Critical Dynamic Stability Assessment of UAV Helicopter

Republic of Iraq Ministry of Higher Education and Scientific Research University of Technology Mechanical Engineering Department

A Study of the Dynamic Stability of Unmanned Aerial Vehicle Helicopter A Thesis Submitted to the Department of Mechanical Engineering of the University of Technology in a Partial Fulfillment of the Requirements for the Degree of Ph.D. of Science in Mechanical Engineering

By

Akeel Ali Wannas

Supervisors Asst. Prof. Dr. Emad Natiq Abdulwahab

Asst. Prof. Dr. Mohammed Idrees Mohsin

2016

I

Dedication

To everyone I love I dedicate this modest effort

Akeel Ali Wannas 2016

II

Supervisor’s Certification

We certify that this thesis entitled “Critical Dynamic Stability Assessment of UAV Helicopter” was prepared by “Akeel Ali Wannas” under our supervision at the Mechanical Engineering Department, University of Technology in a partial fulfillment of the requirements for the Degree of Ph.D. of Science in Mechanical Engineering.

Signature:

Signature:

Asst Prof. Dr. Emad Natiq Abdulwahab

Asst. Prof. Dr. Mohammed Idrees Mohsin

Supervisor

Supervisor

Date

Date

/ 1 / 2016

III

/ 1 / 2016

Linguistic Certification

This to certify that this thesis entitled “Critical Dynamic Stability Assessment of UAV Helicopter” prepared by “Akeel Ali Wannas” was linguistically reviewed. Its language was amended to meet the style of the English language.

Signature: Name: Prof. Dr. Arkan Kh. Husain AL-Taie Title: Linguistic Reviewer Date:

/

/ 2016

IV

Examination Committee Certification

We, the examining committee, certify that we have read this thesis entitled "A Study of Dynamic Stability Assessment of A Small Scale UAV Helicopter" examined the student "Akeel Ali Wannas" in its contents and in what is related to it, and in our opinion, it meets the standard of thesis for the degree of Doctorate of Philosophy in Mechanical Engineering/Applied Mechanics. Signature: Prof. Dr. Arkan Kh. Husain AL-Taie Chairman Date: / / 2016 Signature: Prof. Dr. Karima E. Amori Member Date: / / 2016

Signature: Ass. Prof. Dr. Mauwafak Ali Tawfik Member Date: / / 2016

Signature: Ass. Prof. Dr. Ahmed Abdul Hussein Ali Member Date: / / 2016

Signature: Ass. Prof. Dr. Qasim Abbas Atiyah Member Date: / / 2016

Signature: Ass. Prof. Dr. Emad Natiq Abdulwahab Supervisor Date: / / 2016

Signature: Ass. Prof. Dr. Mohammed Idrees Mohsin Supervisor Date: / / 2016

Approved by Mechanical Engineering Department Signature: Asst. Prof. Dr. Moayed Razoki Hasan Dean of Department Date: / / 2016

V

Acknowledgments

It has been a long journey getting to this point. Throughout this time, I have been very fortunate to have received support, encouragement and assistance from many individuals. There is not enough space in all of the pages in this thesis to express my gratitude towards all of these individuals. To them, I offer a heartfelt thank you! I would like to thank my supervisors Asst Prof. Dr. Emad Natiq Abdulwahab and Asst. Prof. Dr. Mohammed Idrees Mohsin for their guidance, support, and friendship. Their encouragements, insights and patience with me throughout this journey are greatly appreciated. It's my duty to thank the head and staff members of the Mechanical Engineering Department at the University of Technology. I would like to express my gratitude to my family, who had provided me with unconditional support and countless encouragements. Without them, this work would have never come to completion.

Akeel Ali Wannas 2016

VI

Abstract

The helicopter has highly coupled characteristics with MultipleInput Multiple-Output (MIMO) system, which increases the system dynamic complexity. As well, the system dynamics are unstable. This thesis treats the subject of dynamics and control of autonomous UAV (Unmanned Aerial Vehicle) helicopter in hover maneuver flight and presents the reader with a methodology describing the development path from modeling and system analysis over sensor fusion and state estimation to controller synthesis. The focus is directed along two application branches: bring the system from the unstable situation with non-zero state to stable with zero state and find the critical cases that the controller cannot be able to maintain or bring the system in a stable condition. The platform of the thesis is formed by a Trex 500 model helicopter equipped with sensory equipment and on-board computer. A nonlinear generic model of this helicopter is developed using the first principles method and implemented in the MATLAB SIMULINK programming language. Methods for estimating the parameters of this model are developed and an optimal controller is designed on the basis of the parameterized model, linearized in hover equilibrium. A sensor fusion algorithm combining the various on-board sensor information into a measurement vector is developed. These measurement vectors are utilized by a linear Kalman Filter for linear motion and complementary filter for rotational motion, which estimates the states of the model. An interface is defined between the nonlinear model and MATLAB SIMULINK, under which the controller is developed and simulations are performed. Within the simulation environment, the VII

helicopter is able to transit from non-zero states to hover maneuver with zero states and maintains this situation. The mathematical dynamic models were validated for trim conditions in hover flight. Results of navigation algorithms for sensors of linear velocity (with max RMS of error in three axes is 0.1 m/s), angular rate (with max RMS of error in three axes is 0.052 rad/s) and Euler angles (with max RMS of error in three axes is 2.09 degree) give the confidence to integrating it with the controller. In the experimental part we were able to achieve autonomous flight, and the helicopter is capable of performing autonomous flight with hover maneuver. These results give dependability that all stages of the work starting from the modeling ending with the control unit were correct. We found the unstable dynamic cases in the dynamic system, which the controller unit cannot bring the system to the trim point when give the system initial values ( ⁄

⁄

⁄

⁄

), using a simulation program,

which represents a mathematical model of the helicopter with the controller.

VIII

Contents DEDICATION

II

SUPERVISOR’S CERTIFICATION

III

LINGUISTIC CERTIFICATION

IV

EXAMINATION COMMITTEE CERTIFICATION

V

ACKNOWLEDGMENTS

VI

ABSTRACT

VII

CONTENTS

IX

NOTATIONS

XIV

1 CHAPTER ONE: INTRODUCTION

1

1.1 UNMANNED AERIAL VEHICLES

1

1.2 BASIC CONCEPTS IN HELICOPTER DYNAMICS AND CONTROL

2

1.2.1 FRAMES

2

1.2.2 ROTOR HUB

3

1.2.3 CYCLIC INPUT THROUGH SWASH PLATE

4

1.2.4 STABILIZER BAR

9

1.2.5 INPUTS AND HELICOPTER MOVEMENTS

11

1.3 MOTIVATION OF THE STUDY

14

1.4 OBJECTIVES OF THE STUDY

14

2 CHAPTER TWO: LITERATURE SURVEY

16

2.1 HELICOPTER MODELING

16

2.2 ESTIMATION AND SENSOR FUSION

18

2.3 AUTONOMOUS HELICOPTER CONTROL

20

IX

2.4 SUMMARY

24

2.5 CONTRIBUTION OF THE STUDY

25

3 CHAPTER THREE: THEORETICAL ANALYSIS

26

PART I: HELICOPTER DYNAMIC MODELING

26

3.1 MODELING

26

3.2 RIGID BODY EQUATIONS

29

3.3 FORCE AND TORQUE EQUATIONS

31

3.3.1 FORCES

31

3.3.2 TORQUES

32

3.4 FLAPPING AND THRUST EQUATIONS

33

3.4.1 THRUST

34

3.4.2 FLAPPING

37

PART II: SIMULATION

45

3.5 SIMULATION OF THE MATHEMATICAL MODEL

45

3.5.1 FLAPPING AND THRUST

45

3.5.2 FORCES AND TORQUES

48

3.5.3 RIGID BODY

50

3.5.4 VIRTUAL REALITY

50

3.6 TESTING AND VERIFICATION OF THE SIMULATOR

53

3.6.1 TESTING AT HOVER

53

3.6.2 FORWARD MOVEMENT

56

PART III: LINEARIZATION

58

3.7 LINEARIZATION OF MAIN-ROTOR THRUST

58

3.8 LINEARIZATION OF MAIN-ROTOR DRAG

60

3.9 LINEARIZATION OF MAIN-ROTOR AND CONTROL-ROTOR FLAPPING

62

3.10 LINEARIZATION OF LONGITUDINAL ACCELERATION

63

3.11 THE SYSTEM MATRIX FOR THE LINEARIZED MODEL

65

3.12 VERIFICATION OF LINEAR MODEL

69

X

PART IV: CONTROL

70

3.13 DESIGN APPROACH

70

3.14 HELICOPTER INSTRUMENTATION

70

3.14.1 ACTUATORS

71

3.14.2 SENSORS

72

3.15 OPTIMAL CONTROL

73

3.15.1 PERFORMANCE INDEX

73

3.15.2 LINEAR QUADRATIC CONTROL

74

3.16 CONTROL DESIGN

77

3.16.1 DESIGN OF STABILIZING CONTROLLER

77

3.16.2 MODELLING OF KNOWN DISTURBANCES

83

3.16.3 STATE PREDICTOR

89

3.16.4 TUNING OF THE CONTROLLER

91

4 CHAPTER FOUR: EXPERIMENTAL WORK

94

PART I: DESIGN AND CONSTRUCTION OF PLATFORM

94

4.1 HELICOPTER PLATFORM

94

4.2 HARDWARE COMPONENTS SELECTION

97

4.3 RC HELICOPTER

98

4.4 FLIGHT CONTROL COMPUTER

101

4.5 NAVIGATION SENSORS

102

4.6 FAIL-SAFE SERVO CONTROLLER

103

4.7 WIRELESS MODEM

104

4.8 BATTERY

105

4.9 MANUAL CONTROL

106

PART II: MEASUREMENT OF STATE VARIABLES SIGNAL

107

4.10 ATTITUDE AND HEADING REFERENCE SYSTEM FILTER AHRS

108

4.10.1 DESIGN OF ROTATING PLATFORM FOR TESTING FILTER

109

4.10.2 VALIDATION OF FILTER

110

4.11 STATIC CALIBRATION AND VALIDATION OF TRIAXIAL ACCELEROMETER

114

XI

4.12 KALMAN FILTER

118

PART III: PARAMETERS ESTIMATION

126

4.13 STRAIGHTFORWARD EXPERIMENTS

126

4.14 MEDIUM COMPLEXITY EXPERIMENTS

129

4.15 HIGH COMPLEXITY EXPERIMENTS

132

4.16 PARAMETERS

135

5 CHAPTER FIVE: RESULTS AND DISCUSSION

137

PART I: TESTS AND EVALUATIONS

137

5.1 GROUND TEST.

137

5.2 FLIGHT TEST.

138

PART II: CRITICAL DYNAMICS OF CONTROLLER

143

5.3 EFFECT OF THE CRITICAL TRANSLATORY VELOCITIES INDIVIDUALLY ON THE DYNAMIC RESPONSE

144

5.4 EFFECT OF THE CRITICAL EULER ANGLES INDIVIDUALLY ON THE DYNAMIC RESPONSE

149

5.5 EFFECT OF THE CRITICAL TRANSLATORY VELOCITIES TOGETHER ON THE DYNAMIC RESPONSE

154

5.6 EFFECT OF THE CRITICAL EULER ANGLES TOGETHER ON THE DYNAMIC RESPONSE

156

5.7 EFFECT OF THE CRITICAL INITIAL STATES OF THE EULER ANGLES AND TRANSLATORY VELOCITIES TOGETHER ON THE DYNAMIC RESPONSE

156

5.8 DISCUSSION OF TEST RESULTS

158

6 CHAPTER SIX: CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORKS

160

6.1 CONCLUSIONS

160

6.2 RECOMMENDATIONS FOR FUTURE WORKS

161

REFERENCES

163

A APPENDIX A :HELICOPTER DYNAMIC ANALYSIS

A.1

XII

B APPENDIX B: COMPLEMENTARY FILTER DERIVATION

B.1

C APPENDIX C: CALIBRATION PROCEDURE OF ACCELEROMETER

C.1

D

D.1

APPENDIX D: VERIFICATION OF THE LINEAR AND NON-LINEAR MODEL

XIII

Notations Latin Variables Main rotor drag coefficient Rotor disk area State matrix (state space)

Main rotor hinge offset Forces in body frame Forces caused by the gravitational acceleration Forces caused by the main-rotor thrust Forces caused by the tail-rotor thrust

A A A B B B C c D e F F F F

Force component acting on the helicopter in the COG along the x axis

f

Force component acting on the helicopter in the COG along the y axis

f

Main rotor drag coefficient Number of blades Input matrix (state space) Output matrix (state space) Mean blade cord length Feedthrough matrix (state space)

()

Force component acting on the helicopter in the COG along the z axis

f

Objective function

f g H H h h I

Gravitational acceleration Discrete output matrix Angular momentum vector Distance between CG and HP along z axis Distance between CG and tail rotor center along z axis Flapping inertia of a single blade about flapping hinge [

]

Inertia matrix Jacobian function

I

Control rotor linkage gain Swash-plate linkage gain Spring constant of main-rotor blade Main rotor blade to Stabilizer bar linkage gain Kalman gain (Kalman Filter) Distance between CG and main rotor shaft along x axis Torque acting on the helicopter about the x axis Linear state feedback Distance between CG and tail rotor center along x axis Counter-torque caused by drag on the main rotor Torques caused by main rotor Torques caused by tail rotor Flapping hinge moment Main rotor blade weight

[

]

J K K K K K L L

L l M M M M m

Torque acting on the helicopter about the y axis

M

Helicopter mass

m

Torque acting on the helicopter about the z axis

N

Position described by the BF origin relative to the EF

P

XIV

( )

[

]

[

]

Covariance matrix (Kalman Filter)

P

Roll rate of the helicopter stated in BF Main-rotor drag Weighing matrices Variance of Process noise matrix (Kalman Filter)

p

Pitch rate of the helicopter stated in BF Quaternion Distance from the center of the rotor hub to the beginning of the control rotor paddle Distance from the center of the rotor hub to the end of the control rotor Rotation from SF to BF Tail-rotor radius Main Rotor Radius Variance of measurement noise matrix (Kalman Filter)

q

Yaw rate of the helicopter stated in BF Proportional matrix Torques in body frame Optimal input vector Collective control input Lateral control input Longitudinal control input Pedal control input Longitudinal velocity Control input vector velocities defines the SF movement relative to the EF Lateral velocity Measurement noise matrix (Kalman Filter)

r

Roll, Pitch and Yaw rotation rates Vertical velocity Process noise matrix (Kalman Filter) Distance from COG to the main rotor along the by axis

Q Q

Q q R R R R

R R S T u u u u u u u V v v W w w y

Greek Variables

[ ̇

̇

[ ̇ ̇

] ̇]

Vector of Euler rates stated in SF

α β β β β β Γ Γ Θ Θ

Euler Yaw rate of the helicopter stated in SF Blade pitch angle Pitch angle about y-axis Density of air

θ θ θ ρ

Two-dimensional constant lift curve slope Longitudinal flapping angle Lateral flapping angle Control rotor longitudinal TPP tilt Control rotor lateral TPP tilt Coning angle Discrete disturbance matrix (state space) Discrete input matrix (state space) Euler angles

XV

̇ ̇

Abbreviations AHRS BF COG DOF EF HP INSs LMA MARG MEMS MR PWM RMS SF TPP TR UAV

Time constant for longitudinal flapping Time constant for lateral flapping Discrete state matrix (state space) Euler roll rate of the helicopter stated in SF Roll angle about x-axis Euler pitch rate of the helicopter stated in SF Azimuth angle of the main-rotor blades Yaw angle about z-axis Main rotor angular velocity minimum of performance index Performance index

Attitude and Heading Reference System Body Fixed reference frame Center of Gravity Degree Of Freedom Earth Frame Hub Plane Inertial Navigation Systems Levenberg-Marquardt Algorithm Magnetic, Angular Rate, and Gravity Micro Electro-Mechanical System Main Rotor Pulse Width Modulation Root Mean Square value Spatial Frame Tip Path Plane Tail Rotor Unmanned Aerial Vehicle

XVI

τ τ Φ φ φ ψ Ψ ψ Ω

Chapter 1: Introduction

1 CHAPTER ONE: INTRODUCTION

1.1 Unmanned Aerial Vehicles The term Unmanned Aerial Vehicles (UAVs) is used to describe unpiloted flying vessels. This term refers to vehicles that are remotely piloted or autonomously controlled for the execution of a predefined flight task. In both cases the key attributes of these vehicles is the absence of a human pilot on-board. The applicability of UAVs is predominant in the execution of potentially dangerous flight missions or in cases where the small size of the vehicle restricts the presence of a pilot [1]. A major trend within the aerospace industry in recent decades has been a growing interest in unmanned aerial vehicles (UAVs). While their application for military purposes has long been established, there is an increasing number of roles in the civil sector for which UAVs being used, including surveillance for law enforcement, pipeline inspection and crop spraying. The size and performance metrics of UAVs can vary greatly depending on what the vehicle is designed for [2]. UAVs are further classified into two main categories. The first category is fixedwing UAVs (e.g., unmanned airplanes) that require relative velocity for the production of aerodynamic forces and a runaway for take-off and landing [3]. The second category is the rotorcraft UAVs, which also includes helicopters. The advantages of the rotorcraft, the unique flight capabilities have drawn much attention through the years. The primary characteristic attribute of the rotorcraft is the use of rotary wings to produce the thrust force necessary for motion. The main benefit of using a rotorcraft is its ability to move in all directions of the Cartesian space while 1


preserving an independent heading. Therefore, rotorcrafts have an advantage relative to fixed-wing aircrafts because they do not require any relative velocity to produce aerodynamic forces [4] and also due to their vertical flight capability. 1.2 Basic Concepts in Helicopter Dynamics and Control The aim of this section is to familiarize the reader with the terms and notations that arise when working with the complex field of helicopter theory, in order to give the reader a basic knowledge necessary to follow the modeling work presented in this thesis [5]. 1.2.1 Frames To describe the flight dynamics of a helicopter a set of basic frames and notations must be agreed upon. For the equation of motion, a frame with constant inertia of the helicopter is needed. This frame will be denoted as a body fixed reference frame BF. All frames used in this thesis will extend a three dimensional Euclidean space ℝ3, consisting of three orthogonal basis vectors. The basis vector for BF will be denoted by ( bx, by, b

z). This notation will throughout the thesis be used to distinguish between

the unit axes of the different frames and planes. The orientation of the BF is defined as a right hand coordinate system with the origin of the BF in the center of the helicopter mass (denoted CG). Seen from the tail of the helicopter bx-axis points from CG through the nose of the helicopter. The b

y-axis points from CG to the right, and the bz-axis points downwards. The forces and moments acting upon the helicopter body are

predominantly developed by the main rotor of the helicopter. These are transferred through the main rotor hub and rotor shaft to the helicopter body. To describe the forces and moments developed by the rotor disk, a hub plane (HP) in ℝ2 is defined. The HP has the same orientation as the BF, 2


but a different point of origin. The HP has its origin in the center of the rotor hub. Both frames can be seen in Figure ‎1.1.

Figure ‎1.1 Definition of Earth Frame, Body Frame and Hub Plane

In order to describe the position and translateral movement of a body with the use of Newtonian mechanics, a frame that has no accelerations is needed. This inertial frame will be referred to as an earth frame (EF), which is defined with the origin at the airfield from where the helicopter operates. Finally a spatial frame (SF) is defined where the resulting moments and forces described in BF are translated into rotations and translatory movement of the helicopter. The spatial frame has its origin in the helicopter center of mass like the BF and the same orientation as EF. Further notations will be described where it is needed for clarification. 1.2.2 Rotor Hub Figure ‎1.2 shows the main rotor assembly, which enables the feature of thrust vectoring. The servo motors are connected to the 3


nonrotating plate that together with rotating plate make up the swash plate. Through the linkage, the aileron and elevator servos are able to tilt the swash plate in both lateral and longitudinal direction for cyclic input; furthermore the collective pitch servo is able to raise and lower the swash plate for collective input. This can be seen in Figure ‎1.3.

Figure ‎1.2 Mechanical view of the rotor shaft/main rotor assembly

1.2.3 Cyclic Input through Swash Plate In a full size helicopter the cyclic inputs are given by the pilot through a cyclic control stick to the swash plate, which in turn generates the cyclic pitch (input) to the main rotor blades, see Figure ‎1.4. This figure illustrates well how the pilot stick is connected to the pitch of the rotor [6], through the swash plate, when the helicopter rotates counterclockwise. The non-rotating part of the swash plate on the project helicopter has a 90o shift from the HP axes as shown in Figure ‎1.4. However, it can be seen from Figure ‎1.2 that input from the aileron and pitch servo individually controls 4


the pitch given by the swash plate in the lateral and longitudinal direction respectively. This is also the case for the 90o shift. Since the effect is the same of the two setups, the 90o shift of the swash plate’s tilt will be used in the modeling presented in thesis.

Figure ‎1.3 View of the servo linkage to swash plate

To elaborate on this subject, the blade azimuth, denoted by ψ, is defined as the angle with respect to vector The Positive direction of

[

] , see Figure ‎1.5.

is clockwise. Returning to the swash plate, it

can be seen in Figure ‎1.4 that if no cyclic input is given to the cyclic stick, the pitch angle θ of the main rotor blades, given by the swash plate trough the pitch horn, is constant for varying azimuth. However if a cyclic input is given, with the cyclic stick moved forward, the swash plate will also tilt in the same direction. The pitch horn, that gives the input to the main rotor blades, is attached to the swash plate rotating part 90 o ahead of the blades. 5


Therefore, a blade at ψ = 0 o will have no pitch contributions . As the blade traverses around the hub its pitch will gradually be decreasing until it reaches ψ = 90o. Where the pitch will have its lowest value. Thereafter the pitch will gradually increase until the blade reaches a position 90 o later, i.e. ψ = 180o, where again there are no cyclic pitch contributions. As the blade continues its rotation around the hub the pitch will continue to increase until the blade reaches a position, ψ = 270o, where the blade pitch will have its highest value. Thereafter it will begin decreasing until position ψ = 0o is reached thus beginning a new cycle, hence the name cyclic pitch. In order to understand the effects of the cyclic pitch, the lift force generated by the blades as a function of ψ is considered. Evaluating the case in Figure ‎1.4, it can be seen that negative lift will be generated by the blades in the interval

, and positive lift in interval

, assuming that no collective pitch is added. This will force the blades to physically bend (flap), upward in the positive lift interval, and downwards in the negative lift interval. However, due to dynamic effects of the rotor blades the bending will appear 90o later. In the given case the maximal upward bending, or flapping, will be present at ψ = 0o and maximal downward bending at ψ=180o. The result will be a oneper-rev sinusoidal variation of the main blade pitch angle θ, governed by the tilt of the swatch plate, which in turn will result in a one-per-rev sinusoidal variation to the flapping β with a 90o azimuth delay from θ.

6


Figure ‎1.4 Swash plate configuration and mechanical linkage

Figure ‎1.5 Definition of azimuth angle

7


This bending will effectively tilt the rotor blade tip with a given angle with respect to the hub plane, and thus forming a new plane defined as the tip path plane. Tip path plane can be seen as the effective plane that the main rotor blade span. Figure ‎1.6 presents the TPP of a helicopter in forward flight. The TPP is defined by a coning angle β0 longitudinal angle, β1c, and lateral angle, β1s with respect to the hub plane.

Figure ‎1.6 Representation of tip path plane and its defining angles β1c and β1s

Assuming that the thrust generated by the main rotor blades is perpendicular to the rotor disk, i.e. the TPP, the thrust vector will have a longitudinal component seen from the HP. This component will therefore generate a torque about the by, due to the height difference between the HP and BF, which in turn will make the helicopter rotate forward, added with a translational movement in the same direction. In the case, shown in Figure ‎1.4, a longitudinal stick input is given, however, should a lateral stick input be given the same effects would occur only in the lateral direction. i.e. the lateral stick input will lead to lateral tilt of the swash plate, which in turn, will lead to lateral tilt of the TPP. This will result in a 8


manipulation of the thrust in the lateral direction and thereby a lateral rotation and translation is achieved. Moreover the same principals in helicopter movement can be applied to a model scaled helicopter of the present thesis; the only difference is that the inputs are no longer given by the pilot in the cockpit but from electrical servos on the helicopter. If collective and cyclic inputs are applied at the same time, the result will be a TPP with both a coning angle β0, lateral and longitudinal angles (β1s, β1c). Figure ‎1.7 and Figure ‎1.8 represent the coning angle together with lateral and longitudinal flapping respectively.

Figure ‎1.7 Lateral flapping angles and coning angles

1.2.4 Stabilizer Bar Input from the swash plate does not constitute the entire pitch input applied to the main rotor blades. If the cyclic inputs were to be fed directly to the main rotor a large force would be needed to maintain the input due to the inertia of the blades. This would result in an inhibiting effect on the controls; therefore to compensate for this, most helicopters are fitted with stabilizer bars as can be seen in Figure ‎1.2.

9


The stabilizer bar produces no noticeable lift, however a part of the cyclic inputs are fed mechanically to the main rotor blades through the stabilizer bar. At the ends of the stabilizer bar, a pair of paddles is attached which behaves in the same manner as the main rotor blade to cyclic input, see Figure ‎1.2. Cyclic inputs from the flight stick are fed to the stabilizer paddles as a pitch alteration. This makes the paddles flap as described in the rotor blade description. In turn, this flapping is fed to the rotor blades as a pitch input.

Figure ‎1.8 Longitudinal and coning angles

Figure ‎1.9 shows flap and pitch of the main rotor and the stabilizer bar [1]. From the figure it can clearly be seen that, the flap of the stabilizer bar coincides with the pitch of the main rotor. A part of the cyclic input is fed to the main rotor blades through the stabilizer bar; this is often referred to as the Hiller part of the stabilizing system. Another part of the cyclic input is fed directly to the main rotor blades and this is called the Bell part. An increase in the Bell factor will make the helicopter more aggressive in its handling, but too much will make it unstable. An increase in Hiller factor will on the other side make the helicopter easier to control in small maneuver such as in a hover. However, the too large Hiller part 10


will make the control of the system more sluggish. In this thesis, and in helicopter literature, the stabilizer bar is also referred to as the control rotor or Bell-Hiller bar.

Figure ‎1.9 Cyclic pitch and flap curves [1]

1.2.5 Inputs and Helicopter Movements The helicopter is an aircraft that uses rotating wings to provide lift and control. To describe this, the terms used for actuator inputs, and the responding movements of the helicopter will be defined. The position of the helicopter is described by the BF origin relative to the EF, and is denoted as

[

]

be aware that b and

e indexes defines the BF and EF, respectively. The indexes can be read as, body frame relative to earth frame. The derived translateral position is defined

[

] , where the velocities u, v and w define the SF

movement relative to the EF in the longitudinal, lateral and vertical 11


direction, respectively. The velocity vector bVbe can be read as, the velocity of the body frame relative to the earth frame, described in body coordinates system. Describing the attitude of the helicopter we used Euler angles, which describe the difference between SF and BF coordinates system with the notations [

as

[

] . The derived angular rates are defined

] . Where: p, q and r define the Roll, Pitch and Yaw

rotation rates of the helicopter, respectively. The respective velocities angles and rates can be seen in Figure ‎1.10, Figure ‎1.11 and Figure ‎1.12. For simplicity, when u, v, w,

, θ, ψ , p, q and r are stated without any

superscripts or subscripts, they refer to bube, bvbe, bwbe, s b

bs

, sθbs , sψbs , bpbs ,

qbs , brbs , respectively.

