THE PENNSYLVANIA STATE UNIVERSITY SCHREYER HONORS COLLEGE

DEPARTMENT OF AEROSPACE ENGINEERING

DESIGN AND TRIM OPTIMATION OF A FLYING WING UAV

JOHN F. QUINDLEN Spring 2010

A thesis submitted in partial fulfillment of the requirements for a baccalaureate degree in Aerospace Engineering with honors in Aerospace Engineering

Reviewed and approved* by the following:

Jack W. Langelaan Assistant Professor Thesis Supervisor

Mark D. Maughmer Professor Honors Advisor

George A Lesieutre Professor Head of Aerospace Engineering

* Signatures are on file in the Schreyer Honors College.

i

ABSTRACT

Camber changing plain flaps offer tailless sailplanes two potential benefits: a lower minimum speed, ideal for climbing in thermals, and a higher lift to drag ratio to improve glideslope in between thermals. With these two goals in mind, an inboard plain flap is designed for an existing flying wing unmanned aerial vehicles (UAV). The flap places new demands on the aircraft autopilot which must be alleviated with a new flap module. In order to do so, the aerodynamic derivatives of the original aircraft are linearized. Then an inboard flap is designed to require little change in elevon deflection to trim the aircraft for any flap deflection. Next, a nonlinear longitudinal dynamics model is created using the new linearized aerodynamic derivatives of the flapped aircraft. These nonlinear dynamics are then linearized to find the transfer functions of airspeed to pitch angle and pitch angle to elevon deflection. From these two equations of motion, an airspeed controller is designed and optimized using root locus method for a proportional-integral (PI) controller and a proportional-integral-derivative (PID) controller. Just as expected, the inboard flap improves the aircraft performance. After implementing the flap, the lift to drag ratio of the flapped aircraft improves slightly versus the original aircraft. The new configuration offers an average increase in lift to drag ratio of 1% at cruise from 10 m/s to 15 m/s with a maximum improvement of 2% at 12 m/s. Likewise, the aircraft sees roughly a 22% decrease in minimum airspeed at a 15° angle of attack using a 28° flap deflection versus the original aircraft at the same angle of attack. Further simulation results of the new flapped configuration are then analyzed and compared with the original aircraft. The flap module performs well within the cruise range but is of limited effectiveness at either end of the airspeed range.

ii TABLE OF CONTENTS

LIST OF FIGURES ...... iv

LIST OF TABLES...... vi

ACKNOWLEDGEMENTS...... vii

Chapter 1 Introduction ...... 1

Unmodified Zagi Aircraft ...... 2 Aircraft Configuration...... 3

Chapter 2 Flap Design ...... 5

Flap Optimization ...... 8 Aircraft Configuration...... 11 Flap Scheduling...... 12

Chapter 3 Nonlinear Longitudinal Dynamics Modeling...... 18

Aerodynamic Coefficients ...... 19 Nonlinear Dynamics ...... 21 Nonlinear Kinematics ...... 22

Chapter 4 Airspeed Control ...... 25

Transfer Functions ...... 25 Controllers...... 26 Implementation ...... 26

Chapter 5 Flap Module ...... 28

Flap and Elevon Scheduling...... 28 Overall Architecture...... 30

Chapter 6 Simulation Results...... 31

Cruise Range...... 31 Low Speed...... 35 High Speed...... 39 Challenges...... 43

Chapter 7 Conclusion...... 45

References...... 48

Appendix A Aircraft Parameters ...... 49 iii Appendix B Nomenclature ...... 50

iv

LIST OF FIGURES

Figure 1-1: Zagi-5C thermal/hand launched R/C glider. Source: Trick R/C...... 3

Figure 1-2: Top-view of the unmodified Zagi configuration...... 4

Figure 2-1: Straight trailing-edge flap configuration...... 6

Figure 2-2: Constant flap chord to section chord ratio flap configuration...... 7

Figure 2-3: Constant chord flap configuration...... 7

Figure 2-4: The SWIFT flying wing sailplane equipped with a self-trimming flap. Source: Kroo and Beckman, Stanford University...... 9

Figure 2-5: Top-view of the new Zagi configuration...... 12

Figure 2-6: Process used to calculate aircraft parameters for flap scheduling...... 13

Figure 2-7: L/D against airspeed for a range of flap deflections...... 14

Figure 2-8: Best L/D versus airspeed for original and flapped Zagi...... 15

Figure 2-9: Minimum airspeed versus flap deflection for the original and flapped Zagi...... 16

Figure 2-10: Optimal flap deflection against airspeed for minimum speed, best L/D, and the polynomial fit for those two criteria...... 17

Figure 3-1: Architecture of the nonlinear longitudinal dynamics model...... 18

Figure 3-2: Architecture of the aerodynamic coefficients block...... 20

Figure 3-3: Architecture of the nonlinear dynamics block...... 22

Figure 3-4: Architecture of the nonlinear kinematics block...... 24

Figure 4-1: Airspeed control architecture with PI and PID controllers...... 27

Figure 5-1: Flap and elevon scheduling block...... 29

Figure 5-2: Overall control architecture of the entire flap module...... 30

Figure 6-1: Airspeed versus time plot with a 1 m/s increment and decrement in desired airspeed at cruise...... 32 v Figure 6-2: Pitch angle versus time plot with a 1 m/s increment and decrement in desired airspeed at cruise...... 32

Figure 6-3: Flap deflection versus time plot with a 1 m/s increment and decrement in desired airspeed at cruise...... 33

Figure 6-4: Elevon deflection required to trim versus time plot with a 1 m/s increment and decrement in desired airspeed at cruise...... 33

Figure 6-5: Glideslope plot with a 1 m/s increment and decrement in desired airspeed at cruise...... 34

Figure 6-6: Airspeed versus time plot with a 1 m/s decrement in desired airspeed at lower speeds...... 36

Figure 6-7: Pitch angle versus time plot with a 1 m/s decrement in desired airspeed at lower speeds...... 37

Figure 6-8: Flap deflection versus time plot with a 1 m/s decrement in desired airspeed at lower speeds...... 37

Figure 6-9: Elevon deflection required to trim versus time plot with a 1 m/s decrement in desired airspeed at lower speeds...... 38

Figure 6-10: Glideslope plot with a 1 m/s decrement in desired airspeed at lower speeds....38

Figure 6-11: Airspeed versus time plot with a 1 m/s increment in desired airspeed at higher speeds...... 40

Figure 6-12: Pitch angle versus time plot with a 1 m/s increment in desired airspeed at higher speeds...... 41

Figure 6-13: Flap deflection versus time plot with a 1 m/s increment in desired airspeed at higher speeds...... 41

Figure 6-14: Elevon deflection required to trim versus time plot with a 1 m/s increment in desired airspeed at higher speeds...... 42

Figure 6-15: Glideslope plot with a 1 m/s increment in desired airspeed at higher speeds. ..42

