modelling and control of the simicon uav

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Master of Engineering Thesis

Modelling and Control of the SiMiCon UAV

By Andrew Ross

Department of Electronics and Electrical Engineering

January, 2003


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AbstractThe onset of high-speed computing has benefited the field of aerospace engineering

greatly. Vast computing resources, available at a minimal cost, can be effectively

applied to every area of the design, testing, and manufacture of advanced aircraft.

Additionally, modern control techniques mean that performance and stability are nolonger mutually exclusive: both can be achieved. In that vein, this thesis focuses on

the very initial design, aerodynamic modelling, control, and graphical display, of a

novel aircraft concept: the SiMiCon Rotor-Craft (SRC).

Within is an analysis of a novel, disc-shaped hybrid unmanned aerial vehicle concept.

The analysis can broadly be divided into three parts: firstly, the use of easily available

computer tools to evaluate the aerodynamic properties of the aircraft; secondly, the

construction of an advanced simulation environment; and thirdly, the design and

simulation of an effective control system.

The software used to model the aircraft is the USAF Digital DATCOM, augmented

by various other programs. The aircraft is evaluated comprehensively from very low

up to transonic velocities. This gathered data is formed into a large set of easily

accessible lookup tables.

The modelling is carried out using Matlab and Simulink. Included in the simulation

environment are detailed atmospheric, gravitic models, and actuator models.

The control design is of a linear quadratic (LQ) architecture. Various manoeuvres are

carried out, including steady level flight, altitude changes, and turns. With these

results presented, the controller developed further by augmenting it with integralaction. Simulations are repeated, and improvements noted.

Suggestions of further work are given, before the thesis is concluded with a short

overview of the important results derived through the work in the thesis.

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iii

AcknowledgementsI would like to take this opportunity to express my gratitude to those who have

assisted me, to varying degrees, in writing this thesis.

My primary thanks go to Professor Thor-Inge Fossen, my supervisor for the duration

of my stay in Trondheim. He found the perfect balance between ensuring that I work

independently, and offering assistance when needed. Furthermore, the opportunity to

aid him in checking his latest book properly introduced me to the exciting field of

marine cybernetics, for which I am extremely appreciative.

SimiCon, the company that commissioned this thesis, has been an incredible help at

all times. Their concept is the driving force behind this thesis, and I am grateful for

their helpful assistance throughout: especially the friendly support by my main

contact at Simicon, Vegard Hovstein.

MarinTek is also due my grateful thanks for providing me with office space. I would

especially like to thank Dr. Svein Peder Berge for cultivating a relaxed, friendly, and

efficient working area. Also at MarinTek, both Sonia Moi and Nicolai Husteli offered

help and support at many times, and for this I offer my gratitude.


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Nomenclature

a- speed of sound ( ms )2

b- wingspan (m)c - mean wing chord (m)

CD- drag coefficient

Cl- rolling moment coefficient

CL- lift coefficient

CM- pitching moment coefficient

CN- yawing moment coefficient

CY- sideforce coefficient

D- drag (N)

L- lift (N)

L - rolling moment (Nm)

M- pitching moment (Nm)N- yawing moment (Nm)

P- vector of body angular velocities

P- body rolling angular velocity ( )/rad s

Q- body pitching angular velocity ( )/rad s

R- body yawing angular velocity ( rad )/sb

aR - rotation matrix from frame-a to frame-b

Vt- airspeed ( ms )2

q- dynamic pressure ( )2Nm

S- wing area ( ms )2

bU - vector of body velocities

Ub- forward body velocity (2ms )

Vb- sidewards body velocity (2ms )

Wb- downwards body velocity (2ms )

X- axial force (N)

eX - NED position vector (m)

Xe- NED x-axis position (m)

Y- sideforce (N)

Ye- NED y-axis position (m)

Z- vertical force (N)Ze- NED z-axis position (m)

- angle of attack ()

- angle of sideslip ()

- vector of euler angles

- roll angle (rad)

- pitch angle (rad)

- density ( /kg )3m

- yaw angle (rad)

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Table of Contents1 Introduction............................................................................................................1

2 The SRC Aircraft ...................................................................................................3

3 Modelling the SRC ................................................................................................5

3.1 Aerodynamic Forces and Moments ...............................................................5

3.1.1 Aerodynamic Coefficients at Varying Flight Conditions ......................6

3.2 Calculation of Aerodynamic Data ...............................................................10

3.2.1 Xfoil .....................................................................................................10

3.2.2 USAF Digital DATCOM.....................................................................13

3.2.3 Tornado................................................................................................16

3.3 Importing into MATLAB from the Digital DATCOM ...............................18

4 Creating the Simulation Model............................................................................20

4.1 Completing the Aircraft Model....................................................................20

4.1.1 Coordinate Systems .............................................................................20

4.1.2 Lookup Tables .....................................................................................22

4.1.3 Coefficients to Body Forces.................................................................234.2 Simulation Environment ..............................................................................24

4.2.1 Atmosphere Model...............................................................................24

4.2.2 Wind Turbulence Model ......................................................................25

4.2.3 Gravity Model......................................................................................26

4.2.4 Engine Model.......................................................................................26

4.2.5 Flight Condition Calculation................................................................26

4.2.6 Equations of Motion ............................................................................28

4.3 Model Verification.......................................................................................28

4.3.1 Lookup Table Data ..............................................................................28

4.3.2 Poles, Zeros and Eigenvalues ..............................................................29

4.3.3 Uncontrolled Simulations ....................................................................305 Controlling the SRC.............................................................................................33

5.1 The LQ Controller........................................................................................33

5.1.1 Weighting Matrix Selection.................................................................34

5.1.2 Defining Q ...........................................................................................34

5.1.3 Defining R............................................................................................35

5.2 Applying the LQ Controller.........................................................................35

5.2.1 Redefining the System Input................................................................36

5.2.2 Control Allocation ...............................................................................36

5.3 Deriving the Linear Model...........................................................................37

5.4 Linear Model Analysis.................................................................................38

5.4.1 The A-Matrix .......................................................................................385.4.2 B-Matrix Analysis................................................................................42

5.5 Input Matrix Generation ..............................................................................42

5.6 B-Matrix Analysis........................................................................................43

5.7 Pseudo-Inverse of B.....................................................................................44

5.8 Simulink Model of Controller......................................................................44

6 Simulation Results ...............................................................................................46

6.1 Force Input Controller..................................................................................46

6.1.1 Altitude Hold .......................................................................................46

6.1.2 Constant Descent .................................................................................48

6.1.3 Thrust Requirements............................................................................50

6.2 Instant Actuators ..........................................................................................51

6.2.1 Altitude Hold .......................................................................................51

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6.2.3 300 Second Constant Rate Turn ..........................................................57

6.3 Realistic Actuators .......................................................................................60

6.3.1 Altitude Hold .......................................................................................60

6.3.2 Descent Performance ...........................................................................62

7 Improving the Control System.............................................................................647.1 Applying Integral Action to the LQ Controller ...........................................64

7.2 Improved Controller Simulation Results .....................................................65

7.2.1 Altitude Hold .......................................................................................65


8 Graphical Output..................................................................................................69

8.1 Creating the VRML Model..........................................................................69

8.2 Screenshots ..................................................................................................70

9 Future Work.........................................................................................................71

9.1 Aircraft Model .............................................................................................71

9.2 Control System.............................................................................................71

9.3 Graphical System.........................................................................................7110 Conclusion .......................................................................................................73

References....................................................................................................................74

Appendices...................................................................................................................76

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Table of FiguresFigure 2.1 Side View of Simicon Rotor-Craft...............................................................3

Figure 2.2 Front View of SRC With Rotor Extended....................................................3

Figure 2.3 Control surfaces of the SRC .........................................................................4

Figure 3.1 SRC Aerofoil- N0011SC............................................................................11

Figure 3.2 Xfoil Solution of N0011SC at M0.08, 4 ...................................................12Figure 3.3 Lift and Drag Coefficients of N0011SC (M0.08) ......................................13

Figure 3.4 SRC Geometric Approximation .................................................................14

Figure 3.5 SRC Model in Tornado ..............................................................................17

Figure 3.6 Rudder Effects on Yawing Moment from Tornado Simulation.................18

Figure 4.1 Lookup Table Example (CN) .....................................................................23

Figure 4.2 From Wind to Body Forces ........................................................................24

Figure 4.3 Rotation From Wind to Body Frame..........................................................24

Figure 4.4 Gravity Model and Rotation to Body Frame..............................................26