Figure ‎1.10 Pitch motion that makes the helicopter move in the longitudinal direction

A helicopter has four control inputs to control attitude and movement, these are: 1. ulat Control signal for the Aileron Servo that controls the lateral cyclic pitch on the main rotor which results in roll motion that makes the helicopter move in the lateral direction. 12


2. ulong Control signal for the Pitch Servo that controls the longitudinal cyclic pitch on the main rotor which results in pitch motion that makes the helicopter move in the longitudinal direction. 3. ucol Control signal for the Elevator Servo that controls the collective pitch on the main rotor which results in heave motion that makes the helicopter move in the vertical direction. 4. uped Control signal for the Rudder Servo that controls the collective pitch on the tail rotor which results in yaw motion. Thus, the input vector is reduced to:



U  ulat

ulong

ucol

v

p

u ped

T ,

(‎1.1)

and output vector is: X  u

w

q

r T ,

(‎1.2)

Figure ‎1.11 Roll motion that makes the helicopter move in the lateral direction

13


Figure ‎1.12 Yaw motion

1.3 Motivation of the study Piloting a helicopter is a complex task which should only be performed by a qualified operator, so a stated goal in the project proposal, is to adapt the helicopters for autonomous flight. An autonomous vehicle can be operated by giving the helicopter a set of waypoints to follow, and thereby eliminating the need for highly skilled pilots.

1.4 Objectives of the study To specify and state the work progress for this research a list of objectives is constructed in a prioritized order.

14


1. Building

of

a

minimum-complexity

helicopter

simulation

mathematical model and implementation in the simulation environment SIMULINK. 2. The radio controlled helicopter will be modified for data acquisition and autonomous control. For this attitude sensing, sensors will be mounted on the helicopter and a flight computer will be coupled to the servos and the sensors on the helicopter. Furthermore software for data acquisition on the system will be designed by using MATLAB SIMULINK. 3. Finding the parameters for helicopter model. These parameters will be found both through flight tests with the helicopter and through stationary tests performed on individual parts of the helicopter on the ground. 4. Design a controller, using linear optimal control, to stabilize the model in a hover maneuver. 5. Implementation of the controller on the flight computer system and to test it. The test involves handling of the stabilized hover maneuver at a constant height. The controller also, has to have the ability to withstand disturbances. 6. After verification of simulated model and controller, we find critical dynamic stability from simulation, by changing initial conditions of helicopter (attitude, linear velocity and angular velocity), that the controller cannot keep the helicopter in a hover.

15

Chapter 2: Literature Survey

2 CHAPTER TWO: LITERATURE SURVEY

In this chapter, a review of existing literature on the topics of this thesis is presented. At first, we look at the literature on helicopter dynamic modeling. Helicopter modeling for control purposes is a well-established topic in the literature. We then look at estimation and control literature. 2.1 Helicopter Modeling Helicopter modeling has been a major research topic for more than 80 years, ever since the first auto-gyros in the early twentieth century. The modeling efforts have been concentrated in the first-principle direction ranging from simple models based on momentum theory to highly advanced FEM modeling of the rotor wake. The large range of different modeling techniques is initiated by the many different scopes of the research work. Some examples are: Models created for a simulation tool used in pilot training, models for structural analysis, models for stability analysis of a new helicopter design and models intended for control purposes. There are a number of different books available which treat the theory of helicopters. A good starting point for understanding of helicopter theory is Raymond [6]. It focuses on rather simple modeling intended for understanding the helicopter rather than simulating it. It includes simple force equation and quasi steady state flapping equations. Bramwell Likewise [7] gives a good introduction to the basic equations needed for understanding the theory behind helicopter operation, but does not discuss more advanced theory.

16


A very comprehensive treatment of helicopter modeling is Johnson [8], which treats everything from simple hover equations to rotary wake modeling. It presents the theory in a high details level and covers almost all interesting parts of helicopter modeling like autorotation, stall etc. Due to its comprehensiveness it can be somewhat difficult to extract the important parts for a model, but it is an excellent starting point for the modeling. In Padfield [9] the focus is on modeling directed at simulation and control and a good presentation of the basic equation is given, whereas the more advanced parts of the modeling are only briefly touched upon. Parallel with the books a quite large number of articles and reports have been published that often focuses on specific parts of the helicopter theory. A comprehensive treatment of the flapping dynamics is given in Chen [10], which uses the blade element theory for the derivation of the second order dynamic equations. These flapping equations are used in Chen [11] and Talbot, et al. [12], where a complete helicopter model for simulation purposes is presented. Blade element theory is used to derive the forces and moments. In Heffley, Robert and Mnich [13] a simple model is presented which focuses on improving computational effort. It includes first order modeling of the flapping and thrust generation from main and tail rotor as well as simple aerodynamic drag from all surfaces. One of the most widely used non-uniform dynamic inflow models can be found in Pitt and Peters [14] who treated the inflow as a three state first order system. A useful survey of this as well as other nonuniform inflow models can be found in Chen [15]. In Kim, et al. [16] a higher order model is created, including flaps, lag, and inflow dynamics and the model are verified using frequency response data. One of the more recent helicopter models published is Civita [17], which is mainly based on the previously discussed literature. It calculates the flapping and rotor 17


forces and moments using blade element theory and uses non-uniform dynamic inflow from Pitt and Peters [14]. Parameters are identified using frequency response data. In Gavrilets, et al. [18] a simple model is derived form aggressive flight with uniform inflow and first order flapping dynamics. A frequency response system identification approach is taken in Mettler, et al. [19] where simple first principle relations are used to determine a parametrized model for a helicopter that includes the influence of the stabilizer bar. Flybarless helicopters are famous for their high agility and maneuverability, which makes them suitable platforms in many challenging applications. Al-Sharman, et al. [20] is concerned with the problem of estimating the attitude and flapping angles of a flybarless, small-scale, single-rotor helicopter. This work utilizes a nonlinear model for the Maxi Joker 3 helicopter.

2.2 Estimation and Sensor Fusion State estimators and sensor fusion algorithms for autonomous helicopters have been the subject of quite some research for the past two decades. For helicopter state estimation, a number of different approaches have been tried over the years, using both “model free” and “model based” methods. However, most work has been based on a model free approach as helicopter models are often highly non-linear, quite complex to work with, and have rather high computational requirements. In Rock, et al. [21] attitude, velocity and position were determined using a four antennas carrier-phase differential GPS (CDGPS) setup. The high precision of the (CDGPS) made it possible to achieve an accuracy of 3 cm with position estimated and an accuracy of a couple of 18


degrees with attitude estimated. The estimating attitude of a system based on inclinometers and rate gyroscopes fused through a complementary filter is shown in Baerveldt, et al. [22]. In complementary filter, which fuses high frequency band information from the gyroscopes with low frequency band information from the Inclinometers, accuracy of attitude for a couple of degrees is achieved. There is another model given in Jun, et al. [23], which described Kalman filter driven by rates measured and accelerations by an IMU with data from a compass as sensor inputs and GPS. The idea was to use a simple kinematic model driven by the rates measured for attitude estimation and a simple dynamic model for rigid body driven by the measured acceleration for velocity and position estimation. Other approaches are discussed in Gavrilets, et al. [24] and Saripoalli, et al. [25], where bias estimation of gyro is added to the Kalman filter. Ta, et al. [26] used a factor graph framework to solve both estimation and deterministic optimal control problems, and apply it to an obstacle avoidance task on Unmanned Aerial Vehicles (UAVs). Factor graphs allow to consistently using the same optimization method, system dynamics, uncertainty models and other internal and external parameters, which potentially improves the UAV performance as a whole. To this end, this procedure extended the modeling capabilities of factor graphs to represent nonlinear dynamics using constraint factors. For inference; this work formulated Sequential Quadratic Programming as an optimization algorithm on a factor graph with nonlinear constraints. Fault diagnosis and recovery are essential tools for the development of autonomous agents that can operate in hazardous environments. This can be effectively approached from a model-based perspective, where sensor faults are explicitly taken into account in a hybrid model with switching dynamics. However, practical hybrid filters 19


are required to manage an exponential growth in the number of discrete mode sequences, also known as hypotheses. Inspired by an attitude estimation application for a quadrotor UAV with faulty sensors, Santana, et al., [27] introduces the IP-MHMF. A novel filter for hybrid systems that generalizes the well-known IMM and introduces a more informed hypothesis-pruning step than previous algorithms. 2.3 Autonomous Helicopter Control Control of helicopters is by no means a new research area and several research groups around the world have attempted different control strategies during the past years. But, the focus is on improving the performance of control, to address all the possibilities and access to the best accuracy in performance. A review of the literature on control system design for autonomous helicopters shows that a very wide range of control design techniques have been tried. However, if the different and most successful strategies are reviewed, a common trait quickly becomes apparent. An adaptive trajectory tracking controller is presented by Johnson and Kannan [28], where approximate inverse dynamics together with a neural network being used to cancel system dynamics. The design is done by using a cascaded principle with a translational controlling outer loop and an attitude controlling inner loop. A technique called pesudocontrol hedging is used to protect the adaptation process from actuator saturation. The controller was tested on a wide range of flying helicopters and has shown excellent tracking performance in wide flight situations.

20


Nonlinear Model Predictive Control adopts neural network with State following Riccati Equation control, Wan and Bogdanov [29], for high bandwidth helicopter control. The State-Dependent Riccati Equation (SDRE) controller provides robust stabilization of the helicopter while the neural MPC provides high performance. The use of Nonlinear Model Predictive Control and also State Dependent Riccati Equations as a mean of stabilizing the helicopter has the disadvantage that a highly intensive computational effort is needed for real time implementation of the control scheme. It can indeed be quite difficult to achieve control in real time with high-bandwidth of a small-scale helicopter with such methods. Therefore, Bogdanov, et al. [30] proposed a control system based only on StateDependent Riccati Equation (SDRE), but with a nonlinear feedforward compensation to account for model simplifications and good tracking performance, using two different helicopters. Gain scheduled robust control is presented by Civita [17], where loop shaping theory was used to design a set of linear controllers scheduled, based on the system gain margin. The controller was verified in a number of flight maneuvers and showed good tracking performance in a wide flight envelope. Nonlinear Model Predictive Control based on a simple first principle model is presented in Shim, et al. [31] and through the use of an efficient optimization formulation a real time implementation of the controller was achieved. The controller was verified in different flight scenarios and showed good performance. Andrew, et al. [32] reinforced learning techniques used to first learn a helicopter model and then taught a controller that successfully performed sustained inverted hovering. Shim, et al. [33] gives a review of three different approaches for helicopter control: Robust linear MIMO control, fuzzy control with evolutionary training, and nonlinear tracking 21


control based on feedback linearization. Not surprisingly they concluded that the nonlinear controller showed better performance away from hover. The design of the nonlinear controller was elaborated in Koo and Sastry [34]. Aggressive aerobatics flight is shown in Gavrilets, et al. [35] where the control approach was inspired by recorded maneuvers flown by human pilot. A hybrid control strategy was used to switch between open loops prerecorded aerobatic maneuvers and LQ designed close loop controllers for cruise between maneuvers. An attitude control technique on coaxial-helicopter was introduced by Sugawara and Shimada [36]. A high performance computer (HPC) has been widely used with UAV. The controller of the UAV used advanced mathematics that was implemented by HPC. However, there are several problems when using HPC, such as processing delays in calculation and increased cost. This technique performs the same control performance when using a low performance computer. Limited pole placement (LPP) was used to implement this control technique for a prototype of electric motor driven helicopter. Then, it was fit to find the critical delay time. Initial results of a research project were presented by Frye and Provence [37] who investigated the application of the Direct Inverse Control technique to the problem of the Autonomous Hover of a UAV Helicopter. The goal of the project was to investigate the effectiveness of the Direct Inverse Control technique using an Artificial Neural Network to learn and then cancel out the Hover dynamics of the UAV Helicopter under various environmental conditions during a hover mode. Helicopters are under actuated nonlinear mechanical systems; their high-performance controller design presents a challenge. Nodland, et al. [38] presented an optimal controller design for trajectory tracking of a helicopter UAV using a neural network (NN). The state-feedback 22


control system

utilized

the

backstepping

methodology,

employing

kinematic and dynamic controllers. Highly coupled and complex dynamics of the helicopter is a naturally complicated process of modeling and design of controllers. In Tang and Li [39], the helicopter system comprehensive nonlinear model was derived from the first-principles modeling, who uses system identification approach to verify the model parameters. The derived nonlinear model with the high-fidelity linearized model and modest level of complexity was adequate for flight control system design. Helicopters are inherently unstable. It demands efficient and accurate control algorithms to control the attitude of the helicopter. The controller design uses a full-state feedback control.

A reduced-order observer is used to estimate the

unmeasured states.

Integral state augmentation and Linear Quadratic

Regulator methodology were adapted to the desired performance of the control system. A Flight control system design for the yaw channel of a UAV helicopter using a newly developed composite nonlinear feedback (CNF) control technique was presented by Cai, et al. [40]. From the actual flight tests on UAV helicopter, it has been found that the commonly used yaw dynamical model of the UAV helicopter proposed in the literatures was rough and inaccurate. These motivated authors to first obtain a comprehensive model for the yaw channel of our UAV helicopter. The primary focus of Cai, et al. [41] was to design and implement a robust automatic flight control system for a small-scale unmanned helicopter. A comprehensive nonlinear model for the unmanned system, which was constructed by a research team at the National University of Singapore, was first derived and presented. Based on its

23


linearized model, a robust automatic flight control system was then designed by employing Hinfint control and dynamic inversion techniques.

2.4 Summary Mathematical modeling is the essence of the design and simulation as well as control. Therefore, researchers focused at the beginning of the last century and up to now on this part. The development of mathematical models is based on the development of mathematics and speed of computers. Mathematical models by researchers ranged from simple models, based on the theory of momentum to complex models, that take into account most of the factors and influences and the diversity of main rotor (coaxial Rotor, flybar rotor, flybarless rotor, and so on). Many researchers have focused on State estimators and sensor fusion algorithms in their work during the past two decades. Because of the importance of this topic and its entry in the navigation and positioning devices, system identification used sensor fusion algorithms to find the state space of the system and feedback it to the controller. Several mathematical and statistical methods were used to overcome the problems like uncertainty and noise in this area, such as Kalman filter, complementary filter and so on. One of the main research topics in the field of aircraft, especially UAV is the control. Therefore, the researchers mobilized all their efforts in this area to overcome the problems. The development of several models of control was depending on the previous two topics. The previous three paragraphs refer to the essence of UAV research, where one of which is complementary to the other. Where one cannot go to control without passing through modeling and sensor fusion. 24


Finally, the research in this thesis finds it’s significance in the establishment of a systematic methodology of development of autopilot system for helicopter UAV by using commercially available components such as radio control (RC) helicopter, navigation sensors, computers and communication devices. The helicopter has been chosen as UAV platform because of its unique flight capabilities compared to fixed wing aircraft. This research aimed to achieve full autonomous hovering flight and helicopter attitude stabilization capability. These involved several steps such as helicopter dynamic modeling, stability and control analysis, hardware and software integration and flight test. 2.5 Contribution of the study The helicopter is a highly unstable system, for which reason a controller is needed. The purpose of this work is to build a mathematical model of a radio controlled helicopter and devise a controller. The controller should make the helicopter model hover in a simulation and prototype. The definition of hover being that the translatory velocities of the helicopter relative to the earth are zero. As the main objective is to control the dynamic behavior of the helicopter and find the limitations of the controller or critical dynamic cases, it is necessary to derive a representative model that reacts in the same manner as a real helicopter.

25

Chapter 3: Theoretical Analysis

3 CHAPTER THREE: THEORETICAL ANALYSIS Part I:

HELICOPTER DYNAMIC MODELING

The objective of this part is to provide the basic equations of motion of the helicopter, when the helicopter is treated as a rigid body. The Align Rotorcraft TREX 500 shown in Figure ‎3.1 is used in this research study. Analytical model has been constructed in this part, describing the dynamics of the vehicle with the control rotor. It is divided into three sections describing the flapping and thrust, forces and torques and rigid body equations.

Figure ‎3.1 Rotorcraft ALIGN TREX500 used in research

3.1 Modeling The modeling of the helicopter will be performed using a topdown principle. The entire model consists of three boxes in which the equations used in the model are derived; this is sketched in Figure ‎3.2. The first box to be described is the box containing the rigid body equations. 26


These equations describe the position and the Transitional movement of the COG relative to the EF (P and V respectively). Furthermore the attitude, ϴ, and the angular velocity, ω, are described here.

ulong ucol

Flapping and thrust

uped

e

TMR

ulat

TTR β1s β1c

P

b

Force and

b

T

V

Rigid b

F

body

Torque

ϴ ω

Figure ‎3.2 The three parts of the modeling with inputs and outputs.

These equations are all derived from the torques, M, and forces, F, affecting the helicopter. These are computed in the box labeled Force and torque equations. To derive the forces and torques, it is necessary to compute equations for the thrust generated by the main rotor, TMR, and the tail rotor, TTR. To obtain knowledge of the direction of the main-rotor thrust, the flapping of the rotor blades, β1c and β1s, are considered. This is done using the inputs from the swash plate, ulat, ulong and ucol, and the input to the tail rotor, uped. The three parts are described in details in the following sections, starting with the rigid body equations [42]. The following assumptions are made to reduce the complexity of the helicopter system [43]: Teetering rotor: All rotors are modelled as teetering rotors, which means that the blades flap are hingeless in the center rotor hub and that it does not bend. This assumption can be made because of the amplitude of the flapping motion, in response to gusts and control inputs, similar to the different types of rotors [44]. 27


Helicopter induced velocity: A helicopter in translatory movement will generate more lift where the rotor blade advances into the wind than on the retreating side due to relative air speed. This will be neglected, since it only occurs outside the operating point of hover. Blade twist: Blade twist means that the blade is twisted along the length of the blade to compensate for uneven lift. All the blades on the helicopter are without blade twist, Density of air: The thrusts generated by the rotors are dependent on the density of the air. As the altitude of the operating envelope is limited, the density of the air is considered constant. Drag on fuselage: Because the helicopter is in hover and the wind velocity is defined to be zero, any drag on the fuselage will be neglected. Moment of inertia constant over time: It is assumed that the helicopter's moment of inertia is constant, i.e. it does not change due to disturbances such as decreasing fuel amount. COG constant over time: It is presumed that the COG is also constant over time. Rotor angular velocity: Thrust generation is dependent on the angular velocity of the rotor blades. This will be simplified by use of an engine governor, which maintains a constant rotor angular-velocity.

28


3.2 Rigid Body Equations In this section the motion of the helicopter is described. The helicopter is considered as a rigid body, which means that Newton's second law and Euler's rotational equations of motion can be applied [45]. Figure ‎3.3 illustrates the inputs and outputs of the rigid body equations. The forces and torques produced by the main and tail rotor are used to determine the motion of the helicopter. The rigid body equations are defined for fixed-wing aircrafts, but are also applicable for rotary-wing aircraft. For a complete derivation of relations concerning the current section, see Appendix ‎A. e

P

b

M

b

F

b

Rigid body

V

ϴ ω

Figure ‎3.3 Inputs and outputs of the rigid body equations.

From Appendix ‎A assembling (‎A.13), (‎A.20) and (‎A.21) yields the matrix describing the motion of the rigid body  bV   (1 m).b F  b V      Psb ().    ,  1 b     I ( T    ( I . ))     These equations can be extended to:

(‎3.1)

29


 b fx b   v.r  bw.q    b u   b m  f   y b  V   b v     bu.r  bw. p  , m  b w   b     fz b b  u.q  v. p  m  

where

[

] ,

[

] ,

(‎3.2)

[

]

and m is

the mass of the helicopter,       p  sin( ). tan( ).q  cos( ). tan( ).r          cos( ).q  sin( ).r    , sin( ) cos( )     .q  .r     cos( ) cos( )

where

[

] and

[

] and,

 ( I yy  I zz ).q.r  L    I xx   p    ( I zz  I xx ). p.r  M       q    , I yy   r    ( I xx  I yy ).q. p  N    I zz  

where

[

] ,

[  I xx I  0   0

(‎3.3)

(‎3.4)

] and 0 I yy 0

0 0 .  I zz 

(‎3.5)

Where (ϕ, ϴ and ψ) are Euler angles. (

) are angular velocities about x,y and z respectively .

(

) are linear velocities along x,y and z respectively .

(

) are moments about x,y and z respectively .

(

) are moment of inertia about x,y and z respectively.

30


3.3 Force and Torque Equations This section deals with the derivation of the equations, describing the forces and torques acting on the helicopter. Figure ‎3.4 illustrates the inputs and outputs of the force and torque equations, the inputs being the flapping angles β1c and β1s. The thrust generated by the main rotor, TMR, and tail rotor, TTR and the outputs being the forces and torques described in the BF bF and bT , respectively. The forces acting on the helicopter create both translatory and rotary movements because the forces do not act in the COG. They are decomposed into force acting in the COG resulting in translatory movement, and torque acting about the COG resulting in a rotary movement. TMR TTR β1s

Force and

b

T

b

F

Torque

β1c

Figure ‎3.4 Inputs and outputs of the force and torque equations.

3.3.1 Forces The resulting force, bF, stated in the BF, is decomposed along the three axes bfx; bfy and bfz. These forces consist of:b

FMR: Forces caused by the main-rotor thrust

b

FTR: Forces caused by the tail-rotor thrust

b

Fg: Forces caused by the gravitational acceleration

31


which are included in the modeling. The main-rotor and tail-rotor thrust act in the center of the main-rotor disc and tail-rotor disc respectively, while the gravitational force acts in the COG. For a complete derivation of relations concerning the current section, see Appendix ‎A. Assembling the derived equations (‎A.25), (‎A.27) and (‎A.29) in Appendix ‎A, for bFMR, bFTR and bFg, yields the complete force matrix  b fx    b F   b f y  bFMR  bFTR  bFg ,  b fz     TMR.sin(1c )    0    sin( ).m.g    T   sin( ).cos().m.g   TMR.sin(1s )    TR     TMR. cos(1s ).cos(1c )  0  cos().sin( ).m.g   TMR.sin(1c )  sin( ).m.g     TMR.sin(1s )  TTR  sin( ).cos().m.g  .    TMR. cos(1s ).cos(1c )  cos().sin( ).m.g 

(‎3.6)

(‎3.7)

(‎3.8)

3.3.2 Torques The torques about the three axes (bx, by, bz) are primarily caused by three components: b

MMR: Torques caused by the main rotor,

b

MTR: Torques caused by the tail rotor, and

b

MD: Counter-torque caused by drag on the main rotor.

The torque generated by the tail-rotor drag is disregarded due to the relative small influence it has on the model. The torques are defined positive in the clockwise direction. The torque vector, bT , containing the three torque components b

L, bM and bN about bx, by and bz, respectively, are described by

32

Chapter 3: Theoretical Analysis  bL    b T   b M   bM MR  bM TR  bM D  bN   

 b LMR   b LTR   b LD          b M MR    b M TR    b M D  ,  b N MR   b NTR   b N D       

(‎3.9)

(‎3.10)

where bLMR, bMMR , bNMR are the torques caused by the mainrotor, bLTR, bMTR, bNTR torques caused by the tail rotor, and bLD, bMD, bND are the counter torques caused by drag on the main rotor. The derivation of the equations describing the torques caused by the forces from the main and tail rotor is based upon:T  F .d

(‎3.11)

where F is the force and d the distance, from where this force attacks, to the COG. The distances used throughout this section are illustrated in Figure ‎A.3, and the values can be found in Table ‎4.4. The complete torque matrix is the sum of the torques generated by the main-rotor thrust, the tail-rotor thrust and the torques generated by drag on the main rotor. For more details see Appendix ‎A.  b L   b LMR   b LTR   b LD , MR          b T   b M    b M MR    b M TR    b M D , MR  ,  b N   b N MR   b N TR   b N D , MR          b  f y ,MR .hm  bf z ,MR . ym  bf y ,TR .ht  bQMR . sin( 1c )      bf x ,MR .hm  bf z ,MR .lm  bQMR . sin( 1s )   f x ,MR . ym  bf y ,MR .lm  bf y ,TR .lt  bQMR . cos( 1s ). cos( 1c )  

(‎3.12)

(‎3.13)

Where bQMR is drag moment of main rotor 3.4 Flapping and Thrust Equations In this section the equations for the thrust generated by the main and tail rotor are derived, as well as the blade-flapping equations describing 33


the motion of the main-rotor blades when applying an input to the swash plate. The inputs and outputs for this part of the system can be seen in Figure ‎3.5. The inputs to the system are ulat, ulong, ucol and uped. β1s and β1c are the values describing the flapping of the main rotor. TMR is the thrust generated by the main rotor, and TTR is the thrust generated by the tail rotor. ulat

TMR

ulong

TTR

Flapping ucol uped

and thrust

β1s β1c

Figure ‎3.5 Inputs and outputs of the thrust and flapping equations.

3.4.1 Thrust The main-rotor thrust equation is based on [43], and the tail-rotor thrust is derived by defining that the torque bN must equal zero. To determine thrust magnitude, two complimentary methods are used; these are momentum and blade element theory. The first method presented is momentum theory which yields an expression for thrust based on the induced velocity through the rotor disk. Because this is a single equation with two unknowns a second expression is needed to make a solvable set of equations. The second equation is generated using blade element theory, which is based on the development of thrust by each blade element. The result of this section is a set of thrust equations that is solved by a recursive algorithm.

34


3.4.1.1 Main-Rotor Thrust Equation The main rotor thrust can be described by the following equations (see Appendix A) TMR  wb  v1 .