Figure 7-1: Initial construction of the Zagi UAV incorporating the inboard trailing-edge flaps...... 46

Figure 7-2: Servos and electronics used on the Zagi test aircraft...... 47

vi

LIST OF TABLES

Table 2-1: Allowable span for self-trimming flap against aspect ratio. Source: Pitkin and Meggin, NACA...... 10

Table 4-1: PI and PID controller gains...... 26

Table A-1: Linearized aerodynamic derivatives of both the original and flapped Zagi UAVs...... 49

Table A-2: Additional properties of both the original and flapped Zagi UAVs...... 49

vii

ACKNOWLEDGEMENTS

I would like to thank my thesis advisor, Dr. Jack Langelaan, for all his help and guidance

on my thesis research. He helped me pick a topic that matched my varying interests and that

really made my research an incredible and very satisfying learning experience. Even though it

may not have seemed like it at some points, I definitely found my thesis research to be a positive

and enjoyable experience. This genuine interest in my research really made me want to work on

my thesis, rather than simply have to work on it. I am also very thankful for his guidance on my

research. He made sure I avoided a number of hang-ups and kept me on track to finish. Without

him, I probably would have wasted large amount of time on trivial problems.

I would also like to thank my honors advisor, Dr. Mark Maughmer, for his help. I certainly “picked his brain” on multiple occasions, especially when I was going through the flap design and optimization process. His library of stability and control papers were extremely useful for analyzing the flap parameters and researching self-trimming flaps.

Lastly, I would like to thank the Penn State faculty, staff, and graduate students who gave me input and feedback, no matter how small. Anjan Chakrabarty, Nate Depenbusch, Sean Quinn

Marlow, Ben Pipenburg, and Shane Tierney all gave me valuable advice that I certainly put to good use. 1

Chapter 1

Introduction

Dr. Jack Langelaan’s Autonomous Vehicles Laboratory (AVL) currently possesses a number of small, unpowered flying wing gliders. Some of these are instrumented to include a navigation capable autopilot, while others are yet to be built. The particular aircraft platform used is Trick R/C’s Zagi-5C thermal/hand launched glider (Zagi) designed for slope soaring.

Penn State AVL would like to use these small aircraft for light thermal soaring research, which requires a number of performance parameters this flying wing sailplane is not optimized for.

Thermal soaring places unique constraints on an aircraft in that it needs two different foci of flight for the most effective use of the aircraft. Thermal soaring uses updrafts in the air to allow an aircraft flying through the thermal to gain altitude. This updraft gives the aircraft a climb rate and enables it to stay aloft much longer. In order to most effectively use these thermals, an aircraft should have a low speed and sink rate so that it spends as much time as possible in the thermal and so it can circle around in the thermal to gain more and more altitude.

When flying in between thermals, a glider should have a higher lift-to-drag ratio (L/D) to improve the glideslope. The best L/D should occur at a higher speed than the lower speeds for thermaling, so the aircraft cannot be purely optimized for low speeds.

Even though the speed of best L/D improves glideslope, the optimum interthermal glide speed actually occurs at a higher airspeed. According to Thomas [TH-99], the airspeed for best

L/D and the associated minimal loss in altitude does not constitute the optimum speed for interthermal flight; rather, the best average cross-country airspeed should instead occur at a higher airspeed that trades off altitude for faster times between the thermals. At this optimum glide speed, the aircraft reaches the thermal at a lower altitude than the slower airspeed, but at a 2 much faster time, leaving more time to climb in the thermal. Although the optimum interthermal

speed occurs at a slightly higher airspeed, the speed of best L/D is used to constitute the criteria

for improving interthermal flight performance. The focus is placed on minimizing altitude lost,

not best average speed.

A trailing-edge flap offers the best solution to improving the performance of the Zagi for

thermal soaring research. A flap will improve the aircraft L/D for interthermal flight and will

dramatically decrease the minimum speed for more effective use of thermals. A single inboard

flap also limits the redesign effort needed to improve the Zagi for the research. The emphasis of

the project is on improving the existing Zagis, not designing an entirely new aircraft. A

completely new design could certainly possess better performance, but this new design would

also take a large amount of effort and would not make use of what is currently available. Small

modifications to the aircraft are easy to reproduce on multiple aircraft and keep the scope of the

project on a manageable scale. This trailing-edge flap expands the aircraft performance range

while minimizing the amount of time and effort needed to modify the aircraft.

Unmodified Zagi Aircraft

The Zagi is a small flying wing radio-controlled (R/C) aircraft made of expanded polypropylene foam. Since Penn State AVL currently owns a few of these aircraft, its instant availability enables multiple test beds for equipping the flap and measuring aircraft mass properties. The availability also brings a familiarity with the model aircraft, its construction, and its flight behavior. A picture of the stock Zagi glider is seen in Figure 1-1. Penn State AVL has

modified the R/C Zagi to include an ArduPilot autopilot system with add-on boards for a dynamic

pressure sensor and a transmitter to send out telemetry data. The ArduPilot is an Arduino based 3 microcontroller developed from C++ to enable rapid development of autopilot components, such

as a flap module for the aircraft.

Figure 1-1: Zagi-5C thermal/hand launched R/C glider. Source: Trick R/C.

Aircraft Configuration

The Zagi UAV configuration is a rather simplistic flying wing design. The aircraft has an aspect ratio of 5 with a total wing span of 1.2 m (4 ft) and a wing area of 0.29 m2 (3.2 ft2). The aircraft has a straight taper with a taper ratio of 0.52 and a mean aerodynamic chord (MAC) of

0.25 m (0.82 ft). With all the additional modifications, the Zagi UAV weight increased from 4.4

N (1 lb) to 8.9 N (2 lb). According to Trick R/C [TK-09], the Zagi airfoil is a Zagi 101.3 shape, but it can be accurately approximated as Martin Hepperle’s MH-64 airfoil design [HM-09] for tailless R/C aircraft.

One particularly interesting aspect of the Zagi is the control system consisting of two symmetrical elevons. These elevons run nearly full span on the stock Zagi-5C and are assumed to run full span on Penn State Zagi UAV. The elevons have a constant chord of 0.04 m (1.5 in.) across the entire span, meaning the elevon chord to wing chord ratio increases linearly from 12% at the root to 23% at the tip. The wing and elevon planform is illustrated in Figure 1-2. These elevons are controlled by one Hitec HS-81 micro servo each. 4

Figure 1-2: Top-view of the unmodified Zagi configuration.

5

Chapter 2

Flap Design

The trailing-edge plain flap on the Zagi is designed to improve both the L/D at cruise and the minimum airspeed possible. To meet the additional goal of ease of construction, the flap consists of a single segment on each side controlled by one micro servo, similar to the existing control system. The single section limitation prevents a system of camber flaps that could be differentially deflected to produce an elliptical lift distribution. This would allow much better

L/D performance than a single plain flap could produce, but it would also require extensive modifications to the Zagi airframe to incorporate more servos, adding weight near the wingtips.