Figure 4.5 Flight Conditions Calculation Block ..........................................................27

Figure 4.6 Equations of Motion Solver........................................................................28Figure 4.7 Cm Against Altitude, and Alpha ................................................................29

Figure 4.8 A Pole-Zero Map of the Linearised Model ................................................29

Figure 4.9 Uncontrolled SRC Response with Low Elevon Angle...............................31

Figure 4.10 Uncontrolled SRC Response with High Elevon Angle............................32

Figure 5.1 Schematic of State Space System and Controller.......................................33

Figure 5.2 State Space System with Redefined Input..................................................36

Figure 5.3 Schematic of State Space Model with Control Allocation.........................37

Figure 5.4 Model Used as Linearisation Target...........................................................37

Figure 5.5 Internals of Linearisation Target Model.....................................................38

Figure 5.6 Input Target Linearisation Block................................................................42

Figure 5.7 Internals of Input Linearisation Target Block ............................................43Figure 6.1 Altitude Hold of Force Controlled System.................................................46

Figure 6.2 Comparison of Altitude Hold in Calm and Windy Conditions ..................47

Figure 6.3 The Effects of Vertical Gusts on Angle of Attack .....................................48

Figure 6.4 Pitch Angle During Descent .......................................................................49

Figure 6.5 Altitude Error During a Constant Rate Descent.........................................50

Figure 6.6 Changes in Thrust Demands after a Change in Altitude ............................51

Figure 6.7 Altitude Hold using Realistic Inputs ..........................................................52

Figure 6.8 Altitude Hold with Realistic Input in Windy Conditions...........................53

Figure 6.9 Descent of Actuated System.......................................................................54

Figure 6.10 Error During Descent of Instantly Actuated System................................55

Figure 6.11 Descent of Actuated System in Presence of Wind...................................56Figure 6.12 Error During Descent in Wind of Actuated System.................................56

Figure 6.13 A 300 Second Coordinated Turn ..............................................................57

Figure 6.14 Error During Coordinated Turn................................................................58

Figure 6.15 Absolute Error During 300 Second Coordinated Turn.............................59

Figure 6.16 Euler Angles During 300 Second Coordinated Turn ...............................60

Figure 6.17 Realistic Actuator Altitude Hold..............................................................61

Figure 6.18 Realistic Actuators 2.7m/s descent...........................................................62

Figure 6.19 Altitude Error During Descent in Realistic Actuated System..................63

Figure 7.1 Altitude Hold Performance Instant Actuators ............................................65

Figure 7.2 Integral Action Altitude Hold in Windy Atmosphere ................................66

Figure 7.3 Constant Descent Maneouvre.....................................................................67

Figure 7.4 Descent Through Windy Atmosphere with Integral Action.......................68

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Figure 8.1 VRML Model of SRC ................................................................................70

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1 IntroductionDesigning an aircraft takes an incredible amount of time and resources. An aircraft

design usually begins as a sketch on a notepad, or through an artists impression. At

these early stages, the aircraft has no basis in reality, and so can be altered at will. The

shape and size of the aircraft can be changed; the location of the engine or engines is

not set. The ability to quickly and accurately evaluate a design at this conceptual stage

is invaluable. As an aircraft passes from concept into prototype, the design is

essentially set. It is almost impossible to add or remove or alter drastically one of the

major sections of the aircraft. The changes that were easy to make while the aircraft

was only a concept, are now difficult to alter. This increases the value of any method

that quickly, cheaply and effectively evaluates the performance of a concept.

The onset of high-speed computing has benefited the field of aerospace engineering

greatly. Vast computing resources, available at a minimal cost, can be effectively

applied to every area of the design, testing, and manufacture of advanced aircraft.Additionally, modern control techniques mean that performance and stability are no

longer mutually exclusive: both can be achieved. In that vein, this thesis focuses on

the very initial design, aerodynamic modelling, control, and graphical display, of a

novel aircraft concept: the SiMiCon Rotor-Craft (SRC). Specifically, the effective

usage of software in the preliminary design of this aircraft is described in detail.

This SiMiCon design is a revolutionary concept for a hybrid unmanned aerial vehicle

(UAV). The SRC has two distinct modes of propulsion: in one, it functions as a

helicopter, using a rotor-disc, with a vectored jet-engine providing counter-torque; in

the other, it retracts the rotor blades and functions as a conventional jet. Combined,

these modes offer the advantages inherent in a jet-aircraft, as well the advantagesinherent in a helicopter. That is; the high speed and long range of a jet; and the

vertical take-off and landing (VTOL), and other versatile manoeuvres usually only

capable of being performed by a helicopter. Essentially, the aircraft can offer high

performance and efficiency from a velocity of zero, up to nearly transonic speeds.

This thesis focuses purely on the jet-powered section of flight. This begins at around

M0.08, or roughly 150km/h, and extends upwards to the limits of the aircraft.

The SRC consists of a disc shaped wing/body, a vertical tail, and horizontal tail. This

offers a very sleek profile, and makes the aircraft an ideal platform for a hybrid

design. At this early stage of design, the disc has a diameter of three metres, and the

aircraft is expected to have a mass between five and seven hundred kilograms. Thecontrol surfaces are: elevons mounted on the horizontal tail for roll and pitch control;

a rudder mounted on the vertical tail for yaw control; and a split flap on the lower

surface of the disc for low speed flight.

Chapter 2 concentrates on explaining the general concept of the SRC, along with

some preliminary design decisions. In chapter 3, using cheap or free software, the

SRCs aerodynamic parameters are calculated, and a complete aerodynamic database

is created. This includes the coding of extensive scripts to import data into MATLAB.

Chapter 4 integrates this database into a realistic flight simulation environment. This

involves atmospheric, gravity and actuator models. The simulator is coded in Matlab

and Simulink, using various toolboxes. Following on from the model construction,

Chapter 5 develops a control system for the Matlab SRC model. It first details linear

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quadratic theory, and then explains how a linear controller of this type is created to

stabilise and control the aircraft model. Chapter 6 simulates the performance of the

aircraft in several manoeuvres including steady, level flight and coordinated turns.

Chapter 7 moves on to improve the method of control by adding integral action.

Simulation results are then presented to show the effects of integral action,

particularly the effects on steady state errors. Chapter 8 details the graphical modelbuilt to display the SRC concept. Chapter 9 briefly offers opinions on work that could

follow on from this thesis. Chapter 10 concludes the thesis by summarising the

important results derived.

In summary, the aim of this thesis is to build a viable simulation model of the Simicon

Rotor Craft.

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2 The SRC AircraftAs explained during the introduction, the Simicon Rotor-Craft is a concept for an

unmanned aerial vehicle. That is, it combines the low speed of a helicopter, with the

high speed and range of a jet. Artists impressions of the design are shown throughout

this chapter.

Figure 2.1 Side View of Simicon Rotor-Craft

In this thesis, a fuselage diameter of 3m is taken as the standard. The mass of the

aircraft is set at 700kg, with a centre of gravity at or close to the centre of the disc.

The rotor itself utilises a new idea: the Spin-Off rotor concept. This type of helicopter

has no rotor hub: an extremely heavy, complicated, and vital part of a conventional

helicopter. Instead, the SRC uses what is called a Spin-Off rotor. The craft actually

varies the position of the rotor itself. That is, it moves the rotor from its standard

position the x-y plane. This has the effect of making the rotor thrust act through a

point distant from the centre of gravity, and so can introduce large and controllable

rolling and pitching moments.

Figure 2.2 Front View of SRC With Rotor Extended

A benefit of this new rotor design is that the rotor can be reduced in weight

considerably, while the drawback is that a complex control system required. Radio-

controlled models of this concept have already been demonstrated. Early analytical

work on the control during the helicopter mode of flight can be found in Hovstein

(2001)1.

The jet engine on the lower half of the aircraft uses thrust vectoring to counter the

torque of the rotor. This jet engine gradually takes over from the helicopter blades topower the aircraft as airspeed increases, finally taking over completely at a velocity of

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roughly 150km/h. The conceived maximum static thrust of this engine at sea level is

taken to be 3500N.

Although the rotor provides primary control at very low speeds, there are several

control surfaces on the aircraft that provide effective control at all higher speeds. The

surfaces present on the SRC are a rudder on the vertical tailplane, elevons on thehorizontal tailplane, and a split flap on the lower section of the fuselage. Room for an

externally blown flap has been left in the centreline, although that device does not

enter into this thesis.

