NONLINEAR SIMULATION OF A MICRO AIR VEHICLE

NONLINEAR SIMULATION OF A MICRO AIR VEHICLE By JASON JOSEPH JACKOWSKI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2004

I dedicate this work to my wonderful and lovely fiance Ellen Bozarth. Without her love and support, I could not have done this.

ACKNOWLEDGMENTS I thank the AFRL-MN and Eglin Air Force for funding this project. I want to thank the entire flight controls lab which include Ryan Causey, Kristin Fitzpatrick, Joe Kehoe, Mujahid Abdulrahim, Anukul Goel, and Robert Eick. I would like to thank Jason Grzywna and Jason Plew for their help and expertise in soldering. I would like to thank Dr. Rick Lind for his advisement and guidance during this project. A big thanks goes to my brother Jeff Jackowski who helped me make a resolution-enhancing circuit board for our altimeter during Christmas break. iii

TABLE OF CONTENTS page ACKNOWLEDGMENTS............................. iii LIST OF TABLES................................. vi LIST OF FIGURES................................ vii ABSTRACT.................................... ix CHAPTER 1 INTRODUCTION.............................. 1 1.1 Motivation............................... 1 1.2 Background.............................. 2 1.2.1 Micro Air Vehicles....................... 2 1.2.2 AVCAAF Vehicle....................... 4 1.2.3 AVCAAF Autopilot...................... 5 1.3 Overview................................ 9 2 SIMULATION ARCHITECTURE..................... 11 2.1 Simulation Overview......................... 11 2.2 Non-Linear Dynamics Plant..................... 11 3 NONLINEAR EQUATIONS OF MOTION................. 14 3.1 Frames of Reference.......................... 14 3.2 Rotations............................... 14 3.3 Kinematic Equations......................... 16 3.4 Force and Moment Calculations................... 20 3.5 Calculation of States......................... 21 4 CHARACTERIZATION METHODS.................... 22 4.1 Physical Measurements........................ 22 4.2 Finite Element Methods....................... 22 4.3 Wind Tunnel............................. 22 4.4 Computational Fluid Dynamics................... 23 4.5 Flight Testing............................. 23 iv

5 AVCAAF CHARACTERIZATION..................... 24 5.1 Overview................................ 24 5.2 Experimental Aerodynamics..................... 24 5.2.1 Testing............................. 24 5.2.2 Results............................. 26 5.3 Analytical Inertias.......................... 27 5.4 Analytical Aerodynamics....................... 29 5.5 Model Integration........................... 31 5.5.1 Wind Tunnel Data Analysis................. 31 5.5.2 Aerodynamics......................... 33 5.6 Linearized Dynamics......................... 34 5.6.1 Longitudinal.......................... 34 5.6.2 Lateral-Directional...................... 37 5.7 Modeling Results........................... 41 6 AVCAAF SUBSYSTEMS.......................... 43 6.1 Sensor Subsystem........................... 43 6.1.1 Camera Subsystem...................... 43 6.1.2 GPS Subsystem........................ 48 6.1.3 Altitude Subsystem...................... 49 6.2 Actuator Subsystem......................... 49 6.3 Controller Subsystem......................... 50 7 RESULTS AND CONCLUSIONS...................... 51 7.1 Results................................. 51 7.2 Conclusion............................... 51 8 RECCOMENDATIONS........................... 53 8.1 Overview................................ 53 8.2 Wind Tunnel Characterization.................... 53 8.3 Computational Fluid Dynamics Characterization.......... 54 8.4 Streamlining MAV Design to CFD Characterization Process... 54 8.5 Miscellaneous Reccomendations................... 54 REFERENCES................................... 56 BIOGRAPHICAL SKETCH............................ 60 v

Table LIST OF TABLES page 1 1 AVCAAF vehicle general properties................... 5 2 1 Standard atmosphere air densities.................... 12 5 1 AVCAAF vehicle component masses................... 28 5 2 Analytical inertia properties....................... 29 5 3 Estimated dynamic derivatives...................... 30 5 4 Analytical and experimental stability derivatives............ 34 5 5 Longitudinal derivatives.......................... 36 5 6 Longitudinal eigenvalues.......................... 36 5 7 Longitudinal eigenvectors......................... 37 5 8 Lateral directional derivatives....................... 39 5 9 Lateral-directional eigenvalues...................... 39 5 10 Lateral-directional eigenvectors...................... 40 5 11 Lateral-directional eigenvector...................... 40 vi

Figure LIST OF FIGURES page 1 1 Flexible wing 6 in MAV......................... 3 1 2 MAV.................................... 4 1 3 Horizon Detection Example....................... 6 1 4 Lateral stability augmentation system.................. 7 1 5 Longitudinal stability augmentation system............... 7 1 6 Directional control system........................ 8 1 7 Altitude control system.......................... 8 1 8 Successful AVCAAF waypoint navigation................ 9 2 1 Micro Air Vehicle simulation architecture................ 11 2 2 Nonlinear dynamics plant........................ 12 3 1 Earth-fixed and body-fixed frames of reference............. 15 3 2 Set of rotations through the Euler angles................ 16 5 1 AVCAAF model in test section..................... 25 5 2 C L versus angle of attack......................... 26 5 3 C L versus C D............................... 27 5 4 Analytical model............................. 27 5 5 Exploded view............................... 28 5 6 Geometry of panels............................ 30 5 7 Wind tunnel data and fitted curve of C L................ 31 5 8 Wind tunnel data and fitted curve of C D................ 31 5 9 Wind tunnel data and fitted curve of C m................ 32 5 10 Measured values of side force....................... 33 5 11 Variation in dutch roll frequency..................... 41 vii

6 1 AVCAAF sensors subsystem....................... 43 6 2 Image projection and pitch percentage................. 44 6 3 Triangular and trapezoidal ground areas................ 45 6 4 NTSC camera image........................... 46 6 5 NTSC image box and ground intersection................ 47 6 6 Simulated horizon from camera subsystem............... 49 viii

Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science NONLINEAR SIMULATION OF A MICRO AIR VEHICLE By Jason Joseph Jackowski December 2004 Chair: Richard C. Lind, Jr. Major Department: Mechanical and Aerospace Engineering Simulations of micro air vehicles are required for tasks related to mission planning such as control design and flight path optimization. The flight dynamics of these vehicles are difficult to model because of their small size and low airspeeds. This modelling difficulty makes the process of designing controllers for such aircraft difficult and often done by trial and error. This thesis presents a procedure to create a simulation of a micro air vehicle. Methods for characterizing the aircraft are presented and discussed. The resulting model is simulated with a set of nonlinear equations of motion. The simulation will be used for future autopilot development, mission planning, and morphing aircraft controller design. An example of characterizing a micro air vehicle is presented in this thesis. Characterizing the micro air vehicle is performed using a combination of physical measurements, finite element methods, wind tunnel data and computational methods. This characterization includes designing Simulink subsystems to represent the sensors, hardware, and controllers used on the micro air vehicle. Accurate ix

characterization of the components of the aircraft, harware, and sensors should provide a simulation suited for controller design and analysis. x