To avoid such complications, the flap is limited to a single segment on each side of the wing but radical improvements in L/D will not be possible.

With the single segment flap limitation, multiple configurations using a single segment flap are considered. The elevon chord near the wing tips is left unchanged and only the inboard flap chord is altered. One design considered is a straight trailing-edge flap, pictured in Figure 2-

1. This flap takes the chord-wise x-location of the flap trailing-edge at the outboard edge of the flap and keeps that x-value constant across the flap until it reaches the root chord. The leading- edge x-location of the flap changes while the trailing-edge x-location remains fixed, causing flap chord to increase closer to the root section. Another flap design considered is a constant chord ratio flap, seen in Figure 2-2. Starting at specified span-wise y-location, the flap chord to wing section chord ratio is kept constant. The flap chord will increase closer to the root section, but at the same rate as section chord increases, so the overall ratio will remain constant. The third flap configuration is a simple break in the existing elevon, found in Figure 2-3. This flap follows the exact same shape of the current single elevon control surface, but breaks the elevon to produce a 6 constant chord inboard flap and outboard elevon. The break simply places a cut in the existing

0.04 m chord elevon at a specified y-location along the wing.

Figure 2-1: Straight trailing-edge flap configuration.

7

Figure 2-2: Constant flap chord to section chord ratio flap configuration.

Figure 2-3: Constant chord flap configuration. 8

From these flap designs, the constant chord flap configuration offers the best solution for a plain flap. Although any flap deflection will alter the lift distribution, the simple break will minimize the amount of change in lift distribution at small flap deflections. More importantly, this design also requires the least amount of redesign effort to implement the flap. The goal of the project is to enhance an existing flying wing design, not create a completely new one. This flap design does not require any changes in the airfoil shape along the span of the flap. Likewise, it also minimizes the change in aircraft parameters and mass properties. The other designs would have required the construction of an entirely new wing rather than simple cuts in the shape, as the pre-cut EPP foam cannot be easily modified.

Flap Optimization

The flap is designed to require as little elevon deflection as possible in order to trim the aircraft. Possible inboard flap configurations exist which require no elevon deflection to trim the aircraft. These flaps, called self-trimming flaps, automatically trim the aircraft at a new airspeed resulting from a certain flap deflection. Multiple flying wing sailplanes have used this flap design before, including the Flair 30 and SWIFT. According to Kroo and Beckman [KB-91], the SWIFT gets its name from the control surface (Swept Wing Inboard Flap for Trim) and was specifically designed to include a self-trimming flap in order to operate as a foot-launched sailplane. The

SWIFT in flight is pictured in Figure 2-4.

9

Figure 2-4: The SWIFT flying wing sailplane equipped with a self-trimming flap. Source: Kroo and Beckman, Stanford University.

The design of a self-trimming flap requires careful consideration of the aerodynamics of the 3-D wing and flap system. From Donlan’s method [DC-44], a moment neutral flap is created by setting the center of pressure of the additional lift due to the flap to the same x-location as the aerodynamic center of the unflapped wing, depicted below:

= xx , flapcp ,wingac (2-1)

As the flap is deflected, the additional lift from the flap will act at the wing aerodynamic center and cause no additional pitching moment about the wing aerodynamic center. For trimmed flight, the lift force must equal the weight of the aircraft, so the lift force remains constant. The pitching moment coefficient about the aerodynamic center will change with flap deflections, but the total pitching moment about the aerodynamic center will remain constant. The magnitude of the pitching moment coefficient will change as the flap deflects, but the airspeed will also change with flap deflection. The total pitching moment about the aerodynamic center will remain 10 constant if the change in pitching moment coefficient due to flap deflection is balanced with the change in airspeed due to flap deflection.

1 = ρ 2 ScCvM (2-2) ac 2 ,acm

= + ( − )LxxMM cg ac ,wingaccg (2-3)

The equation considered for the dimensional pitching moment about the center of gravity of the aircraft is given in Eq. (2-3). Since the lift force, pitching moment about the aerodynamic center, and the lift force moment arm remain constant, there will be no change in the total pitching moment about the center of gravity of the flying wing.

This flap design method will only work for a certain range of aspect ratios, sweep angles, and taper ratios. If the aspect ratio or sweep angle is too low, then a self-trimming flap may not be possible. Table 2-1 shows different experimental runs on tailless aircraft split flap configurations done by Pitkin and Meggin [PM-44] for the National Advisory Committee for

Aeronautics (NACA). According to the data, the Zagi sweep angle is well within the acceptable range, but the aspect ratio of 5 is below their experimental minimum of 5.5 allowed for a self- trimming flap. This means that an ideal self-trimming flap does not exist for the Zagi unless changes are made to increase its aspect ratio.

Table 2-1: Allowable span for self-trimming flap against aspect ratio. Source: Pitkin and Meggin, NACA.

11 Since the Zagi aspect ratio is too small to allow a true self-trimming flap, an approximate formula is used to design a flap with minimal effect on pitching moment. This method for approximating the design of a self-trimming flap is described by Nickel and Wolfhardt [NW-94].

This method uses the same theory to keep pitching moment constant by setting the center of pressure of the additional lift due to the flap at the aerodynamic center of the entire aircraft, but it also approximates several parameters to express the center of pressure as a function of flap chord and span. The main assumption is that both the wing and flap sections have constant chord and constant sweep.

bF cF x , flapcp tan 0 ++Λ= cQ F 2 4 (2-4)

Equation (2-4) writes x-location of the flap aerodynamic center as a function of flap span, leading-edge sweep, MAC of the flap section, and a variable Q. Q is the factor by which the addition lift due to the flap acts behind the quarter chord line and it is a direct function of flap-to- chord ratio. The span of the flap section can now be found by setting xF = xE, the x-location of the total wing aerodynamic center.

Aircraft Configuration

Using the approximate method for the flap design, the resulting flap covers 61% of the

span of the wing, or 0.37 m (1.2 ft) for each half wing, and shortens the span of each elevon to

0.24 m (0.78 ft). The wing chord at the flap break is 0.22 m (0.74 ft), meaning the flap ratio

increases from 12% at the root to 17% at the break. Everything else about the aircraft remains

unchanged from the original unflapped aircraft configuration. The new aircraft configuration

with the flap included is depicted in Figure 2-5.

12

Figure 2-5: Top-view of the new Zagi configuration.

Flap Scheduling

The flap design is then modeled in the WingsX nonlinear lifting-line program [WX-09] provided by Nicholas Alley. The aircraft model and the WingsX program are then run in the

MATLAB programming environment [MT-09] to quickly execute multiple simulation run cases.