The rudder is conventional, and is used for yaw control. Elevons were chosen for use

to save space and weight. An elevon is a generally uncommon control surface.

Elevons conjoin the tasks of pitch and roll control. Whereas the primary method of

pitch control was through the use of elevators on the tail, and the primary method of

roll control was through the use of ailerons on the wing, elevons merge these into a

single set of control surfaces: the word itself being a merging of its predecessors. On

the SRC, elevons on the tail carry out all rolling and pitching manoeuvres. As will beseen later, this complicates the task of controlling the aircraft. A picture of the SRC

with control surface deflection is depicted in Figure 2.3.

Figure 2.3 Control surfaces of the SRC

Although some preliminary data and simulations exist for the very low speed section

of the flight envelope, no previous work exists which deals with the higher velocity,

jet-powered part of the flight envelope. That is, no wind-tunnel data has been

gathered. Nor has any computational fluid dynamics modelling (CFD) been done.

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3 Modelling the SRCRepresenting an aircraft model in a computer has been the focus of a great deal of

attention over the past thirty years. Carrying out this task successfully allows one to

evaluate the stability and performance of the aircraft in a fraction of the time and at a

fraction of the cost previously required. Additionally, it is an easier task to design a

control system for a system that has been fully quantified. Additionally, it becomes

possible to build more exotic and high performance control systems around this

computer model.

3.1 Aerodynamic Forces and Moments

The task of simulating an aircraft, as with so many other systems from ships to

spacecraft, is one of solving Newtons equations of motion:

abs

F ma= (3.1)

O

abs

I = (3.2)

This task requires that all forces and moments in the six degrees of freedom be known

in an inertial frame. In the field of aerodynamics, forces and moments are generally

expressed in terms of their aerodynamic coefficients, shown below:

2

2

2

2

2

2

1, drag

2

1, lift2

1, side force

2

1, rolling moment

2

1, pitching moment

2

1, yawing moment

2

D

L

Y

l

M

N

D V SC

L V SC

Y V SC

L V SbC

M V ScC

N V SbC

=

=

=

=

=

=

(3.3)

where CD, CLand CYare the drag coefficient, lift coefficient, and sideforce

coefficient respectively: the force coefficients; while Cl, CM, and CNare the rolling,

pitching and yawing moment coefficients respectively. Therefore to go from the

coefficient to the force or moment itself, one multiplies by the dynamic pressure and

wing area. For the moments, another multiplier, either the wingspan or mean chord

length, is also needed. The aerodynamic forces and moments are generated by the

airflow over the aircraft, and so are inherently expressed in terms of the wind-axes.

Each coefficient varies according to terms such as angle of attack, sideslip angle,

altitude, Mach number, control surface deflection and many others. Engine thrust,

though it varies with airspeed and the angles of attack and sideslip, is inherently

expressed in the body-fixed axes, as it always acts through the same point on the

body.

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With knowledge of all wind-forces and moments, and after a rotation into an inertial

frame, the equations of motion can quickly and easily be solved. This leads onto the

main task of creating an aircraft model: the task of evaluating the aerodynamic

coefficients CL, CD, CN at any flight condition.

3.1.1 Aerodynamic Coefficients at Varying Flight Conditions

Each aerodynamic coefficient is dependent on an array of parameters. For example,

the lift coefficient is primarily dependant on angle of attack, with factors such as flap

deflection and elevon deflection also having large influences. In addition to these

dependencies, the lift coefficient also varies with Mach number, altitude, and pitch

rate. Evaluating the lift coefficient with any accuracy requires the estimation of the

effects of all the factors mentioned. All six of the force and moment coefficients share

such complexity, although to varying degrees.

Once the components of each aerodynamic coefficient are collated, the six

coefficients of forces and moments can then be calculated. In general this is simply a

case of summating the coefficients. However, the components that are dependent on

time derivatives of some other parameters: the dynamic coefficients, are inserted

using the following equation:

dynamic contribution2

T

bC rate

V= (3.4)

Where b is the wingspan of the aircraft; VTis the airspeed; and C is the dynamic

derivative in question.

3.1.1.1 Evaluating CL

The factors which affect CLhave already been mentioned in 3.1.1, but will be here.

CL() - lift coefficient dependence on angle of attackCL(elv) - the lifting force from elevon deflectionCL(wf) - the lifting force from the wing flapCLQ - the effect on lifting force of the pitch rate

CL() arises due to the lifting force produced when the aircraft passes through the air

at some angle of attack (for uncambered wings). This value must be looked up atparticular flight conditions to compensate for the variations with Mach number and

altitude.

CL(elv) and CL(wf) are present due to the effect the surfaces have on the liftingforce when lying at some deflection. Each coefficient varies according to its own

deflection, altitude, mach number, and also the aircraft angle of attack.

CLQ is present due to the effect a pitch rate has on the aircraft, primarily the tail. A

positive pitch rate increases the angle of attack of the tail, resulting in a larger lifting

force from the tail. This also occurs with the wing, but is a much smaller effect,

usually only about ten percent of the tail value. Both are calculated at once in the latersimulation. The formula to evaluate CL is then

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( ) ( ) ( )2

L L L L LQ

T

bC C C elv C wflap C Q

V= + + + (3.5)

3.1.1.2 Evaluating CD

The drag coefficient is again broadly dependent on the angle of attack of the aircraft,summed with the effects of the tail and the various effects of any other control

surfaces on the aircraft, in the SRCs case, the wingflap. The effects of these must be

scaled with Mach number and altitude.

The effectiveness of the tailplane is dependent on the airflow over it. As the aircrafts

angle of attack increases, the quality of airflow over the tailplane decreases. This must

be factored into the calculation of the drag coefficient.

Factors such as the contributions of the engine cowling and the landing gear would

normally be included in any complete simulation, but are not included in this

simulation. In addition, the most serious weakness in the evaluation of CDis the lackof CD, a term which compensates for the increased drag arising from sideslip. No

suitable method was found for estimating this parameter.

Therefore the important parameters are:

CD() - the drag as a function of alphaCD(elv) - the drag of the elevon deflectionCD(wf) - the drag of the wing flap

These merge to form the overall drag coefficient in the following manner:

( ) ( ) ( )D D D DC C C elv C wflap= + + (3.6)

3.1.1.3 Evaluating CY

The sideforce coefficient varies mostly with sideslip, due to the effects of the tail. In

addition, it is affected by the rudder deflection. These terms together dwarf any

others, but the effects of a rolling velocity (CYP) are also included in the later model.

The effects of Mach number and altitude are accounted for through the lookup table

system. So the important parameters are:

CD() - the drag as a function of sideslipCY(rud) - the sideforce effects of the rudderCYP - the effects of rolling velocity on sideforce

The sideforce coefficient is calculated according to the following equation:

( ) ( )2

Y Y Y YP

T

bC C C rud C

V= + + P (3.7)

3.1.1.4 EvaluatingL

C

The rolling moment coefficient is dependent on the following factors:

Cl() - the rolling moment as a function of sideslip

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Cl(rud) - the rolling moment as a function of rudder deflection

Cl(elev) - the rolling moment as a function of elevon deflection

ClP - the effects of roll rate on the rolling moment

ClR - the effects of yaw rate on the rolling moment

The dependence on sideslip is the result of the airflow into the tailplane, and apositive angle of sideslip causes a positive roll, that is, right wing downwards. The

rudder dependence is a modification of above effect. It is generally an undesired

effect, as it is cross couples yaw control to roll control. It would be advantageous to

use the rudder to control yaw alone, but this is not possible on most aircraft, due to the

positioning of the rudder above the centre of gravity of the aircraft.

Differential deflection of the elevons is the primary method of controlling the roll of

the aircraft. Lowering the lift production of one elevon and raising the other causes

the shifting of the lifting force to one side, causing a rolling moment. The convention

of leading edge up, trailing edge down giving a positive elevon deflection is used

throughout.

The two dynamic coefficients ClPand ClRare expressions of how the rolling moment

is altered by rolling and yawing rates. All these effects are modified for Mach

number, altitude and angle of attack through the lookup tables.