CHAPTER 1 INTRODUCTION 1.1 Motivation Micro Air Vehicles, typically called MAVs, have been gaining interest in the research community. MAVs are a class of aircraft whose largest dimension ranges from 6-30 inches [9] and operate at speeds up to 30 mph [1]. The small dimensions and light weight of the MAVs make them very portable to remote locations. MAVs can be designed for high agility to operate in urban environments or used to deploy from munitions to assess damage inflicted on a target. These MAVs can be equipped with quiet electric motors, cameras, GPS, and other sensors. The wide range of possible payloads leads to a plethora of uses for MAVs. Autonomous MAVs are highly attractive to the military for battlefield reconnaisance missions. A MAV could become a standard piece of equipment for special forces teams or advanced scout forces. These soldiers could quickly launch an autonomous MAV to scout a potentially hazardous area without endangering themselves. Currently sattelites or larger Unmanned Air Vehicles take time to deploy or re-position to give ground troops the imagery they need. MAVs present a cheap and quick alternative for battlefield reconnaisance. Hazardous chemical spills require expensive equipment for humans to venture into, map, and clean up the infected area. A MAV can be equipped with a sensor to detect harmful chemicals and relay that information to a hazardous materials team. Autonomous MAVs can be used to map out the area and locate trapped people. The MAV can assist in finding an escape route around the chemical spill. This objective can be done very quickly and without risk to the hazardous material teams. 1

2 These scenarios highlight how further research in autonomous MAVs could be beneficial. A simulation capability as presented in this thesis, will assist the development and path planning of the MAVs. 1.2 Background 1.2.1 Micro Air Vehicles The University of Florida has been actively pursuing MAV research for over six years. Over these years the size, design, and payload capacity have seen significant improvements. Practical uses for MAVs would not be possible without recent technological developments. The small dimensions of MAVs have driven a need for the miniaturization of many Radio Controlled airplane electronics. The cellular phone industry has also been beneficial to MAVs by providing miniaturization of batteries. These advances along with other computer, communications, and video camera technologies have allowed MAVs to start having on-board computer processing and autonomy. Aerovironment s Black Widow was the first mission capable MAV [16]. The Black Widow has a wing span of 6 in, is electrically powered, weighs 80 g, has a range of 1.8 km, and has an endurance of 30 min. The Black Widow is a rigid-wing platform with three vertical stabilizers. On-board systems include a custom-made video camera, control system, video transmitter, pitot-static tube, magnetometer, and data logger. The control system can perform altitude hold, airspeed hold, heading hold, and yaw damping. The latter is important as MAVs tend to have high Dutch roll oscillations. The yaw damper reduces these oscillations and helps stabilize the video camera image. The video camera system allows a pilot to control the MAV by video alone. The Black Widow demonstrated that MAVs can make practical platforms for various missions.

3 The research at the University of Florida started with Dr. Peter Ifju s work on a flexible wing MAV [17]. The wing consisted of a latex rubber stretched over a carbon fiber structure of a leading edge and battens. The idea was based on sail powered vessels using sail twist to produce a more constant thrust over a wider range of wind conditions [40]. An example of a 6 in MAV is presented in Figure 1 1. Figure 1 1: Flexible wing 6 in MAV An inherent feature to this flexible wing is adaptive washout, allowing the wing to deform when encountering a gust. This adaptive washout helps to stabilize the MAV during flight by adjusting to the airflow [17, 34, 39]. At low Reynolds numbers, airflow around the wing surface has a faster separation which increases drag. The flexible wing allows the airflow to remain attached longer than a rigid wing. Dr. Ifju has established a rapid protyping facility for MAVs at the University of Florida. This facility allows computer based design of the wings. The computer software can then use a CNC machine to create a hard foam tool to manufacture the wing [21]. These tools are used to shape carbon fiber pre-preg and wing material for curing in a vaccum bag in an oven. Construction of MAVs can take from two to ten days based on the design [21]. Processes in streamlining this

4 construction process are continually implemented to obtain stricter tolerances between the same type of MAV and to reduce production time. The first characterization of a MAV was performed on a 6 in MAV from the Univeristy of Florida in the Basic Aerodynamics Research Tunnel (BART) at NASA Langley Research Center [39]. Simulations of the MAV were then created using the wind tunnel data [40]. This work resulted in a set of linearized dynamic models characterizing the MAV at different flight conditions. The simulations were also used to design controllers based on a dynamic non-linear inversion approach. 1.2.2 AVCAAF Vehicle The micro air vehicle used for simulation is shown in Figure 1 2. The vehicle is a variant of a baseline type which has been designed at the University of Florida [17]. In this case, the MAV is 21 in in length and 24 in in wingspan. The total weight of the vehicle, including all instrumentation, is approximately 540 grams. The basic properties are given in Table 1.2.2. This MAV is the flight test-bed for the Active Vision Control of Agile Autonomous Flight (AVCAAF) project at the University of Florida and is called the AVCAAF vehicle. Figure 1 2: MAV The airframe is constructed almost entirely of composite and nylon. The fuselage is constructed from layers of woven carbon fiber which are cured to form

5 Table 1 1: AVCAAF vehicle general properties Property Value Maximum Takeoff Mass 540 grams Wing Span 60.96 cm Wing Area 5.67 cm 2 Mean Aerodynamic Chord 9.3 cm Static Thrust 3.2 N Payload Capacity 200 grams a rigid structure. The thin, under-cambered wing consists of a carbon fiber sparand-batten skeleton that is covered with a nylon wing skin. The AVCAAF wing was derived from a tail-less MAV wing and has a relflexed airfoil [21]. The original purpose of the reflexed airfoil was to provide stabilization usually generated by the tail. A tail empannage, also constructed of composite and nylon, is connected to the fuselage by a carbon-fiber boom that runs concentrically through the pusherprop disc. Control is accomplished using a set of control surfaces on the tail. Specifically, a rudder along with a pair of independent elevators can be actuated by commands to separate servos. The rudder obviously affects the lateral-directional dynamics response while the elevators can be moved symmetrically to affect the longitudinal dynamics and differentially to affect the lateral-directional dynamics. The on-board sensors consist of a GPS unit, an altimeter, and a video camera. The GPS unit is mounted horizontally on the top of the nose hatch. The altimeter, which actually measures pressure, is mounted inside the fuselage under the nose hatch. The video camera is fixed to point directly out the nose of the aircraft. 1.2.3 AVCAAF Autopilot The autopilot operates on an off-board ground station at the rate of 50 Hz. This ground station essentially consists of a laptop with communication links. Separate streams for video and inertial measurements are sent using transceivers

