been extracted, but this work is still in the development phase. The


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SISO models are transfer function models, while the MIMO models are obtained in state-space form. The pitch rate to longitudinal cyclic and the roll rate to lateral cyclic transferfunctions are described in equations 15 and 16, respectively.

47 48 49 50 51 52 53 54-40

-30

-20

-10

0

10

20

30

40

P i t c

h R a t e ( d e g

/ s )

Time(s)

SimulationFlight DataCIFER

Figure 8: Pitch Rate Comparison Between 6-DoF Simulation, Flight Dataand CIFER for a Pitch Sweep

ses BsQ 0020.0

1 )()(

= (15)

ses A

s R 0037.0

1 )()(

=(16)

The characteristics of the above transfer functions, such as the gains and biases,are listed in Table 3 and Table 4 respectively. The characteristics listed are the timeconstant, cost function, gain, and bias. The time constants and the cost functions wereobtained from CIFER, while the gains and biases were selected on a trial and evaluationbasis.


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12. Lederbogen, Philipp, Flight Test Evaluation and Modeling of a Raptor 50 V2 UAV using CIFER , Department of Aerospace Engineering, University of Kansas,Lawrence, Kansas, December 2004.

13. Lederbogen, Kowalchuk, Holly, and Colgren, Flight Testing of a Raptor 50 V2 Helicopter for Parameter Identification , 35 th Annual Society of Flight Test Engineers

Symposium, 2004.14. Roskam, Jan, Airplane Flight Dynamics and Automatic Flight Controls , Lawrence,Kansas, DAR Corporation, 1998.

15. Ogata, Katsuhiko, Modern Control Engineering 3 rd Edition, Prentice Hall, NewJersey, 1997.

16. Curtiss, H. C., Aerodynamic Models and the Off-Axis Response , American HelicopterSociety, 55 th Annual Forum Proceedings, Volume 2, Montreal, Canada, May 1999.

17. Keller, J. D., An Investigation of Helicopter Dynamic Coupling Using an Analytical Model , Journal of the American Helicopter Society, Volume 41, October 1996.

18. Keller, J. D., and Curtiss, H. C., The Effect of Inflow on the Dynamic Response of Helicopters, American Helicopter Society , 52nd Annual Forum Proceedings ,

Washington D. C., June 1996.19. Padfield, Gareth, Helicopter Flight Dynamics: The Theory and Application of FlyingQualities and Simulation Modeling , American Institute of Aeronautics andAstronautics, Washington D.C., 1996.

20. Mettler, Bernard, Identification Modeling and Characteristics of Miniature Rotorcraft , Kluwer Academic Publishers, Boston, 2003.

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