 . .R 2 .a.b.c 4

,

(‎3.14)

where ρ is the density of the air, Ω is the rotor angular rate, R is rotor radius, a is the constant lift curve slope, b is number of blades, c is chord of blade, wb is the velocity of the main rotor blade relative to the air and v1 is the induced wind velocity through the TPP described by 2

 vˆ 2   T  vˆ 2 v1      MR   ,  2   2. . A  2

(‎3.15)

where the expressions for the variables in (‎3.15) and (‎3.14) are:2 3   wb  wr  ..R ucol   tw  , 3 4  

(‎3.16)

wr b wbu.1c bv.1s ,

(‎3.17)

vˆ 2 b u 2 bv 2  wr ( wr  2.v1 ) ,

(‎3.18)

where wr is the velocity of the main rotor disc relative to the air due to translatory velocities bu, bv and bw, ucol is the collective input from pilot (or θcol), θtw is the blade twist angle. Blade twist (θtw) is neglected, which simplifies the above equation for wb to:2 wb  wr  ..R.ucol , 3

(‎3.19)

The main rotor thrust equations are recursively defined, with TMR depending on v1 and vice versa, so the main rotor thrust, TMR, is calculated by the use of a numerical method. Figure ‎3.6 illustrates the principle of how TMR and v1 are calculated. The basic idea is that the initial values of ̂ , wb, TMR and v1 denoted ̂ , wb,0, TMR,0, and v1,0 respectively, used to calculate new values of TMR and v1. The new values are then again used to calculate a 35


set of new values for TMR and v1. This cycle is repeated until the values for TMR and v1 have settled. Approximately 5 iterations is enough to ensure that the values are settled [46]

Figure ‎3.6 Principle diagram of how the main-rotor thrust and the induced velocity are calculated [42]

3.4.1.2 Tail-Rotor Thrust Equation The main purpose of the tail rotor is to compensate for the torque about bz, bN, determined in Section ‎3.3.2. The helicopter is equipped with a gyroscope which, by the yaw controller, controls the pitch of tail-rotor blades, so that the tail-rotor thrust, TTR(t) = bfy,TR, maintains the helicopter's angular velocity r at zero. That is, it adjusts the thrust of the tail rotor so the helicopter does not rotate about bz. This suggests that the helicopter is unable to yaw, but an input uped can be used to give the tail rotor an extra force, which in turn makes the helicopter yaw. The relationship between the torque about bz and the tail-rotor thrust is derived in Section ‎3.3.1 as b

N  f x, MR . ym  bf y , MR .lm  bf y ,TR .lt  bQMR . cos(1s ).cos(1c ) ,

(‎3.20)

It is assumed that the helicopter's angular velocity r is stabilized by the use of a gyroscope, that is bN = 0, so the tail-rotor thrust is found by isolating b

fy,TR in (‎3.20) 36

Chapter 3: Theoretical Analysis TTR 

f x , MR. ym bf y , MR.lm bQMR. cos(1s ).cos(1c ) lt

 u ped ,

(‎3.21)

where uped is the additional thrust input, introduced to be able to perform a yaw motion. As TTR is a force, uped is also a force, which means that it differs greatly from the three other inputs which are angles. 3.4.2 Flapping This section describes the steps taken to develop the flapping equations for the main rotor and the control rotor. The cyclic input signals Asp and Bsp, are used to control the flapping or tilting of the TPP. Figure ‎A.7 shows how the tip path drawn from the trajectory of the rotor blade tip. It also establishes the positive TPP angles. Described earlier, the thrust vector is perpendicular to the tip path plane and by controlling the flapping action of the TPP we also control the forces affecting the helicopter body. 3.4.2.1 Flapping control The lateral and longitudinal inputs to the swash plate are controlled by the pilot, and fed to both the main rotor and the control rotor. A part of the input is fed directly to the main rotor, while the other part of the input is fed to the main rotor through the control rotor. The mixing of the input is illustrated in Figure ‎3.7. The inputs from the swash plate, Asp and Bsp, result in lateral and longitudinal blade flapping on the main rotor denoted as β1c and β1s. The control rotor has a mechanical linkage gain to the main rotor denoted by KCR. The input results in a flapping motion of the control rotor of which the angles are expressed by βCR,1c and βCR,1s. The mechanical linkage gain from the swash-plate to the main rotor is defined as KMR. The mixer system from swash plate input to main rotor input is described by [47], [48], [49] as 37


AMR  K MR . Asp  KCR .CR,1s ,

(‎3.22)

BMR  K MR .Bsp  KCR .CR,1c ,

(‎3.23)

where AMR and BMR are the cyclic-input's contribution to the pitch of the blades.

Figure ‎3.7 Bell Mixer system [42]

3.4.2.2 Types of Hub Systems A rotor hub can be divided into three types: Teetering, hingeless and articulated. The detailed analysis of their respective dynamic behavior, is given in Appendix ‎A . It can be seen from equations (‎A.105), (‎A.102) and (‎A.91), that all three rotor types can be modeled as a hinged rotor. A fundamental result of rotor dynamics emerges from the above analysis, that the flapping response is approximately 90◦ out of phase with the applied cyclic pitch, i.e., θ1s (lateral cyclic input angle of swash plate) gives β1c, and θ1c (longitudinal cyclic input angle of swash plate) gives β1s . For blades freely 38


articulated at the center of rotation, or teetering rotors, the response is lagged by exactly 90◦ in hover; for hingeless rotors. The phase delay is a result of the rotor being aerodynamically forced, through cyclic pitch, close to resonance, i.e., one-per-rev. [5]. Rotor resonance can be compared with resonance of a mass-spring system as shown on Figure ‎A.9 ; therefore it will be used in the following analysis. 3.4.2.3 Thrust Vectoring The main rotor and control rotor flapping equations will be derived. The output of these equations are the TPP inclination described by the angles β1s, β1c, βcr,1s and βcr,1c, for the main rotor and control rotor respectively. The inputs to these equations are the moments affecting each blade element about the effective flapping hinge. The moments comprise of the following [49] 1. Gyroscopic Moments 2. Aerodynamic Moment 3. Centrifugal Moments 4. Spring Moments 5. Inertial Moments 6. Gravitational Moments

By finding the equilibrium point between all moments affecting the blade, TPP angles can be found. The moments can be split in to internal moments, which consist of the inertia of the rotor blades, and the external moments consisting of the other five moments. M I  M A  M cf  M gyro  M g  M s ,

(‎3.24)

For the full derivations of the moments above see Appendix ‎A. The equations for main rotor flapping can be derived by equating the increments of hinge moments due to centrifugal, inertial, weight, elasticity, aerodynamic and gyroscopic forces to zero 39


M A  M cf  M I  M g  M gyro  M s  0

(‎3.25)

Again rewriting the equation as components of constant sine and cosine terms M  M const  M cos cos( )  M sin sin( ) , M const

(‎3.26)

 1 1 u2 v2 e  2 2 2    ac( R  e)  R      o 2 2 2 2  2 6R  4 R  4 4 R u  ve   ue  v     1c    1s 2 6R   6R  6R 6R 2  u ue  v ve         B1     A1 2  3R 6 R    3R 6 R 2  e 1     2  6 R  3R 

qve we pu    v1  2 2 2 12 R  6 R  6 R 2 

 w pue qv  eM b      2  I b  o  2 2 2 3R 12 R  g  6 R    M b  K o

M cos 

(‎3.27)

  ve 1 2v  ac( R  e) 2  2 R 2     o 2 2 3R    3R  uvB1 u2 e 1 e         1s  2 2 2 2  4 R  8 R 6R  4 6R  v1v 3v 2 1 1  wv   e    A1    q  2 2 2 2 2 2 8 R 4 2 R  6R 4  2 R   eM b   1c  K1c  I b  2 1c  2 pI b   2  I b  g  

M sin 

(‎3.28)

 ue 1 2u  ac( R  e) 2  2 R 2    o 2 2 3 R  3  R   uvA1 e  v2 e 1        1c  2 2 2 2  6R 4 R  4 6R   8 R 2 v1u 3u 1 1  wu   e    B1     p 2 2 2 2 4 8 R 2 R 2 2 R 2   6R 4   eM b   2  I b  g 

  1s  K1s  I b  2 1s  2 pI b 

(‎3.29)

As it can be seen we acquire three equations with three unknowns, i.e. βo, β1c and β1s, that can be solved analytically. However, 40


since sine and cosine terms of the hinge moment do not contain the βo term, which is derived from the Mconst, will result in β1c and β1s being independent of βo. Therefore Mconst term will be discarded from further analysis. To get TPP flapping equations, Solve equations (‎3.28) and (‎3.29) simultaneously with respect to β1c and β1s , when Mcos = 0 and Msin = 0. 3.4.2.4 Control Rotor Flapping When modeling flapping

dynamics it was described that the

blade time constant ψ63% is relatively low. From an example, a helicopter with a hinge offset e/R=0.05 will have a time constant of ψ63%=130◦ azimuth. This implies that a steady state flapping angle will be achieved after about two revolutions. The scaling from a full sized to model sized helicopter will result in a higher number of revolutions per minute. This will result in faster dynamics. To be able to control a model helicopter, a control augmentation has to be implemented which reduces this flapping bandwidth. The control rotor acts as a lagged rate feedback in the pitch and roll axes, reducing the bandwidth and control sensitivity to cyclic inputs [5]. Figure ‎3.8 shows the connection between the fuselage and the rotor disk. The control rotor consists of a teetering rotor mounted on the same shaft as the main rotor. From the swash plate the control rotor receives longitudinal and lateral inputs, much like the main rotor. But unlike the main rotor it does not receive any collective input, and thus do not produce any lift which would result in a coning angle of the control rotor. The control rotor tip path plane flapping can be modeled by [49]  cr   cr ,1c cos( )   cr ,1s sin( ) ,

(‎3.30)

The procedure for obtaining the flapping angles are the same for the control rotor as was followed by the main rotor. By equating the all moments that act on each paddle or rotor blade it is possible to find the 41


blade flapping angle. The natural frequency ωn of a teetering rotor without a flapping restraint in the form of a spring hinge, will be the same as the angular frequency Ω [5]. This results in that there is practically no cross coupling between longitudinal and lateral axes.

Figure ‎3.8 Main rotor coupled to the control rotor dynamics [42]

The moment from the weight of the blades will have constant effect on the flapping angle with regard to azimuth (ψ). The moments that have an effect on flapping angle are aerodynamic, inertial, centrifugal and gyroscopic. The moments about the teetering hinge are [49] M i  M a  M cf  M gyro ,

(‎3.31)

To find the flapping angle, the moments are equated to zero M a  M cf  M gyro  M i  0 .

(‎3.32)

Full derivation of above the equation can be seen in Appendix ‎A. Next step is to collect sine and cosine terms in equation (‎A.185) (in Appendix A) and group them into differential equation for lateral (βCR,1s) and longitudinal (βCR,1c) flapping.

42


T1 T T 2 2  Bsp  2 p  q  1  CR ,1c   2 2  2  CR ,1c T 1  2 CR ,1s  2 CR ,1s 2 

(‎3.33)

T1 T T 2 2  Asp  2 q  p  1  CR ,1s   2 2  2  CR ,1s T 1  2 CR ,1c  2 CR ,1c 2 

(‎3.34)

M sin 

M cos 

where T1 

4   RCR

4 4  RCR ,P 

 T2  4

  1 ,  

 R4   CR  1 ,  R4   CR , P 

(‎3.35)

(‎3.36)

From the longitudinal (Mcos) and lateral (Msin) moments two differential equations, describing longitudinal and lateral flapping are isolated. This gives the final result of the control rotor flapping equations. 1 1 1  T1 Asp  p   T2 q    CR ,1s 4 4 2 1 1    T1 CR ,1c   CR ,1c 4 2

(‎3.37)

1 1 1  T1Bsp  q   T2 p    CR ,1c 4 4 2 1 1    T1 CR ,1s   CR ,1s 4 2

(‎3.38)

CR ,1s 

CR ,1c 

The resulting equations describe a MIMO system where ̇

and ̇

are the outputs and the shaft motion p, q and swash plate

angles Bsp, Asp are inputs. There is also a coupling between longitudinal and lateral flapping which can be seen in the last two terms in both flapping equations. This coupling is a result of the gyroscopic moments.

43


The flapping of the control rotor in equations (‎3.22) and (‎3.23) is mixed together with the input from the swash plate (Bsp, Asp) to form the complete pitch input of the main rotor (see section ‎3.4.2.1). The mixing ratio between swash plate input and the control rotor input is governed by the mechanical links connecting the swash plate, control rotor and the main rotor. These gains (KMR and KCR) can be found by measurements of the pitch responds to a swash plate tilt or a control rotor flapping angle.

44


Part II:

SIMULATION

3.5 Simulation of the mathematical model A helicopter in flight is an inherently unstable system, and if a pilot, even for a second, where to relinquish control it would probably result in a crash. The development of a control system is an iterative procedure, and testing the controller designs on a real helicopter could be a costly and dangerous experience. The goal of simulator [50] [51] [52] is to mimic the complex behavior of a helicopter in flight so that it can be used in controller tests without wasting money. To accomplish this nonlinear model in the modeling sections was implemented by SIMULINK, using the parameters from our own project helicopter. The final simulator structure follows roughly the same subdivision that of the model description. This implies that it is possible to follow a chain of events from piloted inputs to the final response of the rigid body, but also testing subsystems such as translatory movement’s impact on thrust magnitude. The equations presented in ‎Part I: needs to be implemented in SIMULINK for testing and simulation purposes. The structure of this SIMULINK model follows the structure shown previously in Figure ‎3.2. This structure divides the entire non-linear model into three parts, flapping and thrust, forces and torques and rigid body dynamics, as can be seen in Figure ‎3.9. The farthest block to the right in the same figure is called "VR system". This block animates the helicopter movement. Following is a description of each block: 3.5.1 Flapping and thrust This block was implemented using the recursion method from section ‎3.4.1. The way of obtaining a solution to the equation set through 45


Figure ‎3.9 Overview of Simulink model

46


a recursive scheme, for the induced velocity/thrust problem, was first described by a NASA report [53]. This was implemented using thrust equations, proved numerically unstable. The problem is that the induced velocity and thrust do not converge if the collective pitch θo falls below 4◦. However, a helicopter in hover has a pitch of about θo=8o, and a pitch of 4◦ would imply that the helicopter is descending rapidly. A rapid decent lie outside our flight envelope, so no alteration to the NASA scheme were made. The diagram in Figure ‎3.10 shows the details of the model diagram in Figure ‎3.5. This is presented as a number of sub components in the simulator. Main rotor flapping (β1c, β1s) can be obtained after solving equations (‎3.28) and (‎3.29) simultaneously as shown below 1c  -0.0000118A1u 2  0.0008434v - 0.00004037B1v 2  0.0029u - 0.000121B1u 2 - 1.09087B1 - 0.00008075uvA1 + 0.010842p - 0.02258q - 0.0000354A1v 2 - 0.0000236uvB1 - .31858A1

(‎3.39)

1s = - 0.0000118B1v 2 + 0.000843u - 0.00003540B1u 2 - 0.31858B1 + 0.00004037A1u 2 - 0.0000236uvA1 - 0.001987p - 0.001558q + 0.000121A1v 2 - 0.0029v + 0.00008075uvB1 + 1.09087A1

(‎3.40)

These equations are presented in main rotor flap sub-system. Figure ‎3.11 shows how we can employ Simulink to model equations (‎3.37) and (‎3.38). Figure ‎3.10 shows main rotor thrust model as a sub-system of flapping and thrust diagram. It is using S-function tool to solve thrust equations in section ‎3.4.1. The details of inputs and outputs are described as a diagram in Figure ‎3.12.

47


Figure ‎3.10 Sub-systems (flapping and thrust) diagram

3.5.2 Forces and Torques The block in Figure ‎3.9 is implemented with the force and moment equations presented in section ‎3.3.

48


Figure ‎3.11 Control-Rotor Flap Simulink Model

49


Figure ‎3.12 Thrust Simulink model

3.5.3 Rigid Body In Figure ‎3.13, the implementation of rigid body equations in SIMULINK is presented. Figure ‎3.13 shows an example of how the equations are implemented in SIMULINK by the use of S-functions which is a box in SIMULINK executing a user-defined script giving the possibility of a simpler model. The box labeled "Rigid" in Figure ‎3.13 is such a S-function, and contains the equations describing the rigid body dynamics (see section ‎3.2). 3.5.4 Virtual reality Farthest to the right in Figure ‎3.9, a box called "VR system" is placed. This box animates the helicopter movement. This is achieved by giving the VR-model a three dimensional vector containing coordinates of the position and a four dimensional vector for eigenaxis rotation describing the attitude. A screen dump of the helicopter animation is shown in Figure ‎3.14. The equations used in the three parts have been described in ‎Part I: of this chapter.

50


Figure ‎3.13 SIMULINK implementation of the rigid body

51


Figure ‎3.14 Virtual reality (helicopter animation)

52


3.6 Testing and Verification of the Simulator This section presents a number of tests conducted on the simulator, and analysis of these results. The test presented here does not constitute the entire number of test conducted of the simulator, but it is a descriptive set of the total tests made and the test presents properties that are characteristics of the helicopter. 3.6.1 Testing at Hover To verify the non-linear model, first an analysis of the expected movement of an uncontrolled helicopter in hover is carried out. This analysis is based on causes and effective behavior of the states in the nonlinear model. Thereafter, a simulation of the helicopter model in hover, with no input given, will be performed. Due to drag on the main rotor, the helicopter would rotate about the bz axis, if not for the tail rotor counteracting this rotation. To counteract the torque caused by drag on the main rotor, the tail rotor produces a force in negative by direction. This causes an acceleration of the helicopter in this direction and thereby an increase of the velocity in the negative by direction. This velocity causes the blades to flap positive lateral, i.e. β1s becomes positive due to the dihedral effect [54]. When the lateral flapping becomes positive, the longitudinal flapping β1c becomes negative due to the cross couplings of the lateral and longitudinal blade flapping. The lateral flapping also makes the helicopter rotate about bx, i.e. p becomes positive and forces the helicopter velocity along by to become positive, that is v becomes positive. The negative longitudinal flapping causes the helicopter to rotate negatively about by, q becomes negative, and thereby giving the helicopter a positive translatory velocity along bx that is u becomes positive. This is the cause and effective movement, which must not be 53


confused with being sequential, the hovering helicopter would perform if it is not controlled, as illustrated in Figure ‎3.15. In Figure ‎3.16, Figure ‎3.17and Figure ‎3.18 simulation results for the flapping, translatory velocities and the angular velocities, respectively, are presented.

v

negative

β1s positive

p

v

positive

positive

β1c

q

u

negative

negative

positive

Figure ‎3.15 Illustration of the desired movement of the unstable helicopter model in hover

Figure ‎3.16 Longitudinal and lateral flapping in hover

It can be seen that, at time t = 0, the translatory velocity along by becomes negative and the lateral flapping β1s becomes positive. Almost immediately after this, the longitudinal flapping β1c becomes negative. In Figure ‎3.18 it can be seen that p becomes positive almost immediately after the time t = 0 and shortly after q becomes negative. The effect of this can be seen in 54

Chapter 3: Theoretical Analysis 0.1

0

-0.1

u,v,w [m/s]

-0.2 u v w

-0.3

-0.4

-0.5

-0.6

-0.7

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

time

Figure ‎3.17 translatory velocities in hover -3

4

x 10

p q r

3

2

p,q,r [rad/s]

1

0

-1

-2

-3

-4

-5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

Figure ‎3.18 angular velocities in hover

Figure ‎3.17 as changes in v and u. The negative movement along by (v) becomes positive as an effect of p becoming positive and u becomes 55


positive as an effect of q becoming negative. This movement matches the desired movement of Figure ‎3.15 and thereby the qualitative movement of the helicopter is considered to be verified. 3.6.2 Forward movement The longitudinal cyclic input (ulong) is given a short positive input. The result of this input is that the helicopter tilts forward (nose direction) and accumulates velocity, after this no other input is given. Because of this accumulated velocity, a dihedral effect should influence flapping which in turn will affect the velocity.

Figure ‎3.19 Helicopter response for longitudinal input

56


Figure ‎3.19(a) presents the swash plate lateral and longitudinal input respectively. It is difficult to obtain the desired helicopter through a generated signal, where the operator is given feedback from the 3D graphics display. The goal of the operator is only giving the simulator an input in the longitudinal direction, to obtain a forward momentum. The input figure shows that although ulat is given a zero input, the ulong is given a primary input pitch, as intended. This results in a positive flapping angle β1c longitudinally Figure ‎3.19 (b). The β1s in Figure ‎3.19 (b) graph shows a similar pattern to β1c, and suggest that is, mainly governed by the cross coupling. The body velocity in x and y directions respectively, are presented in Figure ‎3.19 (c).The result of the initial positive flapping β1c is that the helicopter attains backward velocity, because of the cross coupling. It is supposed to be a positive lateral velocity, but because the tail force in negative y direction is greater than positive force generated by β1c, the velocity becomes negative. After about 8 seconds the velocities starts to decline and finally reverse. This is due to the wind dihedral effect, where the rotor disk tries to pitch up and away from the wind. This effect can be seen in both velocity directions. The dihedral effect can be seen on the flapping angles, when the helicopter moves backward, the angle β1c are positive and when it moves forwards becomes negative.

57


Part III:

LINEARIZATION

The non-linear equations in Part I are linearized. Different linearization methods are used throughout the linearization. The equations with operating points in zero are linearized with a Taylor approximation, except for the flapping equations, which are linearized through a black-box method. Table ‎3.1 summarizes the operating points and linearization methods for the different equations. In the following sections, the actual linearization is described. Only one of the Taylor approximations is described in this section. Variables are divided into large-scale and small-scale values, with the notation. d (t )  0d (t )   d (t ) ,

(‎3.41)

where 0d(t) is the operating point, and δq(t) is the small-signal value. Table ‎3.1 Summary of the operating points and linearization’s methods for the different equations. Only the main-rotor thrust and main-rotor drag are linearized in an operating point different from zero. Equation

Operating point

Linearization method

QMR (t ),TMR (t ) 1s (t ), 1c (t )

c

Numerical Method

0

Black box method

 (t ) u (t ), v(t ), w  (t ), q (t ), r(t ) p (t ),(t ), (t )

0 0 0

Taylor approximation Taylor approximation Taylor approximation

3.7 Linearization of Main-Rotor Thrust The solution is to calculate and plot TMR(t) for varying values of ucol(t) for every value of ucol, five iterations of TMR(t) and v1 are calculated to ensure convergence. This is shown in Figure ‎3.20. An operating point for ucol is chosen, so that the value of ucol exactly cancels out the 58


gravitational force, hence, the helicopter remains in the same vertical position. The value is calculated by a MATLAB script. This yields the linear expression TMR (t )  374.071.ucol  10.6435 .

(‎3.42)

This is expanded into large-scale and small-scale values 0TMR





  TMR  374.071. 0ucol   ucol  10.6435 ,

(‎3.43)

which yields

0TMR

 TMR  374.071.  ucol ,

(‎3.44)

 374.071. 0ucol  10.6435 .

(‎3.45)

Assuming ucol is zero, the operating point for ucol can be determined, knowing that the helicopter will remain in hover if the mainrotor thrust is equal to the gravitational force. Hence, 0TMR

 374.071. 0ucol  10.6435  m.g , 0 ucol



m.g  10.6435 . 374.071

(‎3.46) (‎3.47)

Substituting this expression into (‎3.43), yields the linearized thrust-equation TMR (t )  374.071. ucol  10.6435

 374.071.  ucol  oucol   10.6435   m.g  10.6435    374.071.   ucol      10.6435 374.071     374.071.  ucol  m.g

(‎3.48)

59


Figure ‎3.20 The main-rotor thrust as a function of the collective pitch. The mainrotor thrust is linearized in the operating point where the main rotor thrust is equal to the gravitational force.

3.8 Linearization of Main-Rotor Drag The linearization of the main-rotor drag is done similarly to the linearization of the main rotor thrust. The relationship between the mainrotor thrust and the main-rotor drag is defined as [49] 1.5 QMR (t )  AQ, MR .TMR  BQ, MR ,

(‎3.49)

This equation is plotted in Figure ‎3.21, together with the operating point of the main-rotor thrust and the slope of the main-rotor drag/thrust equation in the operating point. The operating point of TMR, was chosen in the previous section to cancel out the gravitational force, hence the helicopter is remaining in the same vertical position. Thus, the equation for the drag/thrust function in Figure ‎3.21 yields QMR (t) = 0.0208033TMR (t) + 0.31914436 .

(‎3.50)

This is expanded into large-scale and small-scale values 60




o QMR (t)   QMR (t) = 0.0208 o TMR (t )   TMR (t )

 + 0.3191 ,

(‎3.51)

which yields o QMR (t) = 0.0208 TMR (t ) + 0.3191,

(‎3.52)

 QMR (t) = 0.0208  TMR (t ) ,

(‎3.53)

Figure ‎3.21 The relationship between the main-rotor drag and main-rotor thrust is linearized around the operating point where the main-rotor thrust is equal to the gravitational force.

As previously described, the operating point must equal the gravitational force  QMR (t) = 0.0208  m.g + 0.31914436

 0.0208* 31.7844 + 0.31914436  0.9806 N .m.

(‎3.54)

The equation for the linearized drag is hereby described as QMR (t)   QMR (t)  oQMR (t)  0.0208  TMR (t )  0.9806.

(‎3.55)

61


3.9 Linearization of Main-Rotor and Control-Rotor Flapping The relationship between the angle of the swash-plate and the flapping of the main rotor and the control rotor, is described by equations (‎3.27), (‎3.28), (‎3.29), (‎3.37) and (‎3.38) in Section ‎3.4.2. These equations are rather complex and difficult to linearize. In order to ease this linearization, it has been decided to use a black box method. The expression is approximated with first-order system equation, in which cross-coupling between lateral and longitudinal flapping of the main rotor is neglected. The non-linear equations are implemented in a SIMULINK model, and a step is applied to the inputs ASP and BSP, to determine the time constant from the swash-plate input to the main rotor flapping. The simulation setup is shown in Figure ‎3.22. The step responses are shown in Figure ‎3.22, from where the flapping angles β1s and β1c, and the time constants are found. As can be seen, the cross-couplings are small, and the responses are approximating that of a first order system [55]. The time constants are similar, and determined to be τ1s= τ1c = 0.0067sec, which are used directly in the linearized model. Hence, the linearized flapping equations are described by 

1s ( s). s  



1c ( s). s  

1    Asp ( s) ,  1s 

(‎3.56)

1    Bsp ( s) ,  1c 

(‎3.57)

. 1s (t )  Asp (t ) ,

(‎3.58)

. 1c (t )  Bsp (t ) .

(‎3.59)

which yields 1s (t )  

1c (t )  

1

 1s 1

 1c

62


Figure ‎3.22 Simulation setup for determining the time constant for the transfer functions from ASP to β1s and BSP to β1c.

Figure ‎3.23 Flapping response as a consequence of a step input to the swash plate

3.10 Linearization of Longitudinal Acceleration This section describes the linearization of the longitudinal acceleration, based on a Taylor approximation. The approximations stated 63


in Table ‎3.2 are used in this linearization. The equation for the state derivative ̇ ( ) is defined as b

f x (t ) b (‎3.60)  v(t ).r (t )  bw(t ).q(t ) , m Table ‎3.2 Approximations of nonlinear expressions when operating point is zero. b

b

u (t ) 

Nonlinear expression

Linear expression

a.sin( x(t )) a. cos(x(t )) a.x(t )  b

a.  x(t )

a a.  x(t )

fx(t) is derived in chapter ‎3 as b

f x (t )  TMR (t ).sin( 1c )  m.g.sin( (t )) ,

(‎3.61)

which is substituted into (‎3.60) b

u (t ) 

 TMR (t ). sin( 1c )  m.g. sin( (t )) b  v(t ).r (t )  bw(t ).q(t ) . m

(‎3.62)

Because the operating point for θ(t) is zero, small angle approximations are valid and thus applied for the sine functions b

u (t ) 

 TMR (t ).1c (t )  m.g.  (t ) m

 bv(t ).r (t )  bw(t ).q(t ) ,

(‎3.63)

The expression for TMR(t) is linearized in Section ‎3.7 to TMR (t )  374.071.  ucol  m.g ,

(‎3.64)

β1c(t), θ(t), q(t), bω(t), r(t) and bv(t) are expanded into large-scale and smallscale values

1c (t )  o 1c (t )   1c (t ) ,

(‎3.65)

 (t )  o (t )    (t ) ,

(‎3.66)

q(t )  oq(t )   q(t ) ,

(‎3.67)

r (t )  or (t )   r (t ) ,

(‎3.68)

 (t )  ob (t )   b (t ) ,

(‎3.69)

v(t )  obv(t )   bv(t ) .

(‎3.70)

b

b

64


Since the large signal values of (‎3.65) to (‎3.70) equals zero, the equations can be reduced to

1c (t )   1c (t ) ,

(‎3.71)

 (t )    (t ) ,

(‎3.72)

q(t )   q(t ) ,

(‎3.73)

r (t )   r (t ) ,

(‎3.74)

 (t ) 

 (t ) ,

(‎3.75)

v(t )   bv(t ) .

(‎3.76)

b

b



b

(‎3.64) and (‎3.71) to (‎3.76) are substituted into (‎3.63) b

u (t )  

374.071.  ucol .  1c (t )  g (   (t )  1c (t )) m

  b v(t ).  r (t )   b w(t ).  q(t ).

(‎3.77)

Since the product of small signal values are considered small enough to be neglected, hence equation (‎3.77) yields b

u (t )   g (   (t )  1c (t )) .

