DEGREE PROJECT IN VEHICLE ENGINEERING, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2020

Actuator Saturation Analysis of a Fly-By-Wire Control System for a Delta-Canard Aircraft

ERIK LJUDÉN

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES Author

Erik Ljudén School of Engineering Sciences KTH Royal Institute of Technology

Place

Linköping, Sweden Saab

Examiner

Ulf Ringertz Stockholm KTH Royal Institute of Technology

Supervisor

Peter Jason Linköping Saab Abstract Actuator saturation is a well studied subject regarding control theory. However, little research exist regarding aircraft behavior during actuator saturation. This paper aims to identify flight mechanical parameters that can be useful when analyzing actuator saturation.

The studied aircraft is an unstable delta-canard aircraft. By varying the aircraft’s center-of- gravity and applying a square wave input in pitch, saturated actuators have been found and investigated closer using moment coefficients as well as other flight mechanical parameters.

The studied flight mechanical parameters has proven to be highly relevant when analyzing actuator saturation, and a simple connection between saturated actuators and moment coefficients has been found. One can for example look for sudden changes in the moment coefficients during saturated actuators in order to find potentially dangerous flight cases.

In addition, the studied parameters can be used for robustness analysis, but needs to be further investigated. Lastly, the studied pitch square wave input shows no risk of aircraft departure with saturated elevons during flight, provided non-saturated canards, and that the free-stream velocity is high enough to be flyable.

i Sammanfattning Styrdonsmättning är ett välstuderat ämne inom kontrollteorin. Däremot existerar det lite forskning gällande flygplansbeteende vid styrdonsmättning. Syftet med den här rap- porten är att identifiera flygmekaniska parametrar som kan vara användbara vid analys av styrdonsmättning av ett instabilt delta-canard flygplan. Genom att variera flygplan- ets tyngdpunkt och applicera en pulsinmatning i tippled har styrdonsmättning hittats och undersökts närmare med momentkoefficienter, men även med andra flygmekaniska parametrar. De studerade parametrarna har visat sig vara mycket relevanta vid analys av styrdonsmättning och ett enkelt samband mellan mättade styrdon och momentcoefficienter har hittats. Det går till exempel att leta efter plötsliga ändringar i momentkoefficienterna under mättning av styrdon för att hitta potentiellt farliga flygsituationer. De studerade parametrarna kan användas i en robusthetsanalys, men vidare forskning krävs. Den studerade pulsinmatningen i tippled visar även att så länge canarderna inte ligger i mät- tning, trots att elevonerna ligger i mättning, så är det ingen fara att flyga, förutsatt att

flyghastigheten är tillräckligt hög.

ii Contents

1 Introduction 1

1.1 Background ...... 1

1.2 Problem ...... 1

1.3 Purpose ...... 2

1.4 Goal ...... 2

1.5 Method ...... 2

1.6 Limitations ...... 2

1.7 Outline ...... 2

2 Theory 4

2.1 Desktop Simulation ...... 4

2.2 Nonlinearities of Control Systems ...... 4

2.3 Definition of Lift & Positive Pitching Moment of an Aircraft ...... 4

2.4 Stability ...... 5

2.5 Longitudinal Stability ...... 7

2.6 The Aft-Tail & Tail-First Arrangement ...... 8

2.7 The Modern Fighter Aircraft ...... 9

2.8 Control Surface Arrangements ...... 10

2.9 Aircraft Moment Equation ...... 11

3 Demonstrator 13

3.1 Deciding Maneuvers ...... 13

3.2 Angles & Total Moment on The Studied Aircraft ...... 13

3.3 Moment Coefficients on The Studied Aircraft ...... 14

3.4 Saturation Prediction ...... 18

4 Result 19

4.1 The Studied Square Wave Input ...... 19

4.2 Saturated Region 1 ...... 20

4.3 Saturated Region 2 ...... 23

4.4 Two Different Centers of Gravity ...... 25

5 Discussion 30

6 Conclusion 33

7 Recommendations For Further Research 34

iii Nomenclature

, = weight

= moment of inertia

! = lift

" = total pitching moment

! = lift coefficient

!,<0G = maximum lift coefficient

< = total pitching moment coefficient d = air density

+ = airspeed

+∞ = free-stream velocity

( = wing area

2¯ = wing mean aerodynamic chord

@∞ = free-stream dynamic pressure

; = distance

6 = gravitational acceleration

< = mass

U = angle of attack

U2A8C = stall angle X = deflection angle

\ = pitch angle

W = flight path angle

@ = pitch angle velocity

@¤ = pitch angle acceleration

4;4 = elevons

3;84 = deflection left inner elevon

3;>4 = deflection left outer elevon

3A84 = deflection right inner elevon

3A>4 = deflection right outer elevon

3A = rudder

20= = canards

A4BC = collection name for control surface interference, inertia, pitch damping, and aerodynamic

leading edge flap effects

iv Acronym List

FBW Fly-By-Wire c.g center of gravity

v 1. Introduction

1.1 Background

During the late 19th century some basic theory of aircraft stability appeared and proposed the tail plane to be a

basic element of pitch balance, stability, and control. In 1903 the Wright brothers performed the first flight of a

motor powered aircraft called the ‘Flyer’. This aircraft was difficult to fly due to pitch instability resulting in major

handling difficulties.

The Wright brothers realized that the plane would have to bank in a turn and thus decided to twist the Flyer’s wing during flight, causing one half of the wing to produce less lift and the other side to produce more. The aircraft was controlled using wires connected to a hip cradle which moved the rudder and twisted the wing. The elevator

control was operated using a lever, which adjusted the pitch of the plane [1].

For several years the standard was to use a cable-operated system. A stick was used to operate both elevators

and ailerons through a series of cables and pulleys, while the rudder was moved by foot pedals. With increasing

speed and aircraft weight the physical limitations of pilots began to be realized. The only requirement for the

cable-operated system was the physical strength of the pilot to control the control surfaces. This problem was solved

by integration of technology such as mechanical boosters to help move the control surfaces of large aircraft, and

later, hydro mechanical flight control systems got integrated [2].

With the introduction of Fly-By-Wire (FBW) technology in the 1970’s it became possible to use electrical

signals for control. Stick signals were converted to electrical signals and transmitted via electrical cables to an

on-board computer. In the on-board computer the stick signals together with measured flight condition data got

processed to determine the movement of each control surface actuator to provide the ordered flight response [3].

FBW technology is today used frequently within the flight industry. Due to the on-board computer, unintended

increases in angle of attack and sideslip can rapidly be detected and automatically be resolved by marginally

deflecting the control surfaces in the opposite way while the problem is still small [4]. This has made it possible to

purposely build unstable aircraft to gain the advantages of greater agility, as well as shorter take-off and landing

distances due to an overall increase in lift compared to a stable aircraft.

