Comprehensive System Identification of Ducted Fan UAVs - Cal Poly

Comprehensive System Identification ofDucted Fan UAVsA ThesisPresented to the Faculty ofCalifornia Polytechnic State UniversitySan Luis ObispoIn Partial Fulfillmentof the Requirements for the Degree ofMaster of Science in Aerospace Engineeringby:Daniel N. SalluceJanuary 2004

© Copyright 2004Daniel SalluceALL RIGHTS RESERVEDii

APROVAL PAGETITLE:AUTHOR:Comprehensive System Identification of Ducted Fan UAVsDaniel N. SalluceDATE 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

ABSTRACTThe increase of military operations in urbanized terrain has changed the nature ofwarfare and the battlefield itself. A need for a unique class of vehicles now exists. Thesevehicles must be able to accurately maintain position in space, be robust in the event ofcollisions, relay strategic situational awareness, and operate on an organic troop level in acompletely autonomous fashion. The operational demands of these vehicles mandateaccurate control systems and simulation testing. These needs stress the importance ofsystem identification and modeling throughout the design process. This research focuseson 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 ofidentification techniques can be combined to comprehensively model this family ofvehicles and reveals the unique challenges involved. The result is a high fidelity modelavailable for the purposes of control system design and simulation.iv

ACKNOWLEDGMENTSThe 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 RotorcraftDivision at Moffett Field proved to be invaluable resources and facilitators in thecompletion of this project.v

TABLE OF CONTENTSLIST OF TABLES........................................................................................................... viiiLIST OF FIGURES ........................................................................................................... ixNOMENCLATURE ......................................................................................................... xiiCHAPTER 1 – Introduction and Motivation1.1 Vehicles Examined ............................................................................................11.2 Scope..................................................................................................................8CHAPTER 2 – Dynamic Model Identification Methods and Techniques2.1 Identification Methods.................................................................................... 112.2 CIFER ............................................................................................................. 122.2.1 Flight Test Techniques..................................................................... 132.2.2 Bench Test Techniques.....................................................................142.3 Manufacturer Specifications ............................................................................142.4 Wind Tunnel Tests...........................................................................................15CHAPTER 3 – Vehicle Identification3.1 Areas of Identification .....................................................................................163.2 Bare-Airframe ID.............................................................................................173.2.1 Aerovironment/Honeywell OAV......................................................173.2.2 Allied Aerospace MAV ....................................................................353.2.3 Trek Aerospace Solotrek...................................................................463.2.4 Hiller Flying Platform.......................................................................483.2.5 Vehicle Scaling Laws and Comparisons...........................................523.3 Servo Actuator Identification...........................................................................563.4 Sensor Identification ........................................................................................943.4.1 Accelerometer Identification ............................................................953.4.2 Rate Gyro Identification ...................................................................963.4.3 GPS Receiver Identification .............................................................98vi

3.4.4 Magnetometer Identification...........................................................1013.4.5 Pressure Altimeter Identification ....................................................102CHAPTER 4 – Flight Simulation4.1 Simulated Sweeps ..........................................................................................1044.2 Matlab Linear Model Determination .............................................................110CHAPTER 5 – Conclusions.............................................................................................119BIBLIOGRAPHY............................................................................................................120APPENDIX A – OAV Proposal State Space Form.........................................................123APPENDIX B – Frequency Response Bode Plots for all Actuator Cases ......................124APPENDIX C – Actuator Generated TF Model Bode Plot Verification ........................135APPENDIX D – Actuator Time Domain Verification of Final Models..........................157vii

LIST OF TABLES3.1 – OAV Measured Parameters during Flight Testing .......................................................183.2 – OAV Frequency Range of Good Coherence (rad/sec) .................................................193.3 – OAV Control Derivatives Extracted from Transfer Function Fits ...............................203.4 – OAV DERIVID Identified parameters and Certainties ................................................233.5 – OAV DERIVID Frequency Response Costs ................................................................233.6 – OAV Eigenvalues and Associated Eigenvectors of [F]................................................243.7 – MAV Physical Properties .............................................................................................353.8 – MAV Identified Stability Derivatives...........................................................................393.9 – MAV Identified Control Derivatives............................................................................403.10 – Final Flight Test Identified MAV Derivatives............................................................423.11 – MAV Wind Tunnel Identified Derivatives and Flight Test Results...........................443.12 – Pitching Moment Derivatives and Solotrek Fan Speed ..............................................473.13 – Pitching Moment Coefficient Summary .....................................................................533.14 – Pitching Moment with Blade Chord Summary...........................................................543.15 – Manufacturer Specifications for Servo Actuators Tested...........................................573.16 – Actuator Linkage Geometries.....................................................................................603.17 – Actuator Calibration Factors for Input and Output Channels to Degrees...................613.18 – Frequency Sweep Used for all Actuators....................................................................623.19 – Square Wave Parameters ............................................................................................633.20 – Actuator Bench Test Matrix........................................................................................653.21 – Actuator NAVFIT Frequency Ranges for CIFER Cases............................................673.22 – Actuator NAVFIT Results for all Cases .....................................................................683.23 – Actuator Nonlinear Characteristic Summary..............................................................744.1 – Linmod, Wind Tunnel, and Flight Test Results for i-Star 9”........................................114viii

LIST OF FIGURESFigure 1.1 – Land Warrior OAV Concept ..........................................................................1Figure 1.2 – Hiller Helicopters Flying Platform – 1958.....................................................3Figure 1.3 – Aerovironment / Honeywell DARPA Phase I OAV – 2001 ..........................3Figure 1.4 – Trek Aerospace Solotrek Ducted Fan – 2001.................................................4Figure 1.5 – Allied Aerospace i-Star MAV 9” Vehicle – 2003..........................................4Figure 1.6 – Detailed view of 9” MAV Design ..................................................................5Figure 1.7 – MAV Stator and Vanes...................................................................................6Figure 1.8 – Helicopter Body Axes System........................................................................7Figure 1.9 – Helicopter Body Axes System Applied to the Ducted Fan ............................7Figure 1.10 – Block Diagram of Basic DFCS Architecture ...............................................8Figure 1.11 – Comprehensive Identification Schematic.....................................................9Figure 2.1 – Sample Frequency Sweep Flight Test Command.........................................13Figure 3.1 – Roll rate response frequency domain verification........................................26Figure 3.2 – Pitch rate response frequency domain verification.......................................27Figure 3.3 – Yaw response frequency domain verification ..............................................29Figure 3.4 – Roll response time history verification.........................................................30Figure 3.5 – Pitch response time history verification .......................................................31Figure 3.6 – Yaw response time history verification........................................................32Figure 3.7 – Techsburg Wind Tunnel Setup for OAV......................................................33Figure 3.8 – Techsburg OAV Pitching Moment to Airspeed ...........................................34Figure 3.9 – On and Off Axis MAV Roll Frequency Responses .....................................36Figure 3.10 – On and Off Axis MAV Pitch Frequency Responses ..................................37Figure 3.11 – MAV Lateral Acceleration and Roll Rate Response to Roll Input ............40Figure 3.12 – MAV Longitudinal Acceleration and Pitch Response ...............................41Figure 3.13 – Pitching Moment Wind Tunnel Test Data for i-Star 9” .............................43Figure 3.14 – Solotrek Wind Tunnel Test Results for Pitching Moment .........................46Figure 3.15 – Hiller Flying Platform Pitching Moment Data...........................................48Figure 3.16 – Drag over a Flat Plate Perpendicular to Flow.............................................49Figure 3.17 – Results of Removing Dummy Moment from Hiller Platform Test............50ix

Figure 3.18 – Actuators Tested and Relative Sizes ..........................................................57Figure 3.19 – Actuator Test Stand Apparatus...................................................................58Figure 3.20 – Cirrus CS-10BB Mounted on Wooden Strip..............................................58Figure 3.21 – Schematic Detailing Linkage Geometry.....................................................59Figure 3.22 – Sample Chirp Input, Response, and Square Wave Time History...............64Figure 3.23 – HS512MG Responses Illustrating Difference between 5V and 6V ...........70Figure 3.24 – Sample Square Wave Response .................................................................72Figure 3.25 – Linear Fit for Max Rate Determination......................................................73Figure 3.26 – CS-10BB at 5V TH Illustrating Erratic Response at High Frequency.......75Figure 3.27 – 94091 at 6V TH Illustrating Erratic Response at High Frequency.............75Figure 3.28 – 94091 at 5V TH not Showing Erratic Response.........................................76Figure 3.29 – DS8417 FR Illustrating Mismatch in Linear Model...................................77Figure 3.30 – DS8417 TH Comparison to 1995 STI Findings .........................................78Figure 3.31 – Magnitude Comparison for Linear & Nonlinear Model to Bench Test .....81Figure 3.32 – Phase Comparison for Linear & Nonlinear Model to Bench Test .............82Figure 3.33 – Error Function Fr and NAVFIT Transfer Function Fit ..............................83Figure 3.34 – Rise Time Ratio Phase Lag Relationship ...................................................85Figure 3.35 – Rise Time for Linear Model of DS8417 at 5V...........................................86Figure 3.36 – Sweep Amplitude and Natural Frequency with Rate Limiting ..................87Figure 3.37 – Simulink Actuator Blockset .......................................................................88Figure 3.38 – Configurable Actuator Parameters .............................................................89Figure 3.39 – 2 nd Order Actuator Dynamics behind Mask ...............................................90Figure 3.40 – DS8417 at 5V Time Domain Validation ....................................................91Figure 3.41 – Accelerometer Model .................................................................................95Figure 3.42 – Accelerometer Stationary Noise Model .....................................................96Figure 3.43 – Rate Gyro Model ........................................................................................97Figure 3.43 – Rate Gyro Response to Constant 15 deg/sec for 10 sec .............................98Figure 3.44 – GPS Heading and Speed Model .................................................................99Figure 3.45 – GPS Error and Discrete Signal Model......................................................100Figure 3.46 – GPS Model Results...................................................................................101Figure 3.47 – Magnetometer Model ...............................................................................102x

Figure 3.48 – Magnetometer Depiction at 5 Gauss for 5 Seconds .................................102Figure 3.49 – Pressure Altimeter Model.........................................................................103Figure 3.50 – Pressure Altimeter at 15 feet for 5 seconds ..............................................103Figure 4.1 – Simulink MAV Model................................................................................105Figure 4.2 – Custom PC and COTS Simulation Environment .......................................106Figure 4.3 – Simulink Sweep Generator GUI Built for Sweeps.....................................107Figure 4.4 – Simulink GUI Generated Sweep ................................................................108Figure 4.5 – MAV Flight Test Cross Coherence between Pitch and Roll controls ........109Figure 4.6 – Cross Control Decoupling Block Diagram.................................................111Figure 4.7 – LINMOD and Simulated Sweep Roll Frequency Response ......................115Figure 4.8 – Effect of Removing Cross Control Coupling to Response.........................116Figure 4.9 – Coupling Removed Illustrating linmod and Simulated Sweep Results......117Figure 4.10 – Comparison of linmod and Flight Test Pitch Responses..........................118xi

NOMENCLATUREA Area v& Lateral body accelerationa 1 First Fourier Coefficient w Vertical body velocityb 1 Second Fourier Coefficient w& Vertical body accelerationBW Bandwidth Y Lateral Body Forcec Chord x State MatrixC Nondimensional Coefficient X Longitudinal Body ForceCMPA Commanded Roll Rate Z Vertical Body ForceCMQA Commanded Pitch Rate φ Roll attitudeCMRA Commanded Yaw Rate θ Pitch attitudeCR Cramer-Rao Bound ϕ Heading attitudeF Plant Matrix ω n Natural FrequencyG Control Matrix ˆnω Normalized Natural FrequencyH 1 Output Matrix Position Ω Propeller Rotational VelocityH 2 Output Matrix Rate ρ DensityI Inertia σ Propeller Coefficientj Imaginary Variable τ Time ConstantL Rolling Moment ζ Damping RatioM Pitching Moment ∠ Phase AngleN Yawing Momentp Roll body rate Subscriptsp mixer Lateral mixer signalP Period c, δ Commandq Pitch body rate CG Center of Gravityq mixer Longitudinal mixer signal col Collectiver Yaw body rate FS Full ScaleR Radius Lat Lateralr mixer Pedal mixer signal (deg/sec) lon Longitudinals Frequency Domain Variable mixer Mixert Linear : Nonlinear Rise Time ped PedalˆRtR NLRise Time Nonlinear prop PropellertR LRise Time Linear rad Radiansu Longitudinal body velocity xx X-plane in the Direction of Xu Input Control Matrix yy Y-plane in the Direction of Yu& Longitudinal body acceleration zz Z-plane in the Direction of Zv Lateral body velocity dot Time Derivativexii

CHAPTER 1 – INTRODUCTION AND MOTIVATION1.1 Vehicles ExaminedInterest and application of ring-wing type unmanned aerial vehicles (UAVs) hasincreased within recent years. The military and commercial uses for a vehicle capable ofhovering and forward flight while remaining small and unmanned are countless. Militaryoperations on urbanized terrain (MOUT) have become an area of concern for the UnitedStates military within recent years. An increased need for policing and securingurbanized areas has become apparent with the conflicts in Iraq and Mogadishu. It is thistype of environment that dictates the especially challenging design of small-scale UAVs 1 .Because of the nature of MOUT, precise station-keeping requirements and overallincreased risk of collision with obstacles are important. Add to that the need for small andback-pack carried vehicles and it becomes apparent why the ducted fan design isappealing. The Defense Advanced Research Projects Agency (DARPA) advancedconcept technology demonstrator (ACTD) projects yielded submissions which includedthe Kestrel organic air vehicle (OAV) and i-Star micro air vehicle (MAV). Figure 1.1shows 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 lookingfor stable camera and surveillance platforms. Bridge inspection, traffic monitoring, andsearch and rescue in hostile environments all can benefit from use of a small unmannedvehicle capable of hovering flight. A unique class of small rotorcraft UAVs (RUAVs)incorporating all of the characteristics yields a small design with certain designdifficulties. These RUAVs possess the problem of making a small-scale vehicleunmanned along with the inherently unstable nature of rotorcraft dynamics. The ductedfan 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 withlongitudinal and lateral velocity, respectively.These ducted fan RUAVs have low inertias with most of the weight near thecenter of the vehicle. Their small size and weight make for stringent volumetric and massrestrictions. This leads to lower performance subsystems, especially sensors andactuators. High degrees of cross coupling due to strong gyroscopic effects are created bythe fast spinning propellers. The unconventional designs that have little or no knowledgebase established make physics based modeling difficult 2 . Most RUAV types include theability for a wide range of scales to be produced. Because of the relative simplicity ofconstruction, bigger and smaller vehicles alike can be produced. Usually shorter designcycles due to limited funding and demanding project requirements leave these vehicles inneed of accurate models early in the design cycle. Flight vehicles are available very earlyin the design sequence and make for easier flight test based identification. Thesecharacteristics combine to mandate accurate dynamic models. This research work willfocus on the comprehensive identification of these models.- 2 -