(‎3.78)

3.11 The system matrix for the linearized model In this previous sections TMR(t) was linearized numerically to TMR (t )  374.071.  ucol  m.g ,

(‎3.79)

A numerical approach was also used for QMR(t), described as QMR (t)  0.0208  TMR (t )  0.9806 ,

(‎3.80)

A different approach was used for the linearization of β1c(t) and β1s(t). where a black box method was used to measure the time constants, when approximating their step response with a first order system. The time constants were found to have a value of

 1s  1c = 0.0067s ,

(‎3.81)

65


̇ ( ), a Taylor series

For ̇ ( ) ̇ ( ) ̇ ( ) ̇ ( ) ̇ ( ) ̇ ( ) ̇ ( ) ̇ ( ) expansion was used. The equations are expressed as u (t )   g (   (t )  1c (t )) . g (lt  lm ) 1 .  u ped (t )  g.   (t )  . 1c (t ) m lt

v(t )      w

(‎3.82)

7.7806 0.9806 .  ucol (t )  m.lt m.lt

(‎3.83)

374.071 .  ucol (t ) m

(‎3.84)

p (t )  (10.6435  ht .  1s (t )  y m .m.g .lt  10.6435  y m .lt  ht . 1s (t ).m.g  hm .lt . 1s (t ).m.g  10.6435  hm .lt . 1s (t )  0.9806  ht  ht .lt .  u ped (t )  (374.071  y m .lt  7.7806  ht ).  u ped (t )).  0.9806  1c (t ).

q (t ) 

1 I yy

1 I xx .lt

1 I xx

.(10.6435  hm . 1c (t )  lm .m.g  10.6435  lm

 0.9806.  1s (t )  hm.m.g .  1c (t )  374.071 lm .  ucol (t ))

r(t )  

(‎3.85)

(‎3.86)

1 (lt .  u ped (t )  (374.071y m .  u ped (t )  y m .m.g I zz  10.6435  ym). 1c (t )

(‎3.87)

(t ) p(t )

(‎3.88)

(t ) q(t )

(‎3.89)

 (t ) r (t )

(‎3.90)

1s (t )   1c (t )  

1

 1s 1

 1c

. 1s (t )  Asp (t )

(‎3.91)

. 1c (t )  Bsp (t )

(‎3.92)

The constants contribution

66


K v,dist  

0.9806 m.lt

0.9806 3.240  0.585  0.51736



K p , dist  

(‎3.93)

0.9806  ht I xx .lt

0.9806  0.07 0.051224  0.585  2.2907



given to

̇( )

(‎3.94)

̇ ( ) seen in (‎3.83) and (‎3.85) are the constant

acceleration the tail rotor produces along by and the constant angular acceleration the tail rotor produces about bx respectively. Since ym and lm are equal to zero, the constant contribution of this to ̇ ( ) is zero. The above mentioned equations are used to form a linear statespace model of the system. The mentioned constant acceleration from the tail rotor is not possible to directly implement in a linear state-space model. Therefore, these constants of acceleration are seen as a disturbance to the system, giving a system of the form x (t )  As x(t )  Bs u (t )  Bd d (t ) y (t )  C s x(t )  Ds u (t )

(‎3.95)

where As, Bs, Cs, Ds are the system matrices, Bd the system disturbance distribution matrix, d(t) the tail-rotor disturbance introduced in the system, x(t) the state vector x(t )  u(t ) v(t ) w(t )

p(t ) q(t ) r (t )  (t )  (t )  (t ) 1s (t ) 1c (t )T

(‎3.96)

and u(t) the input vector



u (t )  ulat

ulong

ucol

u ped

T

(‎3.97)

The parameters which are a part of (‎3.82) to (‎3.92) can be found in Table ‎4.4. These parameters are input into the equations from which the system matrices are described as:67




1s

0

0

0

0

9.81

0

0

0

0

0

0

0

0

273

0

0

0

0

0

8.74

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

 149.3

0

0

0

0

0

0

0

0

0

u 0 0  0  0 0 As  0  0  0 0  0 0 

v 0

w 0

p 0

q 0

r 0

 0

 9.81

0

0

0

0

0

9.81

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

ulat 0 0  0  0 0 Bs   0 0  0 0  1 0 

ulong 0

ucol 0

0

 4.105

0

 115.5

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

u 1 0  0  0 C s  0  0 0  0 0 

v 0 1 0

w 0 0 1

0 0 0 0

0 0 0 0

0 0

0 0

p 0 0 0 1

q 0 0 0 0

0 0 0 0 0

1 0 0 0 0



u ped 0   0.309   0   19.54   0  0.7757   0  0   0  0   0 

r 0 0 0 0 0 1 0 0 0

 9.81  0   0    19.14 79.94   0  0   0  0   0   149.3

(‎3.98)

(‎3.99)







0 0 0 0

0 0 0

1s

0 0 0 0

0 0

0 0 0 0 1

0 0 0

0 0 1 0 0

1c

0 0 0 0 1 0

0 0

0 0

1s 0 0  0  0 0  0 0  0 0

(‎3.100)

Since the internal disturbances introduced to the system, initialized in the operating point, only the effective v and p, the disturbance distribution matrix, Bd, are written as:68

Chapter 3: Theoretical Analysis 0 1  0  0 0  Bd  0 0  0 0  0 0 

0 0  0  1 0  0 0  0 0  0 0

(‎3.101)

and Ds=0. 3.12 Verification of Linear Model To verify the linear model, it is tested if the trend of the linear states develops as the trend of the non-linear states. This test, is used to check if the models respond to input in a similar way, is done for both positive and negative steps on all of the inputs, respectively. Appendix D presents the details of the linear and non-linear model verifications.

69


Part IV:

CONTROL

This part describes the introductory considerations, before the actual control design is performed. First the objective of the control is elaborated, and the instrumentation of the helicopter is defined. Finally, an overview of optimal control is presented. 3.13 Design Approach The objective of the control is to drive the helicopter to its equilibrium, defined to be where the translatory velocities are zero, and the Euler angles are constant. In this way the helicopter is stabilized into a hover maneuver. Thus, the controller needs to drive the states of the system to values which correspond to the equilibrium. As this cannot be performed within a limited amount of time, certain success criteria need to be established for the control design. As stated in the problem formulation, the control is based on linear optimal control. It has been decided to design a discrete controller, based on the linearized system defined by (‎3.95). The system is discretized by use of MATLAB , with 10 ms as sampling time. The sample time is selected based on an estimate that 10 ms is sufficiently low compared to the dynamics of the mechanical helicopter system. This leads to the discrete counterpart x(k  1)   s x(k )   su (k )  d d (k ) y (k )  H s x(k )

(‎3.102)

of the continuous LTI (Linear Time Interval theory) system. 3.14 Helicopter Instrumentation The controller eventually have to rely on measurable sensor information and actuator output characteristics, so that, it is necessary to 70


have knowledge of the on-board instrumentation during the controller design process. This section describes the actuators and sensors available on the helicopter.

3.14.1 Actuators The four actuators are supplying the control/input signals ulat, ulong, ucol and uped to the system. They are all servos controlling the pitchangle of the blades of either the main or tail rotor. Because it is assumed that these actuators' dynamics are much faster than the system dynamics, their dynamics are omitted during the control design. Actuator saturations, however, are included when simulating the control of the system. For the first three inputs, ulat, ulong and ucol, the saturation limits are based on pitch operating-range data from real-world helicopters [54] or instruction manual of helicopter. The chosen saturation limits can be found in Table ‎3.3. However, the fourth control signal uped is from the modeling part defined as thrust or force, and no direct information about its saturation limits are available. Thus, to be able to set a value of the saturation limit of the tail rotor an assumption is made. It is assumed that the tail rotor is capable of generating the same lift per area of rotor plane as the main rotor. That is the maximum tail-rotor thrust is found by scaling the maximum main-rotor thrust with the relationship between the area of the tail rotor and the area of the main rotor. The maximum producible thrust of the tail rotor is thus [42] TTR ,max  TMR ,max .

2 RTR 2 RMR

 44.

0.12 0.47 2

 2N ,

(‎3.103)

where TMR,max is the maximum producible lift by the main rotor, RTR is the tail-rotor radius and RMR is the main-rotor radius. 71


Table ‎3.3 Estimated servos operating-range Input

Range

ulat ulong ucol uped

⁄ ⁄ ⁄

3.14.2 Sensors There are nine sensors onboard the helicopter: three accelerometers, three gyroscopes and three compasses. The accelerometers measure the translatory acceleration relative to the EF stated in the BF. Assuming that the initial values of these accelerations are zeros, when the control system is initialized, the accelerations can be integrated to yield velocities. These velocities are measured relative to the EF, but based on the assumption made in Section ‎3.1, stating that the wind velocity relative to the EF, equals zeros. They equal the velocity relative to the wind. Thus, the sensors are assumed to measure the translatory velocities bu, bv and bw directly. The gyroscopes measure angular velocities of the BF relative to the SF. These equal the angular velocities p, q and r can thus be used directly. The final sensor, the compass, is used to observe heading of helicopter and correct the orientation of AHRS to give right (corrected) Euler angles to controller. The dynamics of all the sensors are omitted on the same basis as for the actuators. That is, their dynamics are assumed to have no influence on the total system dynamics.

72


3.15 Optimal Control The purpose of this section is to introduce the principles of optimal control, and give a short description of the linear quadratic approach to optimal control.

3.15.1 Performance Index When controlling a dynamic system, the aim is to bring the states from an initial position to a state reference, and keep them at this reference. It is desirable to drive the states to a steady state value as fast as possible, but this task will always be bounded by the amount of actuator power available. By minimizing a performance index of the type [56] ∑ where and

( ( ) ( ))

(‎3.104)

is the performance index, k is the sample number, is a function of x(k) and u(k), with regard to u(k), an optimal input

sequence u(0), u(1), u(2),…… u(N) can be found for bringing the state x(k) to the state reference [56] In this way it is possible to control a state by using minimum input power. By minimizing the performance index , the controller design has been expressed as a minimization task. For minimization of the performance index in equation (‎3.104) a recursive expression is described by [56] ( ( )) where

( )

[ ( ( ) ( ))

( (

))]

(‎3.105)

denotes the minimum of , with regard to the input sequence u(k).

The recursive expression of k = N since

( ( )) has to be calculated backwards from

( ( )) depends on

( (

)). The results of this 73


recursive calculation is an optimal input u* for each value of k for the given performance index . 3.15.2 Linear Quadratic Control To be able to design an expedient controller, the design of

has

to be considered. In order to find the minimum of x(k) and y(k) they are often chosen to be expressed in a quadratic form. Also a weight relation between x(k) and y(k) has to be defined to be able to punish large input signal values and large state values. A widely used approach is the linear quadratic approach. It is based on minimizing a quadratic performance index of the form [56] ∑(

( )

( )

( )

( ) ( )

( ) ( )

( )

( ))

( )

( )

(‎3.106)

or described as ( )

{

( ) (‎3.107)

Both x(k) and u(k) are squared and Q1, Q2 and QN are weighing matrices that respectively punishes large state values, large input values and large end point values. If it is not of interest to punish large end values, different from the desired state values, QN can be included in Q1 by QN = Q1 and instead running the sum in the performance index from

. The

performance index will then be expressed as:∑

( )

( )

( )

( )

(‎3.108)

Consider a linear system of the form (

)

( )

( )

(‎3.109)

with the performance index given by equation (‎3.106), it can be applied, to any time k 74


( )

( ) ( )

( ( ))

(‎3.110)

( ) ( ) ( )

(‎3.111)

( ) is the optimal input vector and proportional to the state vector

where

( ) with the factor

( ), which is the linear state feedback for the ( ( )) is

closed loop system. The minimized performance index proportional to the squared state vector

( ) with the matrix

order to calculate the two proportional matrices

( ). In

( ), and ( ), Riccati

equations needs to be derived and solved. The solutions expresses

( )

and ( ). To derive the Riccati equations, equation (‎3.105) is used as a starting point, and based on (‎3.111) the solution is guessed to be of the quadratic form

( ) ( ) ( ) [56]. This guess is inserted into equation

(‎3.105) which yields ( ( )) ( )

(

( )

[

( )

( ) ( ) ( )

[

( )

( )

( )

( )

( ) ( ))

( )

( ) (

)(

(

) (

) (

)]

( ) ( )

( ))],

(‎3.112)

To find the minimum of equation (‎3.112) at the time k, the expression is differentiated with regard to u(k) and equaled zero [56] ( ( )) ( )

( )

( )

( )

(

( )(

)( ( )

( )

( ))

( ))

(‎3.113)

By isolating u(k), the expression for the optimal input is derived ( )

[

(

) ]

(

)

( )

(‎3.114)

Substituting equation (‎3.110) into equation (‎3.114), the equation expressing ( ) yields ( )

[

(

) ]

(

) ( )

(‎3.115) 75


which is the first Riccati equation. As can be seen, (

), hence an expression for

( ) depends on

( ) is needed. Substituting equation

(‎3.110) into equation (‎3.112) yields:( ( ))

( ) ( ) ( ) ( ) ( ) ( ( ) ( ))

( ( ) [

( )

( ) ( )) ( )

( )

(

( )(

( ) ( )) ( )

( ))

(

(‎3.116)

( ) ( ))

(

)(

( ))] ( )

Isolation of ( ) yields the second Riccati equation ( )

( )

( )

( ))

(

(

)(

( ))

(‎3.117)

which can be simplified to [56] ( )

(

)[

( )]

(‎3.118)

The performance of the controller after calculation of ( ) is only optimal in relation to the chosen performance index. Therefore, the design parameters of the performance index,

and

, has to be chosen wisely

to give the controller the right properties suiting the given control task. This is a trade-off between good control and good economy.

76


3.16 Control Design This section describes the design of the control for the helicopter. As mentioned in the previous section, it is based on linear optimal control. In the first design step, a stabilizing controller is designed, without any performance optimization. In the second design step, the state space is extended to include known linearization errors. Up to now, exact and full state-information is assumed. As mentioned in the previous section, not all states are measurable, for this reason a predictor is designed in the third design step. Finally, for tuning, the controller is exposed to unknown disturbances. 3.16.1 Design of Stabilizing Controller This section describes the design of a controller that stabilizes the helicopter. First, an open loop simulation of the non-linear model, initialized in the operating point, is performed, to examine the behavior of the system. The translatory velocity and the attitude of the helicopter are shown in Figures 3.16 to 3.19. The model is linearized and initialized in an operating point which is not the equilibrium of the system. The simulations confirm this assumption, that is, the simulations show that the helicopter is an unstable system. Due to the force generated by the tail rotor, a negative translatory acceleration is performed along by. The dihedral effect results in a positive roll movement, and an appertaining negative translatory acceleration is performed along

b

y. As illustrated by Figure ‎3.23, the lateral and

longitudinal flapping cross-couples, which results in a positive pitch movement, and an appertaining positive translatory acceleration along bx. It can also be seen that there is a translatory acceleration along bz. This

77


indicates that the collective input calculated to keep the vertical velocity in the operating point is insufficient. The purpose with the first design step is to stabilize the helicopter, i.e. to bring the states describing the attitude, towards constant values. In this step, full state-information is presumed, and no external disturbances are included. As described in the previous section, the control is based on a stationary LQR [56], thus, stationary values of the proportional matrices ( ) and ( ) are used. These values are calculated on the basis of the recursive Riccati equations. The performance index is, according to the previous section, defined as ∑

( )

( )

( )

( )

(‎3.119)

and the closed loop of this autonomous description of the discrete system is shown in Figure ‎3.24. The remaining design parameters are the weight matrices Q1 and Q2. Ref [56] suggests constraining to the use of only the diagonal elements, which reduces the degrees of freedom. On the basis of knowledge about the open-loop system, pacifications of the maximum values of the states and inputs can be used as a starting point for the weighing.

Figure ‎3.24 Closed loop block scheme of the discrete system

78


(

)

(

)

(‎3.120)

In order to make use of this rule of thumb, the state operating-ranges are estimated, and used together with the input saturation defined in Table ‎3.4. Note that they are estimates of the expected operating range, and are not to be confused with requirements. Before the weighing is started, the objective of the controller should be stated. As previously mentioned, the objective of the first design step is stabilization rather than performance optimization. The weighing will therefore be performed in the following order of priority: 1. Reduction of overshoot and settling time for the attitude, to obtain a smooth movement. 2. Reduction of steady-state error on the translatory velocities. The previously mentioned estimated operating-ranges yields the weighing matrices u 10 6   0  0   0  0 Q1     0  0   0   0  0   0

 1

v 0

w 0

p 0

q 0

10 6

0

0

0

0

10 6

0

0

0

0

32.7

0

0

0

0

32.7

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

r 0







0

 1s

0

0

0

0

0

0

0

0

1c

0  0  0 0 0 0 0 0   0 0 0 0 0 0  0 0 0 0 0 0   32.7 0 0 0 0 0  0 0 132.1 0 0 0   0 132.1 0 0 0 0  0 0 0 132.1 0 0   0 0 0 0 14.6 0   0 0 0 0 0 14.6

(‎3.121)

(‎3.122)

79

Chapter 3: Theoretical Analysis ulat 22.8  Q2    0  0   0

ulong 0 22.8

u col 0 0 22.8

0 0

0

u ped 0  0   0   0.25

(‎3.123)

Table ‎3.4 Estimated state operating-range. State

Operating Range

State

Operating Range

u v w p q r

±0.001 m/s ±0.001 m/s ±0.001 m/s ±10 o/s ±10 o/s Unlimited

ϕ θ ψ β1s β1c

±5o ±5o Unlimited ±15o ±15o

where σ is a constant multiplied by

, introduced to weigh the input

relative to the control performance. A closed loop simulation with these weighing matrices yields the results in Figure ‎3.25. Before this simulation, the operating point of the collective input is adjusted to

to

fit the non-linear model. By considering Figure ‎3.25 (a), it can be seen that, the longitudinal and vertical velocities, converge towards a near-zero constant, while the lateral velocity converges towards -0.009 m/s. This steady state error is caused by the tail-rotor thrust and moment about x-axis (As a result of the presence of torque arm ht= 0.07m), which creates a negative force along by. As can be seen in Figure ‎3.25 (b) the helicopter has a positive roll angle, ϕ, which compensates for the negative force generated by the tail rotor and moment about x-axis. By considering the input matrix in (‎3.99), it can be seen that v is affected by uped. Figure ‎3.25 (d) shows that this input is reduced, to reduce the lateral acceleration and the appertaining velocity. This results in a negative yaw movement (seen in Figure ‎3.25 (b)). From Figure ‎3.25 (c), it can be seen that, the collective input is constant, and that also lateral input is used for the stabilization. Furthermore it can be seen

80


that the control signals are not near the outlined saturations (±0.209 rad and ±2N).

Figure ‎3.25 Simulation results based on the weighing matrices in (‎3.121) to (‎3.123)

This simulation shows that the helicopter is stabilized, but with a settling time on the attitude, and a small steady-state error on the translatory velocities, which is longer than desired. These steady-state errors occur because of the tail-rotor thrust and moment about x-axis, which can be considered as a step disturbance. As previously mentioned; the objective of this design step is stabilization rather than performance, hence, the weighing is modified to reduce the overshoot and settling time of the attitude. Figure ‎3.25 (b) shows that ϕ and ψ are the angles with the longest settling time. To reduce this settling time, the states describing the angular

81


motion of these states, needs to be punished. The weight matrices in (‎3.124), (‎3.125) and (‎3.126) yields the results shown in Figure ‎3.26. u 10 6   0  0   0  0 Q1    0  0   0   0  0   0

v 0

w 0

p 0

10 6

0

0

0

10 6

0

0

0

3270

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

q 0

r 0







0

 1s

0

0

0

0

0

0

0

0

0

1c

0  0  0 0 0 0 0 0 0   0 0 0 0 0 0 0  0 0 0 0 327 0 0   100 0 0 0 0 0 0  0 13210 0 0 0 0 0   0 132.1 0 0 0 0 0  0 0 0 1000 0 0 0   0 0 0 0 0 14.6 0   0 0 0 0 0 0 14.6

  100

(‎3.124)

(‎3.125)

ulat 22.8  Q2    0  0   0

ulong 0 22.8 0 0

u col 0 0 22.8 0

u ped 0  0   0   0.25

(‎3.126)

As can be seen in Figure ‎3.26 (b), the settling time for ϕ and ψ is slightly reduced, due to the higher weighing of p, r, ϕ and ψ. Also, σ increases in order to make the controller use more stable and reduce the fluctuation of actuators. This results in a reduced yaw movement, but in the expense of an increased steady-state error on the lateral velocity. By comparing Figure ‎3.25 (d) and Figure ‎3.26 (d), it can be seen that the tailrotor thrust is reduced. The purpose of the first design step was to stabilize the helicopter. Simulations show that the helicopter is stabilized, but with small steady-state errors on the translatory velocities. The lateral velocity is dominant, with a steady-state error of about 0.04m/s. This error is due to the tail-rotor thrust, which will be treated in the next section.

82


Figure ‎3.26 Simulation results with modified weight matrices

3.16.2 Modelling of Known Disturbances The steady-state offset errors on some of the translatory velocities are caused by known linearization errors, which are not included in the design of the controller. It is however, possible to accommodate or at least reduce these errors by introducing a model of the disturbances in the control law. The steady-state error on the translatory velocity w is caused by a slightly offset 0ucol and is reduced by adjusting the operating point to a more suitable value, as mentioned in the previous section. The small steady-state errors on the translatory velocities u and v, besides being very small originate from the same disturbance and cross coupling between β1s and β1c, namely the negative translatory acceleration along by, created by the tail rotor in addition to the torque generated by that force, because of the presence of the arm ht when the helicopter is initialized in its operating 83


point. Refer to the helicopter behavior described in Section ‎3.16.1. Although the disturbance only causes a small drift on the translatory velocities, it will be included in the control law to reduce the error. The Bd matrix in (‎3.95) states that the disturbance contributes as an addition to the derivative of the state v and p. These additions as shown in (‎3.83) and (‎3.85) can be modeled as a constant step addition, which can be expressed by the autonomous state description [56] (

where

)

( )

( )

( )

( )

] (‎3.127)

], because the disturbance ( )

identity matrices [

and

[

only distributes to v and p. The constants (

) has been

computed in the linearization process in Section ‎3.11.Thus, by the linear system description (

)

( )

( )

( )

( )

( ) (‎3.128)

It is possible to form an augmented system description by expanding the state vector ( ) with the new disturbance state [

( (

) ] )

[

( )

[

] [ ][

( ) such that:-

( ) ] ( )

[ ] ( )

( ) ] ( )

Denoting the augmented matrices and vectors

(‎3.129) , ,

, ( ) and (

),

the augmented system may be expressed as:(

) ( )

( ) ( )

( ) (‎3.130)

The performance function, which is to be minimized, can be expressed as (‎3.108) which has the same form as in (‎3.106). The only difference is that 84


the state vector

( ) and the matrix

are augmented to include the

disturbance state and weighing of this state. That is, they have the form ( ) where

[

( ) ] ( )

[

]

(‎3.131)

is the original system state matrix from the previous section, and

is the disturbance weighing matrix. Thus, the general Riccati equations can again be used to compute the feedback ( ). The computed optimal feedback gain ( ) is also augmented, which from the control law ( )

[ ( )

( ) ( )

yields that the system state feedback

( )] [

( ) ] ( )

(‎3.132)

( ) equals the total feedback, if the

disturbance is zero. The system with feedback is illustrated in Figure ‎3.27. The initial diagonal entries of

will not equal the weighings found in

Section ‎3.16.1, we had some adjustments to give the best performance of the controller as shown in (‎3.133) (Among the criteria that have been developed see Table ‎3.4.The single entry of

is set to zero [56], since

this state is considered as a constant external disturbance, which of course cannot be altered by the control signals. The matrix

in (‎3.108) also

remains untouched from Section ‎3.16.1. Thus, the weighing matrices are

85


Figure ‎3.27 The optimal controller including a disturbance model

Q1s

u 10 7   0  0   0  0   0  0   0   0  0   0

xd ,v Q1d  0 0 

v 0

w 0

p 0

q 0

10 6

0

0

0

0

10 6

0

0

0

0

3270

0

0

0

0

327

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

r 0







0

 1s

0

0

0

0

0

0

0

0

0  0  0 0 0 0 0 0   0 0 0 0 0 0  0 0 0 0 0 0   100000 0 0 0 0 0  0 0 13210 0 0 0   0 132.1 0 0 0 0  0 0 0 1000000 0 0   0 0 0 0 14.6 0   0 0 0 0 0 14.6

xd , p 0 0

(‎3.133)

(‎3.134)

  100 ulat 22.8  Q2    0  0   0

 1c

(‎3.135) ulong 0 2280 0 0

ucol 0 0 22.8 0

u ped 0  0   0   0.25

With these values, the feedback gain

(‎3.136) ( ) is calculated.

This feedback is used to control the non-linear SIMULINK model, where the disturbance state

( ) is introduced as: 86


( )

[

]

[

]

(‎3.137)

When simulating the closed loop system without the disturbance state, the plots for the translatory velocities are as shown in Figure ‎3.28, which shows study state error. After taking the disturbance of velocity into consideration in the controller, simulation of closed loop system shows results as translatory velocities as in Figure ‎3.29. Study state error between velocities in two figures is not enough to be close to what we wanted as in Table ‎3.4. So when adding the second effect of disturbance

the results of the steady-state error are

obtained as in Figure ‎3.30.

Figure ‎3.28 Translatory velocities when

are zero

87


Figure ‎3.29 Translatory velocities when is zero

Figure ‎3.30 Translatory velocities when

88


3.16.3 State Predictor Until now during the controller design, full state information is assumed, which in reality is not the case. As mentioned previously, the sensors available are 3D accelerometer, measuring translatory movement, 3D gyroscope to measure the angular velocities and Euler angles of the helicopter and 3D compass to set reference coordinates of Euler angles. This means that, it is only possible to measure 9 of the 11 states needed for the controller. The flapping angles β1s and β1c are not directly measurable states and need to be estimated. Based on the linear model and the controller, which are both discreet, it is necessary to design a discreet observer (Predictor). Since the helicopter is a MIMO system, it is decided to design a two order state space predictor. It is only estimated the two missing states. The principle of a predictor is to estimate states based on the knowledge of inputs for each state (see Equations (‎3.58) and (‎3.59) ). A system of the state space in continuous form is:

[

̇ ( ) ] ̇ ( )

[

[

( ) ] ( )

[

][

][

( ) ] ( )

[

( ) ] ( )

][

( ) ] ( )

(‎3.138)

By using Matlab, the above formula was converted from continuous to discrete model with discrete time 0.01s becomes:

89

Chapter 3: Theoretical Analysis [

( (

[

( ) ] ( )

) ] ) [

[

][ ][

( ) ] ( )

[

][

( ) ] ( )

( ) ] ( )

(‎3.139)

Figure ‎3.31 shows the flapping angles have high estimation error. This is not unexpected since they are not measurable. It is not possible to tune the estimation of the flapping angles since they only depend on the inputs to the model and not of the other states in the linear model. The difference between the actual flapping angle and the estimated is likely to be a result of inaccuracy introduced in the linearization process. This means that, the values of the measurable states have no direct effect on the flapping. But the important thing in Predictor, is that its sense increases and decreases with flapping angles.