Since the pilot inputs does not move the control surface actuators directly, but are processed in the on-board

computer that determines the final control surface movement, one of many challenges is to understand how such a

FBW respond to actuator saturation. In this thesis the behavior of a FBW control system developed for an unstable

delta wing aircraft is analyzed in flight envelope regions where actuator saturation occur. The goal is to identify

flight mechanical parameters of relevance for analysis of the FBW control system during actuator saturation.

1.2 Problem

The control system to be studied is a FBW designed for an unstable delta-canard aircraft. With a saturated

control surface, the maximum deflection angle is set, generating its maximum lift force contribution. For an unstable

aircraft, saturated control surfaces can potentially be of serious nature if the generated force from the control

1 surfaces is not large enough to counteract the instability of the aircraft. Therefore, it is important to understand how and when actuator saturation occur. The control system is today studied using various simulation programs with and without human pilots where a criteria is used to help determine if the occurrence of saturation can imply a problem or not. However, a better understanding of the FBW response and which flight mechanical parameters that are of relevance to study during actuator saturation are requested.

1.3 Purpose

The purpose of this thesis is to identify flight mechanical parameters that are of relevance to study during actuator saturation.

1.4 Goal

In this study the following questions will be investigated

• Does a simple connection of actuator saturation and flight mechanical parameters exist for the FBW?

• Can the flight mechanical parameters be used to determine how robust the aircraft is towards departure from

controlled flight?

• How does one determine if actuator saturation during flight is a problem or not using flight mechanical

parameters?

1.5 Method

The data to be analyzed will be generated using a desktop simulator and is run by a script. As a first step, one needs to find typical maneuvers that can cause control surface saturation. Further, regions of flight where saturation occur during flight has to be found in order to finally analyze each case in detail. Since one is interested in the control system behavior, the control system will be tested for flight setups outside of the intended design.

Different flight mechanical parameters will be investigated closer in order to understand their connection to actuator saturation. All data gathered from the desktop simulation program will be analyzed using Matlab.

1.6 Limitations

In this study, only longitudinal maneuvers are investigated for actuator saturation. The control theory used by the FBW will not be looked into, but will be treated as a black box. Further, the acquired results will not be verified in a real-time simulator.

1.7 Outline

This paper starts by introducing background theory in section 2, regarding the used desktop simulator, aircraft stability, moments acting on an aircraft, moment coefficients, and a section explaining the purpose of different control surfaces. The theory is further implemented in the demonstrator, section 3, where the decision to study a square wave input in pitch is motivated. Further, the angles and moments acting on the studied aircraft is introduced and the connection between control surface deflection angles, moment coefficients and pitch angle is demonstrated.

2 Also, a prediction of the FBW’s behavior is made. Furthermore, the studied square wave input is shown in the result,

section 4. Here, the results for two regions with saturated actuators, as well as a comparison between two different

centers-of-gravity are shown. Additionally, the results are analyzed in the discussion, section 5. The investigated

questions are answered in the conclusion in 6 and finally, suggestions for additional work are presented in 7.

3 2. Theory

This chapter introduces background theory regarding the used desktop simulator, aircraft stability, moments

acting on an aircraft, and a section explaining the purpose of different control surfaces.

2.1 Desktop Simulation

Simulation has proved to be an excellent tool for development and there exist a number of simulation strategies with one of them being batch simulations. Batch simulations are used to study aircraft performance in a highly

repeatable and controlled environment. It consists of a large number of simulations and allows for a vast set of

initial conditions or variables to be set and be rapidly evaluated [5].

A batch simulation desktop program developed by Saab is used throughout this project. It is run by a script where initial conditions such as trim, speed, altitude, center of gravity (c.g), pilot models, and inputs can be applied.

This allows for the script to be rerun multiple times without changing its outcome, and would not be the case for a

real-time simulator where data is produced after real-time stick inputs. The gathered results are further exported to

Matlab for analysis.

2.2 Nonlinearities of Control Systems

Fly-by-wire control systems are integrated in many modern aircraft. Modern unstable combat aircraft are today

impossible to operate without it. These can under certain conditions demonstrate non-linearity in the flight control

system’s behavior. A common source of non-linearity arises from components of the control system such as control

surface actuators which all have amplitude and rate limiting characteristics. The non-linear response becomes

intrusive if the demands on the actuator are limiting [6].

2.3 Definition of Lift & Positive Pitching Moment of an Aircraft

The general equation to describe the lift ! acting on a wing is represented by

1 2 ! = ! d+ ( (1) 2

where ( is the wing area, d is the air density determined by the altitude, + is the airspeed, and ! is the lift

coefficient. The lift coefficient ! depends on the wing incidence relative to the flight path and increases with the

angle of attack U as seen in Figure 1A. The maximum lift coefficient !,<0G is received at the critical angle of

attack U2A8C , and is also called the stall angle. Further, the positive direction of the aircraft’s pitching moment " is shown in Figure 1B.

4 Figure 1. (A) Lift coefficient ! as function of U. The maximum lift coefficient is received at the stall angle U2A8C . (B) Definition of positive pitching moment direction ", angle of attack U, flight path angle W, and pitch angle \.

The total pitch moment around the aircraft’s center of gravity in Figure 1B is described by

" = <@∞(2¯ (2)

where < is the dimensionless moment coefficient for the whole airplane, @∞ is the free-stream dynamic pressure, ( is the aircraft reference wing area, and 2¯ is the wing mean aerodynamic chord. Further, the free-stream dynamic pressure is given by

1 2 @∞ = d+ (3) 2 ∞

where +∞ is the free-stream velocity [7].

2.4 Stability

A stable aircraft is by definition resistant to disturbances, meaning that it automatically will attempt to remain in trimmed equilibrium flight. The objective of trim is to bring forces and moments that are acting on the airplane into an equilibrium state. This means that total side, normal, and axial forces as well as the, pitch, roll and yaw moments are all zero. A stable aircraft will stay in equilibrium until it is disturbed by pilot stick inputs or external influences such as turbulence. The more stable the aircraft is, the greater the pilot control input needs to be in order to maneuver about the trim state. A stable aircraft with too much stability may limit controllability and maneuverability, while too little stability can give rise to an over responsive aircraft [6]. In an unstable aircraft

5 without a FBW control system, the pilot can not release the control stick for long since the aircraft does not converge

back to its trimmed equilibrium. The Wright brothers’ ‘Flyer’ is an example of such an aircraft [1]. Flying the

‘Flyer’ required continuous stick inputs by the pilot where releasing the control stick would end up with the aircraft

diverging to departure.

When trimming an aircraft the problem usually reduces to the task of longitudinal trim only. Lateral directional

trim adjustments are only likely to be required when the aerodynamic symmetry is lost, which as an example could

be after the launch of a missile or the loss of an engine on a multi-engine airplane.