The vehicles examined within the scope of this research are all very similar indesign in that they consist of mainly a ducted fan utilized for lift. The vehicles examinedare shown in Figures 1.2 – 1.5. Although the mission profiles for all of these vehiclesvaries greatly, the two smaller scale surveillance vehicles, the Kestrel and the i-StarMAV are most representative of future military operations on urbanized terrain (MOUT)applications.Figure 1.2 – Hiller Helicopters Flying Platform – 1958Figure 1.3 – Aerovironment / Honeywell DARPA Phase I OAV – 2001- 3 -

Figure 1.4 – Trek Aerospace Solotrek Ducted Fan – 2001Figure 1.5 – Allied Aerospace i-Star MAV 9” Vehicle – 2003Figure 1.2 depicts the Hiller flying platform. This vehicle underwent some testingof the pitching moment characteristics of ducted fans back in 1958 3 . For this purpose itwas included in the study. Figure 1.3 shows the Aerovironment/Honeywell teamed efforttechnology demonstrator for DARPA. This vehicle was used for flight testing andparametric modeling as well as for the identification of sensor packages. Figure 1.4shows 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 thesevehicles were used for actuator identification, flight testing, and simulation as part ofwork for DARPA. Figure 1.6 shows a detailed view of the MAV.Figure 1.6 – Detailed view of 9” MAV DesignThe basic design of the ducted fan UAV incorporates a small COTS power plantthat is centered inside a duct. The flow of air in the duct is passed over stators for flowstraightening and over vanes which allow actuation to generate moments. Figure 1.7shows the vanes and stators on the bottom of the 9” MAV design.- 5 -

DuctStatorsLowerCenterBodyVanesCamera &Proximity SensorFigure 1.7 – MAV Stator and VanesGreat care is needed in specifying proper coordinate systems. It is not uncommonto 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. Thisis a gimbal-lock orientation and is best avoided for standard Euler sequences. Figure 1.8below illustrates the helicopter coordinate system used for this research and Figure 1.9shows it applied to the ducted fan. Unless otherwise specified, all derivatives andmention of moments are referred to in standard helicopter notation.- 6 -

Figure 1.8 – Helicopter Body Axes SystemY BodyX BodyZ BodyFigure 1.9 – Helicopter Body Axes System Applied to the Ducted FanAll moments and forces are represented as positive in the directions shown with momentsbeing applied in accordance with the positive right-hand rule.- 7 -

1.2 ScopeThis 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.CommandedInputsDigital FlightControlServo-ActuatorsBare-AirframeDynamicsVehicleResponseSensorsFigure 1.10 – Block Diagram of Basic DFCS ArchitectureIt can be seen that simply modeling the bare airframe and its dynamics is notenough to capture the whole nature of the vehicle. Due to the small size and limitedperformance actuators and sensor packages, these areas heavily influence the nature offlight. To accurately model the vehicle for flight control and simulation purposes, a moreexpanded diagram would be required. Figure 1.11 represents the identification effort ofthis research.- 8 -

GPSRate GyrosAccelerometersIMUInner-LoopClosuresSensors &TelemetryControl SystemOuter-LoopClosuresActuatorsPressureAltimeterSOURCES OF IDENTIFICATIONCIFERWind Tunnel or Other Empirical DataManufacturer and Bench DataVehicle DynamicsUnique PitchingMomentCharacteristicsRigid BodyDynamicsFigure 1.11 – Comprehensive Identification SchematicFigure 1.11 shows that a number of techniques (described in Chapter 2) applied to a largerange 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 competepicture of the operation of these ring wing UAVs.- 10 -

CHAPTER 2 – METHODS AND TECHNIQUES2.1 Identification MethodsA combination of the characteristics of these small RUAVs makes systemidentification an important and integral part of the design cycle. The need for a highperforming and robust control system is paramount to vehicle survivability and missionperformance. The design of the flight control system requires an accurate model across avariety of operating conditions and input frequencies.As previous work shows 2 , the use of Froude scaling the natural frequencies ofvehicles reveals the natural frequency would increase by the square root of a scale factormeasured in length. For example, making the vehicle 4 times smaller would increase thenatural frequency by 2. So, as vehicles become smaller, they require a higher bandwidthcontrol system. The need to operate at higher frequencies and in more of the availableflight envelope requires accurate models across large ranges of input frequencies. The useof frequency domain techniques lends itself very nicely to accomplishing this modelingchallenge.The NASA/Ames Research Center tool CIFER ® (Comprehensive Identificationfrom Frequency Responses) is primarily used to identify low order equivalent systemsand parametric state-space models required across broad frequency ranges. This tool isused extensively for the modeling of system dynamics in this effort.The reliance on small scale, low performance components and sensors makescharacterizing the errors and inconsistencies of components important. Without exclusive- 11 -

access to hardware inside of test vehicles, manufacturer data must be applied for errorand noise modeling. These tools and techniques combine to represent the comprehensiveidentification of these vehicles.2.2 CIFERCIFER provides an environment and set of programs that perform the varioussteps of the system identification process. Nonparametric modeling, in which no modelstructure or order is assumed; in the form of frequency responses represented as Bodeplots are first extracted with CIFER. This then allows for the parametric modeling.Transfer functions, low order equivalent (LOE) systems, or state-space models withstability and control derivative representation 3 are all used. The identification process canbe summarizes as 4 :1. Nonparametric frequency response calculation from time history datao Use of Chirp-Z Fast Fourier Transforms (FFT) and complex functions to generatethe frequency responses over multiple windows and samples2. Multi-input frequency response conditioningo Off axis control inputs’ contribution to on axis response is removed3. Multi-window averaging of frequency responseso Combination of different window sampling sizes4. Parametric models fit to frequency responseso Transfer function models fit to single input single output (SISO) systemso State-space models fit to all controls and states for parameter extraction5. Time domain verification of parametric modelsWhen complete, this procedure yields accurate models to be applied for a varietyof tasks. CIFER does require flight test time histories in which the vehicle’s modes havebeen excited by frequency rich inputs. It is not limited to vehicle dynamics either. Thistool can be used anywhere frequency domain analysis is needed. CIFER is a powerfultool that incorporates all of the tools to needed to model in the frequency domain.- 12 -

2.2.1 Flight Test TechniquesThere are a number of techniques that need to be applied to ensure that the flighttest of the vehicle is useful and applicable to system identification. While outside thescope of this research, it is sufficient to say that a combination of frequency richmaneuvers as seen in Figure 2.1 and validation maneuvers like doublets are required. Acombination of sensing and telemetry equipment is needed to measure both the inputfrom the actuators and the vehicle response. Access to the IMU and servo signals isrequired.1510RiseTimeSine Frequency SweepFallTimeControl Deflection (%)50-5-10ZeroDurationZeroDuration-150 15 30 45Time (seconds)Figure 2.1 – Sample Frequency Sweep Flight Test Command- 13 -

2.2.2 Bench Test TechniquesBench testing was used in cases where components were to be tested withoutactually installing them on the vehicle or testing them while in flight. This method wasprimarily applied to the testing of the servo actuators. The search for and classification ofactuators meeting the requirements of the vehicles made it impractical to install thenumerous actuators on the vehicle for testing. In this case, the actuators were tested whilehooked up to specific measuring equipment. Frequency domain analysis with CIFER wasapplied to determine the dynamic characteristics of the components.2.3 Manufacturer SpecificationsThe use of commercial off the shelf (COTS) devices and components for thebuildup of inertial measuring units (IMU) on the vehicles provides for manufacturerspecifications and ratings of component performance. This is important when directaccess of the components and hardware in the loop (HIL) bench testing is not available.The identification of the rate gyros, accelerometers, magnetometers, GPS receiver, andactuators all benefited from the provision of manufacturer identified errors andperformance specifications. In general, these specifications are slightly optimistic andreflect the specific measuring procedure applied by the manufacturer. Averages areusually presented by manufacturers while component-specific results are required insome modeling cases. Due to time constraints and availability of hardware for testing,- 14 -

manufacturer specifications are modeled and applied for the majority of telemetry andmeasuring equipment aboard the vehicles.2.4 Wind Tunnel TestsWind tunnel and other empirical data measured from the vehicles themselves playan important role as well. As previously mentioned, these ducted fan RUAVs exhibitunique corrective pitching moment characteristics due to large M u and L v derivatives.Wind tunnel studies help to better characterize this. The need to accurately characterizethe behavior of the ducted fan in translational velocities has put emphasis on accuratewind tunnel modeling. This type of physics-based modeling is used to draw someconclusions regarding the nature of the strong pitching and rolling moment created whenthe vehicle is in forward flight or in a cross-wind. It is also used to compare and correlatethe CIFER identified dynamics. In the case of the Solotrek vehicle, a wind tunnel was notactually used. Similar techniques and methodology was applied to the vehicle although itwas suspended on top of a moving pickup truck. Regardless, wind tunnel tests and datawere used to validate and compare trends for most of the vehicles studied.- 15 -

CHAPTER 3 – VEHICLE IDENTIFICATION3.1 Areas of IdentificationAs mentioned in Chapter 2, the comprehensive identification of these vehiclesrequires modeling and testing of the bare-airframe dynamics as well as all of the systemsand 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 techniquesoutlined in Chapter 2 will be applied to the bare-airframe of the vehicles with conclusionsbeing drawn for scaling and correlation. COTS actuators will then be analyzed for theredynamics and nonlinearities. Finally, all of the sensors and telemetry equipment used inobservation for the control system will be analyzed and modeled.- 16 -

3.2 Bare-Airframe IDThe bare-airframe dynamics are perhaps the most unique aspect of these vehiclesand the way they fly. A small inertia with a large concentration of mass near the center ofthe duct is inherent in the design. Combined with this, there is heavy coupling betweenpitch and roll due to the gyroscopic effects of the fast spinning propeller. All of thevehicles looked at utilize fixed pitch propellers. Figure 1.11 showed that the pitchingmoment characteristics together with the whole of the bare-airframe rigid body dynamicscharacterize the vehicle in uncontrolled flight.3.2.1 Aerovironment/Honeywell OAVThe goal of the CIFER ® system identification was to achieve an accurate Multi-Input Multi-Output (MIMO) state-space model to support flight control development andvehicle sizing for the DARPA Phase I test vehicle. The frequency range of interest was0.1 –10 rad/sec. Frequency response analyses show that the important dynamiccharacteristics in this frequency range are the rigid body dynamics.Examination of the eigenvalues of the identified model reveals low frequencyunstable periodic modes in both the pitch and roll degrees of freedom. Excellent matchesbetween the model and flight data for the on-axis time responses confirm the accuracy ofthe of the identified state-space dynamic model.- 17 -

The CIFER identification is based on a set of flight test data gathered while flyingthe prototype vehicle. The data was recorded at a nominal data rate of 23 Hz and includedvehicle rate and control mixer inputs. These are presented in Table 3.1.Table 3.1 – OAV Measured Parameters during Flight TestingParameterMeasured Valuep mixerq mixerr mixerpqrCMPACMQACMRAPPQQRRFrequency responses were generated with CIFER’s FRESPID tool from the testdata 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 toremove the effect of off-axis control inputs during the sweeps. COMPOSITE was used tocombine the four windows of data into a single response.The frequency ranges used for the dynamic model identification were the rangeswhen the coherence was good (values above 0.6). These frequency ranges are listed inTable 3.2 and are used in the state space model identification in DERIVID. Examinationof the off-axis frequency responses indicates no significant cross-couplings between thelongitudinal and lateral degrees of freedom. These couplings are therefore not included inthe 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)CMPA CMQA CMRAP 1-8 - -Q - 1-8 -R - - 3-10Because no significant cross-coupling between the longitudinal and lateral degrees offreedom was observed, the state-space form would be modeled after the transferfunctions. The identified transfer functions appear as Equations 3.1-3.2.ppmixer=18.68s(s + 0.0032)e−0.0477s(s + 2.0983)[−0.5761,1.7921](Equation 3.1)qqmixer=21.07s 2 e −0.0653 s(s +1.9496)[−0.7616,1.9349](Equation 3.2)rrmixer= 20.81e−0.0718 ss(Equation 3.3)The 3 rd order denominator forms known as a “hovering cubics” (Equations 3.4 and3.5) exemplify the dynamic modes for the longitudinal and lateral directions 5 . The controlderivatives for the state-space model were initially set as the free gain terms in thenumerators of the transfer functions. These values appear in Table 3.3.( )∆ = s + −Y − L s + Y L s− gL(Equation 3.4)3 2lateral−hover v P v P v( )∆ = s + X + M s + X M s− gM(Equation 3.5)3 2longitudinal−hover u q u q u- 19 -