6 on the aircraft. The image processor and controller analyze these streams and transmit commands to the Radio Controlled (RC) transmitter. The RC transmitter mixes the symmetric and anti-symmetric elevator commands into servo commands. The AVCAAF autopilot performs 3-D waypoint navigation using the GPS receiver, altimeter, and video camera [22]. The video signal sent to the ground station is analyzed for pitch percentage and roll angle. The pitch percentage is the percent of ground seen in the image. This is calculated by first detecting the horizon by statistical modeling. The horizon detection algorithm determines the horizon that best divides the image into ground and sky based on previous calibration. The horizon line is also used to determine the current roll angle. An example of the horizon detection algorithm is shown in Figure 1 3. Figure 1 3: Horizon Detection Example The autopilot consists of a lateral and longitudinal stability augmentation system, directional controller, and an altitude control system. The lateral stability augmentation system stabilizes the vehicle and allows tracking of roll commands. The architecture of the lateral stability augmentation system is shown in Figure 1 4. The controller consists of a integral gain K Iφ, porportional gain K φ, a filter, a camera C, and a vision processing element V.

7 environment V C K Iφ MAV K φ Figure 1 4: Lateral stability augmentation system The longitudinal stability augmentation system stabilizes the AVCAAF vehicle and tracks pitch percentage commands. The architecture of the lateral stability augmentation system is shown in Figure 1 5. The controller consists of a porportional K σ, a camera C, and a vision processing element V. environment V C K σ MAV Figure 1 5: Longitudinal stability augmentation system The directional controller is an outer loop of lateral stability augmentation system shown in Figure 1 6. Here K ψ is the porportional gain and S is the GPS sensor. The current longitude and lattitude is provided by the GPS receiver at a 1 Hz rate. The heading is calculated from the current and previous GPS position. This creates a lag in the system to calculate the heading and does not provide real-time GPS coordinates. This issue is addressed by calculating position and heading estimates which update at the 50 Hz rate of the control system. The altitude controller commands the altitude to reach the correct altitude of the current waypoint. Included in this controller is a switching element to use altitude error or pitch percentage error to determine the elevator deflection

8 environment V C K Iφ ψ c K ψ Kφ MAV S Figure 1 6: Directional control system shown in Figure 1 7. Here K Ih is the altitude integral gain, K h is the altitude porportional gain, and K φδe is a gain to couple the longitudinal dynamics with the lateral-directional dynamics. The controller limits the pitch percentage to ensure that there is always a visible horizon for pitch percentage and roll angle calculation. lim K σ V environment C K Ih h c switch MAV K h K φδe S Figure 1 7: Altitude control system The AVCAAF autopilot underwent a series of flight tests to experimentally tune the various gains of the control systems. These flight tests resulted in an autopilot capable of 3-D waypoint navigation. Figure 1 8 shows the flight path of the AVCAAF vehicle as it tracks three waypoints, shown in red boxes, multiple times.

9 Figure 1 8: Successful AVCAAF waypoint navigation 1.3 Overview This thesis presents a MATLAB/Simulink simulation architecture for micro air vehicles. Nonlinear equations of motion are derived to simulate MAVs. These equations do not assume symmetry and can be used for morphing aircraft. Methods of characterizing aircraft are then presented and discussed. An example of characterizing a MAV is presented. The AVCAAF vehicle is characterized using finite element methods, wind tunnel analysis, and computational fluid dynamics analysis. The model is linearized about a trim condition to analyze the aircraft dynamics to quantify the level of confidence in the model. Subsystems in the simulation are created to emulate the sensors, hardware, and control system that was implemented on the AVCAAF vehicle. The subsystems emulate the camera, GPS receiver, and altimeter sensors. The control system used to perform 3-D waypoint navigation on the AVCAAF vehicle is converted from C++ and implemented in the simulation. The control surface actuators are simulated by adding rate and position limits on the control surface commands.

10 The process of characterizing micro air vehicles in this thesis will be used to model future micro air vehicles. This thesis attempts to lay the ground work to allow controller design on future micro air vehicles.

CHAPTER 2 SIMULATION ARCHITECTURE 2.1 Simulation Overview The MAV simulator is a MATLAB/Simulink program that numerically integrates the nonlinear equations of motion of the system. The simulator consists of four major subsystems: Controller K, Actuators A, Nonlinear Dynamics Plant P, and Sensors S as shown in Figure 2 1. Figure 2 1: Micro Air Vehicle simulation architecture The structure of the simulation is built as a top-down architecture. Each subsystem is modular and contains more subsystems which are easily reconfigured. This architecture allows a plug-and-play capability allowing the simulation to simulate different aircraft, controllers, and sensors easily by simply replacing a subsystem. 2.2 Non-Linear Dynamics Plant The structure of the Nonlinear Dynamics Plant, depicted in Figure 2 2, does not change between aircraft platforms. The other subsystems are specifically designed for the AVCAAF vehicle and are described in Chapter 6. 11

12 Figure 2 2: Nonlinear dynamics plant The Air Density Look-Up block finds the air density based on current altitude. The air density is calculated using first-order interpolation between points from the standard atmosphere table. It interpolates between -200 meters to 1000 meters; however, a larger range could be covered if more reference points were added. Table 2 1 shows the altitude and air density values used. Table 2 1: Standard atmosphere air densities Altitude (m) Air Density (kg/m 3 ) -200 1.2487-100 1.2368 0 1.2250 100 1.2133 200 1.2071 300 1.1901 400 1.1786 500 1.1673 600 1.1560 700 1.1448 800 1.1337 900 1.1226 1000 1.1117

13 The NL Dynamics subsystem calculates the forces and moments acting on the aircraft at each time-step. The subsystem s inputs are the control surface deflections, aircraft moments of inertia, aircraft geometry, air density, and 12 states defining the aircraft s position, orientation, linear velocity and angular velocity. The subsystem calculates the new linear and angular accelerations due to the forces and moments acting on the aircraft. This calculation uses the standard 12 equations of motion [31] and polynomial coefficients that characterize the aircraft, described in Section 3.4. These accelerations along with the previous velocities are the output of this subsystem. The Integrator subsystem numerically integrates the velocities and accelerations each time-step of the simulation. A set of initial conditions can be set by the user to define the initial position, orientation, and velocity of the aircraft. The integrator uses these initial values to set the states when the simulation begins. The Angle Limiter subsystem converts the Euler angles to a range between 0 and 360 o. The Body Axis States to Earth Inertial FOR subsystem converts the current aircraft position, orientation, and velocity into the Earth inertial frame of reference. These states are then passed to the Sensor module from Figure 2 1 for sensor use.