Aircraft stability is often divided into two parts. Static stability and dynamic stability. Static stability is the

tendency of the airplane to converge to the initial equilibrium condition after a small disturbance from its trim

condition. Dynamic stability describes the oscillatory or irregular motion involved of recovering equilibrium after a

disturbance. This is illustrated in Figure 2.

Figure 2. Static and dynamic stability illustrated.

In Figure 2A static stability is illustrated with a ball in a bowl. A disturbance of the bowl will make the ball

move and return to its initial position after a certain time. If the bowl on the other hand is flipped up side down with

the ball balanced on top of it, a small disturbance will make the ball roll of the bowl and not return to its initial

position, and thus, illustrates static instability. Dynamic stability is further illustrated in Figure 2B and 2C with an

oscillating aircraft. The aircraft is dynamically stable in 2B since the oscillation is damped out with time while 2C

is dynamically unstable since the oscillation is amplified with time. It shall further be noted that the aircraft in

Figure 2B is both statically and dynamically stable, while Figure 2C is both statically and dynamically unstable [6].

6 2.5 Longitudinal Stability

Longitudinal stability refers to pitch stability. The point at which the airplane is neither stable nor unstable is called the neutral point. The neutral point is located lengthwise along the aircraft and at the neutral point, the moment is zero independently of the angle of attack. With a c.g position aft of the neutral point, the aircraft becomes unstable, while a c.g position in front of the neutral point makes the aircraft stable. The further the c.g is moved forward, the more stable the aircraft becomes. Moving the c.g close to the neutral point maintains stability but makes the aircraft more responsive in pitch maneuvering. In Figure 3, the degree of stability is illustrated showing an upper and lower bound for the c.g position. Within the aft and forward limit the airplane will be stable and have acceptable controllability. The pilot workload is also illustrated and the neutral point is located at the same level as the aft limit [6].

Figure 3. Degree of aircraft stability with moving c.g position.

Now consider a positive pitch disturbance from the trimmed equilibrium flight. This results in an aircraft nose up and thus, an increase in angle of attack which increases the lift coefficient !. In order for an aircraft to be statically stable, the resulting pitching moment acting on the aircraft must be restoring, and thus counteract the pitching disturbance. By plotting the pitching moment " or the pitching moment coefficient < for various U about the trim value U4, the longitudinal static stability can be determined. At U4 the moment coefficient is zero. With an increase in U, the pitching moment coefficient < becomes negative and is thus restoring for a stable aircraft. The total pitch moment coefficient is on the other hand not restoring for an unstable aircraft and amplifies the pitch disturbance instead. This is visualized in Figure 4 where < = <,U such that the total pitching moment coefficient only is dependent of the angle of attack.

7 Figure 4. Visualisation of how the pitching moment coefficient <,U varies with the angle of attack U for different degrees of longitudinal static stability.

With the assumption that the aerodynamic force and moment coefficients are functions of incidence only, the following equation is strictly valid for a longitudinal statically stable aircraft

3< < 0 (4) 3U

and is usually an acceptable approximation for subsonic aircraft. However, at Mach numbers where compress- ibility effects become significant, equation (4) is no longer valid, since the force and moment coefficients depends on both Mach number and incidence. Instead a more general requirement is needed which yields

3< < 0 (5) 3!

where the lift coefficient ! is positive for trimmed equilibrium flight. Referring back to Figure 4 it should be further noted that the slope decreases from (1) to (3) as the c.g is moved closer to the neutral point [6, 7].

2.6 The Aft-Tail & Tail-First Arrangement

There exists a number of different combinations of lifting and stabilizing surfaces. Two of these are the aft-tail and the tail-first arrangement. The conventional aircraft uses the aft-tail lay-out where lift is mainly generated by the wing. Here, the c.g is longitudinally located at the front half of the wing and a force pointing downwards, acting on the horizontal tail, also known as the horizontal stabilizer, is needed to increase the wing incidence and lift.

Initially, the total lift decreases, see Figure 5, and does not increase before the aircraft starts to rotate nose-up, which indicates that pitch control is indirect for this set up. The tail-first aircraft have a horizontal fore-plane, the canard, placed in front of the wing. Due to the fore-plane being destabilizing in pitch, the wing has to be placed further

8 back with the c.g in front of the wing in order to have a statically stable aircraft. The canard generate positive lift to

the extent that that an aft-tail aircraft can be made highly manoeuvrable due to the canard’s direct pitch control [7].

In Figure 5, the aft-tail and tail-first arrangements are shown. Here visualized in a pitch up maneuver where the

positive lift from the canard can be seen for the tail-first arrangement. Further, both aircraft are statically stable

since the wing lift force acts behind the c.g.

Figure 5. The aft-tail and tail-first arrangement visualized. Both aircraft are statically stable with the wing lift force acting behind the c.g. The control surface forces are marked for a pitch up maneuver. Total lift is initially reduced for the aft-tail to increase the angle of attack, thus showing that pitch control is indirect. Direct pitch control is shown by the canard adding additional lift for the tail-first aircraft.

2.7 The Modern Fighter Aircraft

A modern fighter aircraft is usually designed to be statically unstable. For an unstable aircraft the lift from the wing acts in front of the c.g and must continuously be counteracted by forces from the control surfaces. The lift

caused by the control surfaces used for trimming the aircraft acts in the same direction as the wing lifting force, see

figure 6B. This results in a total lift force which allows unstable airplanes to have smaller wings and be more agile

than stable airplanes. A classical stable aircraft, such as the Saab J35 Draken or the Saab AJ 37 Viggen must have a

large wing area to compensate toward the force acting downward on the control surfaces in a turn. This results in a

large drag and energy loss, which reduces turn rate. In addition, the landing speed is also higher due to the fact

that the force acting on the control surfaces decreases the lift of the wing. A fighter such as the JAS 39 Gripen is

a tail-first aircraft. It is statically unstable in subsonic flight and becomes stable in the supersonic region. This

fighter has the benefit of only needing small control surface deflections compared to a classical statically stable

airplane. The classical statically stable airplane becomes more stable in supersonic flight and adds the demand of

9 larger control surface deflections, and since the dynamic pressure is affecting the control surfaces, strong control

servos must be installed in order to handle the deflection demands which increases the weight of the airplane [8].

Figure 6. Showing two tail-first aircraft where (A) is statically stable, and (B) is statically unstable. Further, the forces acting on the aircraft during longitudinal trimmed flight are shown. The statically unstable aircraft (B) is here displayed with smaller wings compared to aircraft (A).

