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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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6.2.2 Constant Descent .................................................................................53
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
7.2.2 Constant Descent .................................................................................67
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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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
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