CHAPTER 3 NONLINEAR EQUATIONS OF MOTION This chapter derives the nonlinear equations of motions used by the Nonlinear Dynamics Subsystem presented in Section 2.2. The equations of motion describe the rigid body dynamics and neglect the structural dynamics. MAVs obey the same equations of motion as any airplane. The following derivation is for the general airplane as detailed in [13, 31] and expanded to include assymetries. 3.1 Frames of Reference The MAV moves in an earth-fixed inertial reference frame E defined by the basis vectors (ê 1, ê 2, ê 3 ). The vector ê 3 points in the same direction as gravity. Vectors ê 1 and ê 2 are positioned to make E a right-handed coordinate system. The earth-fixed reference frame E is shown in Figure 3 1. Six Degrees Of Freedom (DOF) are necessary to fully describe the MAV s position and orientation from a specified point. The first three DOFs define the distance from the MAV to the fixed-reference frame. The other three DOFs are Euler angles and define the rotation between the fixed-reference frame and the MAV body-fixed reference frame. The body-fixed reference frame B has its origin at the MAV center of gravity and is defined by the basis vectors (ˆb1, ˆb 2, ˆb ) 3. The vector ˆb 1 points toward the nose of the MAV, vector ˆb 2 points toward the left wing, and vector ˆb 3 makes B a right-handed coordinate system. This orientation is the standard aircraft body-fixed coordinate frame. 3.2 Rotations A sequence of three rotations can transform a position from one coordinate system to another. This sequence of rotations is done, in order, through the three 14

15 Figure 3 1: Earth-fixed and body-fixed frames of reference Euler angles yaw (Ψ), pitch (Θ), and roll (Φ). This sequence is a standard 3-2-1 rotation. When these rotations are performed, two intermediate reference frames are created with basis vectors (ˆx 1, ŷ 1, ẑ 1 ) and (ˆx 2, ŷ 2, ẑ 2 ). The rotations are performed in the following order: 1. Rotate the E earth-fixed reference frame about ê 3 through yaw angle Ψ to reach intermediate frame (ˆx 1, ŷ 1, ẑ 1 ). 2. Rotate (ˆx 1, ŷ 1, ẑ 1 ) about ŷ 1 through pitch angle Θ to reach intermediate frame (ˆx 2, ŷ 2, ẑ 2 ). 3. Rotate (ˆx 2, ŷ 2, ẑ 2 ) about ˆx 2 through roll angle Φ to obtain body-fixed frame B. Figure 3 2 shows the sequence of rotations graphicly. The rotation sequence can also be shown mathematically Equation 3.1.

16 Figure 3 2: Set of rotations through the Euler angles ˆb1 ˆb2 ˆb 3 1 0 0 CΘ 0 SΘ CΨ SΨ 0 = 0 CΦ SΦ 0 1 0 SΨ CΨ 0 0 SΦ CΦ SΘ 0 CΘ 0 0 1 CΘCΨ CΘSΨ SΘ = CΨSΦSΘ CΦSΨ SΦSΘSΨ + CΨCΦ SΦCΘ CΦSΘCΨ + SΦSΨ CΦSΘSΨ CΨSΦ CΦCΘ ê 1 ê 2 ê 3 ê 1 ê 2 ê 3 (3.1) = E B ê 1 ê 2 ê 3 Converting a vector from the body-fixed frame B back to the earth-fixed frame E is done by inverting the matrix E B and multiplying it by the vector. 3.3 Kinematic Equations The rigid body equations of motion can be derived from Newton s second law. d F = (mv) dt (3.2) d M = dt H (3.3)

17 F is the force applied to the rigid body and H is the angular momentum of the rigid body. The vector components of 3.2 can be decoupled into three components shown in 3.4. F = d dt (mv) c + d dt (mv) ˆb 2 + d dt (mv) ˆb 3 = F xˆb1 + F yˆb2 + F zˆb3 (3.4) F x, F y, and F z are the forces along the {ˆb1, ˆb 2, ˆb } 3 axes respectively. The velocity of the center of gravity (CG) of the MAV is V c. With the definition of V c and the rigid body assumption, velocities at other points on the MAV may be found. Let r be the position vector from the CG to a differential mass element δm. The velocity of this mass element is expressed in 3.5. V = V c + dr dt = V c + E ω B r (3.5) Here E ω B is the angular velocity of B in E defined in 3.6. E ω B = pˆb 1 + qˆb 2 + rˆb 3 (3.6) Assuming the mass is constant, then the body-fixed accelerations can be found using 3.7, 3.8, and 3.9 F x = m ( u + qw rv) (3.7) F y = m ( v + ru pw) (3.8) F z = m (ẇ + pv qu) (3.9) The angular momentum of the mass element is expressed in 3.10.

18 r vc δm + [ r ( E ω B r )] (3.10) Expanding r into its components yields 3.11. H can now be expressed as 3.12. r = xˆb 1 + yˆb 2 + zˆb 3 (3.11) H = ( pˆb 1 + qˆb 2 + rˆb 3 ) ( x 2 + y 2 + z 2) δm ( xˆb 1 + yˆb 2 + zˆb 3 ) (px + qy + rz) δm (3.12) The scalar parts of H are shown in 3.13, 3.14, and 3.15. H x = p ( y 2 + z 2) δm q xy δm r xz δm (3.13) H y = p xy δm + q ( x 2 + z 2) δm r yz δm (3.14) H z = p xz δm q yz δm + r ( x 2 + y 2) δm (3.15) The summations of 3.13 are the moments and products of inertia defined in 3.16, 3.17, 3.18, 3.19, 3.20, and 3.21. The domain of integration for these equations is the entire aircraft body.

19 I xx = I yy = I zz = I xy = I xz = I yz = (y 2 + z 2) δm (3.16) (x 2 + z 2) δm (3.17) (x 2 + y 2) δm (3.18) xy δm (3.19) xz δm (3.20) yz δm (3.21) Using the equations for the moments and products of inertia, the scalar parts of H become 3.22, 3.23, and 3.24. H x = pi xx qi xy ri xz (3.22) H y = pi xy + qi yy ri yz (3.23) H z = pi xz qi yz + ri zz (3.24) Using 3.3, 3.22, 3.23, and 3.24 the moment equations can be expressed 3.25, 3.26, and 3.27. L = Ḣx + qh z rh y (3.25) M = Ḣy + rh x ph z (3.26) N = Ḣz + ph y qh x (3.27) Where L, M, and N are the moments about {ˆb1, ˆb 2, ˆb } 3 axes respectively. Making the assumption that the rate of change of the moments and products of