2.8 Control Surface Arrangements

The change of a control surface position produces moments about three airplane axes. Most lifting surfaces have

hinged flap-like surfaces at their trailing edges. Deflecting the surface up or downwards results in an incremental

lift force. The deflection angle X is considered to be positive when a negative moment about the associated body

axis is generated by the control surface. The main control surfaces on conventional aircraft are the elevator, aileron

and rudder. The elevator is located at the horizontal tail and by changing the lift on the elevator, pitch control is

achieved. Roll control is achieved by deflecting the ailerons on the right and left wing in opposite directions. The

ailerons are located at the trailing edge outboard toward the wing tips. The rudder controls the yaw, and is located

at the fin [7]. Fighter aircraft may be designed with control surfaces that serve dual purpose. An example is to

combine ailerons and the elevator which is called elevons, a second example is a ruddervator which combines the

function of the rudder and elevator, and a third the stabilator, a movable horizontal tail section which combines the

function of the horizontal stabilizer and elevator [9].

An additional control surface that is commonly used among fighter aircraft is the canard. This control surface

replaces the horizontal stabilizer and serves the purpose of increasing the total lift and improve aircraft control

10 compared to an aircraft with a horizontal stabilizer [10]. The Eurofighter Typhoon, the Rutan Long-EZ, and the

JAS39 Gripen are three examples of canard aircraft. In Figure 7 the main control surfaces of a delta-canard aircraft

are marked.

Figure 7. Figure of a delta-canard aircraft with its main control surfaces marked.

2.9 Aircraft Moment Equation

The total moment on an aircraft is a summation of all the forces acting on the airplane causing moments

around the c.g. In order to demonstrate this in a simple manner, assume steady level flight with thrust and drag at

equilibrium acting at the c.g. The lift is placed in the aerodynamic center which by definition is a point where the

pitching moment is independent of the angle of attack. The total moment equation is defined to be positive for a

nose up movement as described in section 2.3, and the total moment equation around the c.g for the aft-tail aircraft

in Figure 8 becomes

"26 = "F + "ℎ + !F ;26 − !ℎ;ℎ (6)

where "F and "ℎ are aerodynamic pitching moments acting on the wing and the horizontal tail caused by the

pressure distribution due to placing the lift forces in the aerodynamic center, !F and !ℎ are the lift acting on the wing and horizontal tail respectively, ;26,F is the distance from the wing’s aerodynamic center to the c.g and ;26,ℎ is the distance from the horizontal tail’s aerodynamic center to the c.g. Equation (6) can be further expressed in

terms of moment coefficients. The aerodynamic pitching moment due to the horizontal tail "ℎ can be assumed to

be small relative the wing aerodynamic pitching moment "F and thus neglected. By using equation (1), (2), and

11 (3) from section 2.3 with equation (6), and assuming the dynamic pressure at the horizontal tail to be equal to that

of the undisturbed airflow (@ℎ = @∞), the total moment coefficient becomes

;26,F (ℎ;26,ℎ <,26 = <,F + !,F − !,ℎ (7) 2¯ (2¯

where (ℎ;ℎ is the horizontal tail volume, and thus one can see how the different parts of the aircraft contributes to the total moment coefficient [7].

Figure 8. Forces and moments acting on an unstable aft-tail aircraft.

12 3. Demonstrator

In this chapter the theory described in chapter 2 is implemented to the project work. The decision to study a square wave input in pitch is motivated. Further, the angles and moments acting on the studied aircraft is introduced and the connection between control surface deflection angle, moment coefficients and pitch angle is demonstrated.

Finally, a prediction of the FBW’s behavior is made.

3.1 Deciding Maneuvers

The studied FBW control system is developed for an unstable delta-canard aircraft and integrated in a desktop simulation program which can be run for a wide range of flight envelopes and maneuvers. As described in section

2.5, moving the c.g further back from the neutral point makes the aircraft more unstable. This forces the control surfaces to larger actuator deflections in trimmed flight, which in return gives the actuators less authority to perform longitudinal maneuvers. The studied aircraft is unstable in subsonic flight and thus, the FBW needs to continuously use the control surfaces to counteract generated aerodynamic forces. Moving the aircraft’s c.g further and further backward from the neutral point should eventually force the FBW to set the control surface actuators in saturated positions during pitching maneuvers. This has further been confirmed by a number of tests in the desktop simulator and the decision to study a square wave input in pitch has been made. By moving the c.g aft the design limit for the

FBW, saturation of several control surfaces has occurred.

The studied aircraft is built to have the wing enter a stall before the canards do. This is preferred since a decrease of wing lift does not lose the pitch stability authority of the canards. If the canards enter a stall before the wing does, the aircraft risk to diverge. Consequently, longitudinal maneuvers give good prerequisites of aircraft departure occurrence during actuator saturation.

3.2 Angles & Total Moment on The Studied Aircraft

The studied aircraft has a delta-canard configuration, or as described in section 2.6, a tail-first arrangement. The positive angles of the elevons X4 and the canards X2 are presented in Figure 9 along with the positive direction of the total moment "26. Further, the flight path angle W, angle of attack U, the canard angle of attack U2, and pitch angle \ are shown.

Figure 9. Defined positive direction of angles and the total moment for the studied aircraft.

The total pitching moment around the c.g can be described using Newton’s second law for a rotating body,

13 which yields

¥ "26 = HH \ (8)

¥ where "26 is the moment, HH is the moment of inertia around the y-axis, and \ = @¤ is the pitch angle acceleration. Inserting equation (2) into equation (8) and rewriting leads to the following expression

"26 <,26@∞(2¯ @¤ = = (9) HH HH

giving an expression of how the aircraft pitch angle acceleration is effected by the total generated moment

coefficient <,26.

3.3 Moment Coefficients on The Studied Aircraft

The moment formula of equation (6) in section 2.9 contains moment contributions from each surface that

generates lift. It has further been shown that dividing the total moment formula with the dynamic pressure @,

reference wing area ( and the mean aerodynamic chord 2¯, gives the total moment coefficient and its dependency of

the different moment coefficients from each lifting surface. In the desktop simulation program the main contributions

to the total pitching moment coefficient <,26 are the moment coefficient contributions from the different control

surfaces and the moment coefficient due to the angle of attack <,U, introduced in section 2.5, and will be named the ’wing moment coefficient’. The studied aircraft has two inner and two outer elevons, two canards, and a rudder.