Table 3.3 – OAV Control Derivatives Extracted from Transfer Function FitsDerivative ValueL δ 0.326M δ 0.343N δ 0.339A state space form comprised of a set of four matrices (F, G, H 1 , and H 2 ) knownas 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 theunits of rad and rad/sec). The three controls were p mixer , q mixer , and r mixer , as seen inEquation 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 multipliedthrough the H 1 matrix (Equation 3.13) and divided through the initial values of thecontrol derivatives (Table 3.3) in the G matrix (Equation 3.12). CIFER then tuned theparameters in the F and G matrices to match the state space model’s frequency responsesto those for the flight test data.x& = Fx + Gu(Equation 3.6)y = H1x+ H2x& (Equation 3.7)⎧ v ⎫⎪p⎪⎪rad⎪⎪φ⎪rad⎪ ⎪x = ⎨ u ⎬⎪q⎪rad⎪ ⎪⎪ θ ⎪⎪r⎪⎩ rad ⎭(Equation 3.8)- 20 -

⎧ p⎫⎪ ⎪y = ⎨q⎬⎪ r ⎪⎩ ⎭(Equation 3.9)u⎧pmixer⎪ ⎪= ⎨qmixer⎬⎪⎩rmixer⎫⎪⎭(Equation 3.10)⎡Yv0 g 0 0 0 0 ⎤⎢LvL 0 0 0 0 0⎥⎢P⎥⎢0 1 0 0 0 0 0 ⎥⎢⎥F = ⎢0 0 0 Xu0 −g0 ⎥⎢0 0 0 MuMq0 0 ⎥⎢⎥⎢0 0 0 0 1 0 0 ⎥⎢0 0 0 0 0 0 N ⎥⎣r ⎦⎡Yp0 0 ⎤mixer⎢L 0 0⎥⎢pmixer⎥⎢ 0 0 0 ⎥⎢⎥G = ⎢ 0 Xq0 ⎥mixer⎢ ⎥⎢0 Mq0mixer ⎥⎢0 0 0⎥⎢⎥⎢⎣0 0 Nr⎥mixer ⎦(Equation 3.11)(Equation 3.12)H1⎡0 57.3 0 0 0 0 0 ⎤=⎢0 0 0 0 57.3 0 0⎥⎢ ⎥⎢⎣0 0 0 0 0 0 57.3⎥⎦(Equation 3.13)It is worthwhile to note that many of the derivatives were set to zero in theidentification process. Because of the lack of acceleration data, the on-axis dampingparameters X u , Y v , and Z w were unable to be determined in the model and were thusremoved 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 modesare heavily reliant on the values of L v and M u , respectively. If these derivatives were theonly ones in the hovering cubic forms (Equations 3.4 and 3.5), the equations wouldreduce to the degenerate forms seen in Equations 3.14 and 3.15. These forms contain onereal and one complex root for negative values of L v and M u . These roots describe thedynamics of the system and show that L v and M u are the dominant terms required todepict the three modes.3∆lateral −hover= s − gLv(Equation 3.14)3∆longitudinal−hover= s − gMu(Equation 3.15)CIFER allows for a measure of merit, or cost, of the final model fit to thefrequency responses. Lower costs are better fits. The final model had an excellentaverage cost of 23.6. For the best possible fit, pure time delays were identified as0.04205, 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 functionfits. 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 toCIFER adjusting the value to make up for inconsistencies in the model or it is due to thepitch 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 theidentification. A comparison with the control derivatives extracted from the transferfunctions (Table 3.3) reveals very close matches.- 22 -

Table 3.4 – OAV DERIVID Identified Parameters and CertaintiesTable 5 shows the cost functions for the transfer functions. They were all very acceptable.Table 3.5 – OAV DERIVID Frequency Response CostsThe asymmetric design of the vehicle accounts for the difference in the valuesbetween L v and M u . Figure 1.3 depicts the fact that the OAV design has nacelles or cargopods making it asymmetric. The ratio of the identified values (L v : M u = 0.7510) reflectsthe relationship of the lateral and longitudinal inertias specified (I yy : I xx = 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 thecorresponding state values which identify the modes. The larger values indicate the stateswhich are dominant in the modes. A value of 1 in the eigenvector indicates which state isthe primary mode. From the eigenvectors and eigenvalues some interesting dynamics canbe noted.Table 3.6 – OAV Eigenvalues and Associated Eigenvectors of [F]Mode #(Aperiodic Yaw Subsidence)Mode #2(Lateral Low Frequency Periodic)Mode #3(Aperiodic Roll Subsidence)real imaginary real imaginary Real imaginary0.00E+00 0.00E+00 9.25E-01 -/+1.60E+00 -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+00P 0.00E+00 0.00E+00 P 1.00E+00 -/+1.13E-08 P 1.00E+00 0.00E+00PHI 0.00E+00 0.00E+00 PHI 2.70E-01 +/-4.68E-01 PHI -5.40E-01 0.00E+00U 0.00E+00 0.00E+00 U 0.00E+00 0.00E+00 U 0.00E+00 0.00E+00Q 0.00E+00 0.00E+00 Q 0.00E+00 0.00E+00 Q 0.00E+00 0.00E+00THETA 0.00E+00 0.00E+00 THETA 0.00E+00 0.00E+00 THETA 0.00E+00 0.00E+00R 1.00E+00 0.00E+00 R 0.00E+00 0.00E+00 R 0.00E+00 0.00E+00Mode #4(Aperiodic Pitch Subsidence)Mode #5(Longitudinal Low Frequency Periodic)real imaginary real imaginary-2.04E+00 0.00E+00 1.02E+00 -/+1.76E+00[zeta, omega][0.000E+00, 0.000E+00][zeta, omega][-.500E+00, 0.204E+01]V 0.00E+00 0.00E+00 V 0.00E+00 0.00E+00P 0.00E+00 0.00E+00 P 0.00E+00 0.00E+00PHI 0.00E+00 0.00E+00 PHI 0.00E+00 0.00E+00U 2.76E-01 0.00E+00 U -1.38E-01 -/+2.39E-01Q -3.55E-02 0.00E+00 Q 1.78E-02 -/+3.08E-02THETA 1.00E+00 0.00E+00 THETA 1.00E+00 +/-2.21E-08R 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 werecomplex pairs, and three real. These 7 eigenvalues depict 5 modes. Mode #1 is the yawmode which was modeled with no yaw damping, thus the value of 1 for the yaw rate state(r). Mode #2 is associated with the 2 nd order periodic denominator term in the hoveringcubic because of the high values for the lateral velocity (v) and roll rate (p) states. This isa low frequency unstable mode. Likewise, Mode #5 is from the 2 nd order term inlongitudinal hovering cubic. This is seen by the larger eigenvectors for the states oflongitudinal velocity (u) and pitch rate (q). The remaining eigenvectors identify the 1 storder, aperiodic subsidence modes for roll (Mode #3) and pitch (Mode #4). Theseeigenvalues 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 thefollowing frequency responses comparing the parametric state space model and the actualflight test data.- 25 -

Figure 3.1 – Roll rate response frequency domain verificationIt can be seen in Figure 3.1 that the roll rate model fits very well in the regions ofgood coherence. Only where there are dips in this signal to noise ratio does the modelstart 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 yieldcloser 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 statespacemodel in regions of good coherence as well. The coherence is the ratio of outputpower 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 theaccuracy of the state-space model for the pitch rate deteriorates quickly at lowerfrequencies.- 28 -

Figure 3.3 – Yaw response frequency domain verificationThe model revealed that there was no natural yaw damping for this vehicle. Theunstable hovering cubic is prevalent in the 1-3 (rad/sec) region. The fit was accurate athigher 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 duringflight testing. It can be seen that the on-axis responses have an excellent match for all 3controls. 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 aroll damping term, the control surface effectiveness term and Lv in the hovering cubicaccurately 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 iscaptured and represented in the state-space model very accurately.Figure 3.6 – Yaw response time history verificationFigure 3.6 shows the accuracy of the yaw degree of freedom. It stays accurateregardless of being modeled as the simple integrator form with no yaw damping.- 32 -

It can be seen that the Aerovironment Proposal prototype OAV was successfullymodeled with a state-space model. The identified model shows good agreement for boththe time and frequency responses. The identified system showed an unstable periodicmode 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 theinertias I yy to I xx . 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 notbe accurately identified. The identified transfer function modes closely match the modesof the identified state space dynamic model.After flight test was completed for the purposes of identification, the OAV designwas further analyzed in the wind tunnel. The vehicle was put into the Virginia TechStability Wind Tunnel by Techsburg, Inc. without the payload nacelles. A photograph ofthe 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, thevehicle was tested in a baseline configuration similar to that seen in Figure 1.3. From thetests, pitching moment information was extracted with varying wind speeds. Figure 3.8shows the results of that test.21.510.5M (ft-lbf)0-50 -40 -30 -20 -10 0 10 20 30 40 50-0.5-1-1.5-2u (fps)Figure 3.8 – Techsburg OAV Pitching Moment to AirspeedAs Figure 3.8 shows, there is a unique pitching moment created when the vehicleexperiences some wind velocity across the duct. This is illustrated by the slope of thetangent line depicted as a dotted line. In this case, the dimensional derivative about thehover condition is 0.011. This is a corrective moment for velocities below some criticalvelocity. A negative pitching moment is then created above this critical speed. In the caseof OAV as tested, this occurs at roughly 10 fps.- 34 -

3.2.2 Allied Aerospace MAVFlight test was performed on the MAV vehicle in a similar manner as wasdescribed in the previous section for the OAV. Table 3.7 below shows the physicalproperties for the vehicle as it was tested.Table 3.7 – MAV Physical PropertiesPhysical QuantityValueMass (slugs) 0.233C.G. (below duct lip - inches) 2.25Propeller Speed (rad/sec) 1884.0I xx (slug-ft^2) 0.021I yy (slug-ft^2) 0.021I zz (slug-ft^2) 0.021I prop (slug-ft^2) 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 theremoval of off-axis control contributions by using the CIFER tool MISOSA.- 35 -

30MAGNITUDE(DB)-10-50250PHASE(DEG)50-1501COHERENCE0.60.20.1 1 10 100FREQUENCY (RAD/SEC)F040P_COM_ABCDE_pcmd_pb - p/latF040P_COM_ABCDE_pcmd_qb - q/latF040P_COM_ABCDE_pcmd_rb - r/latFigure 3.9 – On and Off Axis MAV Roll Frequency ResponsesFigure 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-axisdirection. The roll rate frequency response has a good coherence from 0.5 to 12 rad/secand this portion of the frequency response is used in the identification.- 36 -

30MAGNITUDE(DB)-10-50250PHASE(DEG)50-1501COHERENCE0.60.20.1 1 10 100FREQUENCY (RAD/SEC)F040Q_COM_ABCDE_qcmd_qb - q/lonF040Q_COM_ABCDE_qcmd_pb - p/lonF040Q_COM_ABCDE_qcmd_rb - r/lonFigure 3.10 – On and Off Axis MAV Pitch Frequency ResponsesFigure 3.10 shows the pitch, roll and yaw rate frequency responses to pitchcontrol. As with the roll control responses, there is good coherence for the on-axisresponse, but no coherence for the off-axis responses. This would indicate that there isvery little cross-coupling and the pitch and roll responses are essentially uncoupled. It isuncertain 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 modelwas setup as shown in Equation 3.16.⎧u&⎫ ⎡Xu 0 −g 0 0 0⎤⎧u⎫ ⎡ 0 Xlon⎤⎪q&⎪ ⎢Mu Mq 0 0 Mp 0⎥⎪q⎪ ⎢0 Mlon⎥⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎢ ⎥⎪ & θ⎪⎢ 0 1 0 0 0 0⎥⎪⎪θ ⎪ ⎢ 0 0 ⎥⎧δlat⎫⎨ ⎬= ⎢ ⎥⎨ ⎬+⎢ ⎥⎨ ⎬⎪v &⎪ ⎢ 0 0 0 Yv 0 g⎥⎪v⎪⎢Ylat 0 ⎥ ⎩ δlon⎭⎪p&⎪ ⎢ 0 Lq 0 Lv Lp 0⎥⎪p⎪⎢Llat0 ⎥⎪ ⎪⎪ ⎪⎪ &⎩φ⎪⎭ ⎣⎢ 0 0 0 0 1 0⎦⎩ ⎥⎪φ⎭⎪⎣⎢ 0 0 ⎦⎥(Equation 3.16)The derivatives M p and L q result from the gyroscopic moments produced by therotating inertia of the propeller. This coupling is one of the unique aspects of thevehicle’s dynamics. Taking into account the angular momentum of the spinning propellerand dividing by the inertia of the total vehicle yields the moment produced by thegyroscopic effects. This is shown as equations 3.17 and 3.18.LqIprop= (Equation 3.17)IxxΩMpIprop= (Equation 3.18)IyyΩThe values for M p and L q therefore can be used for the determination of propellerinertia. This is possible because the rotational speed of the propeller remained mostlyconstant and the inertia of the vehicle changed negligibly due to fuel burned. This isuseful because the inertia of the small propeller while spinning is hard to measure in anytype of simple experiment. A time delay was also added to the dynamics to account fortransport delays in the electronics.- 38 -