The complete rolling moment coefficient is formed from these constituent parts as

follows:

(( ) ( ) ( )2

l l l l LP LR

T

bC C C rud C elv C P C R

V= + + + + ) (3.8)

3.1.1.5 Evaluating CM

The pitching moment broadly depends on the angle of attack, elevon deflection and

flap deflection. Also included is the effect of pitch rate. So the terms are:

CM() - pitching moment coefficient dependence on angle of attackCM(elev) - pitching moment coefficient dependence on elevon deflection

CM(wf) - pitching moment coefficient dependence on wing flap

CMQ - the effect of pitch rate on pitching moment

The centre of lift is always towards the front of an aerofoil. For NACA profiles, it

commonly lies very close to the quarter chord point. As the centre of lift does not act

through the centre of gravity, the generation of lift also causes a pitching moment. If

the centre of lift is behind the centre of gravity, then the aircraft is statically stable, but

if the centre of lift if forward of the c.g, then the aircraft is statically unstable in pitch.

This is the case with the SRC, since the centre of gravity is at the half chord point of

the aerofoil.

A symmetrical deflection in the elevons produces both a vertical and horizontal force.

These contribute directly to both lift and drag. The force, being offset from the centre

of gravity, also produces a pitching moment. It is by this method that the aircrafts

pitch is controlled. This term is heavily dependent on the angle of attack of the

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aircraft, as at larger angles of attack, the tail is effectively obscured from the airflow.

The effect of this is modelled and enters the system through the lookup tables.

The wing flap operates along similar principles, although it actually changes the

properties of the main lifting surface to achieve the same effect. A positive

(downwards) deflection of the split flap produces a large increase in lift and drag, anda large nose down pitching moment.

CMQis directly related to the coefficient CLQexplained in Section 3.1.1.1, as they are

part of the same effect. In short, this change in the pitching moment is produced by

time delays in the angle of attack as the aircraft rotates.

It is usual to also include the term C , that is, the relationship between the time

derivative of angle of attack and pitching moment coefficient. This term arises

because an angle of attack change in the main wing is not experienced at the tail

instantaneously due to the, usually large, separation between them. A time delay

inversely proportionally to Mach number is experienced. As the tail is extremely close

to, and actually above, the main wing, the pitching moments dependence on is

neglected in this thesis.

Using the components summarised above, the complete pitching moment coefficients

is evaluated as follows:

( ) ( ) ( ) (2

M M M M MQ

T

bC C C elv C wflap C Q

V= + + + ) (3.9)

3.1.1.6 Evaluating CN

The yawing moment coefficient is primarily influenced by sideslip and the rudder. In

this thesis, CNis modelled with a considerable number of parameters. In addition to

the large scale moments just mentioned, several more are included. The effects of

elevon deflection, roll rate and yaw rate are all included as follows:

CN() - yawing moment coefficient dependence on sideslipCN(rud) - yawing moment coefficient dependence on rudder deflection

CN(elev) - yawing moment coefficient dependence on elevon deflection

CNP - effects of roll rate on yawing moment coefficient

CNR - effects of yaw rate on yawing moment coefficient

Sideslip causes a yawing moment mainly due to the effects of the vertical tail. In

conventional aircraft, the fuselage also has a considerable effect, but there is almost

no fuselage to describe in the SRC. The tail effect in sideslip is called weathercock

stability. It is so-called because the tail tends to yaw the aircraft into the wind,

reducing sideslip. That is, a positive sideslip generates a positive yawing moment and

hence yaw rate. The rudder is the device used to control yaw, and has a very large

effect on the yawing moment.

In addition to these obvious effect, the elevons also introduce yawing moments when

deflected differentially. The elevons contribute to the drag force of the aircraft. When

the elevons are at the same angle of attack, in a symmetrical deflection, this drag forceis equal across the horizontal tail. However, if the elevons are deflected differentially,

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to introduce a rolling moment for example, the elevon with the larger angle of attack

also generates a larger drag force and the other, with a lower angle of attack,

generates less drag. Therefore this imbalance contributes a yawing moment effect,

which is modelled in this thesis.

Finally, two dynamic coefficients factor into the yawing moment coefficient are alsoincluded: CNPand CNR. These express how a roll rate couples to yaw by generating a

yawing moment in CNP, and the damping effects of yaw in CNR.

The yawing moment, then, is:

( ) ( ) ( ) (2

N N N N NP NR

T

bC C C rud C elv C P C R

V= + + + ) (3.10)

This knowledge of how to calculate the aerodynamic coefficients, detailed throughout

this section, leads on to the task of calculating the components of each. This is dealt

with in 3.2.

3.2 Calculation of Aerodynamic Data

No simple methods exist to perfectly predict each value at a given flight condition.

Wind tunnel modelling and computation fluid dynamics modelling (CFD) are the

most accurate methods. They are also the most expensive methods, primarily due to

the energy demands of the former, and the computational power demands of the latter.

Conventionally, these two methods are combined to varying degrees. Another method

available is to predict the values of each aerodynamic parameter using known

techniques. This particular method is the one used in this thesis. The task then

becomes one of using computer software to predict as much as possible about theaircraft without actually delving deeply into a CFD analysis. Three pieces of software

were used in this endeavour: XFOIL, the USAF DIGITAL DATCOM, and Tornado.

Xfoil was used to generate the two dimensional characteristics of the chosen aerofoil,

the NASA-Langley N0011SC. The DATCOM took this Xfoil data and, using a

geometric approximation, calculated very comprehensive sets of data. These sets were

augmented by data generated by another modeller called Tornado. This software

generated rudder data and additional elevon data.

3.2.1 Xfoil

The freeware 2D aerofoil modeller, Xfoil, developed by Mark Drela at MIT, was used

to evaluate the aerodynamic characteristics of the chosen aerofoil, the N0011SC. This

software has several methods available to solve for the pitching moment. The most

advanced, and the one used in this thesis, is a viscous formulation. This incorporates

an advanced model of the boundary layer, and gives accurate results. The boundary

layer models are based on the ISES methods.

The solutions incorporate the Karmen-Tsien compressibility corrections. In the

presence of shocks, these compressibility corrections fail, and so the Xfoil output data

is only valid up to around Mach 0.7. This is perfectly acceptable for the purposes ofthis thesis.

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A full detailing of how the solvers work can be found in Drela and Giles (1987)2,

while the Xfoil programs operation is detailed in Drela (1989)3.

Xfoil can take several forms of input. The one most convenient is to directly load the

coordinates of the aerofoil to be used in the SRC. These coordinates were downloadedfrom the NASG online database. The standard form of these coordinates is one that

expresses the coordinates starting at the trailing edge, proceeding over the upper

surface, and returning over the lower surface. A plot of these coordinates is shown

below. The coordinates are included in full in Appendix A, as is the Matlab plotting

script.

X 1.000 0.9750 0.0020 0.0000 0.0020 0.9750 1.000

Y 0.000 0.0064 0.0092 0.0000 -.0092 -.0064 0.000

Table 3-1 Example of Xfoil Coordinates

In general, the larger the number of coordinates, the better the solution. However,computing times also increases with number of panels. The PANE command in Xfoil

improves the distribution of panels, giving a large number of points at the leading and

trailing edge, with a lower number across the flatter areas of the aerofoil.

Figure 3.1 SRC Aerofoil- N0011SC

The bluntness of the aerofoil pictured above makes it particularly suitable for use on

the SRC. The trailing edge, in order to support a tailplane, cannot be excessively

sharp.

No explanation of the general usage of Xfoil is given here, only an explanation of

which commands were actually used in this thesis is offered. The Xfoil manual offers

a full and detailed explanation of each one of its features. Firstly, the loading and

smoothing of coordinates using the LOAD and PANE commands. The aerofoil

reference point was set to x=0.5, the centre of gravity, instead of the conventional

quarter chord point. Following this, all work was carried out in the OPER section ofthe program. VISC sets the solving type from inviscid to viscous. For a viscous

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solution, both the Reynolds number and Mach number are required. These are set

using the RE and MACH commands respectively. The useful output from Xfoil is the

lift coefficient, the drag coefficient, and the pitching moment coefficient. All will be

useful later on.

The simulations were carried out across a range of velocities M0.08 up to M0.7:roughly from a transition speed of 150km/h at low altitude, up to an almost transonic

speed. Complete tables of the aerodynamic coefficients of the N0011SC are included

in Appendix A.

Once the Mach number and Reynolds number are set, a sweep across a set of angles

of attack is performed using the ASEQ command, and logged to a text file using the

PACC command. A single solution at an alpha of 4 degrees and at a mach number of

0.08 is shown in Figure 3.2.The upper plot shows how the pressure coefficient varies

across the aerofoil. The viscous solution is shown in solid coloured lines, and the

inviscid solution is shown in a broken black line. The lower plot shows the effective

aerofoil: the actual aerofoil and the effects the boundary layers have on it.