Furthermore, the simulation program also contains moment coefficient terms caused by interference between

different control surfaces, moments of inertia, pitch damping, and leading edge flaps used for optimization, which

are added together in a rest term denoted <,A4BC . The total pitch moment coefficient is thus written

<,26 = <,U + <,20= + <,3A84 + <,3A>4 + <,3;84 + <,3;>4 + <,3A + <,A4BC (10)

where index 20= is the canards, 3A84 is the right inner elevon, 3A>4 the right outer elevon, 3;84 is the left

inner elevon, 3;>4 the left outer elevon, and 3A is the rudder. Since the aircraft is built to be unstable, the wing

moment coefficient <,U has a positive slope and thus, amplifies a disturbance (as described in section 2.5). In

order to fly at equilibrium where <26 equals zero, the control surfaces have to counteract the moment generated from the angle of attack. Hence, the moment coefficients for the control surfaces must be of opposite sign than

that of <,U. Since the control surfaces are stabilizing the aircraft, the moment coefficient contribution of the

control surfaces can be described by line (1) in Figure 4, section 2.5. Likewise, <,U can be described by line (4) due to making the aircraft unstable. The four elevons are attached to the main wing and can be added such that

<,4;4 = <,3A84 + <,3A>4 + <,3;84 + <,3;>4. The contribution from the rudder <,3A , can be neglected for a pure pitch maneuver if the roll angle ? is small or zero. As a result, equation (10) can be written as

14 <,26 = <,U + <,20= + <,4;4 + <,A4BC (11)

In order to increase the intuitive understanding of equation (11), consider a step input maneuver. The step

response is presented in Figure 10 with the aircraft pitch angle \, pitch angle velocity @ and pitch angle acceleration

@¤. The step input corresponds to a maximum stick pull back as seen by the constantly increasing pitch angle \. The

mean deflection angles of the canards and the elevons are presented in Figure 11, where the deflection direction is visualized on the vertical axis. According to Figure 9, a positive deflection angle of the canards corresponds to the

leading edge deflecting upward, whereas a positive deflection angle of the elevons corresponds to the trailing edge

deflecting downward. In Figure 11, the elevons’ deflection angle has been plotted as −X4, such that a trailing edge down corresponds to negative values on the vertical axis. Finally, the moment coefficient contributions of all parts

in equation (11), including the total moment coefficient <,26 are shown for the step input maneuver in Figure 12. In common, these three figures have five vertical lines numbered (1-5) representing time events of interest. These

lines are used to describe the step input maneuver step by step with the following list.

• Before the step input is applied, the FBW is trimming the aircraft. The pitch angle \ is constant, meaning that

the pitch angle velocity @ and pitch angle acceleration @¤ are zero. The canards and the elevons are set at a

constant deflection angle with the elevons deflected upward, thus generating a positive moment along with

the wing moment. The positive moment is counteracted by the canards, but also by <,A4BC . Due to small variations, these moments are summed with equation (11) to a value close to zero, which agrees well with

equation (9), giving @¤ = 0 if "26 = 0. • At point (1), the step input is applied. The aircraft is demanded to change its pitch angle and the FBW deflects

the elevons further upward, while also moving the canards’ leading edge upward. This generates a positive

moment coefficient contribution from the elevons, and a reduced counteracting moment coefficient by the

canards. The total moment coefficient <,26 starts increasing and is only counteracted by the <,A4BC term. This initializes a positive pitch angle acceleration @¤, whereupon the pitch angle velocity @ starts to increase.

The control surface deflection forces acting on the aircraft between point (1-2) can be visualized with Figure

13A, showing the positive moment contribution of the elevons and the reduced counteracting moment by the

canards.

• At point (2) the aircraft reaches its largest acceleration for the whole maneuver. From this point, the FBW

starts to decrease both the trailing edge deflection of the elevons and the leading edge deflection of the canards

downward. The positive moment contribution by the elevons is reduced, and the negative counteracting

moment from the canards is increased. The FBW is thus balancing the constantly increasing wing moment

contribution <,U. Between point (2-3) the total moment coefficient <,26 decreases and hence, decelerates the pitch angle velocity @.

• From point (3) and onward <,U keeps increasing in positive value due to an increasing angle of attack. At point (4) the elevons crosses their neutral position and starts to counteract the wing moment together with the

15 canards. The generated deflection forces producing the counteracting moment is shown in Figure 13B. In

addition, the pitch angle velocity starts to reach a constant value between point (4-5).

• From point (5) the pitch angle velocity has reached steady state. Likewise, the moment coefficients <,A4BC ,

<,20=0A 3B, and <,26 have reached steady state. The canards deflection angle are held at a constant position, while the elevons deflection angle keeps increasing in order to produce a negative moment coefficient

counteracting the constantly increasing <,U.

Figure 10. Step input response of pitch angle \, pitch angle velocity @ and pitch angle acceleration @¤ for a step input.

16 Figure 11. Deflection angles of the canards X2 and elevons −X4 for a step input.

Figure 12. All pitching moment coefficients acting on the aircraft during a step input.

17 Figure 13. Picture showing generated forces due to deflected elevons and canards. In (A) the elevons contribute to the nose up movement produced by the wing. In (B) both the canards and elevons counteract the nose up movement produced by the wing.

The connection of the total pitching moment "26 and the pitch angle acceleration according to equation (9) can here be seen by comparing the pitching moment coefficient <,26 in Figure 12 with @¤ in Figure 10. Further, do notice that the FBW uses the elevons in first hand to produce large moments, while the canards mainly are used for stabilization adjustments due to the small moment contribution as seen in Figure 12 during the whole maneuver. In addition, notice that the moment contribution of <,A4BC only varies slightly.

3.4 Saturation Prediction

In the step input maneuver in Figure 12, <,A4BC varies slightly, but its value is small. The FBW mainly uses the elevons to perform the requested maneuver and uses the canards for stability. From point (5) and onward, the elevons are producing a negative moment counteracting the increasing wing lift force due to increasing angle of attack. It is further important to know that the canards have a larger deflection interval than the elevons. This implies that the FBW should prioritize to saturate the elevons before the canards in order to prevent the wing from stalling, while also heaving generated aerodynamic forces during pitch maneuvers.

18 4. Result

This chapter introduces the studied square wave input and shows the results for two regions with saturated

actuators, as well as a comparison between two different centers-of-gravity.

4.1 The Studied Square Wave Input

A square wave input has been performed for the studied aircraft as seen in Figure 14. Starting after trimmed

conditions, a maximum pull back stick is applied, marked with the left black dotted line. After a few seconds the

stick is set to its maximum forward position, and after a few additional seconds, back to full backward position

and so forth. Further, the c.g has been moved beyond the aft limit that the FBW is designed for. This along with a

built-in anti-symmetry causes the aircraft to roll and is attempted to be reduced by an additional closed loop pilot,

serving the purpose to hold the roll angle close to zero during the whole simulation. The influence of the closed loop

pilot can be seen deflecting the stick in the opposite direction of the roll angle. There are two areas marked in Figure

14. These represent the time intervals where actuator saturation occur and will be further investigated. In addition,

the deflection limits on the inner elevons have been lowered in order to trigger more and longer saturated regions.

Figure 14. Pitch angle \, roll angle q, and stick deflection for a square wave input. The marked regions represent time intervals where control surface actuators have saturated.

The dynamic pressure @, angle of attack U and total lift coefficient ! for the whole maneuver is further shown in figure 15. The dynamic pressure is given by equation (3) and has a large drop due to loss in airspeed +, while the

density can be seen as constant due to subsonic flight speed and small variations in altitude. Moreover, the critical

angle of attack is marked with a horizontal red dotted line.

19 Figure 15. Dynamic pressure, angle of attack, and lift coefficient. The stall angle of the wing is marked with a horizontal red dotted line.