Figure ‎3.31 predictor for Flapping angle β1c

90


3.16.4 Tuning of the Controller The Predictor (observer) causes some undesirable effects on the closed loop system response. The translatory and angular velocities carry sustained oscillations at a much larger magnitude than experienced in simulations of the closed loop system without tuning the controller, as can be seen in Figure ‎3.32. Furthermore, a plot of the corresponding input signal illustrates that the actuators are experiencing almost oscillations, refer to Figure ‎3.33. The oscillations in the velocities and input oscillations are obviously inter-dependent, and they originate from the introduction of the Predictor (observer). The reason for this can be found in the fact that the observer estimates the flapping angles β1s and β1c from the linear model. By investigating the system and inputting matrices Asp and Bsp in equations (‎3.58) and (‎3.59), it can be seen that the flapping angles are determined only from their previous values and the control signals ulat and ulong. This is unlike the non-linear model, which has a feedback from both the translatory and angular velocities to the flapping angles. Refer to Figure ‎3.9. Thus, the linear-system dynamics in the observer is limited, in the sense that it only allows the controller to feedback flapping-angle changes, as these are propagated through the system to the measurable angular and translatory velocities, and subsequently fed back in the control signal. This can be seen using a part of the continuous-time linear system description from section ‎3.11. This propagation delay decreases the effect of the controller's response to velocity changes, which then results in the need of a larger value actuator signal until a change in the output is detected. As the change in the output is detected, the controller has overcompensated and the problem occurs again when negative actuator signal is needed. This leads to saturation depicted on the plot of Figure ‎3.33. 91


Figure ‎3.32 Closed loop simulations of the non-linear system initialized in the operating point. When using the predictor

Figure ‎3.33 A closed loop simulation of the inputs to the non-linear system initialized in the operating point. When using the predictor.

92


The saturation can be avoided by changing the value of the diagonal entries of

relative to

. The control signal needs to be

decreased, and this is done by increasing the values of the diagonal entries of

. The value of this increase resulted after several simulation iterations

in a change of the scaling factor

to

Another simulation of the closed loop system is performed and the resulting plots of the velocities along with the control signals are depicted in Figure ‎3.34. The figure shows that the decrease in magnitude of the control resulted in a much more satisfactory closed-loop response of the system. There are no longer sustained oscillations in the velocities which results in a much steadier helicopter movement. We can see from the figure that the translatory velocities u and v are slowly converging towards zero.

Figure ‎3.34 Closed-loop simulation results of the non-linear system. The values of Q2 are increased with a factor of .

93

Chapter 4: Experimental Work

4 CHAPTER FOUR: EXPERIMENTAL WORK Part I: DESIGN AND CONSTRUCTION OF PLATFORM

Constructing a small-scale helicopter UAV is a challenging task. We have to carefully consider issues such as selection of hardware components, weight balance and layout design. This part presents a systematic design methodology for setting up a system of unmanned helicopter. For ease of understanding, we used the procedure of construction of ref. [57]. 4.1 Helicopter Platform The helicopter platform used in this research is a commercial available RC helicopter. The section starts by explaining the components that complete the basic RC helicopter. The basic platform has been modified to make room for the necessary electronics that will enable the helicopter to fly autonomously. Figure ‎4.1 shows the research helicopter. The helicopter is powered by a brushless electric motor 1600 Kv. The power supplied by LiPo battery is used to drive both the main rotor and tail rotor through a set of gears and belt. The stabilizer bar also denoted BellHiller bar is fixed to the same shaft as the main rotor blades and will therefore rotate at the same velocity. The Bell-Hiller bar is a control augmentation used in helicopters to improve handling properties. Placed in the front is the receiver, actuators (servos) and control electronics, that are fitted with a standard RC helicopter. The rate gyro is used to cancel disturbances that affect the heading of the helicopter; this means the 94


direction that the helicopter is pointing ideally. With a rate gyro the only input affecting the heading is piloted inputs. Under the helicopter, mounted on the modified landing gear is the box containing the electronics necessary for autonomous flight. The complete electronic setup is shown in Figure ‎4.2. The computer is based on a PC104 stack equipped with an x86 processor board, sampling board, power supply and a TCP/IP connection board. The operation system installed on a CF disk is MATLAB XPC target.

Figure ‎4.1 Full view of the model scaled helicopter

A six Degrees of freedom (DOF) sensor block is connected to the samplings board. The sensor block contains 3D accelerometers, measuring acceleration in three orthogonal axes. The block also contains 3D rate gyroscopes that measure rotation rate around the same axes. The servos give commands in the form of a PWM (Pulse Width Modulation) signal, sending the PWM signal to the servos and decoding the signal from the receiver. This is done by a so called servo board. The servo board is based 95


on a PIC18 microprocessor that decodes the incoming PWM signal, this signal can then be read via a RS-232 port. When the helicopter operates autonomously the controlling computer can set the actuator position through the servo board. The servo board has the added function of giving control to either the human pilot or the computer. This command delegation is controlled from the ground via the rc radio - receiver system.

Figure ‎4.2 Diagram of the electronics mounted on the helicopter

96


4.2 Hardware Components Selection The hardware configuration illustrated in Figure ‎4.3, in which every block represents a certain device.The essential hardware components contributory in a fully functional helicopter unmanned system are 1. an RC helicopter; 2. an avionic system for gathering inflight, communicating with the ground station and executing automatic control laws; 3. a manual control system consisting of a wireless joystick and a pilot; and 4. a ground station system for communicating with the avionic system and monitoring the flight states of the UAV.

Figure ‎4.3 Hardware configuration of UAV system

97


4.3 RC Helicopter The first component need to be selected is a high-performance RC helicopter, which is also known as model rotorcraft or a hobby-based, ALIGN TREX 500, illustrated in Figure ‎3.1. Some main specifications of ALIGN TREX 500 are listed in Table ‎4.1. The initial reasons for selecting this RC helicopter are the following: 1. High performance and Low cost: Compared with most Type RC helicopters, ALIGN TREX 500 provides reliable structural design and flight performance, at reasonable price. 2. Sufficient payload: From datasheet in Table ‎4.1, we note that the maximum takeoff load for ALIGN TREX 500 is 4 kg. By subtracting its no-load weight, the active payload of ALIGN TREX 500 is more than 2 kg, which is beyond our extra weight, 1.5 kg in total for the devices avionic system. 3. Great maneuverability: ALIGN TREX 500 was designed for acrobatic flight. Its maneuverability and agility is well known in the RC helicopter hobby flight circle. Table ‎4.1 Specification of ALIGN TREX 500

Specification

Description

Length Height Main Blade Length Main Rotor Diameter Tail Rotor Diameter Motor Pinion Gear Main Drive Gear Autorotation Tail Drive Gear Tail Drive Gear Drive Gear Ratio Weight(w/o power system) Maximum takeoff weight

840mm 310mm 425mm 970mm 200mm 13T 162T 145T 31T 1:12.46:4.68 1370g 4000g

98


Figure ‎4.4 Operating principle of ALIGN TREX 500 helicopter

99


The operating principle of ALIGN TREX 500 (see Figure ‎4.4) is standard and widely adopted by the hobby community. Four digital servo actuators and brushless ESC (Governor Mode) are employed to receive the control signal (from either the human pilot or the avionic system) and drive various control surfaces of ALIGN TREX 500:

1. The aileron servo, which produces a driving signal δlat, is in charge of the leftward and rightward tilting motion of the swash plate. Such a movement changes the cyclic pitch angle of the main rotor blades and results in both a rolling motion and lateral translation. 2. The elevator servo with a driving signal δlong is responsible for the forward and backward tilting motion of the swash plate. This tilting also changes the cyclic pitch angle of the main rotor blades but results in a pitching motion and longitudinal translation. 3. The collective pitch servo through its driving signal δcol changes the vertical position of the swash plate. As a result, the collective pitch angle of the main rotor blades is changed to generate heave motion. 4. The rudder servo, which generates a driving signal δped, cooperates with a yaw rate feedback controller to realize the yaw rate and heading control via controlling the collective pitch angle of the tail rotor blade. The yaw rate controller is added to allow a human pilot to control with the over sensitive dynamics of a bare yaw channel. 5. Finally, ESC works with an RC-purpose motor governor to get a constant rotation speed of the main rotor. A Bell-Hiller stabilizer bar is employed to cooperate with the swash plate for achieving desired rotor flapping responses.

100


4.4 Flight Control Computer The flight control computer is the brain of the avionic system. Its primary functions include 1. analyzing various flight data delivered by onboard sensors; 2. executing flight control law; 3. communicating with the ground station; and 4. logging flight data to a compact flash (CF) card for post-flight analysis. Because of the unique characteristics of the UAV systems, in selecting a flight computer, special attention shall be paid to its size, weight, input/output (I/O) ports configuration, expendability, anti-vibration property, and power consumption. We choose an embedded computer board PC/104 PCM-9375 Advantech as shown in Figure ‎4.5

It is

manufactured based on the PC/104 standard, which is defined by the PC/104 Consortium [58] and commonly used in embedded system applications. Advantech is well suited to our UAV application because of the following features: 1. Small size: PC/104 PCM-9375 Advantech is a handheld computer board with 3.5" size. 2. Lightweight: Without wires and cables, the weight is 133 g. 3. Sufficient processing speed: The processing speed of PCM-9375 Advantech is 500 MHz. 4. Rich I/O ports: PCM-9375 Advantech provides rich I/O ports for communicating with external devices, including four RS-232 serial ports, four USB ports, and two 100 Mb Ethernet port. This feature provides sufficient freedom for the sensor selection and facilitates the online program debugging. 5. Anti-vibration capacity. 101


6. Low power consumption: The power consumption for full working load is about 6.5 Watt per hour.

Figure ‎4.5 PC/104 PCM-9375 Advantech computer board

4.5 Navigation Sensors Navigation sensors provide reliable measurement for the flight status of the flying vehicle. Many commercial navigation sensors are available on the market, varying in manufacturing technology, material, estimation algorithm, measuring range, size, and weight. In accordance with the working principle, a complete navigation solution generally falls into one of the following four types: (i) INS (inertial navigation system), (ii) INS/GPS (INS calibrated by GPS), (iii) GPS-aided AHRS (attitude heading

reference

system

aided

by

GPS),

or

(iv)

MEMS

(Microelectromechanical systems) contain accelerometer, gyroscope and magnetometer. We aim to customize low cost and small-size MEMS with sufficient accuracy. In this work we chose YEI 3-Space Sensor 3-axis 9DOF (see Figure ‎4.6). This sensor has 3D accelerometer, 3D gyroscope and 3D magnetometer. It has RS232 serial port and USB interface communication and work with 5V or 3.3V DC. This is enough to make tests and control on UAV helicopter, because all tests are in hover and for a short time.

102


Figure ‎4.6 YEI 3-Space Sensor 3-axis 9DOF

4.6 Fail-Safe Servo Controller The fail-safe servo controller (or servo controller in short) is another important device to guarantee the airborne security of the miniature UAV helicopters. It is mainly responsible for decoding both piloted and computer-generated servo control commands and selecting desired decoded signals to drive multiple servo actuators. In the case of accidents that might occur during an autonomous flight, with the assistance of the servo controller, the human pilot might have a chance to retrieve the UAV platform. A commercial servo controller board, namely an HBC-101, is selected and shown in Figure ‎4.7 . It has the following features: 1. Reliable switching function: At any time, the HBC-101 is capable of realizing smooth switching between automatic control and manual control signals. 2. Sufficient input and output channels: As mentioned earlier, six channels are used in ALIGN TREX 500 helicopter. The HBC-101 provides eight input and eight output channels to fulfill our requirement. 103


Figure ‎4.7 HBC-101 servo controller

3. Signal recording capacity: The HBC-101 has the capacity to sample both the input and output signals up to 100 Hz. The logged data can be sent to the PCM-9375 Advantech computer board through the serial communication protocol. This function is particularly important to flight dynamics modeling and automatic control performance analysis. 4. High resolution: The input-recording and servo-driving can be regarded as specified A/D and D/A conversion. A resolution up to 0.009 deg provided by the HBC-101 can substantially enhances the data quality and practical control performance. 4.7 Wireless Modem A Pair of wireless modems is used to establish communications between the UAV helicopter and the ground control station. Inflight status down link and command uploading is done through this wireless system. A MikroTik RB711UA-2HnD (see Figure ‎4.8) working at a 2.4 GHz bandwidth is used. This communication board has a handheld size and a weight of 70 g. Some of its distinguished features are listed as follows:

104


1. Long range: With a clear line of sight, the maximum communication range is 2 km. 2. Ethernet protocol communication between wireless modem and another device. 3. Sufficient throughput rate: The data throughput of MikroTik RB711UA-2HnD is 300 Mbps which is good enough for our applications.

Figure ‎4.8 MikroTik RB711UA-2HnD

4.8 Battery To supply electric circuit by enough energy we extended the main battery of helicopter from 2000mAh to 3600mAh (Li-Po (6S 22.2V)), and use DC to DC converter to provide 5 volts to all electric circuit (see Figure ‎4.9).

105


Figure ‎4.9 Illustration battery and dc to dc converter connection

4.9 Manual Control Manual control is realized via a high-quality radio controller, namely, a FlySky FS-TH9X. Its robustness is well acknowledged in the RC community circle. FlySky FS-TH9X has 9 programmable signal channels with sufficient resolution. Among them, six channels are assigned to realize the servo-driving scheme introduced in section ‎4.3. To fulfill the requirement on real-time switching between the automatic and manual control modes, one channel of FlySky FS-TH9X is particularly programmed and allocated to send switching signals when necessary.

106


Part II:

MEASUREMENT OF STATE VARIABLES SIGNAL

An attitude and heading reference system (AHRS) is a key element in controlling and modeling UAVs. It consists of sensors to provide measurement signals for the unmanned aerial vehicles. Typical sensors used to form an AHRS are gyroscopes to provide the 3D angular rates, accelerometers to measure the acceleration along the three axes of the helicopter body coordinate, and magnetometers to capture the magnet values of 3D axes of earth frame, and some sensors provide reliable velocity and position information of the helicopter for trajectory tracking, autonomous flight control, navigation, and mission completion. This part is intended to introduce an integration of a low-cost inertial attitude and position reference system for a mini UAV helicopter by utilizing the complementary filter to estimate AHRS and Kalman Filter KF technique to estimate the position and linear velocity. More specifically, we propose a systematic signal enhancement procedure, which is for MEMS sensors adopted to yield a more accurate measurement of the Euler angles, angular rates, positions, and velocities of the unmanned aerial vehicles. The overall procedure is summarized and depicted in Figure ‎4.10, which consists of two independent parts, one for the AHRS and one for the linear motion.

107


Figure ‎4.10 Illustration diagram for attitude and heading reference system and linear navigation system

4.10 Attitude and Heading Reference System Filter AHRS This section presents an orientation filter using a complementary filter applicable to low-cost sensors based on micro-electro-mechanical system (MEMS) [59]. A quaternion representation is used in the filter, allowing magnetometer and accelerometer data to be used in the optimized gradient-descent algorithm and analytically derived to compute the directional error of the gyroscope measurement as a quaternion derivative [60]. Orientation filter that is applicable to MARG (Magnetic, Angular Rate, and Gravity) sensor arrays addressing issues of computational load and parameter tune. The filter employs a quaternion representation of orientation to describe the coupled nature of orientations in threedimensions and is not subject to the problematic singularities associated with an Euler angle representation. A derivation and empirical evaluation of the filter is presented in Appendix B.

108


Figure ‎4.11 illustrates a representation of the complete filter implementation for an MARG sensor array. For a complete derivation of mathematics and relationships see Appendix ‎B

Figure ‎4.11 Block diagram representation of the complete orientation estimation algorithm including magnetic distortion compensation

4.10.1 Design of Rotating Platform for Testing Filter A platform with orientation feedback and one axis of rotation is developed, In order to validate the performance of the AHRS. The platform must be fabricated with non-magnetic materials, because the magnetometer is sensitive to the wires with high current and ferromagnetic materials. Therefore, all the components of the platform are made from plastic, as shown in Figure ‎4.12. The validation of filter depends on the error between real and filter output angles in real time test. In order to achieve this requirement, one axis of rotation is driven by servo motor. One Hitec HS-311 motor was installed on the yaw axis. This servo motor is capable of providing the position feedback with (enough) a resolution of 0.1°. In our case the 109


maximum rate is 400°/s at 6 Volts, from the datasheet of the servo. This servo motor is controlled by servo controller when receiving commands from the PC via the UART interface. A command received from PC includes angular rate and the position. This platform can teste another two angles pitch and roll, by changing the position of sensors on the servo motor.

Figure ‎4.12 Orientation measurement platform

4.10.2 Validation of Filter The purpose of the validation of the AHRS filter is to validate the reliability. Two tests, dynamic test, and static test were conducted to prove the AHRS validation in this study. It is common [61], [62], [63] and [64],

to quantify orientation

sensor performance as the dynamic and static RMS errors in the decoupled parameters of Euler describing the pitch, roll, and yawing components. Representation of an Euler angle has the advantage that the decoupled angles may be more easily visualized and interpreted. 110


Figures ‎4.13, ‎4.14 and ‎4.15 show typical results of experiments for complementary filter MARG implementation. In each figure, the two traces of the upper plot represent the real measured angle by potentiometer of the servo motor and filter estimated angle. The error calculated in each of the estimated was angles represented in the two drawings of the lower part of each plot. The static and dynamic RMS values of φε, θε, and ψε were calculated where a static state was assumed when the measured corresponding angular rate was less than 5°/s, and a dynamic state, at greater than 5°/s. This threshold was chosen to be suitably high enough above the noise floor of the data. Each RMS value was calculated for the period of time framing only the rotation sequence of the corresponding Euler parameter; as indicated in Figures ‎4.13, ‎4.14 and ‎4.15. The results are summarized in Table ‎4.2. Each value represents the mean of experiments. Results indicate that the proposed filter achieves good levels of accuracy to use in UAV

No. 1 2 3

Table ‎4.2 Static and dynamic RMS error Euler Angle RMS (dynamic) φ θ ψ

1.6105 deg 1.4815 deg 2.093 deg

RMS (static) 0.823 deg 0.8041 deg 1.0318 deg

111


Figure ‎4.13 Measured and estimated angle φ

Figure ‎4.14 Measured and estimated angle θ

112


Figure ‎4.15 Measured and estimated angle ψ

113


4.11 Static Calibration and Validation of Triaxial Accelerometer Triaxial accelerometers are used in various applications, such as inertial navigation systems (INSs) and Inclinometers. Such accelerometers must be calibrated as accurately as possible because accelerometers with even small biases could result in a very fast position drift when they are used for INS applications and could result in inaccurate tilt angle measurements. This section presents a calibration method using a mathematical model of nine calibration parameters: three gain factors, three biases, and three non-orthogonality factors. The fundamental principle of the proposed calibration method is that the sum of the triaxial accelerometer outputs is equal to the gravity vector when the accelerometer is stationary. The proposed method requires the triaxial accelerometer to be placed in forty-eight tilt angles to estimate the nine calibration parameters. Since the mathematical model of the calibration parameters is nonlinear, an iterative method is used Levenberg-Marquardt algorithm (LMA). The accelerometer provides information about acceleration, speed, and position of the helicopter, by using statistical filtering. The development of such a system requires the calibration of sensor, [65], [66], [67] the data output depends on: 1. Local acceleration. 2. The sensor attitude. 3. The real gain and bias parameters. In a 3D sensor, each axis i have a different gain noted as αi and a bias noted as βi. Figure ‎4.16 presents a 3D sensor with offset 

  ( x ,  y ,  z )T , gains

 x ,  y ,  z and orthogonality errors S xy , S xz , S yz for

accelerometer; the model of the sensor output is given by [68] see Figure ‎4.17. 114

Chapter 4: Experimental Work 



 a 

   A         a x (t )    x S xy S xz   ax (t )    x   x   a (t )   S  S yz  ay (t )    y    y  , y xy x           a z (t )   S xz S yz  x   az (t )    z   z 

(‎4.1)

Figure ‎4.16 Sensor behavior

where ε is a residual error or noise, Sij represents the orthogonality errors between sensor axes (cosine of angles between axes of sensor x-y,y-z ,z-x where Sxy= Syx , Syz= Szy, and Szx= Sxz). The matrix α is considered symmetrical [68]. See Appendix C to know how to find parameters of equation (‎4.1). Figure ‎4.18 illustrates the graphical user interface of the calibration software, interfaced with a Matlab. az

ay ax





Figure ‎4.17 Acceleration vector a measured as a 

115


Figure ‎4.18 Matlab Gui program to Estimate Calibration Parameters

The calibration procedure has been performed three times successively, according to Appendix ‎C. Each time, 4800 data points are used for the estimation of parameters. The obtained gains, bias and orthogonality are then compared. The estimation algorithm LMA fit performs the alpha, beta and Sij computation instantly, with about less than 100 iterations [68]. The results are verified by comparing the estimated data and corrected data with how far or close from the surface of a sphere that have radius 1g

and center (0,0,0) by calculating the standard deviation. In

Table ‎4.3 the calibration results for a 3D accelerometer sensor are given. The sensor must only measure the static acceleration of gravity, for which the norm is constant. We need to exclude measurements where the sensor is not static, for which dynamic acceleration is present, by observing the

116


temporal variance of the data signal. In the static position, the accelerometer has a low variance. * *

Real Data Corrected Data

Figure ‎4.19 Sphere of 3D accelerometer sensor

Table ‎4.3 Measured Parameters Parameter Name

Test 1

Test 2

Test 3

βx βy βz αx αy αz sxy sxz Syz STD for x axis STD for y axis STD for z axis

-0.0103 0.0010 -0.0186 0.9936 1.0105 0.9914 -0.0067 -0.000096 0.0072 0.001419 0.00156 0.00115

-0.0099 -0.0008525 -0.0186 0.9930 1.0098 0.9917 0.0011 0.000766 0.000528 0.000956 0.00194 0.00123

-0.00101 -0.0094 -0.0186 0.9938 1.0102 0.9920 0.00118 0.000806 0.00055 0.0015438 0.00167 0.00145

Figure ‎4.19 show that the estimated parameters define a sphere according to the model equation (‎C.1) that fits the measurement [69]. The calibration results in Table ‎4.3 shows that the gains, bias and the orthogonality of sensors give very good stability. In some applications, the orthogonality errors cannot be taken into account because of the low impact on the measurements [69]. 117


4.12 Kalman Filter The Kalman filter is a set of mathematical equations that provides an efficient computational (recursive) solution of the least-squares method. The filter is very powerful in several aspects: it supports estimations of past, present, and even future states, and it can do so even when the precise nature of the modeled system is unknown [70] . Since modeling of a helicopter is required, it can be said that the state consists of the helicopter position Pk and velocity Vk. The input uk is the commanded acceleration and the output yk is the measured position. The acceleration can change at an instant and the position can be measured every Δt second. According to the elementary laws of physics, velocity Vk will be governed by the following equation Vk 1  Vk  t.uk ,

(‎4.2)

Similarly, the position pk will be obtained by the equation. Pk 1  Pk  t.Vk 

1 2 t . uk , 2

(‎4.3)

Now, the state vector xk that consists of position and velocity can be defined as: P  xk   k  , Vk 

(‎4.4)

Knowing that the measured output is equal to the position, we can write the linear system equations as follows: xk 1

1  0

 t 2  t  xk   2 uk  wk , 1   t   

(‎4.5)

and the measurement can then be written as: yk  Ixk  vk ,

(‎4.6) 118


Therefore, estimation of noise parameters is very important. The process noise corresponds to the noise related to the sensor. The process noise here includes the sensor output noise and sensor’s zero g offset value. Similarly, the measurement noise corresponds to the noise in the signal when it is measured. This includes the signal variance. Initialize ̂ as a best initial estimate of position and velocity, and P0 as the uncertainty in initial estimate. Then we execute the Kalman filter equations once per time step. From the equations above, the state transition matrix, measurement matrix and control matrix can be defined in a 3D model, as shown below: State transition matrix A 1 0  0  0 0  0

0

0

t

0

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

t 0 0 1 0

0 0  t  ,  0 0  1

(‎4.7)

State vector xk  Px  P   y  Pz  ,   V x  V y    V z 

(‎4.8)

Initialization of State vector x0

119


0  0    0   , 0  0    0 

(‎4.9)

Measurement matrix H 1 0  0  0 0  0

0

0

0

0

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0  0 , 0 0  1

(‎4.10)

Measurement Noise R  var( x)  0   0   0  0   0

 0   0 ,  0  0   var(Vz )

0

0

0

0

var( y )

0

0

0

0

var( z )

0

0

0

0

var(Vx )

0

0

0

0

var(V y )

0

0

0

0

0

0

1 t 4 8

0

1 t 5 20

0

0

1 t 4 8

0

1 t 5 20

0

0

0

0

1 t 3 6

0

1 t 4 8

0

0

1 t 3 6

0

1 t 4 8

0

0

0

(‎4.11)

or R = v2 Discrete-time Process Noise Q 1 5  20 t   0   0 2  99 E  6 * 100 . 1  t 4 8  0    0





   0   1 t 4  8  0   0    1 t 3 6  0

(‎4.12)

Initialization of covariance matrix P0 = R

120


where Δt denotes the time interval between the measurements, Px, Py, Pz denote the position and Vx, Vy, Vz denotes the velocity of the object along x, y and z-axes. Similarly the RMS noise of accelerometer, as specified in the data sheet is 99μg/√Hz, while the refresh rate here is reduced to 100 Hz.

Figure ‎4.20 Complete operation of Kalman filter with equations

Kalman Filter as a complete operation is shown in Figure ‎4.20. Its settings involve the variance matrix Q of parameter noise, and the variance matrix R of observation noise. The initial parameter for error variance Pk is set at the beginning. Since the Kalman algorithm employs the measurement update and time update stages, the parameter variance matrix Pk is adjusted to reflect the quality of the estimated parameter. The variance Q affects Pk in the time update stage, whereas in the measurement update Pk, R and H influence the Kalman Gain K which in turn affects the variance P [71]. Therefore, the Kalman filter performance depends upon setting these quantities to the appropriate values that reflect the real quality of the respective parameters. The standard Kalman filter assumes that measurement errors form a zero-mean Gaussian distribution. It is uncorrelated in time, and so 121


models their standard deviations with the measurement noise covariance matrix, R. The diagonal terms of R are the variances of each measurement, and the off-diagonal terms represent the correlation between the different components of the measurement noise. For most navigation applications, the noise on each component of the measurement vector is independent, so R is a diagonal matrix [71]. For the purpose of filter validation the data are first retrieved from the accelerometer through a serial interface. After conversion into the relevant ‘g’ units, data are processed for further use. Figure ‎4.21 below shows the data from the accelerometer when the device is under no motion and is held with the z-axis of the accelerometer pointing downwards when no filtering is performed. From Figure ‎4.21, it can be seen that z-axis is under gravity and there is no motion along the x and y-axes. The local value of g is taken as 9.8 m/s2. However, even in the absence of any applied force or acceleration along the two axes, there is still some acceleration. This can be accounted for as noise. Also, since the sensitivity is high, so any sudden changes will get caught. To use acceleration data in the dynamic Kalman filter, one must first eliminates the gravity effect from 3D accelerator data (bax, bay, baz). By subtracting gravity vector [0 0 1] after rotating it to body frame from acceleration data vector as shown below b





a b aacc  q* e g  q ,

(‎4.13)

Where ba is acceleration vector in body frame, baacc is output acceleration vector of sensor in body frame, eg is gravity unity vector in earth frame , q is quaternion of rotation from section ‎4.10 and q* is quaternion invers. 122

Acceleration z [g]


1.1 1 0.9

Acceleration y [g]

0.8

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0.2 0.1 0 -0.1 -0.2

Acceleration x [g]

0

0.2 0.1 0 -0.1 -0.2

Samples

Figure ‎4.21 Accelerometer raw data with z-axis under gravity

In order to get the values in all three axes, the sensor is held stationary such that the x-axis measures acceleration in the forward and backward directions, y-axis measures acceleration in the left and right directions while z-axis measures gravity. The sensor is then moved along xaxis for some time. The z-axis is then realized by first tilting the sensor such that it does not measure gravity, and then motion is detected by moving the sensor in the new z-axis. The reason for measuring motion along all axes is because 3D Kalman filter model is designed for this thesis work, and so in order to estimate position along 3 axes, it is better to have motion along all the 3 axes, so the filter’s error minimization can be realized when all axes suffer acceleration. The original set of data is of 100 sec with 100 Hz, and the same data is afterwards repeated along all axes after correction, as can be seen in Figure ‎4.22; any zero acceleration after 100 seconds appears as little increment velocity, and thus distance keeps on increasing. The data is then processed through all the stages defined in the 123


implementation of calibration, orientation calculation and filtering of the data and then the position and the velocity of sensor along all axes is calculated. Figure ‎4.23 shows the calculated position along x-axes in moving situation.