inertia are negligible and assuming the aircraft is not symmetric, the equations can be expanded into 3.28, 3.29, and 3.30. 20 L = ṗi xx qi xy ṙi xz pqi xz + ( r 2 q 2) I yz + qri zz + rpi xy qri yy (3.28) M = ṗi xy + qi yy ṙi yz + rpi xx qri xy + ( p 2 r 2) I xz + pqi yz pri zz (3.29) N = ṗi xz qi yz + ṙi zz + ( q 2 p 2) I xy + pqi xy pri yz pqi xx + qri xz (3.30) Equations 3.7, 3.8, 3.9, 3.28, 3.29, and 3.30 are the equations of motion used in the simulation. 3.4 Force and Moment Calculations The forces and moments the aircraft acting on the aircraft are represented as functions of the flight condition. The flight condition includes angle of attack, slide-slip angle, aircraft velocity, air density, and control surface deflections. The equations for the coefficients of the forces and moments are given in 3.31, 3.32, 3.33, 3.34, 3.35, and 3.36. C L = C Lα 2 α 2 + C Lα α + C L0 + C Lδsym δ sym + C L α α (3.31) C D = C Dα 2 α 2 + C Dα α + C D0 + C Dδsym δ sym (3.32) C Y = C Yp p + C Yr r + C Yδr δ r + C Yδasy δ asy + C Yβ β (3.33) C m = C mα 3 α 3 + C mα 2 α 2 + C mα α + C m0 + C mδsym δ sym + C m α α (3.34) C l = C lp p + C lr r + C lδr δ r + C lδasy δ asy + C lβ β (3.35) C n = C np p + C nr r + C nδr δ r + C nδasy δ asy + C nβ β (3.36)

21 These equations use the methods presented in Chapter 4 to characterize the aircraft. During each timestep of the simulation, the forces and moments acting on the aircraft are calculated at that particular instant. This calculation is done by evaluating 3.31, 3.32, 3.33, 3.34, 3.35, and 3.36 at that particular flight condition. The coefficients are then multiplied by the current dynamic pressure, wing area, and reference length if applicable. 3.5 Calculation of States The body axis linear accelerations are calculated using 3.7, 3.8, and 3.9. The forces were previously calculated and the mass, current angular rates and velocity are known. The body axis angular accelerations are calculated using 3.28, 3.29, and 3.30. The inertial properties, current angular rates, and moments are known. These body axis linear and angular accelerations are rotated into the earthfixed inertial reference frame and integrated each time step.

CHAPTER 4 CHARACTERIZATION METHODS This chapter presents five different methods that can be used to help characterize aircraft. Some methods may not competely characterize the aircraft so data from multiple methods can be combined to generate a model. 4.1 Physical Measurements Physical measurements can be taken directly from the aircraft. Geometric propetries such as wing span, wing area, and the mean aerodynamic chord can be approximated using a ruler. The mass is easily obtained using a scale and the center of gravity can be experimentally located. Torsional pendulums can be used to determine the moments of inertia and principle axes of the aircraft. 4.2 Finite Element Methods A high fidelity finite element model can produce analytical values for mass, moments of inertia, products of inertia, wing area, and other geometric properties. Programs such as ProEngineer [35] can be utilized to create computer models of each airplane component. These can then be assembled in a flight configuration. Subsequent anaylsis can result in values for the aforementioned properties. 4.3 Wind Tunnel A wind tunnel can be used to accurately characterize forces and moments acting on a micro air vehicle. Micro air vehicles can be small enough to mount a full-size MAV model into the wind tunnel test section without requiring scaling laws. This testing allows the wind tunnel data to include the strong viscous forces associated with the low Reynolds numbers encountered by MAVs [3]. 22

23 Accurate results from wind tunnels depend on the instrumentation used to measure the forces and moments. A 6 in MAV can have forces on the order of 0.02 newtons [3]. Such small forces can be easily distorted by noise or poor calibration. Results from wind tunnel testing can include static derivatives and curves for the forces and moments based on angle of attack, side-slip angle, control surface deflection, and thrust. Dynamic derivatives can be difficult to measure in a wind tunnel and is often approximated by equations. 4.4 Computational Fluid Dynamics The use of Computational Fluid Dynamics (CFD) can assist in evaluating hard to determine parameters, such as the dynamic derivatives, as well as easier to determine parameters, such as static derivatives. CFD methods can also be used when a wind tunnel cannot be used. Vortice Lattice Methods (VLM) can be used; however, the complete Navier-Stokes equations will provide more accurate results. Some VLM programs do not include viscid skin friction which becomes a significant force in low Reynolds number flow. 4.5 Flight Testing An aircraft equipped with sensors, such as accelerometers and gyroscopes, can log flight test data for analysis. This data can be used to perform regression analysis to relate gyro measurements of roll rate, pitch rate, and yaw rate to control surface commands. The aircraft dynamics are determined by a least-squares approach to the flight test data [27]. This method can obtain inaccurate results due to noisy sensor measurements.

CHAPTER 5 AVCAAF CHARACTERIZATION 5.1 Overview This chapter presents an example of characterizing a MAV. This example consists of identifying the longitudinal and lateral dynamics of the AVCAAF vehicle using finite element methods, wind tunnel and computational data. The wind tunnel data is used to find the static aerodynamic force and moment coefficients. The aerodynamic software package, Tornado, approximates the dynamic derivatives of the AVCAAF vehicle [29]. The aerodynamic characteristics are integrated into the standard longitudinal and lateral linear dynamics to characterize the AVCAAF at a trim condition. These linearized dynamics are analyzed for modal properties. The analysis is performed to check stability and to quantify the confidence of the model created from the wind tunnel and computational data. The actual implementation of the dynamics will involve regression analysis of the wind tunnel data, supplemented by the computational data, resulting in functions for force and moment coefficients. This implementation allows simulation of the MAV at different flight conditions. 5.2.1 Testing 5.2 Experimental Aerodynamics The aerodynamics associated with the AVCAAF aircraft are experimentally determined using a wind tunnel at the University of Florida. This wind tunnel is a horizontal, open-circuit low-speed facility. The wind tunnel has a bell mouth inlet and several flow straighteners. The test section is square with dimensions 914 mm and a length of 2 m. The fan speed is regulated by a variable frequency controller 24

25 and operated remotely by a computer. The maximum velocity for test purposes is approximately 15 m/s which correlates to a maximum Reynolds number of 100,000. Testing of the tunnel is controlled by a computer. This computer controls the angle of attack of the model along with acquiring data and performing real-time analysis. Figure 5 1: AVCAAF model in test section The vehicle is mounted onto a sting balance and the resulting structure connects to an aluminum arm as in Figure 5 1. The internal sting balance measures five forces and one moment. The forces are converted in real time to the coefficients, {C L, C D, C m, C n, C Y, C l }, using the dynamic pressure and reference data for wing area and reference length. The main potential sources of uncertainty are errors associated with solving the sting balance forces and moments, angle of attack measurements, and the dynamic pressure determination. Additional minor factors include uncertainties in the determination of geometric quantities such as wing area or chord line. In actuality, the MAV generates loads considerably smaller than usual calibration weights so the sting balance is a reasonable expectation as the main source of error [39, 30]. A preliminary estimate of this error was done by running an extensive set of calibration checks [19].