A 0th/2nd order transfer function is included in the identification to take intoaccount the actuator dynamics. The form of this transfer function is as follows:ωn 2TF =s 2 + 2ζω n+ ωn 2The values of the damping and natural frequency of the actuator used wereobtained from bench tests of the actuator dynamics presented in section 3.3 for theAirtronics 94091 servo actuator running at nominally 5 volts. The natural frequency forthis 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 DerivativesCOUP02Derivativ e Param Value CR Bound C.R. (%) Insens.(%)X u -0.1090 0.04395 40.33 10.92M u 0.5014 0.03412 6.805 2.729M q 0.000 + ...... ...... ......M p 0.000 + ...... ...... ......Y v -0.1090 * ...... ...... ......L q 0.000 + ...... ...... ......L v -0.5014 * ...... ...... ......L p 0.000 + ...... ...... ......I pr op 0.000 + ...... ...... ......+ Eliminated during model structure determinationy Fixed value in model* Fixed derivativ e tied to a free derivativ eY v = 1.000E+00* X u ( COUP02 )L v =-1.000E+00* M u ( COUP02 )The value of the rotating inertia (I prop ) was insensitive in the identification andwas dropped from the list of active elements. This is because there was no goodcoherence 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 themodel to become zero as well.The control derivatives were identified as shown in Table 3.9.Table 3.9 - MAV Identified Control DerivativesCOUP02Derivativ e Param Value CR Bound C.R. (%) Insens.(%)X lon -0.2841 0.01692 5.955 2.058M lon -0.2343 0.01103 4.705 2.149Y lat 0.2495 0.01876 7.519 2.544L lat -0.1789 0.01056 5.902 2.614ø lat 0.06767 * ...... ...... ......ø lon 0.06767 4.599E-03 6.796 3.272* Fixed derivativ e tied to a free derivativ eø lat = 1.000E+00* ø lon ( COUP02 )Figure 3.11 shows the identified model’s roll and lateral acceleration responses for theroll sweep.4020p/latMagnitude(DB)200ay/latMagnitude(DB)0-20-20-40-40-60150100500-50-100-150Phase (Deg)100500-50-100-150-200Phase (Deg)0.80.61 Coherence0.80.61 Coherence0.40.20.1 1 10 100Frequency (Rad/Sec)Flight resultsCOUP02 - Identification Results0.40.20.1 1 10Frequency (Rad/Sec)Figure 3.11 – MAV Lateral Acceleration and Roll Rate Response to Roll Input- 40 -

Figure 3.12 shows the same for the longitudinal acceleration and pitch rate response topitch input.4020q/lonMagnitude(DB)200ax/lonMagnitude(DB)0-20-20-40-40-60150 Phase (Deg)100500-50-100-1500.80.61 Coherence-100 Phase (Deg)-150-200-250-300-350-4000.80.61 Coherence0.40.20.1 1 10 100Frequency (Rad/Sec)Flight resultsCOUP02 - Identification Results0.40.20.1 1 10Frequency (Rad/Sec)Figure 3.12 – MAV Longitudinal Acceleration and Pitch Rate Response to Pitch InputThe combination of Figure 3.11 and Figure 3.12 show that the identified modelagrees with the flight test data. There are some inconsistencies, but overall the costs ofthe fits were low and the model agrees with flight test results. The final identifiedparameters are outlined in Table 3.10.- 41 -

Table 3.10 – Final Flight Test Identified MAV DerivativesDerivativ e Param ValueX u -0.1090M u 0.5014M q 0.000 +M p 0.000 +Y v -0.1090 *L q 0.000 +L v -0.5014 *L p 0.000 +I pr op 0.000 +X lon -0.2841M lon -0.2343Y lat 0.2495L lat -0.1789ø lat 0.06767 *ø lon 0.06767+ Eliminated during model structure determinationy Fixed value in model* Fixed derivativ e tied to a free derivativ eM p = 8.971E+04* I pop ( PIT21 )L q =-8.971E+04* I pop ( PIT21 )Y v = 1.000E+00* X uL v =-1.000E+00* M uø lat = 1.000E+00* ø lonThe identification of the MAV vehicle benefited from also having wind tunneltests 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 ofvehicles. Both the larger 29” vehicle and smaller 9” vehicle were put into the wind tunnelwith 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 aswell as lateral, longitudinal, and vertical velocities.There were issues with the 9” wind tunnel results. To illustrate the wind tunnelmethod 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 asummary of the data collected for the pitching moment.i-Star-9 Pitching Moment Characteristics0.4Pitching Moment (ft-lb)0.20-0.2-0.4-0.6-0.8-10 20 40 60 80 100 120 140Shroud 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 isshown. The slope of this line represents the dimensional derivative M u . What is curioushere, and will be discussed in further detail in the next sections, is the nature of thepitching moment response to increases in speed. As the vehicle experiences a cross windin 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 derivativesto 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 -

Table 3.11 – MAV Wind Tunnel Identified Derivatives and Flight Test ResultsI-Star VehicleDerivative29”9”Wind TunnelFlight TestXu- 0.476 - 0.344 -0.1090Yv- 0.476(Fixed to X u )- 0.344(Fixed to X u )-0.1090(Fixed to X u )Zw- 0.349 - 0.212 n/aLv- 0.046(Fixed to –Mu)0.004(Fixed to –Mu)-0.5014(Fixed to –Mu)Lp0 0 0Mu0.046 0.003 0.5014Mq0 0 0Mpn/a n/a 0Lqn/a n/a 0Nw- 0.056 - 0.006 n/aNr0 n/a n/aXlon- 0.190 - 0.157 -0.2841Ylat0.156 0.123 n/aZcol- 0.012 - 0.264/100 n/aLlat- 0.218 - 0.418 n/aMlon- 0.387 - 0.548 -0.2343Nped0.669 0.555 n/aNcol-0.005 - 0.057/100 n/a- 44 -

Table 3.11 shows that all of the dimensional derivatives for the 29” vehicle arelarger than the 9” values. This is to be expected because the larger vehicle shouldexperience larger forces and moments to go with its increased mass and inertias. It alsoshows 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 M u . Wind tunnel testing revealed amuch smaller value for this critical derivative (0.003) than the flight test (0.5014).- 45 -

3.2.3 Trek Aerospace SolotrekAlthough nothing like the other vehicle’s examined, the Trek Aerospace (nowTrek Entertainment, Inc.) Solotrek does possess ducted fan technologies which arecommon to the MAV and OAV. One of the Solotrek’s ducted fans (Figure 1.4) wasinserted into the NASA Ames 7’ x 10’ wind tunnel at Moffett Field for aerodynamictesting. Forces and moments were recorded with various wind tunnel and fan speedswhile the ducted fan was mounted at 90° to the flow.The pitching moment was recorded with varying forward speeds and propellerRPM. The results of that test are shown in Figure 3.14. This data could be used fordetermination of dimensional pitching moment derivatives.200180Pitching Moment (ft-lbs)16014012010080601800 rpm2200 rpm2600 rpm3000 rpm402000 20 40 60 80 100 120Wind Tunnel Speed (fps)Figure 3.14 – Solotrek Wind Tunnel Test Results for Pitching Moment- 46 -

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 pitchingmoment derivative is obtained for this low speed condition. This is shown in Figure 3.14as dashed lines. The slopes of these lines are the dimensional derivatives. They aresummarized in Table 3.12. Figure 3.14 also shows that some critical velocity may existwhen the derivative will actually swing to negative. This is seen in the 1800 RPM case tobe around 70 fps.Table 3.12 – Pitching Moment Derivatives and Solotrek Fan SpeedFan Speed(rpm)Pitching Moment Derivative M u⎛ft-lb⎞⎜ ⎟⎜ft ⎟⎝ sec ⎠1,800 1.0342,200 1.3762,600 1.9333,000 2.589This wind tunnel testing was the extent of identification work completed for the Solotrekvehicle.- 47 -

3.2.4 Hiller Flying PlatformThe Hiller Flying Platform along with a dummy mannequin was attached to thetop of a truck and possessed equipment to measure moments and forces as it was drivenat Moffett Field in 1958. The results of the tests by Sacks 3 are the basis for the pitchingmoment identification.The primary data of concern is that of the pitching moment directly measuredwith increasing truck speed. The results of those runs are presented in Figure 3.15.450400350Pitching Moment (ft-lbs)3002502001501005000 20 40 60 80Speed (fps)Figure 3.15 – Hiller Flying Platform Pitching Moment DataThe truck test was performed with the fan running at the speed required to keepthe 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 isdone by approximating the man as a flat plate (6’ x 2’). While crude, this investigation ismerely to establish a trend with the pitching moment characteristics of ducted fanvehicles.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 FlowWith the approximation in size of the man, a drag coefficient of C D = 1.1 is foundfrom Figure 3.15. It follows that the drag of the man will vary with velocity as inEquation 3.19.D1 v2ρ AC2plate=D (Equation 3.19)- 49 -

It is known that the dummy was placed directly on top of the platform, so it isassumed that the drag will have a moment arm of 3 feet above the platform, or half theheight of the plate used to approximate the drag. This allows the determination ofmoment produced with airspeed due to the dummy. This is calculated and then subtractedfrom the actual data in Figure 3.15 to produce Figure 3.17.Pitching Moment (ft-lbs)450400350300250200150100500Hiller Test ResultsApproximate Dummy MomentApproximate Duct PitchingMomentLinear Fit for 20 knts0 20 40 60 80Speed (fps)Figure 3.17 – Results of Removing Dummy Moment from Hiller Platform TestIt 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 lineto determine the average slope from 0 to 20 knots (33.8 fps). This slope of this dashedline is the dimensional pitching moment derivative, M u .- 50 -

ft-lbMu PLATFORM= 5.11ftsecThis dimensional derivative is naturally much larger than the other values lookedat for the other vehicles. This makes sense because this is a much larger vehicle. It is apositive number for hover. However, it will go negative if the wind velocity reaches somecritical speed. In this case, that velocity is 55 feet per second. This follows the trend ofthe other vehicles.- 51 -

3.2.5 Vehicle Scaling Laws and ComparisonsIt becomes apparent that the ducted fans looked at all share some basiccharacteristics in one way or another. One of the main advantages of the RUAV designsmentioned in Chapter 1 is that these vehicles can hover. Hovering flight leaves thesevehicles highly susceptible to wind in station-keeping applications. Of particular interestis the derivative M u . 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 thenature of the vehicles and fully characterize and identify their flight, some time is neededto understand the pitching moment characteristics.In order to compare the pitching moment characteristics of the four vehicles, M umust be nondimensionalized to take into account the size of the vehicles, the propellereffects, and the ducts themselves. To do this, the nondimensional pitching momentdefinition for rotorcraft is applied:CM=M( Ω ) 2ρ A R RM ~ pitching momentρ ~densityΩ ~ blade rotation speed (rad/sec)R ~ duct radiusA ~ duct areaThis 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, welook at the derivative about zero to 20 knots airspeed for the vehicles. In other words, theslope of a line fit to the pitching moment vs. airspeed data is calculated for only the lowspeed 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 pitchingmoment. In fact, this nondimensionalization by the third power of the radius follows whatwas observed for ducted fans by Sacks 3 .This approximation of the way the pitching moment varies with duct size is usedto compare the three vehicles. The geometries of the vehicles are used here to determinethe dimensional and nondimensional parameters for comparison (Table 3.13). In the caseof the Solotrek fan, the four different fan speeds are presented.Table 3.13 – Pitching Moment Coefficient SummaryVehiclePitching Moment Derivative M u⎛ ⎞ft-lb⎜ ⎟⎜ft ⎟⎝ sec ⎠NondimensionalC MuFlying Platform 5.11 7.95 x 10 -5Wind Tunnel 0.011 1.09 x 10 -5OAVFlight Test 0.00643 6.52 x 10 -51,800 RPM 1.034 3.21 x 10 -52,200 RPM 1.376 2.86 x 10 -5Solotrek2,600 RPM 1.933 2.87 x 10 -53,000 RPM 2.589 2.90 x 10 -5Wind Tunnel 0.00323 1.30 x 10 -6i-Star 9”Flight Test 0.5014 2.01 x 10 -4i-Star 29” 0.11652 1.14 x 10 -6It is evident from Table 3.13 that the values are within the same order ofmagnitude and show positive speed stability for most of the vehicles and methods. Windtunnel values seem to differ from the other values. The largest values are seen with theflight test for MAV and wind tunnel results for OAV. The values for the different fanspeed for the Solotrek duct are all closely related, demonstrating that the same method isnondimensionalizing well for vehicles of varying prop speeds.- 53 -

Table 3.13 reveals that this method may not be accounting for the entirety ofdominant characteristics for ducted fan vehicles. This is seen in the way the Solotrekdiffers from the other smaller chord vehicles. To account for more specific geometries, amethod which better characterizes the propellers was also investigated. Thisnondimensionalization uses the chord and radius of the rotating propellers tonondimensionalize the pitching moment:CM=σ π=M( Ω ) 2ρσ A R RbcRM ~ pitching momentρ ~densityΩ ~ blade rotation speed (rad/sec)R ~ duct radiusA ~ duct areab ~ # of bladesc ~ mean blade chordTable 3.14 represents the results of this method.Table 3.14 – Pitching Moment with Blade Chord SummaryVehiclePitching Moment Derivative M u⎛ ⎞ft-lb⎜ ⎟⎜ft ⎟⎝ sec ⎠NondimensionalC MuFlying Platform 5.11 4.48 x 10 -4Wind Tunnel 0.011 1.03 x 10 -4OAVFlight Test 0.00643 6.15 x 10 -41,800 RPM 1.034 2.90 x 10 -42,200 RPM 1.376 2.58 x 10 -4Solotrek2,600 RPM 1.933 2.60 x 10 -43,000 RPM 2.589 2.61 x 10 -4Wind Tunnel 0.00323 2.45 x 10 -5i-Star 9”Flight Test 0.5014 3.80 x 10 -3i-Star 29” 0.11652 5.20 x 10 -5This method yields values similar to the previous methods in Table 3.13. Thenumbers here are more closely related and show that the nondimensionalization is an- 54 -

adequate way to characterize the different pitching moment characteristics for thesevehicles. It is can be seen that the derivatives for the i-Star class of vehicles differconsiderably from the other ducted fans analyzed. In the case of the wind tunnel resultsfor these two vehicles, the 9” value (2.45 x 10 -5 ) and the 29” value (5.20 x 10 -5 ) are of thesame order of magnitude, but an order lower than all of the other vehicles. This suggeststhat there may be something unique about the i-Star design, or that there was somethingunexplainable happening with the wind tunnel tests of the vehicles. Flight test revealedthat the 9” vehicle actually had a very large value for M u (3.80 x 10 -3 ). This is an orderlarger than the other vehicles, and a full two orders greater than the wind tunnel resultsfor the same vehicle. This could be due to the fact that M u was found to be so dominant inthe identification.To briefly summarize and conclude, all four of the ducted fan vehicles exhibitlikeness in pitching moment characteristics. The only anomaly seen is with the i-Starvehicle which shows relatively higher and lower C Mu values in comparison to the othervehicles and the method of identification.- 55 -