Figure ‎4.22 Position & velocity of x-axis at static situation sensor

To find the parameters of Kalman filter, one needs to compare estimated position with measured position in real time. Best and cheap way is by using ultrasonic distance sensor. Figure ‎4.24 shows the sensor used in test and has accuracy 1 mm according to the datasheet. From the obtained results, it can be trusted with the results of this method and calibration coefficients to fly up to 100 seconds.

124


Figure ‎4.23 Comparison between estimated and measured position

Figure ‎4.24 USB Ultrasonic (MB1443 HRUSB-MaxSonar-EZ4)

125


Part III:

PARAMETERS ESTIMATION

The purpose of parameter estimation is to identify the constants in the equations defining the nonlinear model. If the first principles model was adequately parameterized, it can function as an accurate helicopter simulation model as well as a basis for an accurate linear model from which a linear controller can be designed. Thus, it is of great importance for the application of the model that it was parametrized with accurate parameters. The following sections list a number of experiments necessary for parameter estimation when the helicopter is in a hover like flight condition. This is not to be understood in the way that the parameters are accurate only for hover flight; they are equally accurate for the whole flight envelope. However, the fuselage related drag coefficients are not estimated since they are only important when the helicopter is moving [17] [45] [72]. All experiments are ground-based, and they are grouped in order of increasing complexity, beginning with simple, measurable entities such as masses and ending with lift-curve slopes of the rotors. A step-by-step execution procedure is presented for each experiment. 4.13 Straightforward Experiments Directly measurable with rulers, weigher, etc.. Masses : Mass of Helicopter [kg] m : Mass of Main-Rotor Blades [kg] mb All of the above masses are measured by weighing them. Distances : Radius of Main Rotor [m] R Rt

: Radius of Tail Rotor [m] 126


RCR

: Outer Radius of Stabilizer Bar [m]

RCR,P : Inner Radius of Stabilizer Bar [m] : Main-Rotor Blade Chord [m] c ym

: Distance between CG and main rotor shaft along y axis [m]

ht

: Distance between CG and tail rotor center along z axis [m]

lm

: Distance between CG and main rotor along x axis [m]

lt

: Distance between COG and tail rotor center along x axis [m]

hm

: Distance between CG and HP along z axis [m]

All of the above distances are measured using a ruler or Vernier. Note that the distances relating to the Center of Mass (CG) can only be measured after the location of the CG on the helicopter has been determined.

Bell-Hiller Gains KMR : Swashplate linkage gain [·] KCR : Control rotor linkage gain [·] Ksb

: Main rotor blade to Stabilizer bar linkage gain [·] The purpose of this experiment is to determine the Bell and

Hiller Gains, which are the gains from the swash plate to main-rotor blade pitch and the gain from the stabilizer bar flapping to the pitch of main rotor blade pitch respectively. The experiment is carried out by use of a camera and image analysis software. Airfoil deflection tests aims at determining the parameters related to the Bell-Hiller mixer. For both longitudinal and lateral directions, three sub-experiments should be conducted. Here, we use the longitudinal direction as an example, whose experiment procedure was depicted in Figure ‎4.25 Parameters that need to be determined include KMR, KCR, and Ksb. 127


1.

For KMR determination (see Figure ‎4.25.‎a), we need to:a) adjust and maintain the stabilizer bar to be level; b) issue δ to tilt the swash plate longitudinally; c) record the cyclic pitch deflection θ of the main rotor blade; and d) note that KMR is the ratio of θ to δ.

2.

For KCR (see Figure ‎4.25.‎b), we have to a) adjust the stabilizer bar to be level; b) keep the cyclic pitch of the main rotor blade unchanged in this experiment; c) inject δ to tilt the swash plate longitudinally; d) record the corresponding stabilizer bar deflection; and e) note that KCR is the ratio of the stabilizer bar deflection to δ.

3.

For Ksb (see Figure ‎4.25.c), we proceed to a) adjust the stabilizer bar to be level; b) keep the swash plate balanced in this experiment; c) manually change the stabilizer bar flapping angle cs and record the corresponding θ ; and d) note that Ksb is the ratio of θ to θs. The same experiments are applied to the lateral direction to

identify (KMR)lat and (KCR)lat. We note that we can obtain another numerical value for Ksb through a lateral deflection experiment. The two results are almost identical, which coincides with our expectation that Ksb is a BellHiller mixer setting that is strictly symmetrical to both directions. The obtained results in this step are listed in Table ‎4.4

128


4.14 Medium Complexity Experiments The parameters estimated by these experiments are directly measurable. However, the experiment setup is somewhat more complex than the straightforward experiments. CG: Center of Gravity This experiment is carried out to establish an estimate of the position of the CG on the helicopter. This is done to be able to measure the distances in the previous section. Helicopter in this project has no change in CG because it does not have fuel tank or masses move. Equipment: String, Camera and image analysis software. See Figure ‎4.26 Step Description Suspend helicopter from a string in single points: 1 a: In helicopter fuselage (front) b: In main rotor hub Take photo from at least three angles: 2 a: From fuselage side and from rotor top b: From fuselage side Extend string line on photo in graphics software 3 CG is located at intersection of extended lines 4 Mark CG on helicopter, so distances can be measured 5

Kb: Spring constant of main-rotor blade The purpose of this experiment is to establish an estimate of the spring constant of the main-rotor blade (see Figure ‎4.27) Equipment: Weights, Camera and image analysis software Step 1 2 3 4 5

Description Attach weights to blade tip Take photo to blade deflection with reference Calculate the angle at tip of blade by image analysis software Repeat steps 1 to 3 many times Calculate the average amount of spring constant 129


a. Main rotor blade deflection due to swash plate tilting

b. Stabilizer bar deflection due to swash plate tilting

c. Main rotor blade deflection due to Stabilizer bar tilting

Figure ‎4.25 Airfoil deflection tests applied to Helicopter

130


Figure ‎4.26 Experiment to find CG

131

Chapter 4: Experimental Work 4 y = 19.546x + 0.2567 R² = 0.9865

3.5

Moment [N.m]

3 2.5 2 1.5 1 0.5 0 0

0.05

0.1

0.15

0.2

Angle of deflection [rad]

Figure ‎4.27 Deflection of main rotor blade

4.15 High Complexity Experiments The following experiments are categorized as high complexity. The helicopter needs to have its engine running during the experiments. This makes the experiments more complex. Ixx, Iyy, Izz : Moment of Inertia Measurement An efficient and a simple method, which is named the trifilar pendulum method in [73], is implemented to obtain the inertia moments of the helicopter by numerical method. The experiment procedure is shown in Figure ‎4.28. The helicopter is suspended by three flexible lines of equal lengths l. R1, R2, and R3, which are the distances between the attached points and CG. We then swing the helicopter about the body-frame axes, which is parallel to suspended lines (z-axis in this example), and record the period time tI of torsional oscillation. The following equation is used to find the moment of inertia along any axis 132

Chapter 4: Experimental Work I xx, yy, zz 

mgR1R 2 R3t I 4 2 L

.

R1sin 1  R 2 sin  2  R3 sin  3 R 2 R3 sin 1  R1R3 sin  2  R1R 2 sin  3

,

(‎4.14)

Equipment: Platform of test as shown in Figure ‎4.28 and a stopwatch

Figure ‎4.28 Illustration of trifler pendulum method

AQ,MR, BQ,MR : Drag-coefficients on Main-Rotor The purpose of this experiment is to determine the drag coefficient from the blades of the helicopter. Main rotor drag coefficients are found using a ground platform that can rotate freely, and to calculate moment by electric scale and arm moment length as shown in Figure ‎4.29. Equipment: ground platform (freely turntable), electrical scale, electric board and software to calculate main rotor blade pitch. Step 1 2 3 4 5

Description Attach helicopter to ground platform Set and maintain main rotor blade pitch to zero. Start helicopter Log value of Newton-meter Use Equation (‎A.38) to calculate AQ,MR, BQ,MR 133


Figure ‎4.29 Illustration calculating of drag coefficients on Main-Rotor

α: Lift-Curve Slope for Main Rotor The purpose of this experiment is to determine the lift-curve slope of the main rotor. The coefficient is found by mounting the helicopter on a special platform like balance scale as shown in Figure ‎4.30 and giving a pitch on the main rotor blades, such that air was forced downwards. By

134


logging the weight reading and the pitch angle the coefficient can be calculated. Equipment: platform (like balance scale), electrical scale, electric board and software to calculate main rotor blade pitch

Figure ‎4.30 Illustration calculating of Lift-Curve Slope for Main Rotor

4.16 Parameters Table ‎4.4 contains data for Align TREX 500 helicopter Table ‎4.4 Parameters of Helicopter

Parameter

Description

R=0.47±0.0005 ym=0±0.0005 ht= 0.07±0.0005 lm=0±0.0005 lt=0.585±0.0005 hm=0.225 Ixx= 0.051246±0.00135 Iyy= 0.11223±0.001761 Izz= 0.075415±0.00161 Ω=235±0.053 ρ=1.29 A=0.693978±0.00147 α=5.5±0.14 B=2

Main Rotor Radius [m] Distance between CG and main rotor shaft along y axis [m] Distance between CG and tail rotor center along z axis [m] Distance between CG and main rotor shaft along x axis [m] Distance between CG and tail rotor center along x axis [m] Distance between CG and HP along z axis [m] Inertia in x direction [kg.m2 ] Inertia in y direction [kg.m2 ] Inertia in z direction [kg.m2 ] Main rotor angular velocity [rad/s ] Density of air [kg/m3 ] Rotor disk area [m2] Two-dimensional constant lift curve slope [1/rad ] Number of blades [.]

135

Chapter 4: Experimental Work Mean blade cord length [m] Helicopter mass [kg ] Flapping inertia of a single blade about flapping hinge [kg . Ib= 0.0035884±0.000084 m2 ] Main rotor hinge offset [m] e=0.06±0.0005 Distance from the center of the rotor hub to the beginning of RCR,P=0.19±0.01 the control rotor paddle [m] Distance from the center of the rotor hub to the end of the RCR=0.22±0.01 control rotor [m] Main rotor blade weight [kg ] mb=0.057±0.005 Flapping hinge moment [N.m] Mb = 0.01254±0.0015 Swash-plate linkage gain [.] KMR=0.7±0.01 Control rotor linkage gain [.] KCR=0.683±0.01 AQ,MR= Main rotor drag coefficient (slope) [.] 0.00246±0.00017 Main rotor drag coefficient (constant) [.] BQ,MR=0.53955±0.0289 Main rotor blade to Stabilizer bar linkage gain [·] Ksb=1.4±0.01 Spring constant of main-rotor blade [N.m/rad] Kb=19.45±0.015 c= 0.0418±0.0005 m=3.241±0.005

Uncertainty measuring: sample calculation of the moment of inertia: If the calculated parameter , , then

is a function of the measured value

is said to be a function of

, and it is often

written as ( ). When this is the case, the uncertainty associated with

is

obtained by √( q  I xx , yy , zz 

)

(

)

mgR1 R 2 R3t I R1 sin 1  R 2 sin  2  R 3 sin  3 . 2 R 2 R 3 sin 1  R1 R 3 sin  2  R1 R 2 sin  3 4 L

Error of timing=0.025s Error of length measuring = 0.0005m Error of angle measuring = 0.00174rad Error of mass measuring = 0.005kg Moment of inertia Uncertainty

.

136

Chapter 5: Results and Discussion

5 CHAPTER FIVE: RESULTS AND DISCUSSION Part I:

TESTS AND EVALUATIONS

A ground test and flight test in hover were conducted in this part to evaluate the performance of the overall UAV helicopter system. Mainly the ground test is planned to evaluate the performance of some basic physics and flight tests to check the performance of the general system. 5.1 Ground Test. The main aims of the ground test are to check the stability and the performance of the software system and the wireless communication range and evaluate the endurance limits of the power system like (sufficient current of the battery, maximum flying time, and temperature of electric equipment at maximum loading). In order to evaluate the endurance limits of the power system, the full onboard avionics were powered on as well as the servos and main brushless motor with hover situation (about 60%-70% throttle stick). The PC/104 controller continuously requested data from all sensors, processed and logged the collected important data, in addition, to sending it to a ground station, which was similar to the process of real flight. LiPo battery has been charged fully before the test. The communication between the ground station and the onboard system was stable. The maximum flight time is about 5 minutes. Ground tests were conducted by putting the UAV helicopter on 3D gimbal as shown in Figure ‎5.1. All rotational axes of gimbal pass through the gravity center of UAV helicopter. The objective of this procedure is to make experiences safer and give more freedom of movement (roll, pitch, and yaw) to the UAV helicopter to note reaction 137


during running the main rotor and whole system. In this way, we can see if the controller operating normally or needs some adjustment or invert motion of some servos.

Figure ‎5.1 The installation of the UAV helicopter on 3D gimbal

5.2 Flight Test. After the ground test, actual flight test was conducted under Autopilot (as shown in Figure ‎5.2) to verify the mathematical model, controller, data processing, the avionics and integrated navigation algorithms in hover maneuver. All flight data was logged into SD card after processed and the control signals. As a result, the system worked in a satisfactory way and it is suitable for real flight. Figures ‎5.3 to ‎5.6 show the results of the flying test. The test lasts for 110 s and the data from 30 to 90 s are selected for data illustration. Figure ‎5.3 shows a comparison between experimental angular rates data of real flying and simulation. The angular rates are shown in 138


Figure ‎5.3 as solid lines yhat represent the data filtered by Kalman filter ⁄ ).

(

Be clear that

we get a good matching between the experimental results and the simulation results. The few errors are remaining because the filter can’t remove all noise in addition to the presence of some bias which could not be removed. Figure ‎5.4 shows the comparison between experimental Euler angles data of real flying and simulation. The Euler angles are shown in Figure ‎5.4 as solid lines represent the data filtered by complementary filter (

).

. Be clear

that we get a good matching between the experimental results and the simulation results. The few errors remaining because the complementary filter depends on magnetometer and accelerometer to correct results may be the sensors are not in perfect alignment with the fuselage in addition to the presence of some bias which could not be removed. There are other possibilities of errors like the location of gravity center not compatible with the mathematical model or some trim points were not specified exactly adequately as well as the additional weight of safe landing not taken into consideration when theoretical calculations and parameters estimation. But in general, the results are considered acceptable and good compatibility between practical and theoretical results. Figure ‎5.5 shows the comparison between experimental linear velocities data of actual flying and simulation. The linear velocities are shown in Figure ‎5.5 as solid lines represent the data filtered by Kalman filter (

⁄ ). We got a

good matching between the experimental results and the simulation results. The few errors remaining are because, the Kalman filter depends on measuring accuracy of Process Noise and Measurement Noise as well as 139


growing small errors of the accelerometer to become big errors because of the integration process. But these measurement results are sufficient to make controller work well and keep the UAV helicopter in Hover status. Figure ‎5.6 shows a comparison between experimental input signals data of actual flying and simulation. The inputs signals are shown in Figure ‎5.6 as solid lines represent the data from controller. We got a good matching between the experimental and the simulation results. The small difference between experimental results and the simulation (because of many reasons) are the linearization state at trim point was not perfect, the weight of safety landing gear (change center of gravity), wind and error in measuring of sensors.

Figure ‎5.2 UAV helicopter in hover maneuver when autopilot is running

140


Figure ‎5.3 Comparison of experimental Euler angles data at actual flying (solid line) and simulation data (dashed line)

Figure ‎5.4 Comparison of experimental Euler angles data at actual flying (solid line) and simulation data (dashed line)

141


Figure ‎5.5 Comparison of experimental linear velocites data at actual flying (solid line) and simulation data (dashed line)

Figure ‎5.6 Comparison of experimental inputs data at actual flying (solid line) and simulation data (dashed line)

142


Part II:

CRITICAL DYNAMICS OF CONTROLLER

After making sure that non-linear mathematical model represents helicopter dynamics, since the helicopter is a system that is inherently unstable. In this section we will try to find critical dynamic stability of helicopter that the controller will not be able to keep control on the system. The purpose of the tests is to determine the limitations of the controller when used on the non-linear model. Initially, different disturbance scenarios are considered with the purpose of selecting the appropriate scenarios for the tests. Following this, a sequence of tests using the chosen scenarios is conducted on the closed loop system. At the end of this chapter, the results of these tests are discussed in a broader perspective. This discussion considers the validity of the test scenarios with respect to non-linear model characteristics, as well as their validity with respect to a qualitative evaluation of controller performance. Offset initial state-values are chosen as the test scenario. These equates to the situation, are when the controller is initialized and the helicopter is not hovering. That is, the initial state-values are incrementally increased, while the helicopter performance is examined. With an 11dimensional state vector, a complete test of all combinations would be too extensive. Therefore, it has been chosen to consider only the states describing the translatory velocities, and the attitude and selected combinations of these. A test tree describing the considered test cases is shown in Figure ‎5.7. As the figure shows, the tests are performed in three levels ordered as follows:1. Alteration of only one initial state-value at a time, until the controller is unable to stabilize the helicopter 143


2. Alteration of three initial state-values at one time, until the controller is unable to stabilize the helicopter 3. Alteration of all six initial state-values at one time, until the controller is unable to stabilize the helicopter

Figure ‎5.7 Illustration of the tests which will be conducted.

Common for all the tests is that only positive offset values are applied. Furthermore a test is performed, with the states initialized well within what the controller can handle, to examine the performance under normal circumstances. Normally, actuator-saturation is unwanted, but as the initial states used in these simulations are on the verge of what the controller can handle, it is expected. 5.3 Effect of the critical Translatory Velocities individually on the dynamic response Input saturation stage, prooves that the system began to enter the stage of instability. In Figures ‎5.8, ‎5.9 and 5.10 a simulation of the helicopter with controller can be seen. The initial states of the helicopter 144


are shown in Table ‎5.1. From the plot of the velocities in Figure ‎5.8 it can be seen that the initial velocity of the helicopter in x-axis is 50m/s in body axis directions.

Critical

Stable

case 1. 2. 3. 4. 5. 6.

Table ‎5.1 The initial states of the helicopter in the simulation State Initial Value u, v, w, ϕ, θ and ψ 50 m/s, 0 m/s, 0 m/s, 0 rad, 0 rad and 0 rad u, v, w, ϕ, θ and ψ 0 m/s, 5 m/s, 0 m/s, 0 rad, 0 rad and 0 rad u, v, w, ϕ, θ and ψ 0 m/s, 0 m/s, 10 m/s, 0 rad, 0 rad and 0 rad u, v, w, ϕ, θ and ψ 62 m/s, 0 m/s, 0 m/s, 0 rad, 0 rad and 0 rad u, v, w, ϕ, θ and ψ 0 m/s, 11 m/s, 0 m/s, 0 rad, 0 rad and 0 rad u, v, w, ϕ, θ and ψ 0 m/s, 0 m/s, 200 m/s, 0 rad, 0 rad and 0 rad

Figure ‎5.8 (a) shows even though the longitudinal velocity u is only slowly converging towards zero, the actuator input ulong does not saturate, as is the case with the lateral actuator input ulat. As shown in Figure ‎5.9 (a) the lateral velocity v is brought towards zero slower than the vertical velocity w, but noticeably the lateral velocity v is brought to zero faster than the longitudinal velocity u because the controller is allowed to use larger actuation input on ulat than ulong. The controller quickly brings the vertical velocity w to zero. The vertical velocity is mainly controlled by the collective input ucol, which is saturated by the controller the first two seconds, as seen in Figure ‎5.10(c). The controller is able to bring the vertical velocity towards zero very fast, due to the fact that the main rotor is capable of producing a relatively large force. In Figure ‎5.8 (b), Figure ‎5.9 (b) and Figure ‎5.10 (b) the attitude states of the helicopter are shown. The roll and pitch angles ϕ and ψ are zero at time t=0 and increase in magnitude as within the next second. This is a logic consequence of the fact that the translatory velocities cannot be brought to zero without rolling and pitching the helicopter.

145


Figure ‎5.8 Dynamic response test when applying initial state value for u is 50 m/s

Figure ‎5.9 Dynamic response test when applying initial state value for v is 5 m/s

146


Figure ‎5.10 Dynamic response test when applying initial state value for w is 10 m/s

The yaw angle is also zero at time t=0, but increases in magnitude, the reason being that the controller can affect the roll rate and the lateral acceleration directly through the yaw input uped. All the above cases are dominated by controller, where it keeps the system in a stable condition. In this section, we will examine the system in critical cases situation, when the controller cannot control the system and bring it to operating point. Figures ‎5.11(a,b) and ‎5.12(a,b) show inability of controller to control dynamic state of system, because the inputs are saturated as shown in Figures ‎5.11(c,d) and ‎5.12(c,d). That means there is no enough power to bring the system to operating point. Figure ‎5.13 (a) shows the vertical velocity reached to a very large value, which may be fictional, however, the controller is able to control helicopter very well in a short time. This goes back to relative velocity through main

147


Figure ‎5.11 Dynamic response test when applying initial state value for u is 62 m/s

Figure ‎5.12 Dynamic response test when applying initial state value for v is 11 m/s

148


Figure ‎5.13 Dynamic response test when applying initial state value for w is 200 m/s

rotor disk is large. This leads to generate high thrust force which greatly helps to control helicopter. Figure ‎5.13 (c,d) shows small time saturation inputs. 5.4 Effect of the critical Euler Angles individually on the dynamic response In Figures ‎5.14, 5.15 and 5.16 show a closed loop simulation with the initial state of the helicopter shown in Table ‎5.2. The helicopter is rotated

rad (

) about each of the three body axis every time. The

yaw angle is driven towards zero with no overshoot as shown in Figure ‎5.16 (b). This is expected, since the yaw angle can be controlled directly by the tail-rotor input uped. In Figure ‎5.14 (b) the roll angle ϕ is driven towards zero quickly compared to the pitch angle θ as shown in Figure ‎5.15 (b). This is also caused by the controller being limited in using 149


the actuator input ulong. From Figure ‎5.14 (c) it can be seen that the lateral input ulat is saturated the first second of the simulation, whereas the longitudinal input ulong does not saturate at any time during the simulation. This shows that the input, ulat, is used more aggressively than the input, ulong. This is expected because of the weighing matrix

which punishes

the use of ulat less than the use of ulong. Figures ‎5.17 and ‎5.18 show a closed loop simulation with the initial state of the helicopter (see Table ‎5.2). These figures show that the controller is unable to control the state of helicopter and keep it in operating point. The reason is due to the saturation of input as shown in Figures ‎5.17 (c) and ‎5.18 (c), leading to a lack of force to maintain stability of system. Also, it can be noted that the critical roll angle is greater than pitch angle, because the weighing matrix

which punishes the use of ulat

is less than the use of ulong and the moment of inertia Ixx less than Iyy that means the helicopter needs more moment to rotate the helicopter in y direction. When the initial yaw angle reaches the maximum magnitude which is equal to

the controller will be able to control helicopter and

keep it within an operating point as shown in Figure ‎5.19. That means; there is no critical dynamic in yaw angle, because it is controlled directly by the tail-rotor input uped.

Critical

Stable

case 1. 2. 3. 4. 5. 6.

Table ‎5.2 The initial states of the helicopter in the simulation State Initial Value u, v, w, ϕ, θ and ψ 0 m/s, 0 m/s, 0 m/s, rad, 0 rad and 0 rad u, v, w, ϕ, θ and ψ 0 m/s, 0 m/s, 0 m/s, 0 rad, rad and 0 rad u, v, w, ϕ, θ and ψ 0 m/s, 0 m/s, 0 m/s, 0 rad, 0 rad and rad u, v, w, ϕ, θ and ψ 0 m/s, 0 m/s, 0 m/s, rad, 0 rad and 0 rad u, v, w, ϕ, θ and ψ 0 m/s, 0 m/s, 0 m/s, 0 rad, rad and 0 rad u, v, w, ϕ, θ and ψ 0 m/s, 0 m/s, 0 m/s, 0 rad, 0 rad and rad

150


Figure ‎5.14 Controller test when applying initial state value for

rad.


rad.

151




rad.

rad.

152




rad.

rad.

153


5.5

Effect of the critical Translatory Velocities together on the dynamic response There are countless cases with regard to the subject of this

section. To illustrate the concept of this section we will take one case only for a stable state and another unstable state. In Figure ‎5.20 a closed loop simulation with the initial state of the helicopter is shown in Table ‎5.3. From this figure we see that the controller is able to bring the system to operating point and keep it in steady state, although there are several initial inputs of velocities u, v and w. Remarkably here that the velocity v in y-axis increased (see Table ‎5.1) without that lead to instability. Because of the effect of initial forward velocity the controller redirects the force of thrust towards the back to generate moment about y-axis that will generate rotational velocity. Figure ‎5.21 shows a closed loop simulation with the initial state of the helicopter (see Table ‎5.3). This figure illustrates that the controller is unable to control state of helicopter and keep it in operating point. The reason is due to the saturation of input as shown in Figure ‎5.21 (c) which is unable to generate enough force and moment to bring the system to operating point.

Table ‎5.3 The initial states of the helicopter in the simulation case Stable Critical

1. 2.

State

Initial Value

u, v, w, ϕ, θ and ψ u, v, w, ϕ, θ and ψ

30 m/s, 15 m/s, 3 m/s 0 rad, 0 rad and 0 rad 55 m/s, 15 m/s, 3 m/s, rad, 0 rad and 0 rad

154


Figure ‎5.20 Controller test when applying initial state values u, v and w are 30m/s, 15m/s and 3m/s respectively .

Figure ‎5.21 Controller test when applying initial state value u, v and w are 55 m/s, 15 m/s, 3 m/s respectively .

155


5.6 Effect of the critical Euler Angles together on the dynamic response In Figure ‎5.22(b) the initial attitude of the helicopter is shown. The helicopter is rotated

rad (

) about each of the three body axis.

The yaw angle ψ is driven towards zero with no overshoot. This is expected, since the yaw angle can be controlled directly by the tail-rotor input uped. The roll angle ϕ is driven towards zero quickly compared to the yaw angle ψ. Figure ‎5.23 shows the controller being unable to bring the system to operating point when initial of all Euler angles are equal to (

rad

). The controller is unable to process all cases, because limited of

inputs as shown in Figure ‎5.23 (c, d). Table ‎5.4 The initial states of the helicopter in the simulation case

State

Initial Value 0 m/s,0 m/s,0 m/s, rad 0 m/s,0 m/s,0 m/s, rad

Stable

1.

u, v, w, ϕ, θ and ψ

Critical

2.