In order to generate a large set of performance results, WingsX is executed for a wide range of angles of attack. MATLAB starts WingsX at α = -4° and iterates angle of attack up to 15°, with a step size of 0.5°. Similarly, the flap deflection angle is also varied from δf = -5° to 30° using a step size of 1°. Given a specific angle of attack and flap deflection pair, WingsX computes the trimmed state of the aircraft at each operating point to create a matrix of parameters for each individual run case. Smaller step sizes could have been used for more precision, but this would greatly increase the execution time of WingsX and MATLAB. The operating points are then related to a specific airspeed, rather than angle of attack and flap deflection, in order to plot the 13 aircraft performance as a function of airspeed. Figure 2-6 demonstrates this process used to calculate L/D and minimum speed. Figure 2-7 plots the L/D performance for each flap deflection against airspeed and merges the best L/D envelope from the range of flap deflections for each airspeed.

Figure 2-6: Process used to calculate aircraft parameters for flap scheduling.

14

Figure 2-7: L/D against airspeed for a range of flap deflections.

The new flap design does improve the L/D performance at cruise, as expected. The

flapped Zagi improves or matches the best L/D performance of the original aircraft at nearly

every airspeed data point, in addition to lower airspeeds at which the original Zagi cannot trim.

The best L/D performance at a given airspeed is found from all the possible flap deflections at

that airspeed and merged into a composite curve for the flapped Zagi best L/D envelope. The

graphs of both the composite best L/D envelope of the flapped Zagi and the best L/D of the

original Zagi are plotted in Figure 2-8. In the cruise regime from 10 m/s to 15 m/s, the flap

increases the L/D ratio by an average of 1% with a maximum improvement of nearly 2% at 12

m/s.

15

Figure 2-8: Best L/D versus airspeed for original and flapped Zagi.

Likewise, the new flap also improves the minimum airspeed of the Zagi. With a 28° flap deflection at a high angle of attack of 15°, the flapped Zagi is able to fly and trim at 6.87 m/s

(13.35 knots) airspeed, while the original Zagi can only reach an airspeed of 8.84 m/s (17.20 knots). This difference represents a 22% decrease in the minimum airspeed with the flapped

Zagi. The entire curve of minimum airspeed against flap deflection is illustrated in Figure 2-9,

with the unflapped Zagi minimum airspeed also included. For comparison, the flapped Zagi

begins to show improvement in minimum airspeed starting at a flap deflection of -3° until it

begins to flatten out at 6.87 m/s.

16

Figure 2-9: Minimum airspeed versus flap deflection for the original and flapped Zagi.

Using the composite L/D curve from Figure 2-6 and the minimum airspeed graph in

Figure 2-9, optimal flap deflections for each performance criteria can be determined. Figure 2-10 plots the optimal flap deflection against airspeed for both minimum airspeed and best L/D.

Because the aircraft can basically only trim for one particular flap setting towards the lower end of the airspeed range, there is little to no difference between the curve for minimum speed and best L/D at this low speed range. They nearly match at this speed range, so the optimal flap deflection for minimum speed can be approximated using the plot of best L/D. The minimum speed curve does begin to diverge from the L/D curve around 7.5 m/s, but the focus has shifted to

L/D performance at that point. A polynomial fit to the best L/D curve will therefore approximate the optimal flap deflection for both performance criteria, not just the best L/D ratio.

17

Figure 2-10: Optimal flap deflection against airspeed for minimum speed, best L/D, and the polynomial fit for those two criteria.

18

Chapter 3

Nonlinear Longitudinal Dynamics Modeling

With the new flapped Zagi configuration, a nonlinear longitudinal dynamics model is created in MATLAB and the Simulink modeling and simulation program [SK-09] to analyze the flight behavior of the Zagi at a given set of flap and elevon deflections and a state vector. This longitudinal dynamics model then determines the new state of the aircraft from these initial inputs. The longitudinal equations of motion of the aircraft are taken from Langelaan [LJ-08] with the wind vector set to zero. These equations of motion use linear aerodynamic coefficients to calculate nonlinear dynamics and kinematics. The overall architecture of the nonlinear longitudinal dynamics model is picture in Figure 3-1.

Figure 3-1: Architecture of the nonlinear longitudinal dynamics model.

19 Aerodynamic Coefficients

The aerodynamic coefficients for lift, drag, and pitching moment are written in terms of

the linearized non-dimensional aerodynamic stability and control derivatives. These aerodynamic

stability and control derivatives were found using WingsX in the flap-scheduling runs. In order

to linearize the derivatives, they need to be set to a specific control point. The point used to

linearize the derivatives is the angle of attack for the best L/D, which corresponds to the best

glide slope. From the linearization of the wing analysis about the angle of attack for best L/D, the

lift curve slope, CLα, is a constant value and does not change with changing angle of attack, hence

∂CLα/∂α is zero, as is the case with all the other derivatives. The aerodynamic coefficients are calculated using the equations given below:

c CCC α ++= ++ CCQC δδ 0 LLL α LQ δe δ f fLeL 2V (3-1)

= )( + δ + CCCfC δ (3-2) δe δ f fDeDLLDD

c CCC α ++= ++ CCQC δδ (3-3) M M 0 Mα 2V MQ δe δ f fMeM

The parameter Cf LLD )( in Eq. (3-2) is a fourth order polynomial function used to calculate the drag coefficient from a given lift coefficient and is derived from relating the untrimmed lift coefficients to corresponding drag coefficients. The function uses CL0 and CLαα to

calculate the drag without contributions from CDδe and CDδf.

The linearized aerodynamic derivatives for the flapped Zagi used in dynamic modeling

are presented in Table A-1 in Appendix A. For comparison, these derivatives are listed alongside

the original, unflapped aerodynamic derivatives. Eqs. (3-1), (3-2), and (3-3) also apply to the

unflapped Zagi aircraft with the aerodynamic derivatives CLδf, CDδf, CMδf all set to zero and different values for CLδe, CDδe, and CMδe. 20 The equations for the aerodynamic coefficients are then modeled in Simulink as a subsystem of the nonlinear longitudinal dynamics block. The aerodynamics coefficients block takes both the elevon and flap deflections as input, as well as the current state vector of pitch rate, airspeed, pitch angle, and angle of attack. It then outputs the non-dimensional lift, drag, and pitching moment coefficients. Figure 3-2 illustrates the architecture of the aerodynamic coefficients block in Simulink.

Figure 3-2: Architecture of the aerodynamic coefficients block. 21 Nonlinear Dynamics

The nonlinear dynamics are written about the velocity vector in the stability axes using the equations presented by Langelaan [LJ-08]. Because the Zagi UAV is unpowered, the thrust and the corresponding thrust coefficient are set to zero, simplifying the equations for the derivatives of airspeed, angle of attack, and pitch rate to the following equations:

S  qV gC −−−= αθ )sin( (3-4) m D

qSC g α Q L −+−= αθ )cos( (3-5) VVm

qScC Q = M (3-6) I yy

The aircraft dynamics are considered nonlinear due to the sine and cosine terms in Eqs.