3.3 Servo Actuator IdentificationThe goal of the actuator test program was to measure a set of data that was used toidentify models of the actuator dynamic response characteristics. These actuator modelsinclude linear transfer functions of the input/output relationships as well as non-linearactuator properties such as actuator rate and position limits.The identification was performed using the CIFER. Linear 0 th /2 nd order transferfunctions capturing the actuator dynamics were identified. Testing allowed for thedetermination of the maximum angular rates and positions using linear curve-fitting ofthe square wave responses. An explanation of the construction of the actuator blockdiagrams built is also included. The actuators are a critical part of the flight controlsystem and it is important to have accurate models of the dynamics and limits of theactuators themselves. Individual blocks were created for each actuator corresponding toeach of the tested 5 volt and 6 volt conditions. This section also includes a time domainvalidation of the actuator models.The goal of bench testing the control surface actuators was to collect a set ofbench test data that will be used to identify the actuator dynamics. This test data was alsoused to determine the position and rate limits of the actuators. The significance of othernon-linear actuator properties, such as hysteresis and stiction, are also evaluated from thebench test data.The bench testing was carried out in accordance with CIFER flight test techniqueswherever possible. Five separate actuators from four manufacturers were tested. The- 56 -

actuators varied in size, weight, cost, and performance. The manufacturers’ specificationsare presented in Table 3.15. Figure 3.18 shows the relative sizes of the actuators tested.Table 3.15 – Manufacturer Specifications for Servo Actuators TestedMODEL NUMBER WEIGHT TORQUE RATE L W D(oz) (oz/in@ 4.8V) (deg/sec) (in) (in) (in)JR PROPO DS8417 2.03 82.0 600.0 0.73 1.52 1.32HITEC HS-512MG 0.80 42.0 352.9 0.39 1.33 1.18JR PROPO DS368 0.80 53.0 285.7 0.50 1.12 1.17AIRTRONICS 94091 0.32 18.0 500.0 0.44 0.91 0.87CIRRUS CS-10BB 0.19 7.0 1000.0 0.37 0.90 0.61Figure 3.18 – Actuators Tested and Relative SizesThe test apparatus was comprised of a rigid aluminum base stand with allowancesfor the actuators to fit inside without moving. For the smaller actuators, small woodenstrips were used to ensure rigid mounting. The actuator horns were connected to horns onpotentiometers using clevises. The potentiometers offered little to no load resistance. Themechanical apparatus can be seen in Figure 3.19.- 57 -

Figure 3.19 – Actuator Test Stand ApparatusA close up of the small Cirrus CS-10BB servo mounted on the test fixture in the woodenstrip 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 arenot the same length. This means that the deflection of the potentiometer horn will not bethe same as the deflection of the servo horn. All attempts were made to keep theselengths the same.Measurements of all the actuators and the various geometries accounting for theaforementioned differences were taken with precision calipers and recorded as seen in theschematic in Figure 3.21.Figure 3.21 – Schematic Detailing Linkage GeometryIt 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 GeometriesSERVOVOLTHORN-HORN(in)SERVOHORN(in)POTHORN(in)CENTER-CENTER(in)MINHORN SERVO INPUT POTMAXMIN(deg)MAX(deg)MIN(10 3 )MAX(10 3 )MINMAXSERVOw/POT @90°DS8417JR94091DS368HS12MGCS-10BB5 3.482 0.994 0.975 3.460 -40° 60° -40.741 34.174 -50 50 1810 3728 91.268°6 3.482 0.994 0.975 3.460 -40° 60° -40.741 34.174 -50 50 1809 3727 91.268°5 3.688 0.757 0.669 3.719 -60° 68° -46.419 30.644 -40 40 1851 3895 87.653°6 3.688 0.757 0.669 3.719 -60° 68° -46.419 30.644 -40 40 1854 3882 87.653°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°The most non-linear case was observed for the HS12MG where problems with thehorns also resulted in binding and interference at larger deflections. For this reason, themaximum commanded deflection was limited to 80% of the maximum actuatordeflection when testing this actuator.The potentiometer apparatus was located next to Allied Aerospace’s HILsimulation test stand. This utilized the ADC and DAC capabilities of the vehiclehardware to feed the actuators the Pulse Width Modulation (PWM) from the stimulusfiles prepared in accordance with CIFER flight test techniques.The two primary measurements required for the CIFER identification were thesweep commanded into the actuator and the potentiometer reading as a result of theactuator moving. Because of the nature of the recording equipment, calibration factorswere required to convert the input and output signals to degrees. These calibration factorswere determined using the geometries shown in Table 3.16 and are presented in Table3.17.- 60 -

Table 3.17 – Actuator Calibration Factors for Input and Output Channels to DegreesSERVOVOLTAGECALIBRATION FACTORIN ChannelOUT Channel(degrees/unit input) (servo deg/POT units)DS8417JR94091DS368HS12MGCS-10BB5 0.000749 0.03916 0.000749 0.03915 0.000963 0.03776 0.000963 0.03805 0.000811 0.0416 0.000811 0.04095 0.000829 0.04166 0.001036 0.04925 0.000836 0.04116 0.000836 0.0411The hardware fed signals from -50,000 to 50,000 to the servos and recordedpotentiometer deflection from roughly 1500 to 4500. The calibration factors in Table 3.17relate these to degrees of command and deflection of the servo. They are a result of thegeometries and readings for each actuator-voltage combination tested.Data was recorded at 50 Hz and there was no filtering of the input and outputchannels. An unidentified glitch was observed in the output signal and showed itself as asignal spike at roughly every 5 samples (0.1 sec). This was evaluated and it wasdetermined to be minor in identifying the dynamics. With that exception, there was verylittle noise in the signals.Frequency sweep actuator commands were used to generate test data from whichfrequency responses of control surface response due to actuator command could beidentified. From these frequency responses, transfer functions of the actuator dynamicswere extracted. The non-linear effects, such as rate and position limits were identified byusing a square-wave command.- 61 -

The time histories of the actuator command signals were computer generatedusing the frequency sweep code that was described for the flight test frequency sweepmaneuvers. The inputs to this code specify the various parameters of the frequencysweep. These parameters are shown in Table 3.18 for the sweeps used in the tests.Table 3.18 – Frequency Sweep Used for all ActuatorsDescription: Units: Value:Control axis - 1Total duration of sine sweep sec 30Duration of zero signal sec 2Time for signal fade-in sec 3Time for signal fade-out sec 1Signal sample rate Hz 50Minimum frequency of sweep Hz 0.1Maximum frequency of sweep Hz 10.0Filter cut-off frequency Hz -1Amplitude of control input % 10, 50, (80),100Maximum allowable amplitude % 100Noise random flag - -1The signal amplitudes used to drive the actuators during the frequency sweep testswere 10, 50 and 100% of the maximum pulse width amplitude and was generated withcomputer code. In the case of some of the smaller actuators (DS368 & HS512MG), the100% input was brought down to 80% because of clevis interference at higherdeflections. White noise is not required in the command signals for actuator testing. Acut-off filter could be included to ensure that the frequency content of the commandsignal does not go beyond a maximum frequency. This is not required for bench testingand 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 computercode.A 100% square wave was used to drive the actuators to their position limits. A50% 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 ParametersDescription: Units: Value:Total duration of wave sec ~30Signal sample rate Hz 50Amplitude of positive step % max 50, 100Positive step hold time sec 0.5Amplitude of negative step % 50, 100Negative step hold time sec 0.5The amplitude of the actuator signal is the percentage of the maximum pulse widthamplitude 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 Input50403020Deflection (deg)1000 5 10 15 20 25 30-10-20-30-40-50Time (sec)Potentiometer Response50403020Deflection (deg)1000 5 10 15 20 25 30-10-20-30-40-50Time (sec)6040Deflection (deg)2000 1 2 3 4 5 6 7 8 9 10-20-40-60Time (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, modelnumber, and conditions of the actuator tested. The CIFER case name for the frequencysweep cases is also listed if identification was completed.Table 3.20 – Actuator Bench Test MatrixCIFERNAMETEXT FILE NAMEMODELNUMBERVOLTAGEAMPLITUDE(% max)SAMPLESRECORDTIME(sec)DS8417_TEST_RUN.TXT JR PROPO DS8417 5 100 n/a n/aDS8417_1 DS8417_100_5.TXT JR PROPO DS8417 5 100 1478 29.56DS8417_2 DS8417_100_5_2.TXT JR PROPO DS8417 5 100 1464 29.28DS8417_3 DS8417_50_5.TXT JR PROPO DS8417 5 50 1461 29.22DS8417_4 DS8417_10_5.TXT JR PROPO DS8417 5 10 1452 29.04DS8417_5 DS8417_100_6.TXT JR PROPO DS8417 6 100 1466 29.32DS8417_6 DS8417_50_6.TXT JR PROPO DS8417 6 50 1464 29.28DS8417_7 DS8417_10_6.TXT JR PROPO DS8417 6 10 n/a n/aDS8417_100_5_square.TXT JR PROPO DS8417 5 100 1067 21.34DS8417_50_5_square.TXT JR PROPO DS8417 5 50 1392 27.84DS8417_10_5_square.TXT JR PROPO DS8417 5 10 1123 22.46DS8417_100_6_square.TXT JR PROPO DS8417 6 100 1380 27.60HS512MG1 HS-512MG_100_5.TXT HITEC HS-512MG 5 100 1462 29.24HS512MG2 HS-512MG_50_5.TXT HITEC HS-512MG 5 50 1482 29.64HS512MG3 HS-512MG_10_5.TXT HITEC HS-512MG 5 10 1496 29.92HS-512MG_100_5_square.TXT HITEC HS-512MG 5 100 1125 22.50HS512MG4 HS-512MG_100_6.TXT HITEC HS-512MG 6 100 1464 29.28HS512MG5 HS-512MG_50_6.TXT HITEC HS-512MG 6 50 1466 29.32HS512MG6 HS-512MG_10_6.TXT HITEC HS-512MG 6 10 1450 29.00HS-512MG_100_6_square.TXT HITEC HS-512MG 6 100 1404 28.08HS-512MG_80_6_square.TXT HITEC HS-512MG 6 80 917 18.34DS368_1 DS368_100_5.TXT JR PROPO DS368 5 100 1459 29.18DS368_2 DS368_50_5.TXT JR PROPO DS368 5 50 1471 29.42DS368_3 DS368_10_5.TXT JR PROPO DS368 5 10 1462 29.24DS368_100_5_square.TXT JR PROPO DS368 5 100 1253 25.06DS368_4 DS368_100_6.TXT JR PROPO DS368 6 100 1458 29.16DS368_5 DS368_50_6.TXT JR PROPO DS368 6 50 1462 29.24DS368_6 DS368_10_6.TXT JR PROPO DS368 6 10 1474 29.48DS368_100_6_square.TXT JR PROPO DS368 6 100 1241 24.8294091_1 94091_80_5.TXT AIRTRONICS 94091 5 80 1467 29.3494091_2 94091_50_5.TXT AIRTRONICS 94091 5 50 1458 29.1694091_3 94091_10_5.TXT AIRTRONICS 94091 5 10 1467 29.3494091_80_5_square.TXT AIRTRONICS 94091 5 80 1417 28.3494091_4 94091_80_6.TXT AIRTRONICS 94091 6 80 1463 29.2694091_5 94091_50_6.TXT AIRTRONICS 94091 6 50 1475 29.5094091_6 94091_10_6.TXT AIRTRONICS 94091 6 10 1440 28.8094091_80_6_square.TXT AIRTRONICS 94091 6 80 1519 30.38CS10BB_1 CS-10BB_100_5.TXT CIRRUS CS-10BB 5 100 1466 29.32CS10BB_2 CS-10BB_50_5.TXT CIRRUS CS-10BB 5 50 1469 29.38CS10BB_3 CS-10BB_10_5.TXT CIRRUS CS-10BB 5 10 1475 29.50CS-10BB_100_5_square.TXT CIRRUS CS-10BB 5 100 1198 23.96CS10BB_4 CS-10BB_100_6.TXT CIRRUS CS-10BB 6 100 1463 29.26CS10BB_5 CS-10BB_50_6.TXT CIRRUS CS-10BB 6 50 1476 29.52CS10BB_6 CS-10BB_10_6.TXT CIRRUS CS-10BB 6 10 1467 29.34CS-10BB_100_6_square.TXT CIRRUS CS-10BB 6 100 1145 22.90- 65 -