26 5.2.2 Results The aerodynamics of the AVCAAF vehicle are determined using a freestream velocity of 13 m/s. The vehicle was mounted wings-level to consider sweeps across angle of attack and mounted at a roll angle of 90 o to consider sweeps across angle of sideslip. A complete set of static derivatives in pitch, roll and yaw are computed [3]. A representative set of this aerodynamic data is given in Figure 5 2 and Figure 5 3. These data consider the aerodynamics at a variety of symmetric deflections for the elevators. Unfortunately, the facility did not allow for measuring the dynamic derivatives. Consequently, the data obtained from the wind tunnel is not sufficient to completely characterize a model of the flight dynamics. The AVCAAF vehicle was also not tested with its motor turned on. Testing the AVCAAF vehicle with the motor on would require an awkward mouning system which does not currently exist. Since the differential elevators and rudder are in the prop-wash during actual flight, the wind tunnel data may not accurately depict the elevator effectiveness. 2 1.5 δ e =0 δ e =10 δ e =30 1 C L 0.5 0 0.5 1 5 0 5 10 15 20 AOA (deg) Figure 5 2: C L versus angle of attack

27 2 1.5 δ e =0 δ e =30 1 C L 0.5 0 0.5 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 C D Figure 5 3: C L versus C D 5.3 Analytical Inertias The inertia properties of the aircraft are estimated from finite element methods. This analysis uses a CAD model, shown in Figure 5 4, and the ProEngineer [35] software package. The model was created by dimensioning each component of the MAV and assembling them in a flight configuration. An exploded view of the ProEngineer model is shown in Figure 5 5 and a list of components is presented in Table 5.3. The estimated inertia properties are given in Table 5.3. Figure 5 4: Analytical model The model used to calculate these values accounts for the dominant mass elements; however, some small errors remain. The mass of the model is 6.5% less

28 Figure 5 5: Exploded view Table 5 1: AVCAAF vehicle component masses Component Mass (grams) Altimeter Board 27.0 Avionics 61.0 Battery 131.7 Camera Mount 5.0 Camera Transmitter 18.2 Fuselage 47.9 Hatch 6.7 Horizontal Stabilizer 8.9 Motor 50.0 Propeller 9.0 Propulsion Gearing 24.6 RC Receiver 8.0 Servo x3 29.7 Servo Mount 7.0 Speed Controller 12.2 Tail Boom 6.5 Video Camera 11.9 Vertical Stabilizer 10.6 Wing 34.0 Total Mass 509.9 than the actual vehicle because small parts, such as wires and control rods, are excluded from the model. The center of gravity is also in error and lies 0.125 in aft

29 Table 5 2: Analytical inertia properties Property kg m 2 I xx I yy I zz I xy I xz I yz 1.127 e 03 6.604 e 03 7.130 e 03 3.920 e 05 3.798 e 04 9.670 e 06 of the actual position. These errors are quite small so the properties in Table 5.3 are accepted with reasonable confidence. These values will also be used by the simulation for the AVCAAF MAV. 5.4 Analytical Aerodynamics A computational analysis is also used to estimate the aerodynamics of the vehicle. In this case, the aerodynamics are estimated using the Tornado software package [29]. This software uses a vortex lattice method to solve for flow over lifting surfaces. The analysis assumes incompressible flow which is certainly appropriate for the flight regime of a MAV. The analysis also assumes inviscid flow which creates some errors in the resulting solution; however, the inviscid pressures are still represented. The analysis represents the lifting surfaces as a set of panels. The geometry of these panels used for the AVCAAF aircraft is shown in Figure 5 6. The software only considers wings and tails so the fuselage, along with its associated aerodynamic contribution, is not modeled. The reference point about which moments are calculated is also shown in Figure 5 6. A set of static and dynamic derivatives are computed from a central difference expansion about a given flight condition. In this case, a trim state associated with straight and level flight is used for the condition. The output contains almost all of the stability derivatives needed to form a set of full-state linearized

30 3 D Wing configuration Wing z coordinate 0.1 0.05 0 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 Wing y coordinate 0.25 0 0.1 0.2 0.3 Wing x coordinate Figure 5 6: Geometry of panels dynamics. Some parameters, such as {C Lu, C mu, C Du }, were neglected because the aerodynamics were assumed to have no variation with Mach. The dynamic derivatives obtained from the analysis are listed in Table 5 3. Table 5 3: Estimated dynamic derivatives Parameter Value C mq -6.0391 C Yp -0.2920 C Yr 0.7587 C np 0.0190 C nr -0.3061 C lp -0.3857 C lr 0.3178 The derivatives in Table 5 3 are accepted with a moderate level of confidence; however, they are recognized to have some level of error. An obvious source of error includes the lack of aerodynamic contribution associated with the fuselage. Another source of error is the effects of a reflexed airfoil which exists on the physical vehicle but is difficult to model with the software. Tornado also assumes the lifting surfaces to be rigid. Since the AVCAAF vehicle has flexible wings, the data obtained from Tornado will not accurately represent the adaptive washout

31 phenomenon. Finally, thrust is not modeled so the analysis assumed thrust equaled the aligned drag component for the trim condition. 5.5 Model Integration 5.5.1 Wind Tunnel Data Analysis Regression analysis is used to obtain static derivatives of the force and moment coefficients from the wind tunnel data. The data was fit to a set of polynomials representing separate parameters such as lift and pitch moment. Figures 5 7, 5 8, and 5 9 show the actual wind tunnel data and the curves fitted to the data. The resulting equations are in radians are 5.1, 5.2, and 5.3. 1.4 1.2 1 Wind Tunnel Data Regression curve 0.8 0.6 C L 0.4 0.2 0 0.2 0.4 0.6 4 2 0 2 4 6 8 10 12 Angle of Attack (deg) Figure 5 7: Wind tunnel data and fitted curve of C L 0.2 0.18 Wind Tunnel Data Regression curve 0.16 0.14 C D 0.12 0.1 0.08 0.06 0.04 0.02 4 2 0 2 4 6 8 10 12 Angle of Attack (deg) Figure 5 8: Wind tunnel data and fitted curve of C D