Figure 3.2 Xfoil Solution of N0011SC at M0.08, 4

The ASEQ operation is performed at increasing Mach numbers until a suitable

amount of data has been gathered. The simulation was run at M0.08, 0.1, 0.2, 0.3, 0.4,

0.5, 0.6 and 0.7. A display of the lift and drag coefficients at the transition speed is

shown in Figure 3.3.

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0 1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

1.5

angle of attack (deg)

CL

Lift Coefficient at M0.08

0 1 2 3 4 5 6 7 8 9 100

0.005

0.01

0.015

angle of attack (deg)

CD

Drag coefficient at M0.08

Figure 3.3 Lift and Drag Coefficients of N0011SC (M0.08)

Using these methods, a complete set of two-dimensional data of the N0011SC

aerofoil is made available. From this point onwards, it becomes a task of convertingthis 2D data into a 3D representation of the actual aircraft.

3.2.2 USAF Digital DATCOM

The USAF Stability and Control DATCOM (Data Compendium)4arose from a

United States Air Force program. It is an expansive document created during the

1960s and expanded upon in the 1970s, and was created to document the methods

for estimating parameters of stability and control of an aircraft. The techniques within

this document extend from predicting the lifting coefficient of a simple aircraft, to

predicting the control parameters of a hypersonic (>Mach 5) flap.

In the late 1970s, these methods were coded in Fortran at Wright AFB. This software

program is called the USAF Digital DATCOM, and was made available to

aeronautical engineers many years ago. Having the DATCOM methods implemented

in software has obvious advantages. The methods can be used to generate vast

quantities of data using only a fraction of the time and effort previously required.

This software is used to generate the majority of the coefficients detailed in Section

3.1.1, and is the source of over 95% of the final aerodynamic database. Again, no

general explanation of the usage of this software is offered. The manual of the Digital

DATCOM can be consulted by the interested reader5. The complete set of input files

is included in Appendix D.

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Six different analyses were performed on the SRC concept. The first is an analysis of

the static coefficients of the SRC. That is, an evaluation of parameter dependence on

angle of attack, and sideslip among others. The second is an analysis of the dynamic

coefficients. This studies the effects that the body rates have on each parameter. For

example, how pitch rate affects the lift coefficient. The third and fourth analyses study

the effects of symmetrical and differential elevon deflection respectively. That is, thepitching and rolling effects of the elevons. The fifth evaluates trim conditions of the

aircraft. That is, how the aircraft can be trimmed in straight and level flight: which

elevon deflection achieves a trim condition. The last studies the effects of the split

flap on the lower surface of the aircraft fuselage/wing.

3.2.2.1 DATCOM Approximations

The DATCOM program allows for a diverse array of aircraft to be simulated, from a

conventional aircraft such as a Cessna 172R, to supersonic aircraft with canard

control surfaces. The program can accept complicated geometries. That stated,

approximations have to be made of more complicated shapes.

The SRC, being disc shaped, is not the easiest shape to input into the DATCOM

program. This software allows for an aircraft shape to be defined in several parts: the

body, the inner wing, and the outer wing. This allows for close approximations to

complex shapes such as the Boeing 777 aircraft to be made. In order to simulate the

disc shaped SRC, it is reduced from a circular to an octagonal wing.

Figure 3.4 SRC Geometric Approximation

Furthermore, it is assumed that the engine cowling fits into the aircraft body, and

therefore does not contribute any more to the drag of the aircraft.

Aside from these simplifications, the aircraft is given a full analysis. The complete

listing of the geometry and other parameters of the SRC is included in Appendix A.

3.2.2.2 Static Analysis

The static analysis of the SRC produces those components dependent on the

orientation of the aircraft. For example, the lift coefficient as a function of angle of

attack, a static property, is evaluated within this section. The coefficients calculated

within this section, all as functions of alpha, altitude and Mach number are:

DC - drag coefficient

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LC - lift coefficient

C - pitching moment coefficient

NC -N

, yawing moment coefficient sideslip derivative

lC -L

, rolling moment coefficient sideslip derivative

3.2.2.3 Dynamic Analysis

The dynamic analysis in the DATCOM evaluates the aerodynamic derivatives. That

is, those derivatives dependent on things such as rolling or pitching rotation rates. The

coefficients evaluated are:

LQC -L

Q

, the lift pitch rate derivative

C -Q

, the pitching moment pitch rate derivative

LPC -L

P

, derivative of rolling moment with respect to rolling velocity

YPC -Y

P

, derivative of sideforce with respect to rolling velocity

NPC -N

P

, derivative of yawing moment with respect to rolling velocity

NRC -N

R

, derivative of yawing moment with respect to yawing velocity

LRC -L

R

, derivative of rolling moment with respect to yawing velocity

These together form an almost complete set of dynamic derivatives. Others are absent,

but those given above generally dwarf these. Each coefficient varies with Mach

number, altitude and angle of attack.

3.2.2.4 Symmetrical Elevon Analysis

The analysis of elevon effects is formed from two sections. One analyses the rolling

effects of asymmetrical deflections, and the other, this section, studies the effects ofsymmetrical deflections. The symmetrical results are derived by defining the entire

tail surface as a flap. The parameters evaluated in this section are as follows:

LC - change in lifting coefficient due to elevon deflection

C - change in pitching moment coefficient due to elevon deflection

DC - change in drag coefficient due to elevon deflection

LC and C are evaluated as functions of Mach number, altitude, and elevon

deflection. DC is also evaluated with the extra dimension of angle of attack.

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3.2.2.5 Differential Elevon Analysis

The differential elevon analysis is achieved by defining the tail surface as an all-

moving horizontal stabiliser. The only output from this analysis is the rolling moment

due to this deflection:

LC - rolling moment coefficient due to asymmetrical elevon deflection

This coefficient varies according to Mach number, altitude and flap deflection. It does

not vary with angle of attack, and this is a weakness in the model. However, high

alpha flight is outwith the remit of this thesis, and so is a tolerable weakness.

3.2.2.6 Split Flap Analysis

This section is very similar to 3.2.2.4 in that it uses the same method of analysis. The

outputs are:

LC - change in lifting coefficient due to flap deflectionC - change in pitching moment coefficient due to flap deflection

DC - change in drag coefficient due to flap deflection

3.2.2.7 Trim Analysis

The trim analysis does a basic trim study by evaluating the necessary tail angle to

achieve a trim condition in pitch. The data derived from this section is used for

analytical purposes. The data are derived at each Mach number, and altitude of note,

and are:

elevon - elevon deflection necessary to trim aircraft

DtC - tail contribution to drag coefficient at trim

LtC - tail contribution to lift coefficient at trim

tC - tail contribution to pitching moment at trim

3.2.3 Tornado

Tornado is a Matlab program developed by Tomas Melin at the Royal Institute of

Technology in Stockholm. A full explanation of the workings of the software can befound in Melin (2000)6. The program uses a vortex lattice method to evaluate

aerodynamic properties of aircraft shapes. In this thesis, it was used to evaluate those

coefficients not calculated by the DATCOM. Primarily, it was used to calculate the

effects of the rudder, and to extend the quality of the elevon model. It also calculated

the CYcoefficient: how the sideforce changes with sideslip. The full list of those

coefficients calculated in Tornado are listed here:

CY- sideforce due to sideslip

- sideforce due to rudder deflectionY rud C

- yawing moment due to rudder deflectionN rudC

- rolling moment due to rudder deflectionl rud C

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l elevC - rolling moment due to differential elevon deflection

N elevC - yawing moment due to differential elevon deflection

3.2.3.1 Defining the SRC Model in TornadoTornado has the capability to define much more complex shapes, but this is time

consuming both in construction and simulation of the model. The DATCOM, as

explained earlier in this chapter, can only take three different sections of the fuselage

nad wing of the aircraft: body, inner wing, and outer wing. Tornado can define a

limitless amount of these. This can be advantageous, as it allows for very complex

geometries to be modelled, but it is time consuming, both in terms of input and

calculation. Once again the disc is approximated by an octagon. The complete

specifications of the Tornado model of the SRC are included in Appendix E.

A plot of the SRC, made from within the Tornado program, is shown in Figure 3.5.

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5

-1

-0.5

0

0.5

1

1.50

0.5

1

Wing x-coord

3-D Wing configuration, Vortex layout.