Three of CIFER’s subprograms were utilized to perform the identification.FRESPID (frequency response identification) was used to generate multiple responses atdifferent 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 theactuator dynamics from the COMPOSITE results. These linear models are required forthe optimization of the control system using CONDUIT. A strong effect of the nonlinearcharacteristics on the responses was observed. Correlation to previous studies onnonlinear actuators is provided which explains some of the inaccuracies in the linearmodel.Following the test matrix yielded 5 actuators with 2 different voltages and 3different sweep magnitudes. These 30 cases were processed in CIFER and frequencyresponses were generated within FRESPID. A single sweep was used for each of theconditions. Five frequency responses were generated for each case based on window sizefor 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. TheCOMPOSITE response is the response used for the transfer function fitting.The responses were analyzed for regions of best coherence in order to ensurefidelity of the responses to be used for linear model fitting within NAVFIT. Plots for eachof the FRESPID generated frequency responses for each case are presented at the end ofthis memo in Appendix B.Table 3.21 shows the responses used for identification and the frequency rangeswhere 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 inTable 3.21.Table 3.21 – Actuator NAVFIT Frequency Ranges for CIFER CasesCIFER NAME MODEL NUMBER VOLTAGENAVFIT FREQ RANGEAMPLITUDE(rad/sec) (rad/sec)(% max) MIN MAXDS8417_1 JR PROPO DS8417 5 100 1 35DS8417_2 JR PROPO DS8417 5 100 1 35DS8417_3 JR PROPO DS8417 5 50 1 45DS8417_4 JR PROPO DS8417 5 10 - -DS8417_5 JR PROPO DS8417 6 100 1 35DS8417_6 JR PROPO DS8417 6 50 1 35DS8417_7 JR PROPO DS8417 6 10 - -HS512MG1 HITEC HS-512MG 5 100 1 25HS512MG2 HITEC HS-512MG 5 50 1 35HS512MG3 HITEC HS-512MG 5 10HS512MG4 HITEC HS-512MG 6 100 1 35HS512MG5 HITEC HS-512MG 6 50 1 35HS512MG6 HITEC HS-512MG 6 10 - -DS368_1 JR PROPO DS368 5 100 1 25DS368_2 JR PROPO DS368 5 50 1 30DS368_3 JR PROPO DS368 5 10 - -DS368_4 JR PROPO DS368 6 100 1 25DS368_5 JR PROPO DS368 6 50 1 30DS368_6 JR PROPO DS368 6 10 - -94091_1 AIRTRONICS 94091 5 100 1 3594091_2 AIRTRONICS 94091 5 50 1 3594091_3 AIRTRONICS 94091 5 10 - -94091_4 AIRTRONICS 94091 6 100 1 3594091_5 AIRTRONICS 94091 6 50 1 3594091_6 AIRTRONICS 94091 6 10 - -CS10BB_1 CIRRUS CS-10BB 5 100 1 35CS10BB_2 CIRRUS CS-10BB 5 50 1 35CS10BB_3 CIRRUS CS-10BB 5 10 - -CS10BB_4 CIRRUS CS-10BB 6 100 1 35CS10BB_5 CIRRUS CS-10BB 6 50 1 35CS10BB_6 CIRRUS CS-10BB 6 10 - -It became apparent after generating responses for the 10% max deflection casesthat the signals were not adequate for system identification work. Although the coherencewas 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 0 th /2 ndforms were not valid, these responses were rejected from system identification results.Table 3.22 includes the complete NAVFIT results for natural frequency and dampingratio for each case. The NAVFIT cost function result for each case is also presented.Table 3.22 – Actuator NAVFIT Results for all CasesCIFER NAME MODEL NUMBER VOLTAGEAMPLITUDE(% max)ζω n(rad/sec)τ(sec)COSTDS8417_1 JR PROPO DS8417 5 100 0.4986 20.4054 0.0110 37.804DS8417_2 JR PROPO DS8417 5 100 0.5074 20.3759 0.0067 33.020DS8417_3 JR PROPO DS8417 5 50 0.5166 33.0836 0.0055 17.862DS8417_4 JR PROPO DS8417 5 10 - - - -DS8417_5 JR PROPO DS8417 6 100 0.5034 22.5944 0.0054 26.364DS8417_6 JR PROPO DS8417 6 50 0.6556 50.9824 0.0155 1.324DS8417_7 JR PROPO DS8417 6 10 - - - -HS512MG1 HITEC HS-512MG 5 100 0.5472 13.9439 0.0079 26.653HS512MG2 HITEC HS-512MG 5 50 0.5606 21.9789 0.0121 11.352HS512MG3 HITEC HS-512MG 5 10HS512MG4 HITEC HS-512MG 6 100 0.5352 16.4602 0.0132 42.696HS512MG5 HITEC HS-512MG 6 50 0.5243 22.9373 0.0127 12.813HS512MG6 HITEC HS-512MG 6 10 - - - -DS368_1 JR PROPO DS368 5 100 0.5920 11.2955 0.009 65.993DS368_2 JR PROPO DS368 5 50 0.5136 16.4615 0.0042 32.826DS368_3 JR PROPO DS368 5 10 - - - -DS368_4 JR PROPO DS368 6 100 0.5168 12.1254 0.006 59.259DS368_5 JR PROPO DS368 6 50 0.5039 18.0762 0.0117 31.008DS368_6 JR PROPO DS368 6 10 - - - -94091_1 AIRTRONICS 94091 5 100 0.5446 17.429 0.0054 27.49094091_2 AIRTRONICS 94091 5 50 0.5108 21.3608 0.0048 9.59394091_3 AIRTRONICS 94091 5 10 - - - -94091_4 AIRTRONICS 94091 6 100 0.5302 18.8425 0.010 19.96494091_5 AIRTRONICS 94091 6 50 0.5489 23.4087 0.0073 13.92294091_6 AIRTRONICS 94091 6 10 - - - -CS10BB_1 CIRRUS CS-10BB 5 100 0.5345 18.3889 0.0036 19.862CS10BB_2 CIRRUS CS-10BB 5 50 0.5019 26.2309 0.0045 6.294CS10BB_3 CIRRUS CS-10BB 5 10 - - - -CS10BB_4 CIRRUS CS-10BB 6 100 0.5273 21.0582 0.0077 16.154CS10BB_5 CIRRUS CS-10BB 6 50 0.5192 29.0844 0.0069 5.641CS10BB_6 CIRRUS CS-10BB 6 10 - - - -- 68 -

Table 3.22 shows that for the first actuator tested, the same 100% sweep at 5Vwas applied. The NAVFIT results for these same sweeps show nearly identical results.This was done to ensure repeatability and consistency of the test. The frequencyresponses and the transfer function fits are presented by CIFER name (referenced inTable 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 higherdamping ratio. This is because the smaller deflections allow the actuator to reach higherfrequencies before the rate limit is reached. This is evident in the frequency responses forall the actuators as illustrated for the HS512MG in Figure 3.23.- 69 -

6 Volts5 VoltsFigure 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 numberscannot be taken as the pure transport delay because some of the delay is being absorbedin 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 andhigher natural frequencies for all of the actuators.The costs for each fit seem to be very reasonable and show that the second ordermodel is quite valid for the responses exhibited by all of the actuators. The only responsethat may have been the subject of error is the 50% deflection sweep at 6V on the DS8417which shows a really low cost and a noticeably higher natural frequency than the rest ofthe responses. This is due to the fact that for the frequency range analyzed, the magnitudedid not break sharply. This left a relatively flat response for which a second order formwas easily fitted.More constrictive frequency ranges were chosen to study the effects this rangehad on the fit presented by NAVFIT. It was determined that the frequency range had littleeffect on the transfer function fit unless it went below the break frequency. As thefrequency responses show, all of the responses demonstrate clean breaks at their naturalfrequencies and 0-dB gain at low frequencies with the exception of the 50% deflection onthe DS8417 at 6V.The primary tool used for the determination of the nonlinear properties of theactuators was the square wave shown in Figure 3.22. The square wave commanded a nearinstantaneous change from maximum to minimum deflection. Using the geometriccalibration 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 receivinga 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 thechange 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 responsedue to the dynamics of the actuator that were ignored. Most fits actually started at up to0.1 second after the commanded change. Figures 3.24 and 3.25 illustrate this for the JRPROPO DS8417 for full 100% deflection at 6V. The response data in Figure 3.24 showsthe measurement spike every fifth data point that was described earlier. The presence ofthis spike does not have a significant effect on the identification results.6040ResponseCommand2000 0.2 0.4 0.6 0.8 1 1.2-20-40-60Figure 3.24 – Sample Square Wave Response- 72 -

400035003000250020001500y = -13429x + 9284.1R 2 = 0.9958100050000.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58Figure 3.25 – Linear Fit for Max Rate DeterminationThe square wave commanded a maximum and minimum deflection. This isshown in Figure 3.24 as 50 and -50 degrees, respectively because the actual limits werenot known during testing.Figure 3.24 clearly shows that the servo was saturated. It also shows the transientresponse. The figure shows how the maximum positions can be read from the plot. It isasymmetric 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 tobe symmetric about zero degrees.Figure 3.25 shows how nicely a linear curve fit could be accomplished. By usingthe slope from that line and applying the calibration factor from Table 3.17, themaximum rate in degrees/second was found. This was repeated for each of the actuatorsto yield the final nonlinear characteristics for the actuators as shown in Table 3.23.- 73 -

Table 3.23 – Actuator Nonlinear Characteristic SummarySERVOVOLTSPOSITION(deg)RATE(deg/sec)MIN MAXDS8417JR94091DS368HS12MGCS10BB5 -437.2 437.1+/- 37.45716-524.5 534.95 -402.3 422.0+/- 38.53156-435.2 467.85 -219.8 220.5+/- 40.57446-264.6 265.75 -328.1 317.6+/- 41.43946-376.5 404.75 -474.7 442.1+/- 41.79926-567.2 536.4It is apparent from Table 3.23 that there are different rates for different directionson 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 lighterCS-10BB showed to be more asymmetric. Many factors can contribute to thisasymmetry. Because there is a motor with an armature inside, the brushes on the motormay 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 wassaturated to fit a valid linear curve at this deflection. For this reason, all results used a full100% 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 anyidentifiable stiction or hysteresis. Although they undoubtedly exist, the methodsemployed here did not reveal any substantial findings. More accurate potentiometers,- 74 -

higher data rates, and tighter tolerances on the test equipment may have revealed thesenonlinearities.It should be mentioned that observing the smaller actuators like the CS-10BB and94091 revealed that at very high frequencies the actuator demonstrated output not directlycorrelated to the input. This sporadic output is visible in the time responses shown inFigures 3.26 and 3.27.504030Deflection (deg)201000 5 10 15 20 25 30-10-20-30-40-50Time (sec)Figure 3.26 – CS-10BB at 5V Time History Illustrating Erratic Response at HighFrequency403020Deflection (deg)1000 5 10 15 20 25 30-10-20-30-40-50Time (sec)Figure 3.27 – 94091 at 6V Time History Illustrating Erratic Response at High Frequency- 75 -

Deflection (deg)504030201000 5 10 15 20 25 30-10-20-30-40-50Time (sec)Figure 3.28 – 94091 at 5V Time History not Showing Erratic ResponseInterestingly, the 94091 at 5V did not display this asymmetric response to theextent that the 6V case did (Figure 3.28). The coherence for these actuators in thisfrequency range still remains relatively high, indicating that the output is correlated withthe input. What the time histories reveal though is that the oscillations do not occur about0°. These smaller actuators have issues tracking the input symmetrically at highfrequencies. The nature of the sporadic response was observed in all of the actuators tosome extent, but not more so than in the 94091 at 6V and CS-10BB at 5V and 6V. Theerrors in tracking the input signal at high frequencies associated with these smallactuators must be a consideration when selecting an actuator for high bandwidthapplications.It is known that the nonlinear characteristics of the actuators, especially ratelimiting, will have an effect on the accuracy of the linear transfer function models. It wasobserved 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 characterizethe response. This was investigated further in an attempt to add fidelity to the model. TheDS8417 showed the worst correlation between the linear model response and the- 76 -

esponse and the response obtained from test data. Figure 3.29 shows the phase of theDS8417 at 5V with a 100% sweep.Figure 3.29 – DS8417 Frequency Response Illustrating Mismatch in Linear ModelAs Figure 14 illustrates, the phase is not fully characterized by the second orderfit at frequencies beyond 10 rad/sec. The mismatch shows itself as more time delay rolloff at higher frequencies.Previous work completed by STI during investigation of PIOs due to nonlinearvehicle characteristics 5 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 ofthe DS8417 at 5V is presented as Figure 3.30.Chirp Input5040Chirp InputActuator Response3020Deflection (deg)10020 20.5 21 21.5 22-10-20-30-40-50Time (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 fitthe data in the frequency domain for a given frequency range, but not for all frequencyranges. For a more accurate match over broader frequency ranges, an exact sinusoidaldescribing function is required. To compute this function, the Fourier integrals are firstcomputed 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) beingeither the input or output periodic forcing function. For our case these are both sinusoidswith period P, so the input describing function’s a 1 term is always zero. Using these tocharacterize the magnitude and phase of the describing functions yields the followingrelationships 5 .In these equations, δ c is the actuator deflection commanded and δ is the actualoutput after rate saturation. Having these open frequency domain representations allowsus 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 benchtest data.That method was effectively applied for the DS8417 actuator by applying a ratelimiter on the identified models within Simulink and using FRESPID to then generate afrequency 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 TestFigure 3.31 shows that as expected, the addition of the rate limiter in the modelcauses the response to break sooner; this yields a lower natural frequency. The additionof the rate limit increases the magnitude accuracy of the model over the linear NAVFITresult. 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 ofmatching the bench test data better. This is most likely due to the fact that othernonlinearities exist and become more influential at higher frequencies. More accurate testequipment 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 thefrequency response quotient of the rate saturated response to the identified linearNAVFIT model to be generated. NAVFIT was then used to try to characterize the errorwith a linear transfer function. This resulted in the responses shown in Figure 3.33 for theerror 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 32degrees at approximately 14 rad/sec. However, the maximum error is an importantparameter because it is directly related to the ratio of linear and nonlinear rise times ( t ˆR) 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 thatthe inclusion of the rate limit in the model accurately models the magnitude as was seenpreviously in Figure 3.31. The loss of phase fidelity starting at around 12 rad/sec is arelatively high frequency for control system design and shows that the linear model withthe rate limiter would be fairly accurate for simulation purposes.- 84 -

Results from STI utilizing the exact describing function yielded Figure 3.34. Thefrequency is normalized by the actuator bandwidth ( ωn) to represent the ratio ˆnω . Thisgenerates the family of curves relating the difference in phase to the ratio of linear risetime to nonlinear rise time ( tˆRtRL= ).tRNLFigure 3.34 – Rise Time Ratio Phase Lag RelationshipFor the maximum phase error of 32 degrees seen in Figure 3.33, Figure 3.34predicts a rise time ratio of t ˆR= 0.17 at a normalized frequency of ˆnω = 0.6. Looking atthe step response of the linear NAVFIT results without rate limiting, we see the rise timeto betR= 0.08 sec, as shown in Figure 3.35.L- 85 -