5.7

rad,

rad and

rad,

rad and

Effect of the critical initial States of the Euler Angles and Translatory Velocities together on the dynamic response In Figure ‎5.23 a simulation is performed with the initial state of

the helicopter shown in Table ‎5.5. In Figure ‎5.23, the closed loop simulation is performed with initial values of u, v and w equal to 8 m/s and initial values of ϕ, θ and ψ equal to

rad. The simulation is similar to

the previous sections in the way that the translatory velocities v and w are driven towards zero relatively fast, whereas the longitudinal velocity u is relatively slowly converging towards zero. This is, as in the previous tests, a consequence of the limited use of ulong and the moment of inertia Ixx less 156


than Iyy. Figure ‎5.23 (c) shows the actuator use, and it is the lateral input ulat and longitudinal input ulong that are not saturated, that means it can give more initial of velocity in all directions. Increasing the longitudinal velocity u as shown in Table ‎5.5, the system reaches to instability as shown in Figure ‎5.23. The controller is unable to bring the system to the operating point because of saturation of inputs as shown Figure ‎5.23 (c, d). Table ‎5.5 The initial states of the helicopter in the simulation case

State

Initial Value 8 m/s, 8 m/s, 8 m/s, rad 55 m/s, 8 m/s, 8 m/s, rad

Stable

3.


Critical

4.


rad, rad,

rad and rad and

Figure ‎5.22 Controller test when applying initial state value for ϕ, θ and ψ of rad .

157


Figure ‎5.23 Controller test when applying initial state value for ϕ, θ and ψ of rad

5.8 Discussion of Test Results When considering robust performance, in the sense that the controller is able to stabilize the system when separate initial state-values are different from zero, it can be concluded that the controller shows satisfactory performance. It was able to stabilize the system with initial translatory velocities of up to u =50m/s, v =5m/s and w=10m/s, respectively. Considering the application of the controller on a real model helicopter, these situations correspond to switching on the controller when the helicopter has the above-mentioned velocities, one at a time. Within this context, the obtained results must be regarded as acceptable. Furthermore, when testing the system with initial values of the Euler angles one by one, the controller displayed surprisingly good performance. It was able to stabilize the system when values of

rad,

rad and 158


rad were applied. Again, when considering the application of the controller on a real model helicopter, these results are satisfactory When setting multiple initial states to values different from zero, the test for maximum robustness cannot be performed, since the combinations of initial values are endless. However, the results from tests in sections ‎5.5, ‎5.6 and ‎5.7 give an indication of how well the controller is able to stabilize the system. Test in section ‎5.7, the controller proved able to handle simultaneous initial values of

m/s and

rad. Despite the values here are somewhat lower than the results from test in previous sections for velocities, obtained when applying the initial values one at a time, they are still satisfactory. When considering speed performance, expressed as the time it takes for the controller to bring the states sufficiently near the equilibrium that is hover; it is more difficult to reach a conclusion. The results from tests in previous sections state that the helicopter reaches hover within varying times. These results cannot be properly evaluated because no requirements were set up in this project to the maximum allowable value of this time, because the main subject is the study of critical situations of the helicopter. However, the values seem reasonable. As soon as we provide work requirements of the helicopter, by simulation we can identify all the critical dynamic situations of a helicopter with the controller, without the need for expensive practical tests that may lead to the destruction of helicopter and equipment.

159

Chapter 6: Conclusions and Recommendations for future works

6 CHAPTER SIX: CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORKS

6.1 Conclusions In this thesis, a minimum-complexity helicopter simulation math model has been derived with the purpose of describing the dynamic behavior of a helicopter. The purpose of the project was to develope a helicopter model, and to design an optimal controller able to stabilize the unstable helicopter model in a hover maneuver. From the results have been obtained from this work, the following remarks can be listed: • The non-linear model was carried out. As the modeling was complex, assumptions were made to simplify it. • Three ways used Taylor, numerical and the black-box to linearize the non-linear model enough to meet the needs of the controller design. • LQR suitable way to maintain system in trim point (equilibrium point). • Tests were then conducted to examine the performance of the controller. Different test scenarios were considered, with the purpose of selecting an appropriate scenario. The selected scenarios introduced initial state values different from zero as disturbances to the system. • Orientation filter, applicable to Inertial Measurement Units and Magnetic, Angular Rate, and Gravity sensor arrays that significantly ameliorate the computational load and parameter tuning burdens are associated with conventional Kalman-based approaches.

160

Chapter 6: Conclusions and Recommendations for future works

• The numerical calibration of 3D accelerometers sensors is not an expensive solution and can be implemented with no mechanical means. 6.2 Recommendations for future works During this thesis a number of different issues that requires further consideration have been brought to attention and these are summarized here as suggestions for future work. The different suggestions range from specific improvements for the developed algorithms to general suggestions for research topics that could be investigated. 1

Study different mathematical control ways like H-infinity or PID, to choose the optimal method, which fits the case of the UAV helicopter and cover more number of critical situations and give best performance.

2

Using system identification to find parameters of state space of system.

3

Study the effect of using extended Kalman filter to find full state on control performance.

4

An investigation to study effect of changing of center of gravity, moment of inertia of UAV helicopter and weight on each of the mathematical model and the control.

5

Study mathematical model of nonlinear control without linearizing using time marching scheme.

6

Study how to expand the region of stability.

7

Test LQR controller for forward flight.

161

References

References [1] Raptis, Ioannis A. and Valavanis, Kimon P., Linear and Nonlinear Control of Small-Scale Unmanned Helicopters, 1st ed., Professor S.G. Tzafestas, Ed. New York: Springer, 2011. [2] Carnduff, Stephen, System Identification of Unmanned Aerial Vehicles, 1st ed.: Cranfield University Ph.D. Thesis, 2009. [3] Tischler, M.B. and Remple, R.K., Aircraft and Rotorcraft System Identification, 2nd ed. NY: Amer Inst of Aeronautics, 2006. [4] M. Krstic, I; Kanellakopoulos and Kokotovic, P.V., Nonlinear and Adaptive Control Design, 1st ed. New York: Wiley–Interscience, 1995. [5] Gareth D. Padfield, HELICOPTER FLIGHT DYNAMICS: The Theory and Application of Flying Qualities and Simulation Modelling, Second Edition ed.: Blackwell, 2007. [6] Raymond, W. Prouty, Helicopter Performance, Stability and Control, 1st ed.: PWS Publishers, 1989. [7] A. R. S. Bramwell, Helicopter Dynamics, 1st ed.: Edward Arnold Ltd, 1976. [8] Wayne Johnson, Helicopter Theory, 1st ed.: Dover Publications Inc, 1980. [9] Gareth D. Padfield, Helicopter Flight Dynamics: The Theory and Application of Flying Qualities and Simulation Modeling, 1st ed.: AIAA, 1996. [10] Chen, Robert T. N., Effects of Primary Rotors Parameters on Flapping Dynamics, 1st ed. Ames Research Center; CA, United States: NASA-TP1431, A-7777, 1980. [11] Chen, Robert T. N., "A Simplified Rotor System Mathematical Model for Piloted Flight Dynamics Simulation," Ames Research Center, California, 1979.

163

References [12] Talbot, P. D.; Tinling, B. E.; Decker, W. A. and Chen, R. T. N., A Mathematical Model of a Single Main Rotor Helicopter for Piloted Simulation, First, Ed. Ames Research Center, California: NASA Technical Memorandum 84281, 1982. [13] Robert K. Heffley and Marc A. Mnich, Minimum-Complexity Helicopter Simulation Math Model, 1st ed.: Nasa, 1988. [14] Pitt, Dale M. and Peters, David A., "Theoretical Prediction of DynamicInflow Derivatives," Vertica, vol. 5, pp. 21-34, 1981. [15] Chen, Robert T. N., "A Survey of Non-Uniform Inflow Models For Rotorcraft Flight Dynamics and Control Applications," Vertica, vol. 14, no. 2, pp. 147-184, 1990. [16] Kim, Frederick D.; Celi, Roberto and Tischler, Mark B., "Forward Flight Trim and Frequency Response Validation of a Helicopter Simulation Model," Journal of Aircraft, vol. 30, no. 6, pp. 854-863, 1993. [17] Civita, Marco La, Integrated Modeling and Robust Control for FullEnvelope Flight of Robotic Helicopters, 1st ed. PhD Thesis: Carnegie Mellon University, 2002. [18] Gavrilets, V.; Mettler, B. and Feron, E., "Nonlinear Model for a SmallSize Acrobatic Helicopter," in AIAA Guidance, Navigation, and Control Conference and Exhibit, 2001. [19] Mettler, Bernard; Tischler, Mark B. and Kanade, Takeo, "System Identification Modeling of a Small-Scale Unmanned Rotorcraft for Flight Control Design," Journal of the American Helicopter Society, vol. 47, pp. 50-63, 2002. [20] Al-Sharman, M.K.S.; Abdel-Hafez, M.F. and Al-Omari, M.A.R.I., "Attitude and Flapping Angles Estimation for a Small-Scale Flybarless Helicopter Using a Kalman Filter," Sensors Journal, IEEE, vol. 15, no. 4, pp. 2114 - 2122, April 2015. [21] Stephen M. Rock, Eric W. Frew, Hank Jones, and Edward A. and

164

References Woodley, Bruce R. LeMaster, "Combined CDGPS and Vision-Based Control of a Small Autonomous Helicopter," in Proceedings of the American Control Conference, 1998, pp. 694-698. [22] Baerveldt, Albert-Jan and Klang, Robert, "A Low-cost and Low-weight Attitude Estimation System for an Autonomous Helicopter," in IEEE International Conference on Inteligent Engineering Systmes, 1997, pp. 391-395. [23] Jun, Myungsoo; Roumeliotis, Stergios I. and Sukhatme, Gaurav S., "State Estimation of an Autonomous Helicopter Using Kalman Filtering," in International Conference on Intelligent Robots and Systems, 1999. [24] Gavrilets, V.; Shterenberg, A. and Dahleh, M. A. and Feron, E., "Avionics System For A Small Unmanned Helicopter Performing Agressive Maneuvers," in IEEE Aerospace and Electronic Systems Magazine, 2001. [25] Saripoalli, Srikanth; Roberts, Jonathan M.; Corke, Peter I. and Buskey, Gregg, "A Tale of Two Helicopters," in Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems, 2003, pp. 805-810. [26] Ta, Duy-Nguyen; Kobilarov, M. and Dellaert, F., "A factor graph approach to estimation and model predictive control on Unmanned Aerial Vehicles," in Unmanned Aircraft Systems (ICUAS),IEEE, International Conference on, Orlando, FL, 27-30 May 2014, pp. 181 - 188. [27] Santana, P; Lopes, R.; Amui, B.; Borges, G.; Ishihara, J. and Williams, B., "A new filter for hybrid systems and its applications to robust attitude estimation," in Decision and Control (CDC), IEEE 53rd Annual Conference, Los Angeles, CA, 15-17 Dec. 2014, pp. 759 - 764. [28] Johnson, Eric N. and Kannan, Suresh K., "Adaptive Trajectory Control for Autonomous Helicopters," AIAA Journal of Guidance, Control, and Dynamics, vol. 28, no. 3, pp. 524-538, 2005. [29] Wan, E. A. and Bogdanov, A. A., "Model predictive neural control with applications to a 6 DoF helicopter model," in American Control

165

References Conference, 2001. [30] Bogdanov, A.; E.Wan and Harvey, G., "SDRE Flight Control for X-Cell and R-Max Autonomous Helicopters," in The 43rd IEEE Conference on Decision and Control, 2004. [31] Shim, D. H.; Kim, H. J. and Sastry, S., "Nonlinear Model Predictive Tracking Control for Rotorcraft-based Unmanned Aerial Vehicles," in American Control Conference, 2002. [32] Ng, Andrew Y.; Coates, Adam; Diel, Mark; Ganapathi, Varun; Schulte, Jamie; Tse, Ben; Berger, Eric and Liang, Eric, "Inverted autonomous helicopter flight via reinforcement learning," in International Symposium on Experimental Robotics, 2004. [33] Shim, H.; Koo, T. J.; Hoffmann, F. and Sastry, S., "A Comprehensive Study of Control Design for an Autonomous Helicopter," in The 37th IEEE Conference on Decision and Control, 1998. [34] Koo, T. John and Sastry, S., "Output Tracking Control design of a Helicopter Model Based on Approximate Linearization," in The 37th IEEE Conference on Decision and Control, 1998. [35] Gavrilets, V.; Martinos, I. and Mettler, B.; Feron, E., "Control Logic for Automated Aerobatic Flight of a Miniature Helicopter Monterey, California," in AIAA Guidance, Navigation, and Control Conference and Exhibit, 2002. [36] Y. Sugawara and A. Shimada, "Altitude control using limited pole placement

on

coaxial-helicopter,"

in

Advanced

Motion

Control

(AMC),2014 IEEE 13th International Workshop on, Yokohama, 14-16 March 2014, pp. 506 - 511. [37] Frye, M.T. and Provence, R.S., "Direct Inverse Control using an Artificial Neural Network for the Autonomous Hover of a Helicopter," in Systems, Man and Cybernetics (SMC), 2014 IEEE International Conference, San Diego, CA, 5-8 Oct. 2014, pp. 4121 - 4122.

166

References [38] Nodland, D.; Zargarzadeh, H. and Jagannathan, S., "Neural network-based optimal control for trajectory tracking of a helicopter UAV," in Decision and Control and European Control Conference (CDC-ECC), 50th IEEE Conference on, Orlando, FL, 12-15 Dec. 2011, pp. 3876 - 3881. [39] Tang, Yi-Rui and Li, Yangmin, "Design of an optimal flight control system with integral augmented compensator for a nonlinear UAV helicopter," in Intelligent Control and Automation (WCICA), 10th World Congress on, Beijing, 6-8 July 2012, pp. 3927 - 3932. [40] Cai, Guowei; Chen, B.M.; Peng, Kemao; Dong, Miaobo; Lee, T.H., "Design and implementation of a nonlinear flight control law for the yaw channel of a UAV helicopter," in Decision and Control, 46th IEEE Conference, New Orleans, LA, 12-14 Dec. 2007, pp. 1963 - 1968. [41] Cai, Guowei; Chen, B.M. and Lee, T.H., "Design and implementation of robust flight control system for a small-scale UAV helicopter," in Asian Control Conference,IEEE, ASCC 2009. 7th, Hong Kong, 27-29 Aug. 2009, pp. 691 - 697. [42] Hald, Ulrik B.; Hesselbæk, Mikkel V.; Holmgaard, Jacob T.; Jensen, Christian S.; Jakobsen, Stefan L. and Siegumfeldt, Martin, "Autonomous Helicopter-Modelling

and

Control,"

AALBORG

UNIVERSITY-

Department of Control Engineering, 2005. [43] Munzinger, C., Development of a Real-time Flight Simulator For an Experimental Model Helicopter, 1st ed.: Georgia Institute of Technology School of Aerospace Engineering, 1998. [44] Mario Innocenti , "Helicopter Flight Dynamics: The Theory and Application of Flying Qualities and Simulation Modeling," Journal of Guidance, Control, and Dynamics, vol. 22, no. 2, pp. 383-384., 1999. [45] Mettler, B., Identification Modelling and Characteristics of Miniature Rotorcraft, 1st ed.: Kluwer, 2003. [46] Heffley, R. K. and Mnich, M. A, Minimum-Complexity Helicopter

167

References Simulation Math Model, 1st ed. Army Research and Technology Labs.; Manudyne Systems, Inc.; Moffett Field, CA, United States: NAS2-11665; RTOP 992-21-01, Apr 01, 1988. [47] A. M. YOUNG, "Aircraft," Patent US2,384,516, sep. 11, 1945. [48] Cunha, Rita and Silvestre, Carlos, "Dynamic Modeling And Stability Analysis Of Model-Scale Helicopters With Bell-Hiller Stabilizing Bar," in AIAA Guidance Navigation and Control Conference, Texas, USA, 2003. [49] Mustafic, E.; Fogh, M. and Pettersen, R., Non-Linear Control of a Rotary Unmanned Aerial Vehicle, 1st ed. Dep. of control engineering: Aalborg University, 2005. [50] Banks, J.; Carson, J.; Nelson, B. and Nicol, D., Discrete-Event System Simulation, 1st ed.: Prentice Hall, 2001. [51] Sokolowski, J.A. and Banks, C.M., Principles of Modeling and Simulation, 1st ed.: NJ: Wiley, 2009. [52] Waltman, Gene L. and Nasa History Division, Black Magic and Gremlins: Analog

Flight

Simulations,

1st

ed.:

WWW.Militarybookshop.CompanyUK, 2011. [53] A. Mnich Marc and K. Heffley Robert, "Minimum-Complexity Helicopter Simulation Math Model," NASA, 1988. [54] R.W. Prouty, Helicopter Performance, Stability, and Control, 1st ed.: PWS Publishers, 1986. [55] Norman S. Nise, CONTROL SYSTEMS ENGINEERING, Sixth Edition ed. California State Polytechnic University, Pomona: John Wiley & Sons , 2011. [56] Hendricks, Elbert; Jannerup, Ole; Sørensen and Paul Haase, Linear Systems Control, Deterministic and Stochastic Methods, 1st ed.: SpringerVerlag Berlin Heidelberg, 2008. [57] Cai, Guowei; Chen, Ben M. and Lee, Tong Heng, Unmanned Rotorcraft

168

References Systems, 1st ed. London: Springer, 2011. [58] (

2010,

Aug)

PC/104

embedded

consortium.

[Online].

http://www.pc104.org [59] Yoo, Tae Suk; Hong, Sung Kyung; Yoon, Hyok Min and Park, Sungsu, "Gain-Scheduled Complementary Filter Design for a MEMS Based Attitude and Heading Reference System," Sensors, vol. 11, pp. 38163830, 2011. [60] Alam, Fakhri; ZhaiHe, Zhou and JiaJia, Hu, "A Comparative Analysis of Orientation Estimation Filters using MEMS based IMU," pp. 86-91, March 21-22, 2014. [61] Marins, J. L.; Yun, Xiaoping; Bachmann, E. R.; B., R. and Zyda, M. J., "An extended kalman filter for quaternion-based orientation estimation using marg sensors," Proc. IEEE/RSJ International Conference on Intelligent Robots and Systems, vol. 4, pp. 2003-2011, 29 October - 3 November, 2001. [62] Xsens Technologies B.V. MTi and MTx, "User Manual and Technical Documentation," Pantheon 6a, 7521 PR Enschede, The Netherlands, May 2009. [63] Micro Strain Inc., "3DM-GX3 -25 Miniature Attitude Heading Reference Sensor," 459 Hurricane Lane, Suite 102, Williston, VT 05495 USA, 1.04 edition, 2009,. [64] VectorNav Technologies, "LLC. VN -100 User Manual. College Station," TX 77840 USA, preliminary edition, 2009. ,. [65] Alonso, Roberto and Shuster, Malcolm D., "Complete Linear AttitudeIndependent Magnetometer," Jornal of the Astronautical Sciences, vol. 50, no. 4, pp. 477-490, December 2002. [66] Pylvänäinen, Timo, "Automatic and adaptative calibration of 3D field sensors," Applied Mathematical Modelling ,scienceDirect, vol. 32, no. 4, pp. 575–587, April 2008.

169

References [67] Press, William H.; Teukolsky, Saul A.; Vetterling, William T. and Flannery, Brian P., Numerical Recipes in C The Art of Scientific Computing, 1st ed. Cambridge: Cambridge University Press, 1992. [68] Camps, F.; Harasse, S. and Monin, A., "Numerical calibration for 3-axis accelerometers and magnetometers," Electro/Information Technology, 2009. eit '09. IEEE International Conference on, Windsor, pp. 217 - 221, June 2009. [69] Batista, P.; Silvestre, C.; Oliveira, P. and Cardeira, B., "Accelerometer Calibration and Dynamic Bias and Gravity Estimation: Analysis, Design, and Experimental Evaluation," Control Systems Technology, vol. 19, no. 5, pp. 1128 - 1137, Sept 2011. [70] R. E. Kalman, "A New Approach to Linear Filtering and Prediction," Transaction of the ASME—Journal of Basic Engineering, pp. 35-45, 1960. [71] Paul D. Groves, Principles of GNSS, Inertial, and Multisensor Integrated Navigation System, 1st ed.: Artech House, 2008. [72] Tischler, M. B. and Cauffman, M. G, "Frequency-response method for rotorcraft system identification: Flight application to bo-105 coupled rotor/fuselage dynamics," Journal of the American Helicopter Society, vol. 37, no. 3, pp. 3-17, 1992. [73] Piersol, Allan G. and Paez, Thomas L., Harris’ Shock And Vibration Handbook, Sixth Edition ed.: McGraw-Hill, 2010. [74] T.

Bak.

(2002)

MechanicsI.

[Online].

http://www.control.auc.dk/~jan/undervisning/MechanicsI/mechbook.pdf [75] Wie, B., Space Vehicle Dynamics and Contro, 1st ed.: AIAA, 1998. [76] Craig, J. J., Introduction to Robotics Mechanics and Control, 2nd ed.: Addison-Wesley, 1989. [77] Koo, T. J.; Ma, Y. and Sastry, S. S. (2001) Nonlinear Control of a Helicopter

Based

Unmanned

Aerial

Vehicle

Model.

[Online].

http://robotics.eecs.berkeley.edu/~sastry/pubs/SastryAllJuly07/SastryALL

170

References /2002/KooNonlinear2002.pdf [78] Prouty, Raymond W., Helicopter Performance, Stability and Control, 2nd ed.: Krieger, 1990. [79] (2015)

Angular

momentum.

[Online].

http://en.wikipedia.org/wiki/Angular_momentum [80] Madgwick, O. H.; Harrison, J.L. and Vaidyanathan, Ravi, "Estimation of IMU and MARG orientation using a gradient descent algorithm," in IEEE International Conference on Rehabilitation Robotics. Rehab Week Zurich, ETH Zurich Science City, Switzerland, June 29 - July 1, 2011, pp. 1 - 7. [81] Cooke, Joseph M.; Zyda, Michael J.; Pratt, David R.; and Mcghee, Robert B., "Npsnet: Flight simulation dynamic modelling using quaternions," Presence, vol. 1, pp. 404-420, 1992. [82] Jacobs, J. A., "The earth’s core," International geophysics series, 1987. [83] E. R. Bachmann, Xiaoping Yun, and C. W. Peterson, "An investigation of the effects of magnetic variations on inertial/magnetic orientation sensors," In Proc. IEEE International Conference on Robotics and Automation ICRA '04, vol. 2, pp. 1115-1122, April 2004. [84] Vasconcelos, J. F.; Elkaim, G.; Silvestre, C.; Oliveira, P. and Cardeira, B., "A geometric approach to strapdown magnetometer calibration in sensor frame," Navigation, Guidance and Control of Underwater Vehicles, vol. 2, 2008. [85] Kian Sek Tee, Mohammed Awad; Dehghani, Abbas; Moser, David and Zahedi,

Saeed,

"Triaxial

Accelerometer

Static

Calibration,"

in

Proceedings of the World Congress on Engineering 2011 Vol III, vol. 1, London, U.K., January July 6 - 8, 2011, p. 1309. [86] Mark, K. Transtrum; James and P. Sethna, "Improvements to the Levenberg-Marquardt

algorithm

for

nonlinear

least-squares

minimization," Journal of Computational Physics, January 30 2012. [87] Jr., George B. Thomas; Weir, Maurice D. and Hass, Joel R., Thomas'

171

References Calculus, Multivariable, 12th ed.: Pearson, 2009. [88] Marquardt, D.W., "An algorithm for least-squares estimation of nonlinear parameters," Journal of the Society for Industrial and Applied Mathematics, vol. 11, no. 2, pp. 431-441, 1963. [89] YEI Technology. (2011) yei 3-space embedded sensor. [Online]. http://www.yeitechnology.com/sites/default/files/3Space_Sensor_Users_Manual_Embedded_1.1_r20_18Oct2012_0.pdf [90] J. E. Bortz, "A new mathematical formulation for strapdown inertial navigation," Aerospace and Electronic Systems, IEEE Transactions on, vol. AES-7, no. 1, pp. 61-66, 1971. [91] Wang, M.; Yang, Y.C.; Hatch, R.R. and Zhang, Y.H., "Adaptive filter for a miniature MEMS based attitude and heading reference system," in Proceedings of IEEE Position Location and Navigation Symposium, Monterey, CA, USA, April 26–29, 2004, pp. 193–200. [92] Ojeda, L. and Borenstein, J., "FLEXnav: fuzzy logic expert rule-based position estimation for mobile robots on rugged terrain," in Robotics and Automation, Proceedings. ICRA '02. IEEE International Conference on, vol. 1, May 11-15, 2002, pp. 317-322. [93] Luinge, H. J.; Veltink, P. H. and Baten, C. T. M., "Estimation of orientation with gyroscopes and accelerometers," in In Proc. First Joint [Engineering in Medicine and Biology 21st Annual Conf. and the 1999 Annual Fall Meeting of the Biomedical Engineering Soc.] BMES/EMBS Conference, vol. 2, October 13-16, 1999, p. 844. [94] Foxlin, E., "Inertial head-tracker sensor fusion by a complementary separate-bias kalman filter," Proc. Virtual Reality Annual International Symposium the IEEE, vol. 267, pp. 185-194, March 30-April 3, 1996. [95] Yazdi, N.; Ayazi, F. and Najafi, K., "Micromachined inertial sensors," Proceedings of the IEEE, vol. 86, no. 8, pp. 1640-1659, August 1998. [96] Mahony, R.; Hamel, T. and Pimlin, J.M., "Nonlinear complementary

172

References filters on the special orthogonal group," Automatic Control, IEEE, vol. 53, no. 5, pp. 1203 -1218, June 2008. [97] Beauregard, S., "Omnidirectional pedestrian navigation for first responders," in Proc. 4th Workshop on Positioning, Navigation and Communication WPNC '07, March 22-22, 2007, pp. 33-36. [98] Sabatini, A. M., "Quaternion-based extended kalman filter for determining orientation by inertial and magnetic sensing," Biomedical Engineering, IEEE, vol. 53, no. 7, pp. 1346-1356, July 2006. [99] Heinz, E. A.; Kunze, K. S.; Gruber, M.; Bannach, D.; and Lukowicz, P., "Using wearable sensors for real-time recognition tasks in games of martial arts - an initial experiment," in Proc. IEEE Symposium on Computational Intelligence and Games, May 22-24, 2006, pp. 98-102. [100] Beer, F. P.; jr., E. R. J. and Clausen, W. E., Vector Mechanics for Engineers, Dynamics, 7th ed.: McGraw-Hill, 2004. [101] W. Prouty Raymond, Helicopter Performance, Stability and Control.: PWS Publishers, 1989.

173

APPENDIX A: Helicopter Dynamic Analysis

A APPENDIX A :Helicopter Dynamic Analysis A.1. Rigid Body Equations A.1.1. Euler Angles The forces and torques acting on the helicopter are stated in the BF. The position of the COG is stated in the EF, why the forces and torques need to be transformed into the SF (which has the same orientation as the EF). To describe the rotation of a frame about one of its axes, a rotation matrix is needed. This rotation matrix is dependent on the axes by which the rotation is performed. Matrices describing the rotations about the three axes are described by [74] as 0 1 Rx ( )  0 cos  0  sin 

cos Ry ( )   0   sin   cos Rz ( )   sin   0

0  sin    cos 

(‎A.1)

0  sin   1 0   0 cos 

(‎A.2)

sin cos 0

0 0  1

(‎A.3)

The Euler angles, ϕ, ϴ and ψ, are used to describe the angles between the SF and BF. Using these Euler angles, it is possible to derive a matrix describing the rotation of the BF relative to the SF. The derivation is done with the BF having the same attitude as the SF as a starting point. The standard rotational sequence for aircrafts is the so-called z-y-x [45] ; yaw (angle ψ about bz), pitch (angle ϴ about by) and roll (angle ϕ about bx). The successive rotation from SF to BF is hereby given as A.1


Rbs ()  Rx ( ) Ry ( ) Rz ( )

c c  Rbs ( )   s s c  c s  c s c  s s

c s s s s  c c c s c  s c

(‎A.4)

 s  s c   (‎A.5) c c 

where sin and cos are abbreviated with s and c, respectively. As the rotation matrix is orthonormal [74] T Rsb ()  Rbs () .