32 0.15 0.1 Wind Tunnel Data Regression curve 0.05 0 C m 0.05 0.1 0.15 0.2 4 2 0 2 4 6 8 10 12 Angle of Attack (deg) Figure 5 9: Wind tunnel data and fitted curve of C m C D = 5.5475α 2 1.0587α + 0.0814 (5.1) C L = 10.5381α 2 + 5.0342α 0.1514 (5.2) C m = 49.2329α 3 + 10.7473α 2 0.8158α 0.0002 (5.3) Derivatives of these polynomials were taken with respect to α. Other derivatives that are independent of α were determined by finding a linear relationship with respect to the associated parameter (β, δ r, δ sym, etc). Finally, the derivatives were evaluated at the flight condition used to compute the analytical aerodynamics corresponding to angle of attack of 5 o. This angle of attack is considered a trim condition and is analyzed only to determine aircraft stability at this condition. Some anomalies are noted in the experimental aerodynamics. In particular, the side force shown in Figure 5 10 seems erroneous. The side force during a sweep through angle of attack should be nearly zero but is measured with a significant magnitude. The side force during the longitudinal test was actually the same order of magnitude as that measured during lateral-directional testing that varied rudder and angle of sideslip. This anomaly is not fully explained but may be caused from

33 the asymmetry of the model, the alignment of the model in the tunnel, or the calibration of the sting balance. 0.06 0.04 0.02 C y 0 0.02 0.04 0.06 4 2 0 2 4 6 8 10 12 α (deg) Figure 5 10: Measured values of side force 5.5.2 Aerodynamics The aerodynamics used for analyzing the flight dynamics of the AVCAAF vehicle are extracted from Table 5 4. These values present both experimental estimates and analytical estimates. The values in bold font are the actual values used in formulating the model. The values extracted from Table 5 4 are divided between the experimental estimates and analytical estimates. The experimental estimates would normally be preferred but some anomalies, such as the side force, resulted in higher confidence being associated with some analytical values. The derivatives with respect to β were taken from Tornado data due to the wind tunnel data set containing two values of β at zero and five degrees. The physical mounting of the AVCAAF aircraft in the wind tunnel currently limits the range of side-slip that can be measured. Also, the values of C m α and C L α were not obtained from either experimental analysis or analytical analysis. The value for this parameter was estimated from a published value for a different vehicle [5].

34 Table 5 4: Analytical and experimental stability derivatives Stability Derivative Tornado Wind Tunnel C Lα 4.4486 8.7162 C mα -0.3519-0.3347 C Dα 0.4639-0.2329 C Yα 0 0.4596 C Lo 0.5204-0.2443 C Do 0.0166 0.0836 C mo 0.1112 0.0144 C Lδsym 0.9009 0.7833 C mδsym -1.6461-1.4177 C Dδsym 0.0274-0.1064 C Yδr 0.6929 0.1377 C lδr 0.0284 0.0753 C nδr -0.3239-0.3312 C Yδasy 0.6859 0.1556 C lδasy -0.0280-0.0675 C nδasy -0.2883-0.2920 C Yβ -0.7121 0.4488 C lβ -0.0769 2.2056 C nβ 0.31274 7.6871 The simulation uses the data selected in Table 5 4 as well as the dynamic derivatives in 5 3 to supplement the wind tunnel data. 5.6 Linearized Dynamics These modes are initial estimates of the flight dynamics and must be accepted with caution. The aerodynamics used to generate the model showed discrepancies between experimental and analytical estimates so the model is inherently questionable. The aircraft is undergoing flight testing but the sensor package does not yet measure parameters sufficient for extensive modeling [22]. 5.6.1 Longitudinal The flight dynamics describing longitudinal maneuvers around the trim condition are computed by combining data from Table 5.3, Table 5 3, and Table 5 4. The resulting model represents the linearized dynamics for which longitudinal

and lateral-directional components are decoupled. The dynamics are realized as a state-space expression [31]. 35 u ẇ q θ = A lon u w q θ + B lon δ sym Where the A lon and B lon matrices are comprised of longitudinal derivatives. The longitudinal derivatives are defined in Table 5 5. A lon = X u X w 0 g Z u Z w u 0 0 M u + MẇZ u M w + MẇZ w M q + Mẇu 0 0 0 0 1 0 X δsym Z δsym B lon = M u + MẇZ δsym 0 α = 5 o. The following are the A lon and B lon matrices for the AVCAAF vehicle at A lon = 0.1799 0.4617 0 9.81 1.1198 9.4678 13 0 0.0942 1.8271 3.2945 0 0 0 1 0

36 Table 5 5: Longitudinal derivatives Parameter Value X u X w Z u Z w M u (C Du +2C D0 )QS mu 0 (C Dα +2C L0 )QS mu 0 (C Lu +2C L0 )QS mu 0 (C Lα +2C D0 )QS mu 0 (QS c) C mu u 0 I y Mẇ C m α c 2u 0 QS c u o I y M w C mα (QS c) u 0 I y M q C mq c 2u 0 QS c I y X δsym Z δsym QS C Dδsym m QS C Lδsym m B lon = 0.3828 10.9559 144.5432 0 The eigenvalues of this model relate the natural frequencies and dampings of the flight modes. These properties are presented in Table 5 6 and indicate a pair of oscillatory modes are present in the dynamics. Table 5 6: Longitudinal eigenvalues Mode Frequency (rad/s) Damping phugoid 0.7173 0.0664 short period 7.4846 0.8582

37 The eigenvectors associated with these eigenvalues are given in polar form in Table 5 7. Note these are in terms of non-dimensional states. Table 5 7: Longitudinal eigenvectors Short Period Mode Phugoid Mode Magnitude Phase Magnitude Phase û 0.1518 8.22 o 1.0264 102.05 o ŵ 1.5590 98.00 o 0.0478 69.12 o ˆq 0.0268 149.12 o 0.0026 93.81 o θ 1.0000 0.00 o 1.0000 0.00 o A mode is described as phugoid mode because of its relationship between pitch angle and airspeed. The mode has a small natural frequency and is lightly damped. As such, the mode has characteristics which are classically associated with a phugoid mode. The remaining mode is described as a short period mode. This mode has a close relationship between angle of attack and pitch rate. Also, the natural frequency of this mode is an order of magnitude higher than the phugoid mode. Consequently, this mode is similar in nature to the classic definition of a short period mode. 5.6.2 Lateral-Directional The flight dynamics associated with lateral-directional maneuvers around trim are also computed using data from Table 5.3, Table 5 3, and Table 5 4. The dynamics are again realized as a state-space expression [31]. β ṗ ṙ φ = A lat β p r φ + B lat δ asy δ rud

38 Where the A lat and B lat matricies are comprised of lateral directional derivatives, defined in Table 5 8. A lat = Y β u 0 Y p u 0 ( ) 1 Yr u 0 g cos θ 0 u 0 L β L p L r 0 N β N p N r 0 0 1 0 0 B lat = 0 L δasy N δasy Y δr u 0 L δr N δr 0 0 α = 5 o. The following are the A lat and B lat matrices for the AVCAAF vehicle at 0.7661 0.0074 0.0578 0.7546 111.8602 13.1591 10.8419 0 A lat = 157.6609 0.2247 3.6182 0 0 1 0 0 0 0.7455 34.0289 109.5044 B lat = 147.1914 166.9433 0 0 The modal parameters are computed from the eigenvalues and eigenvectors of A lat. The natural frequencies and dampings resulting from the eigenvalues are given in Table 5 9. In this case, the lateral-directional dynamics have a divergence, a convergence, and an oscillatory mode.