Figure 3.35 – Rise Time for Linear Model of DS8417 at 5VDetermining the rise time from the nonlinear, rate-limited model wasaccomplished by analyzing the square wave time responses and found to betR= 0.192NLsec. Comparing this rise time to the linear rise time reveals a ratio of t ˆR= 0.38. Althoughnot exactly the predicted 0.17, the only nonlinearity that was included in this model wasthe rate limiting.As mentioned previously from Figure 3.34, the predicted maximum difference inphase lag would be expected at a normalized frequency of ˆnω = 0.6. The bandwidth of theDS8417 is approximately ωn= 20 rad/sec (Table 8). The error function in Figure 3.33shows the maximum additional lag to occur at 14 rad/sec. This corresponds to anormalized frequency of ˆnω = 0.7. This is very close to the predicted frequency where theadditional 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 factthat all the natural frequencies and damping ratios were higher for the 50% sweeps thanthe 100% ones. Plotting this trend as in Figure 3.36 shows that as expected, the naturalfrequency drops with increased sweep amplitude. This trend is also evident in Figure 16where the addition of the rate limit effectively causes the response to break sooner andillustrates how much an effect the rate limit has on the response.1.210.80.60.40.200% 20% 40% 60% 80% 100%Amplitude of Sweep (% of max deflection)Figure 3.36 – Sweep Amplitude and Natural Frequency with Rate LimitingThe result of the comparison to the STI data is that the general trends of the dataare correct. The addition of the rate limit in the model effectively corrects the magnitudeof the response. According to the error function in Figure 3.33, the model should losefidelity in the phase of the response around 14 rad/sec where the error is at a maximum.- 87 -

With the nature of the linear and nonlinear characteristics of the actuatorsdetermined modeling and validation of the actuators was performed. The modeling wasdone in a way which could be used for control system optimization and simulation. Themodel is built within Simulink and includes the linear 0 th /2 nd transfer function form andthe identified nonlinear characteristics of rate and position limits. The validation of themodels is accomplished in the time domain by feeding the models the same chirp inputused in the test and comparing the responses to bench test responses.With the actuator models identified, Simulink block diagrams were created to beused in the inner loop block diagrams for MAV control system optimization andsimulation. The blockset can be seen in Figure 3.37.Figure 3.37 - Simulink Actuator BlocksetEach block is configurable when double clicked, but reflects the CIFER identifiedresults for each voltage based on the results presented in Table 3.22 for the dynamics and- 88 -

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 thecondition and max deflections being used for the application, as seen in Figure 3.38.Figure 3.38 – Configurable Actuator ParametersThe physical characteristics of the actuator from the manufacturer are alsopresented in the header of the block parameters dialogue. A Matlab (HTML-based) helpfile is also accessible through the parameters dialogue.- 89 -

The blocks are all masks with the same underlying block diagram as shown in Figure3.39.Figure 3.39 - 2 nd Order Actuator Dynamics behind MaskIt 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 forthis parameter, but it can still be specified within the parameters dialogue.- 90 -

Verification of the identified models was accomplished by using the same sweepinput fed into the actuators during bench testing. A typical result is shown for the DS8417in Figure 3.40 with all actuator model validations appearing at the end of this memowithin Appendix D.Figure 3.40 – DS8417 at 5V Time Domain ValidationFigure 3.40 shows that the response has been captured in the model whichincludes the rate and position limits. The non-linearties not accounted for and theasymmetric response, begin to show as a loss of fidelity beyond approximately 13 ~ 16- 91 -

ad/sec. The total deflection of the actuator and phase are not fully modeled at thesehigher frequencies, as seen when zooming in on the response in Figure 3.40. From theerror function presented previously in Figure 3.33, we see that the maximum differencein phase shows itself at 14 rad/sec (2.2 Hz). This corresponds to what is seen here in thetime domain. Any accurate modeling beyond 5 Hz would require more accurate test anddata 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 toidentify models of the actuator dynamic response characteristics. These actuator modelsinclude linear transfer functions of the input/output relationships as well as non-linearactuator properties such as actuator rate and position limits.The responses of the actuators were modeled by using CIFER to generatefrequency responses and then fit 0 th /2 nd order transfer functions. The position and ratelimits of the actuators were determined by analyzing the response to the square waveinput. It was found that the phase characteristics for some of the actuators were not fullycaptured with the linear models. Comparing to known theory revealed the extent to whichthe maximum rate of the actuator affects the response. The inclusion of the rate limit inthe model significantly improved the accuracy of the magnitude but some differences arestill seen at higher frequency due to nonlinear effects that are not included.The identified actuator dynamics and nonlinear rate and position limits were usedto construct a set of Simulink actuator blocks. These blocks are customizable and includethe manufacturer specifications. A time domain validation of the models showed them tobe accurate up to the highest frequency range of interest for flight control work. Whencomparing the manufacturer listed rate limit specifications (3.15) with those obtained- 92 -

from testing (3.22), it was found that the true actuator rate limits were lower than thosequoted. All of the actuators demonstrated increased bandwidth, damping ratios, and ratelimits when powered at 6V instead of 5V. The smallest and fastest actuators have issuestracking the input at high frequencies. The CS-10BB at 5V and 6V and the 94091 at 6Vexhibited these characteristics. Based on bandwidth, maximum rate, weight, and size theAirtronics 94091 is the best performing when run at 5V. It is one of fastest actuatorstested while remaining the 2 nd lightest. Its performance is comparable to the much largerand heavier JR DS8417 while being much smaller.The manufacturers’ specified maximum torques of the actuators tested variedconsiderably. This is an important factor because the application will drive the amount oftorque required. All bench tests were conducted with the actuators unloaded and noconclusions could be made about the effect of load on the actuator response.- 93 -

3.4 Sensor IdentificationThe identification of the sensors and their respective errors is an area that requiressome attention. Because these vehicles are unmanned they usually utilize their controlsystems in a conservative manner. Expanding the envelope of operation would bebeneficial to the overall performance and mission success. However, the small size of thevehicles leaves them susceptible to low performance sensors. Knowing the limitation ofthe 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 isusually comprised of proportional, rate, and integral feedback. This PID controller isusually 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 tohigh coupling and large propeller inertias. In flight test however, this provedunwarranted. The reliance on the highest performing, small-packaged, rate gyro is high.Magnetometers are used for heading determination. The accelerometers are needed fordetermination of lateral and longitudinal speed as well as vertical speed. This iscomplimented with a pressure altimeter. Ultimately, machine or synthetic vision, laserranging equipment, and other advanced telemetry would be needed for accurate positionand landing requirements. GPS with selective availability (SA) off working nominally at1 Hz was used for outer loop position control. All of these areas need to be modeled tohave a working model of the entire system (1.11).- 94 -

3.4.1 Accelerometer IdentificationModeling of typical accelerometers was done with the representative CrossbowCXL04LP3. This is the accelerometer present on the Honeywell OAV. Theaccelerometers were modeled with white noise and random bias. Figure 3.41 shows howthis was done. According to the manufacturer, the modules could report up 0.2 g of maxbias. This would be erratic and slowly switching between positive and negative. Arandom 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 systemwas identified as 10 mg RMS.Figure 3.41 – Accelerometer ModelFigure 3.42 shows the noise and nonlinear effects the model has while the sensor isstationary over a period of 10 minutes.- 95 -

Figure 3.42 – Accelerometer Stationary Noise Model3.4.2 Rate Gyro IdentificationIdentification 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) weremodeled in a similar fashion as the accelerometers. The parameters are different; they arebased on Inertial Science specifications. The description from Inertial Science specifiedthe noise as a function of the bandwidth at which the gyros were run. The expressionwas:Noise =deg0.01 secBWIt 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 themanufacturer as well as Hysteresis and a slow drift modeled as a sine wave of lowfrequency.Figure 3.43 – Rate Gyro ModelThe hysteresis was identified as a 0.1 wide dead zone, and the max bias specifiedwas 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 sec3.4.3 GPS Receiver IdentificationTo model the GPS error and characteristics, a lot of tie was spent studying thenature of the test data provided for the µ−BLOX GPS-MS1E receiver used on theHoneywell OAV. The GPS manufacturer supplied detailed metrics as well as actual testdata to verify the accuracy of the model. Figure 3.44 shows how the manufacturer’sspecifications were implemented in Simulink. The actual positions north and east in feetare biased by a low frequency random number that sweeps the position about the originto a max error of 10 feet. The random numbers are set to a variance to closely meet the 5meter Circular Error Probability (CEP 50%) specification provided by µ−BLOX whichquantifies 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 caseof Selective Availability (SA) off.The module was running at a 1 Hz sampling rate. This was modeled with a zeroorder hold. The speed calculation was modeled by applying a unit delay and taking thedifference of the positions and dividing by the sample time. Figure 3.44 appears as themain green block in Figure 3.45, which shows how the speeds were combined and theheading 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 assumingthe sensor is stationary at (0,0).Figure 3.46 – GPS Model Results3.4.4 Magnetometer IdentificationIdentification of the magnetometers used for heading determination was performed onthe Honeywell HMC 2003 used on the OAV. The magnetometers were modeled with amax noise of 0.001 gauss, and a small Hysteresis 0.002 gauss wide. The only otherspecification modeled was the 40 microgauss resolution specified by Honeywell. Figure3.47 shows the model, while Figure 3.48 depicts a 5 sec reading at 5 gauss.- 101 -

Figure 3.47 – Magnetometer ModelFigure 3.48 – Magnetometer Depiction at 5 Gauss for 5 Seconds3.4.4 Pressure Altimeter IdentificationIdentification was performed on the Motorola MPX 4115A based on manufacturerspecifications. Motorola specified a max noise error of 0.03 inches of Hg. This wasscaled to an approximate linear relationship in the standard troposphere relating pressureto altitude. Figure 3.49 depicts the final model.- 102 -

Figure 3.49 – Pressure Altimeter ModelFigure 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 SimulationThe wealth of identification information and models were applied to a full nonlinearsimulation. This model was used to extract a linear state-space model about hover as wellas investigate certain flying qualities. Automated sweeps were fed through the model inan attempt to simulate flight test sweeps which were unavailable and evaluate the effectsof 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 M u derivatives withthe other vehicles, it was the timeliest and possessed the most information from windtunnel testing, sensors, actuators, and flight control laws. This vehicle was also in a PhaseI DARPA ACTD program at the time of writing.4.1 Simulated Frequency SweepsAn industry supplied Simulink model was used to feed frequency sweeps in ofvarying parameters in order to create time history responses for use in CIFER. Figure 4.1shows the top level Simulink model used.- 104 -

Figure 4.1 – Simulink MAV ModelFigure 4.1 shows the special code written to handle the unique task of real-timesimulation on a PC running COTS equipment. Special code was written to throttleMatlab’s Simulink to run in near real-time. This is seen as the TimeKeeper subsystemblock. Although no guarantee of frame sizes and determinism is made within the timercode, it nevertheless works quite well. Code written to handle joystick input from theLogitech Strike Force 3D USB Joystick is also required. Output for such things asgraphics and sound are provided by special software utilizing a 100 Base-T networkshares the computing load. Together, these subsystems combine to create a unique andpowerful simulation environment shown in Figure 4.2.- 105 -

Figure 4.2 – Custom PC and COTS Simulation EnvironmentWhile outside the scope of this research, it suffices to say that the environmentallows for some unique monitoring and evaluation of the overall simulation. Othersubsystems were built up to handle the flow of state variables and the creation andformatting of CIFER specific time history text files.Special code was also written to handle the sweep of the vehicle. As it wouldbecome apparent, and mentioned in the proper methods to frequency domainidentification, 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 wasaccomplished with special code and a graphical user interface (GUI) which handles thespecification of parameters. This sweep GUI is depicted in Figure 4.3.- 106 -

Figure 4.3 – Simulink Sweep Generator GUI Built for SweepsUsing the GUI and code in Figure 4.3, the sweep of Figure 4.4 was used tosimulate a sweep through the actual vehicle with all of its included sensors and nonlinearactuators.- 107 -

Figure 4.4 – Simulink GUI Generated SweepOf note from Figure 4.4 is that the sweep does not have a fade in and fade outtime associated with it as was seen in Chapter 2, Figure 2.1. This is due primarily to thefact that for a 300 second sweep, the amount of energy going in to the system in the lowfrequency region needs to be high. In a piloted sweep, there is usually plenty of lowerfrequency data due to doublets and natural oscillation by the pilot. The parameters forthis 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 occurringoscillations and energy other than that of the sweep entered. Also, by the nature of thesimulation, all coupling is hard-wired directly into the simulation. This means that theaddition of noise to break up off-axis coupling will still show high degrees of correlationto on-axis inputs.The RUAV class of vehicles analyzed all use spinning propellers inside a duct forlift. With small vehicle inertias and very high speed propellers, gyroscopic couplingoccurs 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 vectoris moved in roll. The reverse is true if moved in pitch; a rolling moment is produced. Thiseffect is apparent in the stability derivatives M p and L q . Due to sign conventions instandard helicopter coordinate systems, M p will be a positive value and L q will benegative. It is these gyroscopic effects that make simulating a sweep through the vehicledifficult. They directly correlate the roll and pitch controls and make it difficult forCIFER, or any identification tool to determine which input is creating which output. Thiswas seen when the MAV vehicle was flight tested. The actual flight test data revealedcorrelation and cross-control coherence between the roll and pitch commands. This isshown in Figure 4.5.1COHERENCE0.60.20.1 1FREQUENCY (RAD/SEC)10 100Cross Coherence between Pitch and RollFigure 4.5 – MAV Flight Test Cross Coherence between Pitch and Roll controlsIt is readily evident that there is a large amount of coherence at some gyroscopic modebetween 2 ~ 7 rad/sec.- 109 -