(3-4) and (3-5). The dynamics equations depend upon the previous aerodynamic coefficients, as well as the current state vector, mass properties, and vehicle parameters. These mass and vehicle properties are found in Table A-2 in Appendix A.

The nonlinear dynamics equations are also modeled in Simulink as a subsystem of the nonlinear longitudinal dynamics block. Figure 3-3 illustrates the architecture of the nonlinear dynamics block. The nonlinear dynamics block takes the lift, drag, and pitching moment coefficients calculated in the aerodynamic coefficients block as input. It calculates the rate of change in airspeed, angle of attack, and pitch rate from the coefficients and the current state variables, and integrates these derivatives using the initial state vector to find the new state variables.

22

Figure 3-3: Architecture of the nonlinear dynamics block.

Nonlinear Kinematics

Since the wind vector is set to zero, the aircraft kinematics from Langelaan [LJ-08] can now be written in terms of airspeed, pitch angle, and angle of attack as: 23

 = Vx θ − α )cos( (3-7)

 = Vz θ − α )sin( (3-8)

θ = Q (3-9)

As with the dynamics, the kinematic equations are considered nonlinear due to the sine and cosine terms in Eqs. (3-7) and (3-8). The kinematic equations calculate the rate of change of

the aircraft horizontal and vertical position based upon airspeed and flight path angle. Equation

(3-9) defines the variable Q as the pitch rate of the aircraft.

As with the previous sets of equations, the kinematic equations are modeled in Simulink as a subsystem of the nonlinear longitudinal dynamics block. The subsystem takes both the current and initial state vectors as input. The current state variables are used to calculate the

derivatives, while the initial state variables are used in the integration of these derivatives to find

horizontal and vertical position and pitch angle. Figure 3-4 depicts the nonlinear kinematics

block in Simulink.

24

Figure 3-4: Architecture of the nonlinear kinematics block. 25

Chapter 4

Airspeed Control

The nonlinear longitudinal dynamics model analyzes the motion of the Zagi based upon the current state of the aircraft. In order to calculate the new state, the dynamics model needs the current trim condition of the aircraft. This trim condition is determined by the elevon deflection, so the airspeed is controlled by a nested loop for elevon deflection. In the nested loop, elevon deflection controls pitch in the inner loop, whereas pitch controls airspeed in the outer loop.

Transfer Functions

Given the longitudinal dynamics model for the flapped Zagi, the transfer functions from airspeed to pitch angle and from pitch angle to elevon deflection are required to design an airspeed controller. These transfer functions are found by linearizing the longitudinal dynamics and then converting these linear equations from state space. By using small angle approximations, the transfer function for airspeed to pitch angle is written as a simple first-order equation:

gV −= θ s (4-1)

Meanwhile, the transfer function for pitch angle to elevon deflection is not as straightforward and is instead written as the fourth-order function given by:

θ 177 2 ss −−− 5.1511723 = δ 4 3 2 ssss ++++ 117657.8722781.11 e (4-2) 26 Controllers

The airspeed control block includes two different controllers to find the elevon deflection given the commanded airspeed. First, airspeed to pitch angle requires a proportional-integral (PI) controller given its plant. The use of a PI controller rather than proportional controller increases the type of the transfer function to a type 2 function with zero steady state error to a step input.

The PI controller gains are found using MATLAB’s rltool function [MT-09] with Eq. (4-1) to minimize rise time and dampen overshoot. The gains for the PI controller are found in Table 4-1.

Unlike the airspeed to pitch controller, the pitch angle to elevon deflection controller requires a more complex approach. Rather than a PI controller, the pitch to elevon controller is a proportional-integral-derivative (PID) controller designed from the plant in Eq. (4-2). The controller gains for the PID controller are also found using rltool with Eq. (4-2). The PID controller gains are found in Table 4-1 below along with the PI controller gains.

Table 4-1: PI and PID controller gains. Gain PI Controller PID Controller KP 0.455 6.172 KI 0.5 8.517 KD - 0.519

Implementation

With both the inner and outer loops of the nested loop designed, the airspeed controller can be modeled in Simulink. The overall design of the controller, including the PI and PID controllers, is illustrated in Figure 4-1. A few other features must be added to the Simulink model in the actual implementation of the controller. 27 A summing block between desired airspeed and actual airspeed gives the error as the

difference of the actual airspeed from the desired airspeed. When desired airspeed matches actual

airspeed, the corresponding error is zero and no additional change in pitch is necessary.

Similarly, the PI controller requires a summing block after the controller to calculate the error

between desired pitch angle and actual pitch angle. This error is then sent to the PID controller to

find the corresponding elevon deflection.

The PID controller also requires a summing block after the controller to correct the

elevon deflection. The elevon deflection outputted by the PID controller only corresponds to the

change in elevon deflection from the initial deflection. Before the elevon deflection is passed to

the longitudinal dynamics model, the summing block corrects the elevon output with the initial

elevon position to give the true elevon position at the current state. Lastly, the saturation block

limits the elevon deflection to a maximum and minimum of +/- 0.5 radians (28.6°). Most elevon

deflections are rather small, but this saturation prevents the airspeed control block from

attempting to trim the aircraft at an unrealistically high elevon deflection.

Figure 4-1: Airspeed control architecture with PI and PID controllers. 28

Chapter 5

Flap Module

The complete airspeed control and longitudinal dynamic stability models are integrated to create the overall flap module for the ArduPilot in Simulink. The flap module represents the complete Simulink controller model which will be directly translated into the autopilot software.

The flap module is set up to autonomously control the trim of the aircraft. The only user interaction with the flap module is a user commanded desired airspeed. The module takes an user input for desired airspeed then trims the aircraft for this desired airspeed. With the module engaged, the pilot of the aircraft becomes more of an operator than a pilot. The entire longitudinal behavior of the aircraft is controlled solely through simple clicks of a button, increasing the autonomy of the UAV.

Flap and Elevon Scheduling

Before the different subsystems can be completely integrated, a new flap and elevon scheduling block had to be created to calculate the flap deflection for the user’s desired airspeed and the elevon deflection required to trim the aircraft at this setting. These control surface deflections are the inputs used by both the airspeed control and nonlinear longitudinal dynamics blocks. Figure 5-1 depicts the flap and elevon scheduling subsystem block diagram.

29

Figure 5-1: Flap and elevon scheduling block.

The flap deflection is directly computed from the desired airspeed using the polynomial fit function previously determined in the flap scheduling results. In order to maintain accuracy, the polynomial fit lookup function is written as a 12th-order polynomial function. This order is high, but is necessary to fit a polynomial function to the shape of the optimal flap deflection results.