Wing y-coord

Wingz-coord

Figure 3.5 SRC Model in Tornado

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-25 -20 -15 -10 -5 0 5 10 15 20 25-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

Rudder Deflection (deg)

Cn

Yawing Moment Coefficient Against Rudder Deflection

Cn

Cn-interp

Figure 3.6 Rudder Effects on Yawing Moment from Tornado Simulation

Figure 3.6 shows a Tornado simulation of the yawing moment properties of the

aircrafts rudder. Superimposed on this in blue, is an interpolation of this inherently

piecewise data. The interpolation displays how the data is given the appearance ofcontinuity through the use of interpolation in the lookup tables.

3.3 Importing into MATLAB from the Digital DATCOM

The data output format of the Digital DATCOM, samples of which can be found in

the Appendices, is extremely unwieldy, and cannot simply be loaded into Matlab. To

do it manually would take many, many hours, and would be fraught with errors. The

data is spread over tens of thousands of lines across tens of output files. Extensive

importation scripts have been written to carry out this task with as little user input as

possible. These scripts are included in Appendix C. The code begins by importing the

data using low-level file operations into the Matlab workspace. Following this, theprogram sorts the vast amounts of data into its constituent sections. Mainly by using

string searches, the numeric data is filtered out, and sorted into suitable forms: usually

three or four-dimensional numeric arrays. Finally, the data tables are stored in a single

structure of the following form.

Aerodatabaseo Static

, , , ,L D M N LC C C C C

o Dynamic

, , , , , ,LQ MQ LP YP NP NR LRC C C C C C C o DiffElevon

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lC

o ElvPitch

, ,L MC C C D

D

o Wingflap

, ,L MC C C o Trim

, , ,elevon Dt Lt Mt C C C

o Tornado

, , , ,Y N elev l rud N rud Y rud C C C C C

The data in this form is ideal for being accessed as lookup tables.

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4 Creating the Simulation ModelWith a detailed model of the aerodynamics of the aircraft being made available

through the methods detailed in Chapter 3, one can proceed to develop a simulation

which integrates this aerodynamic model with the other parts of an aircraft like the

engine and control surface actuators. Additionally, this completed aircraft model can

then be placed within a realistic environment factoring in things such as atmospheric

and gravity models. The end result is a robust model that offers accurate results across

a whole host of flight conditions, allowing for detailed simulations to be carried out.

The software used in this endeavour is Matlab and Simulink, with various other

toolboxes.

What follows is an explanation of the constituent parts of this model. In addition,

some basic theory is explained in order to reveal the purpose of some sections of the

model.

4.1 Completing the Aircraft Model

The aim of this aircraft model is to have a model that can solve for the aerodynamic

forces in body axes, given any reasonable flight condition. That is, the model takes as

inputs such parameters as Mach number, altitude, angle of attack and sideslip, control

deflections and suchlike. From these inputs, the model can collate all forces in a

suitable frame, for the future solving of the equations of motion.

4.1.1 Coordinate SystemsBefore detailing each part of the simulation model, it is necessary to briefly explain

the different coordinate systems used within this thesis. Several coordinate systems

are required, since many particular forces or velocities are naturally considered in

terms of certain a certain frame. The three axes systems that are used are:

1.) Wind axes

2.) Body axes

3.) Earth axes

4.1.1.1 Wind Axes

The wind axes are those axes defined by the orientation of the aircraft relative to the

airflow. Aerodynamic forces and moments, generated by the motion of the aircraft in

through the atmosphere, are naturally evaluated in terms of the wind axes. These axes

are right-handed, and so the X, Y and Z-axes are considered positive into, rightwards,

and downwards of the wind.

By convention, the lift force is taken to be positive upwards, and the drag force

positive backwards. Therefore the actual lift and drag forces in wind axes are L and

D respectively.

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It is possible to use this frame as the basis for solving the equations of motion. Almost

all forces and moments are inherently calculated in the wind axes, and the others can

be rotated into the wind-axes with ease, before then rotating the complete set into an

inertial frame. The problem with this method is that, as the wind-axes are not fixed to

the body, the inertia matrix is not constant. Therefore, to use this method properly, the

system must take time-varying moments of inertia into account. To avoid such a levelof complexity, all forces and moments are considered exclusively in the body axes.

4.1.1.2 Body Axes

The body axes are fixed to, and so translate and rotate with, the aircraft. The

convention is that x, y and z point forwards, starboard, and downwards respectively,

with the origin lying at some reference point on the aircraft. As the engine is fixed to

the body, it is in these axes that the propulsive forces and moments from the engine

are resolved.

For the purposes of collating all forces, the body axes have a significant advantage

over the wind-axes. As mentioned in Section 4.1.1.1,the wind-axes must take time

varying moments of inertia into account. The body frame, being fixed to the aircraft,

has a fixed moment of inertia, neglecting any fuel consumption or release of payload.

It is a more intensive task to collect all forces into the body axes, since almost all are

naturally calculated in terms of the wind axes.

The body and wind axes are related to each other by the angles of attack and sideslip,

and : the aerodynamic angles. Any vector defined in one can be converted into theother by a rotation about these angles. The rotation matrix from body to wind axes is:

cos( )cos( ) sin( ) sin( )cos( )cos( )sin( ) cos( ) sin( )sin( )

sin( ) 0 cos( )

W

BR

=

(4.1)

This matrix is orthogonal, therefore the transform from wind to body axes, BWR , is

simply the transpose of the above. All forces and moments are calculated in terms of

either body or wind axes, and it is trivial to group all together in one frame using (4.1)

or its inverse, but in order to solve the equations of motion, each force and moment

must first be rendered in an inertial frame.

4.1.1.3 NED Frame

An inertial frame is one that does not translate or rotate relative to the fixed stars.

Such a level of complexity is entirely inappropriate in the study of kinematics on

Earth. In aircraft, two frames are typical: an Earth Centred Inertial (ECI) frame, or a

North-East-Down (NED) frame. The former translates, but does not rotate, with the

Earth. Though not perfect, it is closer to an inertial frame than the latter, which

assumes a flat Earth. Navigation plays no part in this study, so the accuracy of the ECI

frame is not required; therefore the NED approximation suffices.

The conversions between body and NED frames can be carried out using either a

three or four variable attitude propagation technique. The orientation of the aircraft inthe NED frame is expressed using the three euler angles , , and ; roll angle, pitch

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angle, and yaw angle respectively. In this thesis, the three variable approach is used.

The weakness of this method is that singularities exist at 90= degrees. Although aprecise value of degrees is rare, the region surrounding these points also give

unreliable results. The position and orientation derivatives are given by7

90

:

( )( )

cos cos cos sin sin sin cos

sin sin cos cos cos

u vn

w

+ = + +

(4.2)

( )

( )

sin cos cos cos sin sin sin

sin sin cos cos sin

u ve

w

+ += +

(4.3)

sin cos sin cos cosd u v w = + + (4.4)

sin tan cos tanp q r = + + (4.5)

cos sinq r = (4.6)sin cos

,cos cos 2q r

= + (4.7)

A four variable propagation method, known as the quaternion method, avoids the

singularity indicated in (4.7).The only problem with this technique, neglecting the

slightly higher computational requirements, is that it is not an optimal method. That

is, four variables are used to express the values of three states: roll, pitch and yaw.

This introduces purely computational modes into the system, which do not exist in the

actual aircraft8.

In this thesis, the three variable method is used. The advantage of removing the

singularity is negligible, since this singularity will not be encountered in the set ofbasic simulations. This thesis purely studies basic manoeuvres such as level flight,

climbing and descending, and turning. To study advanced manoeuvres such as a

loop-the-loop would require the use of the four variable quaternion method.

4.1.2 Lookup Tables

This set of blocks is used to access the data collated in Chapter 3. Through doing this,

the complete set of aerodynamic forces in wind-axes is made available. These forces

can then be rotated to the body axes, and summed with the engine forces. The lookup

table blocks interpolate the data for any value of input. This means that although all

the lookup tables are generated in a piecewise fashion, they interpolations give a goodapproximation of continuity. For example, most data is generated at altitude steps of

anywhere from 100m to 1000m, but the lookup table generates values at any point in

between these steps through cubic interpolation. This feature grows in importance

when high order tables are used. The block is actually capable of carrying out an N-D

interpolation, but in this thesis the lookup table of highest order is one of four

dimensions: altitude, mach number, angle of attack, and flap deflection.

The lookup tables shown below in Figure 4.1 implement Equation (3.10):the

calculation of the yawing moment coefficient.