39 Table 5 8: Lateral directional derivatives Parameter Value Y β Y p Y r L β L p L r N β N p N r Y δr L δasy L δr QSC yβ m QSbC yp 2mu 0 QSbC yr 2mu 0 QSbC lβ I xx QSb 2 C lp 2I xx u 0 QSb 2 C lr 2I xx u 0 QSbC nβ I zz QSb 2 C np 2I zz u 0 QSb 2 C nr 2I zz u 0 QSC yδr m QSbC lδasy I xx QSbC lδr I xx QSbC nδasy N δasy N δr I zz QSbC nδr I zz Table 5 9: Lateral-directional eigenvalues Mode Frequency (rad/s) Damping spiral 2.0888-1.0000 dutch roll 5.7078 0.4526 roll 14.4649 1.0000 The eigenvectors associated with the divergence and convergence are given in Table 5 10. Note these are in terms of non-dimensional states.

40 Table 5 10: Lateral-directional eigenvectors Roll Mode Spiral Mode Magnitude Phase Magnitude Phase ˆv 0.0045 180 o 0.0057 0 o ˆp 0.3392 180 o 0.0220 0 o ˆr 0.0268 0 o 0.0490 0 o φ 1.0000 0 o 0.4502 0 o ψ 0.0791 180 o 1.0000 0 o The stable mode has obvious characteristics associated with the classical definition of roll mode. The response of this mode is predominately a roll motion with only minor variation in angle of sideslip or yaw. The unstable mode is characterized as a spiral divergence but with some reservations. The eigenvector indicates the response resembles a classic spiral mode in that excitation of this mode is essentially yaw with some roll. Conversely, the magnitude of the eigenvalue is quite large to be considered a spiral pole. The remaining mode relates to a dutch roll dynamic as evidenced by its eigenvector in Table 5 11. The motion associated with this mode is a complex relationship between yaw and roll and angle of sideslip. The phases and magnitudes slightly differ from the motions of large aircraft; however, the dynamics are clearly dutch roll. Table 5 11: Lateral-directional eigenvector Dutch Roll Mode Magnitude Phase ˆv 0.0160 117.02 o ˆp 0.1338 116.91 o ˆr 0.1446 162.61 o φ 1.0000 0 o ψ 1.0801 45.70 o Also, the natural frequency associated with the dutch roll agrees with a basic trend. Namely, the magnitude of the natural frequency should increase as wing

41 span decreases. Figure 5 11 indicates the natural frequency estimated for the AVCAAF aircraft lies along a reasonable curve with values from other aircraft. 25 15 cm Micro Air Vehicle Frequency (rad/sec) 20 15 10 Black Widow 5 AVCAAF DragonFly UAV F 16 Boeing 747 0 10 1 10 0 10 1 10 2 Wingspan (m) Figure 5 11: Variation in dutch roll frequency 5.7 Modeling Results This chapter has shown the development of the linearized longitudinal and lateral dynamics of a micro air vehicle using wind tunnel and computational data. The wind tunnel data did not include the dynamic derivatives and included some spurious data. The software package Tornado was used to supplement the wind tunnel data to complete the model. The linearized model has modal properties that are similar to standard aircraft modes. The spiral mode was analyzed as unstable. This instability has not been confirmed by flight testing due to the difficulty of recognizing spiral divergence during flight. It should be stated again that the dynamics presented were for one flight condition. The simulation will encompase a wider range of flight conditions by using functions to solve for the forces and moments the MAV experiences each time step. There is not much confidence in this model accurately characterizing the actual AVCAAF vehicle. This is due to the conflicting results obtained from the Tornado and wind tunnel data. Some sources of inaccuracies include computational

data based on inviscid flow, inaccurate modeling of the flexible wings, difficulty in modeling the reflexed airfoil, and spurious experimental data. 42

CHAPTER 6 AVCAAF SUBSYSTEMS The subsystems designed specifically for the AVCAAF aircraft were the Sensors block, Controller block, and Actuator block in Figure 2 1. This chapter discusses the design of these subsystems. 6.1 Sensor Subsystem The Sensors block from Figure 2 1 is comprised of three subsystems for the AVCAAF vehicle. These subsystems are the Camera, GPS, and Altitude blocks shown in Figure 6 1. Figure 6 1: AVCAAF sensors subsystem 6.1.1 Camera Subsystem The camera subsystem emulates the vision output the controller receives from the goundstation for horizon analysis. The subsystem calculates the pitch 43

44 percentage seen by the camera based on pitch angle, roll angle, and camera view angle. Pitch percentage is the percent of ground seen in the image. It is assumed that when the pitch angle and roll angle are zero the pitch percentage is 50%. This equation can be further modified to account for altitude and distance to the horizon. 6 2 shows a side view of the MAV capturing an image with its camera. This case assumes there is no roll angle. Here γ is the camera view half-angle, θ is the pitch angle, A A is the image plane, and D is the length from the camera to the image plane. L is the length from the camera to the image plane along the camera half-angle γ. h p is the percentage of ground seen in the image plane. The geometric identities 6.1 and 6.2 can be observed. Figure 6 2: Image projection and pitch percentage L = D cos (γ) (6.1) h p = L sin (γ) D tan (θ) (6.2) 6.2 can be expanded into 6.3. h p = D (tan (γ) tan (θ)) (6.3)

45 The pitch percentage can be found be dividing h p by the image plane length, shown in 6.4. The controller for the AVCAAF vehicle requires this value to be between 0 and 1, where 1 correlates to a pitch percentage of 100% ( ground completely fills the image). h p pitch % = 2D tan (γ) D (tan (γ) tan (θ)) = 2D tan (γ) = 1 tan (θ) 2 2 tan (γ) (6.4) 6.4 is only valid for the case where the roll angle is zero. The equation for pitch percentage is more complex when roll is added. There are two different general cases to consider when calculating pitch percentage with roll added. These cases are when the ground area seen by the camera is either a triangle or trapezoid, as shown in Figure 6 3. Figure 6 3: Triangular and trapezoidal ground areas Figure 6 3 also shows the image taken from the camera is not circular. The camera in the AVCAAF transmits a standard NTSC video signal. This video format will also make the pitch percentage calculation more complex. The standard NTSC signal has a ratio of 3:4 for the height and length of the image respectively. The pitch percentage will now be calculated as the percent of ground in the rectangular image plane.