The elevon deflection calculated in this scheduling is actually the initial elevon deflection, not the actual elevon deflection. This initial elevon deflection is utilized by the airspeed control block to determine the actual elevon deflection, which is then used to control the airspeed. The scheduling subsystem uses the optimal flap deflection taken from the polynomial fit function to calculate the lift and pitching moment coefficients contributions of this flap deflection. These contributions are then subtracted from the total lift coefficient required to fly and the total pitching moment coefficient, which is zero since the aircraft is trimmed, to calculate the residual contributions to the lift and moment coefficient that the angle of attack and elevon deflection must provide. The angle of attack and elevon deflection are then found using the inverse of the aerodynamic derivatives matrix multiplied by the residual lift and pitching moment coefficients:

−1 ⎧α ⎫ ⎡ Lα CC δeL ⎤ ⎧C ,residualL ⎫ ⎨ ⎬ = ⎢ ⎥ ⎨ ⎬ (5-1) ⎩δ e ⎭ ⎣ Mα CC δeM ⎦ ⎩C ,residualM ⎭ 30 Overall Architecture

The flap and elevon scheduling subsystem block makes up the last subsystem of the Zagi flap module. These subsystems are then added together and connected to create the working flap module. The longitudinal dynamics and airspeed control subsystems use the flap and elevon deflections from the scheduling subsystem to determine flight behavior. The airspeed control then sends the actual elevon deflection to the longitudinal dynamics block, which then outputs the current state of the aircraft. This state vector of pitch rate, airspeed, pitch angle, and angle of attack is then fed back throughout the system. Figure 5-2 illustrates the highest-level block diagram of the complete flap module.

The complete diagram in Figure 5-2 also depicts a smaller set of vectors and function blocks that have not been described. First, a low pass filter is added to smooth out the discontinuous step in desired airspeed. The low pass filter was analyzed using simulation with the step function and a crossover frequency of 1 rad/s is used for the time constant. Likewise, the initial state vector sent to the longitudinal dynamics block is simply used as the initial conditions for the integrators in the dynamics and kinematics subsystems.

Figure 5-2: Overall control architecture of the entire flap module. 31

Chapter 6

Simulation Results

Multiple simulation cases were run to test the flap module behavior to different airspeeds commanded by the user. The Simulink model checks the stability of the module with different step sizes for both increases and decreases in airspeed. Airspeed, pitch angle, flap and elevon deflection, and glideslope are all plotted on similar axes to compare the behavior of the Zagi at each step change. The simulation cases were run for the aircraft target cruise range, as well as low and high speeds to test off-design response.

Cruise Range

The tests within the cruise range were started at an initial airspeed of 11m/s, which corresponds to the best L/D point found in the WingsX model results. This airspeed corresponds to an angle of attack of 2.5° and a small flap deflection of 1°. For the first simulation test run, the aircraft is trimmed at 11 m/s and allowed to continue gliding for 20 seconds to eliminate any small transient responses caused by slight differences between the WingsX analysis and the

Simulink model. After 20 seconds, a step increase in desired airspeed of 1 m/s is applied and the aircraft transient responses are allowed to dampen out for another 20 seconds. This same process is also applied for a 1 m/s decrease in desired airspeed. With these two test runs, the aircraft airspeed, pitch angle, flap deflection, and elevon deflection are plotted in Figures 6-1 through 6-4.

Figure 6-5 also graphs the aircraft glideslope with the jump near the middle of the plot corresponding to the step change in desired airspeed.

32

Figure 6-1: Airspeed versus time plot with a 1 m/s increment and decrement in desired airspeed at cruise.

Figure 6-2: Pitch angle versus time plot with a 1 m/s increment and decrement in desired airspeed at cruise. 33

Figure 6-3: Flap deflection versus time plot with a 1 m/s increment and decrement in desired airspeed at cruise.

Figure 6-4: Elevon deflection required to trim versus time plot with a 1 m/s increment and 34 decrement in desired airspeed at cruise.

Figure 6-5: Glideslope plot with a 1 m/s increment and decrement in desired airspeed at cruise.

Figures 6-1 through 6-5 all show the stable behavior of the Zagi UAV in the cruise regime of 10 to 15 m/s. All five plots show the aircraft trimmed gliding descent from 0 to 20 seconds. The colors of all the lines reflect the desired airspeed condition with blue representing a decrease in desired airspeed and the red representing an increase. Once the user input is changed to roughly 10 or 12 m/s for a decrease or increase in desired airspeed, the flap module corrects the control surfaces to match this new desired trim setting. Figure 6-1 compares the jump in desired airspeed from the step function with the slower response of the controller’s commanded airspeed.

This smoothed out commanded airspeed prevents a singularity in airspeed as the user input changes. Figures 6-2, 6-3, and 6-4 all show the state behavior as the aircraft responds to the new desired flight speed. Figure 6-4 shows particularly interesting behavior with a large initial jump when the desired airspeed changes that quickly dampens out. This short, but violent transient 35 response could have a large effect upon the model stability. In reality, the elevon servos can not deflect as quickly as in the Simulink model, so strong lags could induce destabilizing conditions.

Figure 6-5 shows the glideslope of the UAV as the desired airspeed changes. In both cases, the aircraft begins at the same flight speed and glideslope so the two colored lines overlap.

With the change in desired airspeed, the aircraft flight performance becomes clear. One particular feature of Figure 6-5 is the difference in glideslope of the decrease in flap setting compared against the initial trimmed glide. The initial glide is supposed to represent the best glideslope of the aircraft, but Figure 6-5 seems to suggest otherwise. The blue line corresponding to the 1 m/s decrease in airspeed suggests that the slower airspeed actually has a better L/D ratio than the initial gliding flight assumed to have the best L/D. More tests will have to be done to completely analyze the Simulink model of the flap module, but most of the general behavior matches its predictions.

Low Speed

The next test case is a run near the minimum speed to test the behavior of the flap module at the lower end of the airspeed range. This region also corresponds to points farther away from the control point used to linearize the aerodynamic derivatives for the aerodynamic coefficients block. This means that the airspeeds and angles predicted in the WingsX nonlinear lifting-line model will be further away from the Simulink result than previously encountered in the cruise range. In fact, the test case used a value of 4° for angle of attack and 12.5° for flap deflection to get an initial airspeed of 9.3 m/s. In the nonlinear model data, this same angle of attack and flap deflection produces an airspeed of 10.3 m/s, an 11% difference.

Just like in the previous test cases, the aircraft starts in trimmed gliding flight before a

step command in airspeed is inputted. In this test case, the airspeed drops by 1 m/s only; an 36 increment in airspeed is not looked at. Figures 6-6 through 6-10 all depict the airspeed, pitch angle, flap deflection, elevon deflection, and glideslope in the same order as in the first simulation test case.

Figure 6-6: Airspeed versus time plot with a 1 m/s decrement in desired airspeed at lower speeds.

37

Figure 6-7: Pitch angle versus time plot with a 1 m/s decrement in desired airspeed at lower speeds.