4.2 Saturated Region 1

The two regions with saturated control surface actuators in Figure 14 and Figure 15 will now be investigated

closer starting with the shaded red region 1. Firstly, the pitch angle, pitch angle velocity and pitch angle acceleration

are presented in Figure 16. Secondly, the deflection angle of the canards, the left and right inner elevons as well

as the left and right outer elevons are shown in Figure 17. Finally the pitching moment coefficients are shown in

Figure 18.

The maneuver starts with the same behavior as for the step input in section 3.3 with \ increasing. The difference

of the two flight cases are a longer simulation time, lower start velocity as well as a c.g behind the designed aft limit.

After the pitch angle increase has started, the FBW is reducing the pitch angle velocity @ by moving the elevons

trailing edge down, while also moving the canards leading edge down. The right inner elevon follows the same path

as the left one, but starts to deflect upward right before the saturation region. Likewise behavior is observed for

the right outer elevon. The deviation of the right elevons is explained by looking at Figure 14, where one can see

how the aircraft starts rolling to the left. The right elevons behavior is thus the FBW’s response to the closed loop

pilot input, trying to steer against the aircraft roll. Further, the left elevons have to be deflected in the opposite

direction. This results with the left inner elevon being the first control surface to saturate, as seen in Figure 17, due

to its lowered maximum deflection limit. The left outer elevon is the second control surface to saturate due to the

aircraft roll angle, and a few time steps later, the right inner elevon becomes the third control surface to saturate.

The saturation of the right inner elevon is however due to the angle of attack approaching the wing stall angle as

seen in Figure 15. After half the time of the saturated region, the roll angle in Figure 14 is decreasing, thus allowing

20 the right outer elevon to deflect its trailing edge further down in order to prevent the wing from stalling. Finally, the

FBW decreases the roll angle such that all three saturated control surfaces can exit their saturated position and start

decreasing the roll rate.

The saturation phenomena can be further seen when looking at the moment coefficients in Figure 18. The total

moment from the elevons <,4;4 becomes restricted by the aircraft roll, whereupon the canards starts generating the

main counteracting negative moment coefficient <,20= to the positive wing moment coefficient <,U. Further

notice that <,A4BC is small in comparison with the moments caused by the control surfaces and the wing moment. Looking at the individual moment coefficients of the elevons one can see a strong indication that the left inner

canard 3;84 is saturated in the whole red shaded region due to its almost constant value, indicating that it does not

move and thus, potentially has saturated. The same behavior can be seen for the left outer elevon 3;>4 which also

has a moment coefficient close to constant. One further interesting observation is the magnitude of the control

surfaces moment contribution. Although the maximum deflection of the inner elevons have been set to a reduced value, they are producing a larger pitch moment coefficient compared to the outer elevons. The inner elevons are

thus primarily used for the aircraft pitch maneuvering, while the outer elevons primarily are used to counteract the

aircraft roll. This is best visualized by looking at the individual moment coefficients of the elevons in Figure 18.

Figure 16. Pitch angle \, pitch angle velocity @ and pitch angle acceleration @¤ for region 1.

21 Figure 17. Deflection angles of the canards X2 and each individual elevon plotted as −X4 for region 1.

Figure 18. All pitching moment coefficients acting on the aircraft, including each individual moment coefficient for the elevons. Plotted for region 1.

22 4.3 Saturated Region 2

The blue shaded region 2 in Figure 14 and Figure 15 will now be investigated closer. In Figure 19, the pitch

angle, pitch angle velocity, and pitch angle acceleration are presented. The deflection angles of the canards as well

as the deflection angles of the elevons are shown i Figure 20. In addition, the total pitching moments are displayed

in Figure 21. The stick starts at full backward position and is later pushed all the way forward. This is marked in all

three figures by the black dotted line.

Before the aircraft enters the saturation region, the stick is set at a full backward position, thus demanding the

aircraft to increase its pitch angle \. The pitch angle is increased as seen in Figure 19. However, the pitch angle velocity @ is decreasing and by further investigating Figure 20, one can see that all control surfaces are approaching

their maximum deflection limit to decelerate @. The inner elevons are the first control surfaces to enter saturation

and right before the stick forward input, the left outer elevon also saturates. At this point in time the FBW is trying

to prevent the wing from crossing the stall angle in Figure 15.

At the time point where the stick is set to full forward position, the aircraft is demanded to decrease its pitch

angle in order to nose down. This puts the right outer elevon into saturation as well, see Figure 20. The slope

of the canards deflection angle becomes much steeper and a few time steps later the canards enter saturation.

Simultaneously, the pitch angle in Figure 19 starts decreasing. However, the stall angle is crossed where a drop

in the total lift coefficient ! can be witnessed in Figure 15. The aircraft manages to recover from the stall, thus allowing the canards in Figure 20 to exit the saturation. A marginal time step later, both the right inner and right

outer elevon exit their saturated condition and start compensating for the induced roll angle seen in Figure 14.

In Figure 21 the moment coefficients can be seen. <,A4BC varies slightly and is small in comparison to the

control surface moment coefficients and the wing moment coefficient <,U. At the time point where both inner elevons enter saturation, and the left outer elevon is close to saturation, the maximum negative moment contribution

of the elevons is reached. From this point, the elevons moment contribution remains close to constant. The negative

moment contribution of the canards now becomes larger than the elevons, which also indicates that the elevons

have reached saturated conditions. Further, the moment coefficient of the canards becomes constant when the

canards enter saturation. The same tendencies of constant moment coefficients during saturated conditions can be

observed when inspecting the individual moment coefficients of the elevons. It is further interesting to note the

similar looking appearance that the canards and the individual moment coefficients have with the control surface

deflections. In addition, observe once again the larger magnitude of the pitch moment coefficients for the inner

elevons compared to the outer ones.

23 Figure 19. Pitch angle \, pitch angle velocity @ and pitch angle acceleration @¤ for region 2.

Figure 20. Deflection angles of the canards X2 and each individual elevon plotted as −X4 for region 2.

24 Figure 21. All pitching moment coefficients acting on the aircraft, including each individual moment coefficient for the elevons. Plotted for region 2.

4.4 Two Different Centers of Gravity

In the studied square wave input, two regions of saturated control surface actuators have been investigated closer.

However, the c.g has not been changed and the aircraft manages to recover from a stalled wing in saturation region

2. In this section, the c.g is moved a marginal step backward, resulting in a stall of the wing which the FBW does not manage to recover from. The old and the new c.g will be denoted CG 1 and CG 2 respectively and are plotted in the following figures for saturation region 2. Firstly, the pitch angle, roll angle, and stick deflection is shown in

Figure 22. Secondly, the dynamic pressure, angle of attack, and Lift coefficient are displayed in Figure 23. The pitch angle, pitch angle velocity, and pitch angle acceleration are further plotted in Figure 24. Furthermore, the deflection angles for the canards and the elevons are shown in Figure 25. Finally, the moment coefficients acting on the aircraft for the two centers of gravity are visualized in Figure 26, and the individual moment coefficients of the elevons are shown in Figure 27.