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1

Cn

1.5

bProduct2

Product1

Product

f(u) Fcn

1-D T(u)

Cn(rud) Lookup

1-D T(u)

Cn(ail) Lookup

3-D T(u)

CNR Lookup

3-D T(u)

CNP Lookup

3-D T(u)

CNB Lookup

9

ElevDiff

8

RudDef

7

Vt

6

R

5

P

4

Beta

3

Machno

2

Alt

1

Alpha

Figure 4.1 Lookup Table Example (CN)

All lookup tables can be viewed in Appendix A.

4.1.3 Coefficients to Body ForcesEach force and moment coefficient is first dimensionalised according to the following

equations:

D

L

Y

l

N

D qSC

L qSC

Y qSC

L qSbC

qScC

N qSbC

=

=

=

=

=

=

(4.8)

Where q is the dynamic pressure 21

2V , and the other terms retain their earlier

definitions. These forces are rendered in the wind axes using convention as the guide:

lift is positive upwards, and so is negative in the wind z-axis, while drag is positive

rearwards, and so is negative along in wind x-axis. These equations are implemented

in the Simulink model shown in Figure 4.2.

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6

Yaw.Mom

5

Pit.Mom

4

Roll,Mom.

3

Lift

2

Sideforce

1

Drag

1.5

cbar

3

b1

3

b

Y

-C-

WingArea

alpha

beta

Sinv

Sinv

Alpha(deg)

Alt

Machno

Beta(deg)

Vt

Rudder Def

Elev Diff Def

ElvPitch

WingFDef

P(deg/s)

Q(deg/s)

R(deg/s)

CD

CY

CL

Cl_

CM

CN

SRC Lookup Table

Reshape1

Reshape

Matrix

Multiply

Product1

MatrixMultiply

Product

N

M

L_

L

f(u)

DynPress

m

m

m

m

D

-1

-L

-1

-D

8

P

7

C

6

Vt

5

Beta

4

Machno

3

Alt

2

AOA

1

rho

Figure 4.2 From Wind to Body Forces

The rotation from wind to body axes is shown below in Figure 4.3.

cos( )cos( ) sin( ) sin( )cos( )

cos( )sin( ) cos( ) sin( )sin( )

sin( ) 0 cos( )

T

B

WS

=

1

Sinv

sin

sin(beta)

sin

sin(alpha)

u[4]sb

u[1]*u[3]

sacb

cos

cos(beta)

cos

cos(alpha)

u[3]

cb

u[2]*u[3]

cacb

u[2]

ca

Vert Cat

Matrix

Concatenation3

Horiz Cat

Matrix

Concatenation2

Horiz Cat

Matrix

Concatenation1

Horiz Cat

Matrix

Concatenation

uT

Math

Function

deg rad

Angl e Conve rsion 1

deg rad

Angl e Conve rsion

0

0

-u[1]*u[4]

-sasb

-u[1]

-sa

-u[2]*u[4]

-casb

2

beta

1

alpha

[3x3][3x3]

[1x3]

[1x3]

[1x3]

[1x3]

[1x3]

4

4

4

4

4

4

4

4

4

4

Figure 4.3 Rotation From Wind to Body Frame

4.2 Simulation Environment

4.2.1 Atmosphere Model

A model of the International Standard Atmosphere (ISA), from the Simulink

Aerospace Blockset, is used in the simulation to generate all atmospheric parameters.

This model is valid up to altitudes of 20km, and so it is well within its capabilities tosimulate the atmosphere in this thesis.

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The block is used to generate parameters such as air density and speed of sound.

These parameters are vital throughout the simulation, and allow for incredibly

accurate calculations of dynamic pressure Mach number.

This model of the atmosphere allows for simulations to be carried out at any sensiblealtitude. Simulations above the tropopause present no problem whatsoever.

The ISA model is dealt with in full in the US Government document entitled U.S.

Standard Atmosphere9.

4.2.2 Wind Turbulence Model

The standard Dryden Wind Turbulence model from the Aerospace Blockset of

Simulink is used to simulate atmospheric turbulence. The Dryden Turbulence model

theorises that turbulence can be modelled by passing white noise through appropriate

filters.

The translational transfer functions from white noise to the velocities in Xe, Yeand Ze

axes in respectively are as follows:

2

2

2 1( )

1

1( )

1

1( )

1

uu u

u

vv v

v

ww w

w

LG s

LVs

V

LG s

V L sV

LG s

V Ls

V

=+

=

+

= +

(4.9)

where the terms are intensity values in the axis denoted by the subscript; the Lterms are turbulence scaling lengths defined as functions of altitude; and V is the

airspeed. For a detailed explanation of the workings of the Dryden Turbulence model,

the reader should consult MIL-F-8785C.

10

The model is perfectly capable of simulating both the translational and rotational

effects of turbulence. In this thesis, however, only the translational effects are

considered. In effect, this assumes that any wind gust occurs over an area much larger

than the aircraft. The SRC is a very small aircraft, with a wingspan of only 3m, and so

this assumption is a good one. Even with this assumption, the wind model still

simulates longitudinal, lateral and vertical wind disturbances, and so represents a very

advanced system.

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4.2.3 Gravity Model

Implemented within the system is a model of the World Global Survey 84 (WGS84)11

gravity model from the Mathworks Aerospace Blockset. The basic Taylor-Series

model is used, at latitude of 45. This model is more accurate than need be: the

simplicity of its inclusion is the only reason it is present. Across altitudes from 0m to

15km, the change in gravitational acceleration is only about 0.5%.

The gravity vector itself lies in the z-axis of the NED frame. To sum this with the

aerodynamic and engine forces, this is rotated into the body frame using a Direction

Cosine Matrix. This is shown in Figure 4.4.

1

XYZwg

0

zip

-K-

mass

45

lambda

WGS84

(Taylo r Series)Height (m)

Latitude (deg)Gravity (m/s^2)

WGS84 Gravity M odel

U( : )

Reshape

Matrix

Multiply

Product

Vert Cat

Matrix

Concatenation

3

height

2

DCM

1

XYZ

[3x1]

3

[3x1]

[3x1]

[3x3]

[3x1] 3

Figure 4.4 Gravity Model and Rotation to Body Frame

4.2.4 Engine Model

The engine of the SRC is modelled using a generic turbofan model from the Simulink

Aerospace Blockset. This model includes a 1storder actuator delay with a time

constant of 1s. The maximum static thrust at sea level was set to 3500N.

4.2.5 Flight Condition Calculation

The key flight condition inputs to the aircraft model are altitude, Mach number, angle

of attack and angle of sideslip. A Simulink block that calculates these can be seen in

Figure 4.5.

1tan ( )W

U = (4.10)

1sin ( )t

V

V = (4.11)

tVMa

= (4.12)

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where is the angle of attack of the aircraft; is the angle of sideslip of the aircraft;M is the Mach number the aircraft is flying at; U, V and W are the body velocities of

the aircraft; Vtis the airspeed of the aircraft: the magnitude of the body velocity

vector; and a is the speed of sound.

The angle of attack, sideslip and Mach number are all the result of the aircrafts

passage through the atmosphere. That is, the flight conditions are calculated according

to the velocity vector through the atmosphere. If the atmosphere is calmed, then this

vector is simply the body velocity. However, the turbulent nature of the atmosphere

necessitates the use of a different vector. These wind velocities are generated in the

NED-frame, and so must be rotated using a Direction Cosine Matrix (DCM), which

rotates any vector from the NED frame into the body fixed frame. Using this method,

an accurate calculation of alpha, beta and airspeed can be made. The new velocity

vector is calculated as follows:

N

w B E

D

W

V V DCM W

W

=

(4.13)

5

windv

4

altitude

3

beta

2

alpha

1

Machno

f(u)

mag(Vb)

rad deg

beta(deg)

f(u)

asin(V)

atan(u[3]/u[1])

alpha(rad)

rad deg

alpha(deg)

Vt/a (M)f(u)

Vt

Terminator3

Terminator2

Terminator1

Terminator

Matrix

Multiply

Product

Height (m)

Temperature (K)

Speed of Sound (m/s)

Air Press ure (N/ m^2)

Air Dens ity (Kg/m^3)

ISA Atmosphere Model

-u[3]

Height

Wind velocity (m/s)

Angular rates (rad/sec)

Al ti tude (m)

Airspeed (m/s)

Dryden Wind Turbulence Model

3

DCM

2

Vb

1

Xe

Figure 4.5 Flight Conditions Calculation Block

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4.2.6 Equations of Motion

The equations of motion are solved using the Simulink block provided with the

Aerospace Blockset. The input to the block is simply the summated body forces. The

parameters specified are the initial conditions: , , ,e bX V P , and the mass and inertia

tensor.