Figure 6-8: Flap deflection versus time plot with a 1 m/s decrement in desired airspeed at lower speeds. 38

Figure 6-9: Elevon deflection required to trim versus time plot with a 1 m/s decrement in desired airspeed at lower speeds.

Figure 6-10: Glideslope plot with a 1 m/s decrement in desired airspeed at lower speeds. 39

The flap module still works for the 1 m/s drop in airspeed, even with a lower starting trim

condition. Figure 6-10 depicts the glideslope of the aircraft and the effect of the step change on

the aircraft flight. Not surprisingly, with the larger downward flap deflection and higher drag, the

aircraft has a noticeably smaller L/D ratio and corresponding glideslope. The rest of the graphs

show the aircraft behavior with this change in airspeed, but Figure 6-9 demonstrates a peculiar

elevon response. The elevon is ultimately able to trim the aircraft in both the original and new

desired airspeed, but it shows a large transient response right when the desired airspeed changes.

The smearing of blue color on the plot is actually caused by a high frequency of elevon

oscillations. This transient response is dangerous to the aircraft as it might indicate that the flap

module might not be stable for the range of airspeeds originally predicted. It is also dangerous to

an elevon servo since the servo will not be able to keep up with any of these incredibly high

frequency oscillations. The oscillations could severely damage the servos or prevent the Zagi

from achieving trimmed flight. With the unpredictable oscillations, the slightest lag in the elevon

deflection could induce destabilizing behavior.

High Speed

The flap module is also tested at the upper edge of its airspeed range. The aircraft is initially trimmed with 1° of angle of attack and 0° of flap deflection, which produces an airspeed of 12.8 m/s. In the WingsX model, this angle of attack and flap deflection corresponds to 13 m/s, giving it a slight difference. This test case has a smaller difference than the low speed test case because the aircraft parameters at these points match up closer to the values at the control point.

Just like in all the other test cases, the aircraft is allowed to trim for 20 seconds and remove any 40 transient responses before a 1 m/s increase in desired speed sent to the aircraft. Figures 6-11 through 6-15 illustrate the behavior of the aircraft for the test case.

Figure 6-11: Airspeed versus time plot with a 1 m/s increment in desired airspeed at higher speeds.

41

Figure 6-12: Pitch angle versus time plot with a 1 m/s increment in desired airspeed at higher speeds.

Figure 6-13: Flap deflection versus time plot with a 1 m/s increment in desired airspeed at higher speeds. 42

Figure 6-14: Elevon deflection required to trim versus time plot with a 1 m/s increment in desired airspeed at higher speeds.

Figure 6-15: Glideslope plot with a 1 m/s increment in desired airspeed at higher speeds. 43

Unlike the previous cases, this test case clearly shows the limitations of the flap module.

Although it might not be readily apparent from the smooth glideslope in Figure 6-15, the aircraft is actually experiencing small oscillations that do not dampen out with time. Figure 6-11 shows the aircraft actual airspeed continuously fluctuating around the aircraft desired airspeed after the step change in airspeed is applied. Figure 6-12 also shows this oscillating response for pitch angle, but both are caused by the unstable behavior of the elevon in Figure 6-14.

The elevon deflection first shows transient oscillations in the initial trim case, but the oscillations quickly dampen out. When the increase in airspeed is applied, the new oscillations in elevon deflection fail to dampen out this time. The magnitudes of the oscillations grow until they reach the saturation limits. With large fluctuations in elevon deflection, the aircraft will not be able to trim at this higher airspeed, even though the WingsX model predicts the aircraft will easily trim there.

Challenges

Despite the overall success of the autopilot module, the multiple test cases reveal a number of challenges with the flap module. The fact that the aircraft equations of motion use aerodynamic derivatives linearized about a specific control point degrades the Simulink model accuracy as it moves farther away from this set point. The flap module is also limited in the magnitude of step size for desired airspeed. New limits on the ArduPilot module will have to address this limitation, which will hamper, but not eliminate, the stability and accuracy of the flap module for the autopilot.

The linearized aerodynamic derivatives from the longitudinal dynamics will cause a difference in control surface deflections between the WingsX and Simulink model as the aircraft 44 trims at the desired airspeed. Since the derivatives are linearized about a specific point, the parameters are most accurate for airspeeds closest to that point. As the airspeed departs the designated 11 m/s, the derivatives used in the nonlinear WingsX model begin to drift away from the constant derivatives of the Simulink model. Flap deflection is calculated using the

polynomial fit function from the nonlinear longitudinal dynamics case. When a flap deflection

for an off-design airspeed is multiplied by the flap derivative from the 11 m/s airspeed, the

resultant lift, drag, or moment contribution differs from WingsX predictions. This explains the

initial transient jumps when the aircraft tries to trim at a given flap setting and airspeed using

WingsX data. The differences force the module to trim the aircraft with a slightly different

elevon deflection than what is predicted with WingsX.

Likewise, the step size also limits the effectiveness of the flap module. The aircraft shows different behavior for both the size and direction of the step change in desired airspeed.

For example, when the aircraft is trimmed at cruise speed of 11 m/s, it will only trim for a maximum decrease in desired airspeed of 1 m/s while it will still trim the aircraft with a step increase of up to 3 m/s. Such small step sizes limit the autopilot module effectiveness. In reality, the actual step size in desired will probably be a larger 1 to 3 m/s change, so it is unclear if the autopilot will be able to trim the aircraft with a large step size. 45

Chapter 7

Conclusion

The simulation test cases demonstrate that the flap module does indeed work as intended.

There is some unstable and transient behavior at either end of the cruise speed regime, but this

should be correctable. Slight tweeking of the controller gains for the PI and PID controllers

might increase the range of trimmable airspeeds. They might also have a beneficial effect on the

transient responses by limiting their magnitude or frequency. Other modifications to the flap and

elevon scheduling block or further analysis of the longitudinal dynamics might help to alleviate

this issue. In general, the simulation results illustrate that the controller does indeed respond to

user commands for desired airspeed and automatically trims the aircraft, but with a more limited

range of acceptable airspeeds than originally anticipated.

With the completed Simulink model, the next step is to test the simulation model with actual flight data. The first task will be to actually implement the flap design on the Zagi UAV and incorporate the flap module into the ArduPilot board. Currently, a test Zagi incorporating the flap is under construction, as seen in Figure 7-1. The breaks in the trailing-edge control surfaces between the inboard flaps and outboard elevons are clearly visible. The servos which previously controlled the original elevons now control the flaps. New servo bays for the outboard servos to control the elevons are outlined in pen, but have not yet been removed.

46

Figure 7-1: Initial construction of the Zagi UAV incorporating the inboard trailing-edge flaps.

The necessary servos, batteries, receivers, and boards are also shown in Figure 7-2. All of these electronics are the same as those found in the unmodified Zagi, except for the extra set of

Hitec HS-55 servos to control the outboard elevons. The ArduPilot board with the dynamic pressure sensor remains unchanged, but no navigation capabilities are included. The navigation chips could be installed at a later time for autonomous soaring between waypoints. The flap module is being programmed into ArduPilot in parallel with the construction of the test aircraft.