Starting with Figure 22, one can observe a larger pitch angle for CG 2, which decays slower over time. The stick input by the closed loop pilot is also larger, as well as the aircraft roll angle. Further, the difference in dynamic pressure in Figure 23 is marginal. However, the wing stall angle is crossed earlier in time and the aircraft does not recover to non-stalled conditions during the rest of the maneuver. In addition, the lift coefficient only varies slightly.

In Figure 24 one can see that the aircraft is slower to decelerate its pitch angle velocity. Interesting differences occur in Figure 25, where the canards and all elevons for CG 2 enter saturated states earlier than for CG 1. Here, the effect of the induced roll angle for CG 2 is seen, with the right outer elevon starting to deflect upward right after the stick forward command has been initialized. A red vertical line is drawn in all six figures. This marks the time point

25 where both the right inner and right outer elevon for CG 2 stop to move.

Right after the canards enter saturation, the difference of the total moment coefficient <,26 between CG 1

and CG 2 starts to increase as seen in Figure 26. The wing moment coefficient <,U is larger for CG 2, however, the total moment coefficient is not. This is due to the reducing moment coefficient contribution of the elevons

CG 2 <,4;4. The canards are actually producing a larger moment with the c.g moved backwards and the <,A4BC

coefficient is not contributing with any more moment to <,U. Looking at Figure 27, the reason of the reduced total moment coefficient emerges. Between the vertical dotted black line and the vertical dotted red line, the left inner

elevon CG 2 3;84 and left outer elevon CG 2 3;>4 highly reduces their moment contribution. The same can be seen

for the right inner elevon CG 2 3A84 and the right outer elevon CG 2 3A>4. Referring back to section 4.2, it was

stated that a moment coefficient with a constant value was giving an indication of the control surface not moving.

For CG 2 however, both inner elevons and the left outer elevon are losing moment while saturated. At the red vertical line, the right inner and right outer elevon stop decreasing their deflection angle upward. This results in the

FBW not managing to counteract the increasing aircraft roll angle, due to also trying to reduce the pitch angle and

thus recover from the stall. This leads to a constantly increasing stall angle and thus, the fact of aircraft departure.

Figure 22. Pitch angle \, roll angle q, and stick deflection for two different centers of gravity.

26 Figure 23. Dynamic pressure, angle of attack, and lift coefficient for two different centers of gravity. The stall angle of the wing is marked with the horizontal red dotted line.

Figure 24. Pitch angle \, pitch angle velocity @ and pitch angle acceleration @¤ for two different centers of gravity.

27 Figure 25. Deflection angles of the canards X2 and each individual elevon −X4 for two different centers of gravity.

Figure 26. All pitching moment coefficients acting on the aircraft for two different centers of gravity.

28 Figure 27. Moment coefficients of each elevon for two different centers of gravity.

29 5. Discussion

The purpose of this thesis is to identify relevant flight mechanical parameters that are of relevance to study

during actuator saturation caused by the FBW. In section 3.3, the studied FBW’s response to a step input is presented where the connection between control surface actuator deflections, moment coefficients, and the aircraft’s pitch

angle is illustrated. First off, the control surface deflections behavior can be seen directly in the moment coefficients.

As an example, an increase of deflection angle gives an increase in the corresponding moment coefficient. Further,

the moment coefficients gives the moment direction and are thus useful in order to determine what counteracts what.

As described in the theory, section 2.9, the moment coefficient from each part of the aircraft can be added to a total

moment coefficient, which in the desktop program simply is done using equation (10). The connection between the

total moment coefficient and the pitch angle acceleration is shown using Newton’s second law for a rotating body

according to equation (9), which shows that <,26 is proportional to @¤. In addition, the step input response clearly display how an acceleration creates a velocity, which in turn changes the pitch angle of the aircraft. Moreover, the wing moment coefficient <,U in Figure 12, shows the stability of the aircraft. The aircraft is unstable due to the constantly increasing positive value after the stick backward input has been applied, giving a disturbance from

trimmed conditions. This agrees with the theory introduced in section 2.5. After point (5), it can be seen in both

Figure 11 and Figure 12 how the elevons are used by the FBW to counteract the wing moment, while the canards

are held at a constant position, and thus producing a close to constant moment coefficient. Since these parameters

show a strong connection to one another, it was decided to use them when investigating saturation closer.

For the studied square wave input, the c.g is placed behind the intended aft limit design. The aircraft starts to roll

to the left, right after the stick backward command has been initiated, and is attempted to be reduced by the closed

loop pilot command. However, the roll causes both the left elevons to saturate (see saturation region 1, Figure 17).

This leads to the pitch velocity not being decelerated fast enough, causing the right inner elevon to saturate as well

in order to avoid stalling the wing, see Figure 15. By further investigating Figure 18, one can see that the right inner

elevon, 3A84, is producing a larger negative moment coefficient compared to the left inner elevon, 3;84, even though

its deflection angle is smaller. This can be due to anti-symmetry and the aircraft orientation, where aerodynamic

effects due to the roll angle might lead to different dynamic pressures on the control surfaces. Recall the observed

behavior of the canards producing a close to constant moment coefficient when held at a constant position for the

FBW’s step input response. Likewise behavior is observed for the saturated individual moment coefficients of

the elevons in Figure 18, thus indicating that the control surfaces are not moving. In addition, the inner elevons

maximum deflection has been lowered. However, the inner elevons are contributing with the largest pitching

moment coefficient, which shows that the inner elevons play an important role for pitch maneuvering. Furthermore,

notice how the summed moment coefficient of the elevons <,4;4 is producing a larger negative moment coefficient compared to the canards up until the left outer elevon , in Figure 17, also enter saturation. Afterward, the canards

start generating a larger moment coefficient, but do however not enter any saturated state. Accordingly, this implies

that the aircraft still has pitch stability control. Without control authority of the canards, the aircraft risks stalling

30 the wing or enter aircraft divergence if the total moment coefficient <,26 becomes positive. In saturation region 2, all control surfaces enter saturation eventually. For this region several interesting

observations can be made. A distinct difference in performance and hardware limitation can here be seen in Figure