8

erates

7

PQRdot

6

PQR

5

Vb

4

DCM

3

euler

2

Xe

1

Ve

Forces X Y Z (N)

Moments L M N (N-m)

Ve (m/s)

Xe (m)

Euler (rad)

DCM

Vb (m/s)

p,q,r (rad/s)

pdot,qdot,rdot (rad/s^2)

Euler Rates (rad/s)

CUSTOM

Euler

Angles

6DoF (Euler Angles)

2

body moments

1

body forces

Figure 4.6 Equations of Motion Solver

4.3 Model Verification

This small section is used to demonstrate the instability of the SRC concept.

4.3.1 Lookup Table Data

The pitching moment coefficient is displayed in a 3D plot in Figure 4.7 as it varies

with angle of attack and altitude. It is clear from this plot that 0MC

>

. The

requirement for longitudinal static stability is that this derivative be negative. This is

intuitively correct. If 0MC

Body..............................................86Figure B.9 Addition of Engine Force/Moments ..........................................................87Figure B.10 Gravity Vector Calculation......................................................................87Figure B.11 Left+Right Elevon-> Symm+Diff Deflections........................................88Figure B.12 2nd Order Nonlinear Input Actuators ......................................................88Figure B.13 K-Gen- Controller Gain Generator..........................................................88Figure B.14 -K*x_error & Reference Value Generators.............................................89Figure B.15 B-Pseudoinverse Calculation...................................................................89

Figure B.16 Flight Condition Calculation Block.........................................................90

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Appendix A - Parameters of the SRC

A.1 Aircraft Dimensions

A.1.1 Wing/BodyDisc diameter= 3mDisc aerofoil = N0011SC

A.1.2 Vertical Tail

Longitudinal location of vertical tail apex = 1.8mRoot chord = 1.2mTip chord = 0.4m

Sweep angle = 16.9Tail height = 1m

Aerofoil = N0011SCRudder chord= 0.2m

Maximum rudder deflection = 20Rudder rate limits = 120/s

A.1.3 Horizontal Tail/ Elevons

Longitudinal location of horizontal tail apex = 2.2mVertical location of horizontal tail = 0.5mRoot chord = 0.8mTip chord = 0.35m

Sweep angle = 30Aerofoil = N0011SC

Maximum elevon deflection = 20

Elevon rate limits = 120/s

A.2 Aircraft Mass, Inertia and C.G

Mass = 700kg

218.53 0 200

Inertia matrix = 0 323.13 0

200 0 477.5

Longitudinal c.g location = 1.5mVertical c.g location = 0m

A.3 Aerofoil (N0011SC) Properties

Table A-1 N0011SC Coordinates

X Zupper X Zlower

1.0000 0 0 0

0.9750 0.0064 0.0020 -0.0092

0.9500 0.0120 0.0065 -0.0158

0.9250 0.0171 0.0125 -0.0203

0.9000 0.0216 0.0250 -0.0262

0.8750 0.0257 0.0375 -0.0302

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Appendix B Matlab Models

B.1 SRC Force Calculation

The blocks displayed within this section together evaluate the body forces on theSRC.

B.1.1 Lookup Tables

1

CL

f(u)

c/(2*Vt)*CLQ*alpha

Switch

1.5

Mean Chord

3-D T(u)

CLQ Lookup

3-D T(u)

CLDelta Wingflap Lookup

3-D T(u)

CLDelta Tailplane Lookup

3-D T(u)

CL Static Lookup

7

Wing Flap Def

6

Q

5

ElvDef

4

Vt

3

mach

2

alt

1

alpha

4

Figure B.1 CL Lookup Tables

1

CD

Switch

4-D T(u)

DeltaCD Tailplane Lookup

4-D T(u)

DeltaCD Split Flap Lookup

3-D T(u)

CD Lookup

5

Wing Flap Def

4

ElvDef

3

mach

2

alt

1

alpha

Figure B.2 CD Lookup Tables

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1CY

3

b

Product1

f(u) Fcn

3-D T(u)

CYP Lookup

3-D T(u)

CY Static Lookup

1-D T(u)

CY Rudder

7 Vt6

P

5

al t

4

rudderdef

3

mach

2

beta

1

alpha

3

Figure B.3 CY Lookup Tables

1

Cl

4-D T(u)

delta Cl Lookup

1.5

bProduct2

Product1

Product

f(u) Fcn

3-D T(u)

CLR Lookup

3-D T(u)

CLP Lookup

3-D T(u)

CLB Lookup

1-D T(u)

CL(rud) Lookup

8

ElDifDef

7

Vt

6

R

5

P

4

Beta

3

Machno

2

Alt

1

Alpha

33

Figure B.4 Cl Lookup Tables

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1

CM

f(u)

c/(2*Vt)*CLQ*Q

Switch

1.5

Mean Chord3-D T(u)

CMQ Lookup

3-D T(u)

CMDelta Wingflap Lookup

3-D T(u)

CMDelta Elevon Lookup

3-D T(u)

CM Lookup

7

Vt

6

Q

5

WingFlapDef

4

ElvPitch

3

Machno

2

Alt

1

Alp ha

Figure B.5 CM Lookup Tables

1

Cn

1.5

bProduct2

Product1

Product

f(u) Fcn

1-D T(u)

Cn(rud) Lookup

1-D T(u)

Cn(ail) Lookup

3-D T(u)

CNR Lookup

3-D T(u)

CNP Lookup

3-D T(u)

CNB Lookup

9

ElevDiff

8

RudDef

7

Vt

6

R

5

P

4

Beta

3

Machno

2

Alt

1

Alpha

3

3

Figure B.6 CN Lookup Tables

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95/145

B.1.3 Other Forces

B.1.3.1 Engine Forces

2

LMN

1

XYZ

Throttle position

Mach

Alti tude (m)

Thrust (N)

Fuel flow (kg/s)

Turbofan Engine System

Trm2

Trm1

Trm

TrAirSp

ControlInptsCT

Input Processor

Height (m)

Temperature (K)

Speed of Sound (m/s)Air Pressure (N/ m^2)

Air Density (Kg/m^3)

ISA Atmosphere Model

0.1

Eng.Displ

rho

AOA

Alt

Machno

Beta

Vt

C

P

Drag

Sideforce

Lift

Roll,Mom.

Pit.Mom

Yaw.Mom

Body ForcesMomnts

6

ControlInpts

5 PQR

4

Beta

3

Alp ha

2

Alt

1

Machno

Figure B.9 Addition of Engine Force/Moments

B.1.3.2 Gravity Forces

1

XYZwg

0

zip

-K-

mass

45

lambda

WGS84

(Taylo r Series)Height (m)

Latitude (deg)Gravity (m/s^2)

WGS84 Gravity M odel

U( : )

Reshape

Matrix

Multiply

Product

Vert Cat

Matrix

Concatenation

3

height

2

DCM

1

XYZ

[3x1]

3

[3x1]

[3x1]

[3x3]

[3x1] 3

Figure B.10 Gravity Vector Calculation

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B.2 Input Calculations

2

ElvPit

1

ElvRoll

0.5

avgpit

2

RElv

1

LElv

Figure B.11 Left+Right Elevon-> Symm+Diff Deflections

1

Cntrl

Ac_dem Ac_ac

Wing Flap Nonlinear Actuator

Ac_dem Ac_ac

Rudder Nonlinear Actuator

Ac_dem Ac_ac

Right Elevon Nonlinear Actuator

Ac_dem Ac_ac

Left Elevon Nonlinear Actuator

rad deg

Ang le Conve rsion7

rad deg

Ang le Conve rsion6

rad deg

Ang le Conve rsion5

rad deg

Ang le Conve rsion4

deg rad

Ang le Conve rsion3

deg rad

Ang le Conve rsion2

deg rad

Ang le Conve rsion1

deg rad

Angl e Con version

5

Throttle

4

WFlap

3

RElv

2

LElv

1

rudder

Figure B.12 2nd Order Nonlinear Input Actuators

B.3 Control System Blocks

1

k

k

To Workspace

MATLAB

Function

MATLAB Fcn

Trigger

2

u

1

x

4{12}

[6x1][6x1]

[18x1]

[6x12]

[6x12]

[6x12]

Figure B.13 K-Gen- Controller Gain Generator

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1

u

xx_errx_ref

level flight

xx_errx_ref

coord turn

xx_errx_ref

constant climb/descend reference

To Workspa

modelling and control of the simicon uav

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