There are no easily accessible ArduPilot airframes available to use for modeling the Zagi in 47 Arduino, meaning there aren’t any existing flying wing templates to use. Without a previous airframe, a completely new airframe model has to be written into ArduPilot or an existing model has to be heavily modified to work correctly.

Figure 7-2: Servos and electronics used on the Zagi test aircraft.

In conclusion, the new flapped Zagi was originally modified to produce a self-trimming flap that required no new additional elevon deflection to trim out the aircraft behavior. Even though the Zagi aspect ratio is too small for exact use, an approximate method is used to create a flap that behaves as close as possible to a self-trimming flap. A new flap module then is designed to autonomously trim the aircraft with only a user input for desired airspeed. Simulation of the flap module indicates that the module does indeed work, but with slight limitations. Flight testing will be performed in the future to validate and verify the flap design and its behavior. 48

References

[DC-44] Donlan, C., “An Interim Report on the Stability and Control of Tailless Airplanes,”

Langley Stability Research Division, NACA TR-796, 1944.

[HM-09] Hepperle, M., “Airfoils for Flying Wings,” MH Airfoils [online database], URL:

http://www.mh-aerotools.de/airfoils/ [cited 29 March 2009].

[KB-91] Kroo, I. and Beckman, E., “Development of the SWIFT – a Tailless Foot-Launched

Sailplane,” Hang Gliding, Jan. 1991.

[LJ-08] Langelaan, J., “Biologically Inspired Flight Techniques for Small and Micro Unmanned

Aerial Vehicles,” AIAA Guidance, Navigation, and Control Conference, Paper No. 2008-

6511, AIAA, 2008.

[MT-09] MATLAB, Software Package, Ver. 7.6.0.324, The MathWorks, Natick, MA, 2008.

[NW-94] Nickel, K. and Wohlfahrt, M., Tailless Aircraft in Theory and Practice, 1st ed., AIAA,

Washington, DC, 1994, Chaps. 2, 7.

[PM-44] Pitkin, M., and Meggin, B., “Analysis of Factors Affecting Net Lift Increment

Attainable With Trailing Edge Split Flaps on Tailless Airplanes,” NACA TR-L4I18,

1944.

[SK-09] Simulink, Software Package, Ver. 7.1, The MathWorks, Natick, MA, 2008.

[TH-99] Thomas, F., Fundamentals of Sailplane Design, 1st ed., College Park Press, College

Park, MD, 1999, pp. 62-72.

[TK-09] Trick R/C, “Zagi 5C Assembly Manual,” Instruction Manual, Trick R/C Products,

Venice, CA, 2009.

[WX-09] WingsX, Software Package, Ver. 1.0, Nicholas Alley, Atlanta, GA, 2009. 49 Appendix A

Aircraft Parameters

Table A-1: Linearized aerodynamic derivatives of both the original and flapped Zagi UAVs. Variable Unmodified Zagi Modified Zagi Description C L0 0.2135 0.2212 CL at α = 0

CLα 4.9025 4.7357 CL / ∂∂ α (1/rad) C LQ 0.0577 0.0029 L / ∂∂ QC (1/rad)

C δeL 2.0822 0.659 C / ∂∂ δ eL (1/rad) (1/rad) C δfL - 1.3816 C / ∂∂ δ fL

x 4 − 4242.05057.0 x 3 x 4 − 3230.11169.1 x 3 2 2 fLD + x + 0145.01365.0 x + x − 0360.05375.0 x = 0 + CCx LL αα + 0118.0 + 0113.0

C δeD 0.1299 0.0442 C / ∂∂ δ eD (1/rad) (1/rad) C δfD - 0.0829 C / ∂∂ δ fD

C M 0 0.0144 0.0108 CM at α = 0

CMα -0.7759 -0.6843 CM / ∂∂ α (1/rad) C MQ -0.7313 -0.7482 M / ∂∂ QC (1/rad)

C δeM -1.3379 -0.5345 C / ∂∂ δ eM (1/rad) C C / ∂∂ δ (1/rad) δfM - -0.8692 fM

Table A-2: Additional properties of both the original and flapped Zagi UAVs. Variable Unmodified Zagi Modified Zagi Units Weight 8.896 8.896 N 2 Wing Area 0.294 0.294 m Span 1.219 1.219 m Chord 0.241 0.241 m 2 Ixx 0.0093 0.0093 kg*m 2 Iyy 0.0099 0.0099 kg*m 2 Izz 0.00063 0.00063 kg*m 50 Appendix B

Nomenclature

bF = flap span c = chord cF = flap chord g = gravitational constant

Iyy = moment of inertia about y axis

L = lift force

Mac = pitching moment about aerodynamic center

Mcg = pitching moment about center of gravity m = mass q = dynamic pressure

Q = pitch rate

S = wing area

V = airspeed xcp,flap = span-wise x-location of center of pressure of additional lift due to flap xac,wing = span-wise x-location of the aerodynamic center of the wing xcg = span-wise x-location of the center of gravity x = horizontal position z = vertical position

CD = drag coefficient

CDδe = ∂CD/∂ δe

CDδf = ∂CD/∂ δf

CL = lift coefficient 51

CLδe = ∂CL/∂ δe

CLδf = ∂CL/∂ δf

CLα = ∂CL/∂ α

CLQ = ∂CL/∂ Q

CL0 = CL at α = 0

CM = pitching moment coefficient

CMac = pitching moment coefficient about wing aerodynamic center

CMδe = ∂CM/∂ δe

CMδf = ∂CM/∂ δf

CMα = ∂CM/∂ α

CMQ = ∂CM/∂ Q

CM0 = CM at α = 0

α = angle of attack

δe = elevon deflection

δf = flap deflection

θ = pitch angle

ρ = density

Λ0 = leading-edge sweep

Academic Vita for John F. Quindlen

John F. Quindlen 16 Hilltop Road Rose Valley, 19086 (610) 574-1389 [email protected]

Education: The Pennsylvania State University, University Park, PA Bachelors of Science in Aerospace Engineering, May 2010 Schreyer Honors College Honors Thesis: Design and Trim Optimization of a Flying Wing UAV Thesis Advisor: Dr. Jack W. Langelaan

Research Projects: Autonomous Vehicles Laboratory (Fall 2009 – Present) Advisor: Dr. Jack W. Langelaan

Zephyrus Human Powered Aircraft (Spring 2007 – Present) Advisor: Dr. Mark Maughmer

Related Experience: US Army Aberdeen Testing Center (Summer 2008) Test Director Intern, Aviation and Foreign Systems Division Supervisor: Ms. Joyce Carter

Awards: Sigma Gamma Tau Tau Beta Pi Dean’s List

Professional Memberships: Student Member, American Institute of Aeronautics and Astronautics