4.1 Matlab Linear Model DeterminationAssuming that there are no other sources of coupling in pitch and roll besidesgyroscopic effects, the coupling could be calculated from what is known about theangular momentum of the propeller and would be the key to modeling and sweeping thesimulation. Equations 3.17 and 3.18 from the MAV bare airframe identification arerepeated here.MLqpI=II=IproppropxxyyΩΩ(Equations 3.17 and 3.18)A look at these equations shows that there would be a linear relationship betweenthe amount of moment received in pitch or roll due to the cross control’s generatedresponse. In fact, as Figure 4.6 shows, the dynamics in pitch and roll can be separatedentirely.- 110 -

Figure 4.6 – Cross Control Decoupling Block DiagramFigure 6 shows that by applying Equations 3.17 and 3.18, equivalent controlinputs are generated from the off-axis responses. For a pitch command, a pitch responseand a roll response are generated. Because the gyroscopic nature is known, it can then beapplied to come up with an equivalent roll command input. The similar approach is usedwith the roll command. To illustrate how this is possible, we look at the linearizationresults 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 statesoccurring as integrators in Simulink) presented in Equations 4.1 – 4.8.x& = Fx + Gu(Equation 4.1)y = H0x+ H1x& (Equation 4.2)⎧p⎫⎪q⎪⎪ ⎪⎪r⎪⎪ ⎪u⎪ ⎪x= y =⎨v⎬⎪w⎪⎪ ⎪⎪φ⎪⎪ ⎪⎪θ⎪⎪⎩ψ⎪⎭⎧δlat⎫δ⎪ lon ⎪u = ⎨ ⎬⎪δcol⎪⎪⎩δ⎪ped ⎭(Equation 4.3)(Equation 4.4)- 112 -

⎡ Lp Lq Lr Lu Lv Lw0 0 0⎤⎢MpMq Mr Mu Mv M 0 0 0⎥⎢w⎥⎢Np Nq Nr Nu Nv N 0 0 0⎥w⎢⎥⎢X pXq Xr Xu Xv Xw0 −g0⎥F = ⎢Yp Yq Yr Yu Yv Ywg 0 0⎥⎢ ⎥⎢Zp Zq Zr Zu Zv Zw0 0 0⎥⎢ 1 0 0 0 0 0 0 0 0⎥⎢⎥⎢ 0 1 0 0 0 0 0 0 0⎥⎢0 0 1 0 0 0 0 0 0⎥⎣⎦⎡ L L L L⎢⎢M M M M⎢N N N N⎢⎢X X X XG Y Y Y Ylat lon col pedlat lon col pedlat lon col pedlat lon col ped= ⎢⎢lat lon col⎥ped⎥⎢Z Z Z Z⎢ 0 0 0 0⎢⎢ 0 0 0 0⎢⎣ 0 0 0 0lat lon col ped⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(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 asTable 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 VehicleDerivativeLINMODWind Tunnel9”Flight TestXu-0.4003 - 0.344 -0.1090Yv-0.4003- 0.344(Fixed to X u )-0.1090(Fixed to X u )Zw-0.1737 - 0.212 n/aLv-0.23730.004(Fixed to –Mu)-0.5014(Fixed to –Mu)Lp0 0 0Mu0.2373 0.003 0.5014Mq0 0 0Mp2.6261 n/a n/aLq-2.6261 n/a n/aNw-0.004 - 0.006 n/aNr0 n/a n/aXlon-0.1554 - 0.157 -0.2841Ylat0.1233 0.123 n/aZcol-0.0027 - 0.00264 n/aLlat-0.412 - 0.418 n/aMlon-0.8361 - 0.548 -0.2343Nped1.1416 0.555 n/aNcol0.0004 - 0.00057 n/a- 114 -

It can be seen right away that the results from linmod agree very well with thewind tunnel results. This is to be expected because the simulation is based on a tablelookup scheme directly based on tables from the wind tunnel data. This shows thatlinmod is working and the vehicle is trimmed in a hover state.Returning now to the simulated sweeps, we can overlay the frequency responsefor the simulated sweep with the results of linmod. This is done in Figure 4.7.Figure 4.7 – LINMOD and Simulated Sweep Roll Frequency ResponseFigure 4.7 illustrates how the simulated sweep breaks down due to the crosscouplingin 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 thegyroscopic mode (2~4 rad/sec) and a stable phase characteristic- which is know to beuntrue. Comparing the linmod results to the FRESIPD case where all the off-axiscontribution is intact reveals a better match, but misses the nature of the response and istainted by the fact that a good amount of energy was put into the system from the pitchcoupling.From Figure 6 it is evident that we can model and validate the system the withoutthe pitch and roll coupling by treating each response as uncoupled. We can thensuperimpose the coupling as linear feedback into the off-axis control. The nature of thecoupling is known already so we can avoid the breakdown in coherence. With thecoupling 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 couplingis removed. This means that once the coupling is removed from the model by removingthe propeller inertia, the coupling all but disappears. This proves that the couplingdiagram in Figure 4.6 would be a valid approach for correction and Figure 4.9 illustrateshow well the results of sweeping the model and the results of linmod agree.Figure 4.9 – Coupling Removed Illustrating linmod and Simulated Sweep ResultsFigure 4.9 shows that the results of linmod and simulated sweep match up verywell and that this method of treating the coupling as an external, linear, effect works froma modeling point of view. A similar approach was used for the pitch response to becompared 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 flighttest data is shown in Figure 4.10.Figure 4.10 – Comparison of linmod and Flight Test Pitch ResponsesFigure 10 shows excellent agreement between the flight test results and the linearmodel 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 fromlinmod, these results are deemed fairly good considering the use of limited flight testdata. It is interesting to note that all the models reveal a lack of fidelity at about 2 ~ 3rad/sec. This is gyroscopic coupling mode.- 118 -

CHAPTER 5 – CONCLUSIONSThe need for accurate simulation models of small scale, ducted-fan, unmanned airvehicles has lead to the development of techniques unique to this class of vehicles. Takenas a whole, this research activity shows that by combining existing industry tools withnew techniques a fairly high fidelity model can be constructed. This modelcomprehensively contains sensor and high fidelity actuator models along with nonlinearbare airframe models.Models and trends were developed by analyzing a number of different vehiclesspanning almost 50 years. All the vehicles showed that the ducted fan is vulnerable to ahigh degree of pitching and translation at slow speeds due to a strong effect of the lateraland longitudinal moment derivatives, Mu and Lv. This class of vehicles also shows thatthe coupling of roll and pitch due to the spinning ducted fan proves troublesome duringidentification. This is avoided by identifying the linear coupling and then removing itfrom the correlated responses. The use of flight test results, simulation analysis, and windtunnel data all may be required to ensure proper modeling techniques.Sensor performance is seen to be less than desirable due to the small packagingand weight of the available components. In the area of actuation, the maximum rate of theservos was seen to have profound effects on high bandwidth performance. This isimportant to consider because almost all MOUT exercises require some sort of higherbandwidth maneuvering.Overall, this research has shed some light on some of the unique tasks andprocedures for the system identification of ducted fan unmanned air vehicles.- 119 -

BIBLIOGRAPHYWorks Cited:1.) James M. McMichael, Col. Michael S. Francis, USAF (ret), “Micro Air Vehicles– Toward a New Dimension in Flight”, DARPA Program Office Web Release:Dec. 1997.2.) Theodore, Colin, M. Tischler, J. Colbourne, “Rapid Frequency Domain ModelingMethods for UAV Flight Control Applications”, AIAA Proceedings, Aug. 2003.3.) Sacks, Alvin H., “The Flying Platform as a Research Vehicle for DuctedPropellers”, Hiller Helicopters, Proceedings for the 26 th Annual Meeting ofInstitute of the Aeronautical Sciences, Jan. 1958.4.) Tischler, Mark, “CIFER User’s Manual Volumes 1-4”, Army/NASA RotorcraftDivision (ARH), Internal Documentation.5.) D. T. McRuer, I. Ashkenas, D. Graham, Aircraft Dynamics and AutomaticControl, Princeton, NJ: Princeton University Press, 1973.6.) Klyde, D.H., McRuer, D.T., Myers, T.T, “Pilot-Induced Oscillation Analysis andPrediction with Actuator Rate Limiting", Journal of Guidance, Control, andDynamics, Vol. 20, No. 1, Jan-Feb 1997.References:1. Graham, D., McRuer, D., Analysis of Nonlinear Control Systems, Wiley andSons, New York, 1961.2. Klyde D., D.T. McRuer, & T. Myers “Unified Pilot-Induced Oscillation Theory,Volume I: PIO Analysis with Linear and Nonlinear Effective VehicleCharacteristics, Including Rate Limiting”; Systems Technology, Inc.; December1995.3. Lazareff, M., “Aerodynamics of Shrouded Propellers”, Nord Aviation, France,Agardograph 126, The Aerodynamics and V/STOL Aircraft, May 1968.4. Lipera, L., Colbourne, J. D., Tischler, M. B., Mansur, M. H., Rotkowitz, M. C.,and Patangui, P., “The Micro Craft iSTAR Micro Air Vehicle: Control SystemDesign and Testing,” Proceedings of the American Helicopter Society 57thAnnual Forum, Washington, DC, May 2001.5. Mansur, M. H. Frye, M., Mettler, B., Montegut, M., “Rapid Prototyping andEvaluation of Control System Designs for Manned and Unmanned Applications,”Proceedings of the American Helicopter Society 56th Annual Forum, VirginiaBeach, VA, May 2000.- 120 -

6. Mettler, B., Tischler, M. B., and Kanade, T., “System Identification of Small-SizeUnmanned Helicopter Dynamics,” Proceedings of the American HelicopterSociety 55th Annual Forum, Montreal, Canada, May 1999.7. Mettler, B., Tischler, M. B., and Kanade, T., “System Identification of a Small-Scale Unmanned Rotorcraft for Flight Control Design,” Journal of the AmericanHelicopter Society, Vol. 47, No. 1, January 2002.8. “Micro Craft Ducted Air Vehicle”, Larry Lipera, American Helicopter SocietyInternational Powered Lift Conference, Arlington, CA, November 2000.9. Nelson, R.C., Flight Stability and Automatic Control, 2nd ed., New York:McGraw-Hill, 1998.10. Nise, N.S., Control Systems Engineering, 2nd Edition, Addison-Wesley 1995.11. Tischler M. B., Colbourne J. D., Morel, M. R, Biezad D. J., Cheung K. K., LevineW. S., and Moldoveanu V., “A Multidisciplinary Flight Control DevelopmentEnvironment and Its Application to a Helicopter,” IEEE Control SystemMagazine, Vol. 19, No. 4, pg 22-33, August 1999.12. Tischler, M.B. and M.G. Cauffman, “Frequency-Response Method for RotorcraftSystem Identification: Flight Application to BO-105 Coupled Rotor/FuselageDynamics.” Journal of the American Helicopter Society, 1992. 37/3: p. 3-17.- 121 -

Manufacturer References:Crossbow CXL04LP3http://www.xbow.com/pdf/Accelerometer/LP/LP%20Accel.pdfCrossbow Technology, Inc.41 Daggett DriveSan Jose, CA 95134-2109Phone:(408) 965-3300Fax:(408) 324-4840Email:info@xbow.comJR Components 8700G Super ServosSaturation identified from World Class Models:http://www.worldclassmodels.com/cgi-bin/agora/agora.cgi?product=servosµ−BLOX GPS-MS1Ehttp://www.u-blox.ch/gps/gps-ms1e/ubloxgps performance.pdfZuercherstrasse 68P/O Box 788800 ThalwilSwitzerlandEmail:info@u-blox.comPhone (UK):+44 (0) 1622 618628Inertial Science RRS75RRS75.pdfPeter MoonInertial Science, Inc.(805) 499-3191, (805) 498-4882 Faxhttp://www.inertialscience.compjmoon@inertialscience.comHoneywell HMC 2003http://www.ssec.honeywell.com/magnetic/datasheets/hmc2003.pdfMotorola MPX 4115Ahttp://e-www.motorola.com/webapp/sps/prod_cat/prod_summary.jsp?code=MPX4115&catId=M98716- 122 -

Appendix AOAV Proposal VehicleIdentified State-Space Quadruple and Formx&= Fx + Guy = H x+ H x&1 2x⎧v⎫⎪p⎪⎪ ⎪⎪φ⎪⎪ ⎪= ⎨u⎬⎪q⎪⎪ ⎪⎪θ⎪⎪ ⎪⎩r⎭y⎧p⎫⎪ ⎪= ⎨q⎬⎪r⎪⎩ ⎭u⎧ pmixer⎪ ⎪= ⎨qmixer⎬⎪⎩rmixer⎫⎪⎭F⎡ 0 0 32.17 0 0 0 0⎤⎢0.197 0 0 0 0 0 0⎥⎢−⎥⎢ 0 1 0 0 0 0 0⎥⎢⎥= ⎢ 0 0 0 0 0 −32.17 0⎥⎢ 0 0 0 .2623 0 0 0⎥⎢⎥⎢ 0 0 0 0 1 0 0⎥⎢ 0 0 0 0 0 0 0⎥⎣⎦⎡ 0 0 0 ⎤⎢.2958 0 0⎥⎢⎥⎢ 0 0 0 ⎥⎢⎥G = ⎢ 0 0 0 ⎥⎢ 0 .3013 0 ⎥⎢⎥⎢ 0 0 0 ⎥⎢ 0 0 .3629⎥⎣⎦H1⎡0 57.3 0 0 0 0 0 ⎤=⎢0 0 0 0 57.3 0 0⎥⎢ ⎥⎢⎣0 0 0 0 0 0 57.3⎥⎦H2= 0- 123 -

Appendix BFrequency 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 CActuator Generated Transfer Function ModelsBode 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 DActuator Time Domain Verification of Final Models- 157 -

Time Domain VerificationDS8417 5VDS8417 6V- 158 -

94091 5V94091 6V- 159 -

CS-10BB 5VCS-10BB 6V- 160 -

DS368 5VDS368 6V- 161 -

HS-512MG 5VHS-512MG 6V- 162 -