20. All inner elevons, except the right outer elevon, enter saturation before the stick input is changed from backward

to forward position. The saturation is caused by performance limitations and the fact that the FBW is decelerating @

too slow. At the point in time where the stick is changed to full forward position, all elevons have saturated, and the

hardware limitation is reached, forcing the FBW to use the canards to change the pitch angle velocity direction, while also, trying to prevent the wing from stalling. The constant value of the moment coefficients observed earlier,

can also here be seen for the saturated control surfaces in Figure 21. Furthermore, how the moment coefficient

contribution by the canards becomes larger compared to the elevons’ contribution can once again be seen. In

this case, pitch stability control authority is totally lost since the canards enter saturation, but the total moment

coefficient <,26 is held negative, giving a negative nose down acceleration. The wing moment coefficient <,U is thus counteracted such that the aircraft does not diverge. However, during this time interval the aircraft enter and

exits wing stall. The stall can be observed in Figure 15 where the lift coefficient ! for the aircraft has a small drop. With a stalling wing, one would also expect the moment coefficients contribution by the elevons to decrease due to

decreased lift acting on them. Looking at Figure 21 does not give much of an indication other than a minor decrease

of the inner elevons’ moment coefficients. Likewise, the same minor decrease can also be seen in saturation region

1 for the left inner elevon 3;84 in Figure 18, where the wing is on the edge to stall. However, these observations

are too uncertain to draw any conclusions about and needs to be investigated further. Nevertheless, the aircraft

barely manages to recover from the stall with all its control surfaces saturated, thus indicating that the set c.g is the

most aft limit possible to avoid stalling the wing for the square wave maneouver. It is further worth to mention that

the canards exit their saturated condition right after the aircraft has recovered from the stall. This can be seen in

Figure 20, and shows that the FBW prioritizes to regain its pitch stability authority even though a full stick forward

command is demanded by the pilot.

Moving on to the results of CG 1 and CG 2, the c.g for CG 2 has been moved marginally behind CG 1. For the

second c.g the aircraft never recovers from the wing stall. The difference in pitch angle in Figure 22 shows that the

aircraft maneuver is slower in pitch for CG 2 and has a larger roll angle. Further, the difference in dynamic pressure

and lift coefficient is only marginal and does not provide any further information regarding what causes the aircraft

to never recover from the stall. The pitch angle velocity in Figure 24 does however give a distinct difference at the

time point where the canards saturate (in Figure 25), showing a slower decay of @¤ for CG 2. A decrease in @¤ can

also be seen at this time point. By investigating the total moment coefficients in Figure 26, there is a significant

difference from the time the canards enter saturated conditions. The canards can further be seen producing a close

to constant moment coefficient for both CG 1 and CG 2. Moreover, from this point and onward, the <,A4BC moment coefficient becomes smaller for CG2 compared to CG1. Consequently, the increasing wing stall angle must be

due to the moment coefficient drop of the elevons, see CG 2 <,4;4. By finally investigating Figure 27 closer it is

31 obvious that all elevons are losing moment and the total moment coefficient is mainly lost by the left inner elevon. It has previously been observed that a moment coefficient is close to constant if the actuator is not moving the control surface. Due to the high stall angle, this implies that there is local stall over the control surfaces where the flow has separated, leading to a local decrease of lift which decreases the moment coefficient.

32 6. Conclusion

This thesis aims to answer three questions regarding saturation. The first question aims to answer if there exists

a simple connection of actuator saturation and flight mechanical parameters. Separating the moment coefficients of

the canards and the elevons provide valuable information regarding the saturation. Not only does it show how the

FBW uses the canards to generate a counteracting moment coefficient to the wing moment coefficient during stalled

elevons, but also shows how time intervals of saturation can be identified by close to constant moment coefficients

during non-stalled conditions.

The second question aims to answer if the found flight mechanical parameters can be used to determine how

robust the aircraft is towards departure from controlled flight. The answer is yes. Here, extra interest should

be put into the angle of attack, the pitch angle velocity, the total lift, and the moment coefficients. However, in

order to determine the robustness using moment coefficients, a better understanding of the minimum total moment

coefficient needed to counteract the momentum without stalling the wing is required. It has further been shown that

the inner elevons provide larger moment coefficients compared to the outer ones during saturated conditions for the

studied pitch maneuver. This implies that decreases in the inner elevons’ moment coefficients, due to for example

local flow separation, infer a larger risk of aircraft departure and should especially be considered when determining

the robustness in pitch.

The last question addresses how one can use the found parameters to determine if saturation implies a problem.

It is shown for the first saturated region that no problem occur with stall due to the canards still having control

authority. It can especially be seen that the canards are providing a larger moment compared to the canards after

the elevons have entered saturation, see Figure 18. Yet, the moment coefficient of the canards keeps increasing

in value, meaning that the canards further can be used to decrease @. This implies that as long as the canards

have not entered saturation, there is no risk of aircraft departure in pitch for the studied square wave input. In the

second saturation region the canards saturate, and the aircraft enters and exits the stall angle. Even though the

FBW manages to handle the situation, this saturation implies problems since all control surfaces are at maximum

deflection, generating a constant moment. Accordingly, whether or not the total moment coefficient is large enough

is determined by the magnitude of the momentum. However, with the c.g moved to CG 2, the problem is obvious,

since all moment coefficients of the elevons drop in value, thus showing that there are separations on the wing and

aircraft departure is a fact. Do note however that the observations are valid for the studied maneuver, and that other

maneuvers possibly can show a different result.

In conclusion, the used flight mechanical parameters are highly relevant when analyzing actuator saturation.

There also exist a simple connection between saturated actuators and moment coefficients. In addition, the studied

parameters can be used for robustness analysis, but needs to be further investigated. Furthermore, during saturated

actuators one can look for sudden changes in the moment coefficients in order to find potentially dangerous flight

cases. Finally, the studied pitch square wave input shows no risk of aircraft departure with saturated elevons during

flight, provided non-saturated canards, and that the free-stream velocity is high enough to be flyable.

33 7. Recommendations For Further Research

Additional work is to find a relationship between the total moment coefficients and the total momentum of the

aircraft. This would acquire a better understanding of how close to departure the aircraft actually is and perhaps

a robustness limit. It would be further interesting to find how much the slope of the moment coefficients can vary during saturated conditions. For CG 2, the elevons lost a considerable amount of moment coefficient value, whereas the moment coefficients remained close to constant for CG 1. An improved understanding of exactly which

parameters that causes the change in the moment coefficients during saturation would therefore also be useful to

investigate further. In combination with knowledge about the momentum, the slope could potentially be used to

predict if the aircraft will be able to recover from a stall or not. Looking closer at the local lift force acting on each

control surface could also help increase the understanding. For saturation region 2, the FBW is allowing a too

large @ for the stick backward position. Therefore, the reason behind the FBW’s decision would be interesting to

understand more in detail by investigating the actual control theory. This has not been done in this thesis due to

safety regulations. Another interesting thing to investigate would be to decrease the maximum allowed deflections

of the control surfaces further. In addition, how the FBW behaves if the canards saturates before the elevons could

also be interesting to know in order to increase the knowledge of the FBW’s behavior. Lastly, the <,A4BC has in this paper been treated, but is not considered to be the reason behind why the wing stalls for CG 2. However, this

coefficient would be interesting to know more about in order to understand how different maneuvers affect its variation.

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35 TRITA SCI-GRU 2020:190

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