(‎A.6)

This matrix is capable of transforming a vector (x) from BF to SF through the relation s

x  Rsb ().b x,

(A.7)

where Θ is the vector of Euler angles. A.1.2. Euler Rates Two different notations are used to describe the angular velocity of the BF with respect to the SF, that is the angular velocity vector ω and the Euler rates ̇ (the time derivative of the Euler angles [74]). Figure ‎A.1 Illustration of the SF and the BF, where the BF has been rotated about

the vector ω0 from its initial position equivalent with the SF. shows the SF and the BF, where the BF has been rotated about the vector ω0. This vector describes the angular velocity, ω, of the BF relative to the SF, and it can be projected onto either the sx, sy and sz axes or onto the bx, b

y and bz axes. It is these Figure ‎A.1 two projections which are denoted ̇

and ω, respectively. The following definitions are used, to distinguish between the two angular velocities ̇ : Euler rates - the angular velocity of the BF (and thereby the helicopter) with respect to the SF projected onto the SF (sω0). A.2


ω: Angular velocity vector - the angular velocity of the BF (and thereby the helicopter) with respect to the SF projected onto the BF (bωo). s

ωo

z

b

z

b

x

b

x

s

y

b

y

Figure ‎A.1 Illustration of the SF and the BF, where the BF has been rotated about the vector ω0 from its initial position equivalent with the SF.

An equation describing the relationship between ̇ and ω is described by [74] as

 p   0 0      q   0  R x ( )   R x ( ) R y ( )  0   r  0   0    0  s     1    0 c s .c    0  s c .c   

  Pbs (). where [ ̇

̇

(‎A.8)

(‎A.9) (‎A.10)

̇ ] are the Euler rates, and Pbs is the transformation matrix

from SF to BF. Inversion of this rotation matrix yields the rotation matrix that transforms the angular velocities (ω) to the Euler rates ( ̇ ) Psb ()  Pbs1 ()

(‎A.11) A.3


 1 s .t  0 c  s 0 c 

 c .t   s  ’ c   c 

(‎A.12)

where sin, cos and tan are abbreviated with s, c and t respectively. Whereas the rotation matrix Rsb maps positions from the BF to the SF, Psb maps angular velocities between the two frames through the relation

  P (). .  sb

(‎A.13)

A.1.3. Angular Acceleration The torque equation of a rigid body about its COG is given by [75] as  dH  H     o  H ,  dt 

(‎A.14)

where H is the angular momentum vector and ̇

is the external torques

acting on the body about its COG [75]. This is given in the BF, why the rest of the vectors also need to be projected onto the BF. Thus, the angular momentum vector is defined as

H  I . ,

(‎A.15)

where I is the inertia matrix of the helicopter, which, for a rigid body free to move in three dimensions, is defined [76]  I xx I  0   0

0 I yy 0

0 0 ,  I zz 

(‎A.16)

under the assumption that the frame in which I is stated, represents the principle axes of the rigid body. By substituting (‎A.15) into (‎A.14), the torque equation becomes

d I .     I .  dt dI d  .  I .    I .  . dt dt

T

b

(‎A.17) (‎A.18) A.4


As the body is considered rigid, the inertia is constant ((dI/dt) = 0). This leads to

T  I .    I .  .

b

(‎A.19)

From this equation the angular acceleration is isolated





  I 1 bT    I . 

(‎A.20)

A.1.4. Translatory Acceleration To describe the translatory movements of the helicopter, the kinematic principles of moving reference frames needs to be taken into consideration. these kinematic equations describes the translatory acceleration as

1 V  .b F  b V m

b

(‎A.21)

where m is the mass of the helicopter, bF is the vector of forces acting on the helicopter stated in the BF, ω is the angular velocity vector and bV is a vector containing the translatory velocities of the helicopter relative to the EF stated in the BF. A.2. Force and Torque Equations A.2.1. Forces Generated by the Main Rotor The main-rotor thrust is oriented perpendicular to the TPP, defined by β and β in lateral and longitudinal direction, respectively, as 1s

1c

shown in Figure ‎A.2 The force component along, bx, depends on the thrust generated by the main rotor, and is the projection of TMR onto the HP in the b

x direction b

f x, MR  TMR.sin( 1c ) ,

(‎A.22)

A.5


The force component along by depends on both the main-rotor and tailrotor thrust. The contribution from the main-rotor thrust is similar to that of b

fx,MR, hence the thrust vector is projected onto the HP in the by direction,

which yields a force equal to b

f y , MR  TMR.sin( 1s ) ,

(‎A.23)

The force component along bz depends also on the main-rotor thrust only. The thrust vector is projected onto the HP in the bz direction, which yields b

f z , MR  TMR. cos(1s ).cos(1c ) ,

(‎A.24)

Now bFMR can be written as b

A.2.1.

FMR

 b f x , MR    TMR . sin(1c )  b      f y , MR   TMR . sin(1s )  . b  f z , MR   TMR . cos(1s ). cos(1c )  

(‎A.25)

Forces Generated by the Tail Rotor The tail-rotor thrust yields a force in the by direction only, thus

b

fx,TR = 0 and bfz,TR = 0. As TTR is defined positive in the by direction, bfy,TR

can be written as b

f y ,TR  TTR .

(‎A.26)

This yield  b f x ,TR   0    b FTR   b f y ,TR   TTR     b f z ,TR   0   

A.2.2.

(‎A.27)

Forces Generated by the Gravitational Acceleration

In the SF the gravitational force only has a component along sz in the positive direction sFg = [0, 0, m.g]T . The forces generated by the main and tail rotor are stated in the BF, why the gravitational force (g) is mapped into the BF, using the derived transformation matrix Rbs

A.6


TPP

-β1s

TMR

HP y

-x

a

z

TPP

TMR

-β1c

HP

x

y

b

z

a.

A front view of the helicopter depicts β1s used to denote the lateral flapping (angle between the HP and TPP).

b.

A side view of the helicopter depicts β1c used to denote the longitudinal flapping (angle between the HP and TPP).

Figure ‎A.2 The angles β1c and β1s between the HP and TPP describe the orientation of the main-rotor thrust in the BF.

 b f x, g    b Fg   b f y , g   Rbs ( ). sFg ,  b f z, g     0    sin( ).m.g  bRbs ( ).  0   sin( ).cos().m.g      m.g  cos().sin( ).m.g 

(‎A.28)

(‎A.29)

where m is the mass of the helicopter, g is the gravitational acceleration and A.2.3.

[

] are the Euler angles

Torques Generated by the Main-Rotor The torques about bx consist of the force, bfy,MR, perpendicular to the distance, hm, and the force, bfz,MR, perpendicular to the distance ym A.7

APPENDIX A: Helicopter Dynamic Analysis b

LMR bf y , MR .hm bf z ,MR . ym .

lt

(‎A.30)

lm ym

x

z

a. y

hm x

-y ht

a.

b.

b. z

Top view of the helicopter. lm is the distance from COG to the main rotor, and lt is the distance from COG to the tail rotor (both along the x axis). ym is the distance from COG to the main rotor along the y axis. Side view of the helicopter. hm is the distance from COG to the main rotor, and ht is the distance from COG to the tail rotor (both along the z axis).

Figure ‎A.3 The distances used to calculate the torques generated by the main and tail rotor.

The torques about by consists of the force, bfx,MR, perpendicular to the distance, hm, and the force, bfz,MR, perpendicular to lm b

M MR bf x, MR .hm bf z , MR .lm ,

(‎A.31)

The torques about bz consists of the force, bfx,MR, perpendicular to the distance, ym, and the force, bfy,MR, perpendicular to lm b

N MR bf x, MR . ym  bf y , MR .lm ,

(‎A.32)

The three torque equations, resulting from the forces, are combined into a complete vector

A.8


 b LMR   b f y , MR .hm  b f z , MR . y m  b   b  b M   f . h  f . l MR x , MR m z , MR m    .  b N MR   b f x , MR . y m  bf y , MR .l m     

(‎A.33)

A.2.4. Torques Generated by the Tail-Rotor The torques about bx consist of the force generated by the tailrotor thrust, bfy,TR, perpendicular to the distance ht b

LTR  bf y ,TR .ht .

(‎A.34)

The tail rotor has no contribution to the torque about by, thus b

M TR  0 .

(‎A.35)

The torques about bz consist of the force generated by the tail-rotor thrust, b

fy,TR, perpendicular to the distance lt b

NTR bf y ,TR .lt .

(‎A.36)

The three torque equations, resulting from the forces, are combined into a vector  b LMR   b f y ,TR .ht  b    M  0 MR    .  b N MR  bf y ,TR .lt     

A.2.5.

(‎A.37)

Torques Generated by Drag on the Main Rotor As the main rotor rotates, torque is generated due to aerodynamic

drag. The accurate drag is complex to model, and therefore a simple model is used. A model of the relationship between the drag and the thrust of the main rotor in hover is described by [77] as 1.5 QMR  ( AQ, MR .TMR  BQ, MR ) ,

(‎A.38)

where AQ,MR is a coefficient expressing the relationship between the mainrotor thrust and the drag, and BQ,MR is the initial drag of the main rotor when the blade pitch is zero. The coefficients can be found in Table ‎4.4.

A.9


The torque generated by the main-rotor drag is perpendicular to the TPP, as shown in Figure ‎A.3 The torque vector is projected into the HP, using the flapping angles β1c and β1s. The main rotor rotates in the clockwise direction, and the torque is defined positive in the clockwise direction. This means that the torque vector will be positive along the positive bz axis. -β1s

TPP

HP y

-x

a.

z QMR TPP

-β1c

HP

x

-y

b.

z QMR

a. Front view b.

Side view.

Figure ‎A.4 The drag on the main-rotor blades results in torque acting on the body axes of the helicopter.

This torque vector is projected into to HP, for decomposition. The component along bx is described by b

LD,MR bQMR. sin(1c ) .

(‎A.39)

Projection of the torque vector QMR along by yields b

M D, MR bQMR .sin(1s ) .

(‎A.40)

The component along bz yields

A.10

APPENDIX A: Helicopter Dynamic Analysis b

N D,MR bQMR . cos(1s ).cos(1c ) .

(‎A.41)

The complete torque matrix, due to drag on the main rotor becomes hereby b  b LD , MR    QMR .sin(1c ) b    b M   Q . sin(  ) MR 1s  D , MR   .  b N D , MR   bQMR . cos(1s ).cos(1c )    

(‎A.42)

A.3. Flapping and Thrust Equations A.3.1. A.3.1.1.

Thrust Momentum Theory Momentum theory uses conservation laws; conservation of mass,

momentum and energy, to give an estimate of the performance of a rotor. To develop a model for the thrust and induced velocity we take a look at a helicopter in hover. Thereafter this model is expanded to include factors that arise in forward flight 1. Momentum theory in hover: Figure ‎A.5 depicts the wake induced by a helicopter in hover. There are three main areas of interest in the rotor wake. First there is station 0, which is the area far above the rotor disk, where influence from the rotor is negligible. Station 1 which is on the rotor disk itself, and the area far below the rotor disk, which is labeled station 2. It is assumed that the air far above the rotor, at station 0, is stagnant. A mass flux is generated through the rotor disk by the work done by the rotor

 mass   Av1 , m  time

(‎A.43)

A.11


Figure ‎A.5 Wake induced by hover

where ρ is the density of air, A is the area if the rotor disk and v1 is the induced velocity of air through the rotor disk. Because of conservation of mass, the mass flux throughout the wake of the helicopter must be constant. The total change in velocity from station 0, to station 2 is

v  v2  v0 ,

(‎A.44)

but because of the assumption that the air above the rotor disk is stagnant, or v0 = 0, will result in

v  v2 ,

(‎A.45)

The relationship describing force is normally written as the acceleration of a body times the mass of the body. Rewritten to describe the thrust developed by a rotor continuously accelerating a mass of air

T  m .v ,

(‎A.46)

Inserting the expression for mass flux ̇ and the total change in velocity Δv yields T  . A.v1.v2

(‎A.47) A.12


 .v2 . m

(‎A.48)

In this form the thrust equation is not very useful. An expression which is only dependent of the velocity at the rotor disk v1 would simplify this expression. Energy conservation equates the work done by the rotor, with, the change in kinetic energy in the wake. Since the mass flux is constant, this can be expressed as 1  .v22 T .v1  . m 2

(‎A.49)

By inserting equation (‎A.48), into equation (‎A.49) it possible to show that the relationship between the velocity in the far wake v2 and the induced velocity at the rotor disk v1 is v2=2·v1. This leaves an expression stating the relationship between the induced velocity v1 and the thrust developed by the rotor T  2.. A.v12 ,

(‎A.50)

rearranging this gives us an expression for the induced velocity T 2.. A . 2. Momentum theory in forward flight v1 

(‎A.51)

When a helicopter gains a forward velocity, the assumption that air far above the rotor disk is stagnant is no longer valid. In Figure ‎A.5 we see that the air at station 0 is no longer stagnant but is moving at the velocity v. The air passing through the rotor disk at station 1 is now a result of the induced velocity, as in the hover case, and the velocity component v from station 0. The mass flux through the rotor disk now becomes   . A V 2  v12 . m

(‎A.52)

A.13


Figure ‎A.6 Wake induced by forward flight

The new thrust equation which takes into account the forward velocity now becomes T  . A. V 2  v12 .2.v1 ,

(‎A.53)

rearanging the equation to [78] 2

v14

V

2

.v12

 T      0 ,  2.. A 

(‎A.54)

and solving for induced velocity gives1 2

2

V 2   T  V2    v1      , 2 2 2 .  . A    

(‎A.55)

Until now the contribution of V has only been seen as the velocity experienced by the helicopter in longitudinal direction. However, the helicopter can have a velocity component in all translatory degrees of

1

This is a forth order equation with four solutions where the negative sign solutions are disregarded, due to the fact that the windmill brake state of the helicopter is not considered by this thesis. For further information on the windmill brake state the reader is refereed to [78]

A.14


freedom, u, v and w, thereby, affecting the mass inflow through the rotor disk. Furthermore, it is assumed that the translational velocity of the HP is the same as BF. Another factor of the inflow volume is the attitude of the rotor disk to the surrounding air. The helicopter disk can have an attitude described by β1c and β1s, or TPP. Figure ‎A.7 show how the velocities experienced by the helicopter which is decomposed to components parallel and perpendicular to the rotor disk, respectively. To solve the thrust/induced velocity problem for all possible velocity components a new set of variables are defined; ̂ which includes all velocity components that lies parallel to the TPP, and wr which includes all velocity components that lies perpendicular to the TPP. Equation (‎A.56) and (‎A.57) shows the calculation of ̂ and wr magnitude2. Vˆ 2 (u. cos(1c ))2  (v. cos(1s ))2  (w.sin( 1c ).sin( 1s ))2

(‎A.56)

wr  u. sin( 1c )  v. sin( 1s )  w. cos(1c ).cos(1s ) .

(‎A.57)

Using small angle approximation the expressions in (‎A.56) and (‎A.57) are reduced to Vˆ 2 u 2  v 2  (w.1c .1s ) 2 ,

(‎A.58)

wr  u.1c  v.1s  w ,

(‎A.59)

The final thrust equation derived from momentum theory becomes 2 T  . A. Vˆ 2  v1  wr  .2.v1 ,

(‎A.60)

Again, partially solving the equation for the induced velocity v1 we get

2

̂ projects the components of the velocity vector in to ℝ (HP), however, we are only interested in the magnitude of the projected vector, not the direction, therefore the Euclidean length of the resulting vector is calculated. In difference to the ̂ , wr projects the velocity vector on to ℝ (TPP z-axis), thus no additional math operations are needed. Positive direction of the wr is opposite of the v1

A.15

APPENDIX A: Helicopter Dynamic Analysis Vˆ 2  wr wr  2v1  v1    2

2  Vˆ 2  wr wr  2v1    T         2    2. . A  2

(‎A.61)

a. Finding the velocity components parallel to TPP

b. Finding the velocity components perpendicular to TPP. Figure ‎A.7 Velocity components

In the final expression both thrust and induced velocity are unknown, in order to solve this, another expression for either thrust or velocity is needed. Therefore, blade element theory is used to get a second expression making it possible to solve the thrust/induced velocity problem. A.3.1.2.

Blade Element The momentum method gives much insight into conditions at the

rotor and its wake; however it does not deal with actual development of thrust, i.e. the lift on each blade element with respect to blade pitch. To determine the lift magnitude, a helicopter wing is considered as a airplane wing. Therefore, the theory by which lift is described on an airplane can be

A.16


modified and used on a helicopter rotor. Thus the governing equation for the lift on the blade element for a helicopter is [78]

L  q.cl .c.r ,

(‎A.62)

where r is distance from the center of rotation, c is the cord of blade, ΔL is the lift experienced by an increment of blade Δr , c1 is local lift coefficient and q is the local dynamic pressure. q is a function of velocity due to rotation of the blade element and can be derived from Bernoulli’s equation 1 q  . .V 2 , 2

(‎A.63)

where V describes the velocity of air rotor blade experiences, and can be described by two mutually perpendicular velocities UP and UT , see Figure ‎A.5. V  U T2  U P2  UT ,

(‎A.64)

Figure ‎A.8 Cross section of a blade element used to identify and define the mutually perpendicular velocities UP and UT.

Tangential velocity UT, is the velocity component of the blade perpendicular to the blade element and parallel to the rotational path of the blade (TPP) see Figure ‎A.8, and is defined as A.17


UT   .r ,

(‎A.65)

The perpendicular velocity of the blade, UP, is the velocity of the blade element perpendicular to the UT U P  v1  wr ,

(‎A.66)

where v1 is the induced velocity and wr is the velocity of the main rotor disc relative to the air due to translatory velocities bu, bv and bw,. This function is derived by decomposing the translational velocities of the helicopter in body coordinate frame, bvse to a plane orthogonal to the TPP, i.e. the opposite direction of v1, Figure ‎A.7 Assuming that UP is much smaller than the UT, equation (‎A.64) can be reduced to V  UT ,

(‎A.67)

1 1 q  . .U T2  . .( .r ) 2 , 2 2

(‎A.68)

therefore:

The local lift coefficient, cl, is defined as a product of local angle of attack α in radians, and the slope of the lift curve a per radians, cl  a. ,

(‎A.69)

local angle of attack, α, is determined by the geometric pitch of the blade, θ, and the local inflow angle, ϕ as shown in Figure ‎A.8

    ,

(‎A.70)

where

  arctan

Up UT .

(‎A.71)

Substituting equation (‎A.71) in to equation (‎A.70), and using small angle approximation results in

  

v1  wr  .r ,

(‎A.72)

and thereafter substituting in to equation for local lift coefficient

A.18


v  wr  cl  a.  1  .r 

 . 

(‎A.73)

Thus the lift on an increment of rotor blade element is



v  wr   .( .r ) 2 . .  1 .c.r . 2  .r  

L 

(‎A.74)

To get total lift along blade L

 2

R

.c.a. .(  r dr  2

2

0

v1  wr



R

 rdr ).,

(‎A.75)

0

the lift per blade is L



 2.R. .  .c.a. .R 2 .  (v1  wr )  , 4 3  

(‎A.76)

total thrust generated by the main rotor blades is T  b.L 



 2.R. .  .c.a.b. .R 2 .  (v1  wr )  , 4 3  

(‎A.77)

where b is the number of the main rotor blades. However most common twist used on the helicopters is not ideal but linear twist such that r

   col  tw , R

(‎A.78)

where θtw is the difference in the pitch of the blade from the root to the tip and θcol is input pitch angle from collective. Therefore to finalize the thrust equation, a compensation for ideal blade twist assumption must be applied. According to [78], collective pitch of a blade with ideal twist can be related to the tip pitch of a blade with ideal twist through the approximate relationship 3

   col  tw , 4

(‎A.79)

Thus the new thrust equation is

A.19


3      2.R. . col   tw    4  2  T  .c.a.b. .R .  (v1  wr )  ,   4 3    

A.3.2.

(‎A.80)

Flapping

A.3.2.1.

Types of Hub Systems A rotor hub can be divided in to three types: Teetering, hingeless

and articulated. Each hub type will here be described, and the difference in their respective dynamic behavior, will later be analyzed. Before presenting each type, a short summation of what defines each rotor hub will be given.  The teetering hub type is based on a frictionless hinge placed in the middle of the shaft, thus leaving the rotor free to pivot without restraint.  A hingeless rotor hub attaches the rotor blades to the rotor shaft rigidly, thus blade flapping is a result of bending of the rotor blade itself.  The articulated (hinged) hub attaches the rotor to the shaft with hinges that lie with an offset from the shaft. These hinges allow for frictionless flapping, feathering and lead-lag motion. A.3.2.2.

Analysis of Different Types of Hub Systems There exists a 90◦ lag between the aerodynamic input and the

flapping output of the TPP, and this is typical of systems in resonance. Rotor resonance can be compared to resonance of a mass-spring system as shown on Figure ‎A.9; therefore it will be used in the following analysis. 1. Teetering Rotor The mass-spring system shown in Figure ‎A.9 will have a natural frequency, ωn, at which mass will oscillate if it is plucked and released. If the system is periodically plucked, for example by an electric motor with A.20


an asymmetric wheel, it can be seen that the amplitude of the resulting oscillation will be greatest if the angular velocity of the electric motor is exactly ωn. The same principal can be applied to a helicopter main rotor, i.e. if the main rotor is forced with its natural frequency the maximal flapping will occur. The natural frequency of a teetering rotors flapping motion is equal to its rotational speed. Therefore the cyclic pitch input applied to the rotor, given by the swash plate together with the stabilizer bar, is exactly the resonance frequency of the flapping motion. This fact is paramount for understanding the 90◦ phase shift between maximal input and maximal flapping, and will be proven shortly. Now, considering the rotating blade element onFigure ‎A.9, the restoring moment about the flapping hinge, Mr, is due to centrifugal force acting through a moment arm, which is a function of β. The centrifugal force acting on a blade element is

Figure ‎A.9 Drawing of a teetering blade and a mass-spring system (right) used to analyses the rotor natural frequency.

Fcf   2 .r.m.r ,

(‎A.81)

therefore an increment in the restoring moment due to the ΔFcf is:

A.21


M r  Fcf .r.sin(  ) ,   2 .r 2 .m.r. sin(  ) .

(‎A.82) (‎A.83)

Total restoring moment is obtained by integration R

M r   2 .sin(  ). m.r 2 .dr , 0

(‎A.84)

but R

I   m.r 2 .dr , 0

(‎A.85)

and small angle approximation in (‎A.84), total restoring moment equation is derived as M r   2 .I . ,

(‎A.86)

Considering the differential equation of an undamped mass-spring system m.x  K .x  Finput ,

(‎A.87)

and relating it to a rotor blade flapping we get I .  M r  M in ,

(‎A.88)

I .   2 .I .  M in ,

(‎A.89)

using Laplace transforming get I . .s 2   2 .I .  M in ,

(‎A.90)

or output/input formula  M in



1I , s 2 2

(‎A.91)

It can be seen a second order equation is derived with two poles at Ω. In order to simplify the analysis, we have neglected to consider the dampening of the system. Due to the fact that the damping, c, does not have any effect on the natural frequency, it will be introduced as an arbitrary value in order to prove the statement of 90◦ phase between input and the maximal output. Now generating a bode plot of the system with A.22


arbitrary values for Ω, m and R, see Figure ‎A.10, it can be seen that at frequency phase is exactly 90◦, and thereby proving the statement made earlier

Figure ‎A.10 Bode plot of an undamped teetering rotor characteristic, where Ω = 235 rad/s, R = 0.46 m, I = 0.00409kg.m2

2. Hinged Rotor Adding a hinge offset to the main rotor will change the natural frequency of the system to one slightly higher than the rotational frequency. This claim will be described momentarily, however in order to gain an intuitive understanding of the effect, that a hinge offset produces, a pendulum example is considered. If a pendulum is exited in the way that it swings back and forth with a cord length, l, it will have a natural frequency denoted by ωl. However if the pendulum cord length is changed, for example made shorter, pendulum systems natural frequency will increase. That is meaning the shorter cord have higher natural frequency. In principal, the same effects apply to the main rotor with respect to hinge offset. Now to prove this claim the same system shown in Figure ‎A.9 is considered, moreover the system is added a flapping hinge, see Figure ‎A.11. The new increment of the restoring moment about the hinge due to centrifugal force is A.23

APPENDIX A: Helicopter Dynamic Analysis M re   2 ..m.r. .r.(r  e) ,

(‎A.92)

Figure ‎A.11 Rotor with hinge offset used to analyze its natural frequency.

Since the portion of the blade outside the flapping hinge is analyzed, it is convenient to redefine the distance to the blade element by the introduction of a new parameter r  r  e ,

(‎A.93)

M re   2 ..m.r . .r .(r   e) ,

(‎A.94)

therefore

Then the total moment is M re   2 . .

R e

0

m.(r   e).r .dr  ,

R e R e M re   2 . .  m.r 2 .dr   e. m.r .dr   , 0  0 

(‎A.95) (‎A.96)

however R e

0

R e

0

m.r 2 .dr   I b ,

(‎A.97)

Mb , g

(‎A.98)

m.r .dr  

where Ib and Mb are moment of inertia and static moment of the blade about the flapping hinge [78],Thus A.24


 M M re   2 . . I b  e. b g 

  . 

(‎A.99)

Again relating the system with an undamped mass-spring system I b .  M r  M in ,  M I b .   2 . I b  e. b g 

 .  M in , 

(‎A.100) (‎A.101)

Laplace transforming it and deriving a transfer function  M in



1 Ib   2 .e.M b s    2  I b .g  2

   

, (‎A.102)

Again a second order function is derived and using the same values of, m and R and from the system without a flapping hinge and using e=0.06m, following bode plot of the system can be seen onFigure ‎A.12. It can be seen the natural frequency of the system is no longer the rotational frequency, ωn= 235rad/s, but a frequency slightly higher, ωn ≈ 255rad/s. Therefore the maximal flapping of the main rotor with hinge offset will occur faster than on the teetering rotor, given that the input to both systems has the same characteristics.

Figure ‎A.12 Bode plot of an undamped hinged rotor characteristic, where Ω=235 rad/s, R=0.46m, Ib = 0.00409kg.m2, Mb=0.11949N.m and e=0.06m

A.25


3. Hingeless Rotor Hingeless rotors flap through a structural bending of the blades, therefore in addition to the moment due to centrifugal force, the restoring moment is added a spring term K.β, therefore the restoring moment equation is M r  ( 2 .I  K ). ,

(‎A.103)

I b .  ( 2 .I  K ).  M in ,

(‎A.104)

the system equation

Figure ‎A.13 Hinged rotor

Laplace transforming it and rearranging the equation in to a transfer function  M in



1 Ib .  2 K 2  s     I b  

(‎A.105)

Again, a second order function is derived and using the same values of , m, and R from the system without a flapping hinge and using an experimental value for, K, following bode plot of the system can be seen onFigure ‎A.14. Again it can be seen that the natural frequency of the rotor is higher than the rotational frequency, therefore the maximal flapping will

A.26


occur faster than in a teetering rotor, given that the input to both systems has the same characteristics.

Figure ‎A.14 Bode plot of an undamped hinged rotor characteristic, where Ω=235 rad/s, R=0.46m, Ib = 0.00409kg.m2 and K=24N.m/rad.

4. Summary It can be seen from equations (‎A.105), (‎A.102) and (‎A.91), that all three rotor types can be modeled as a hinged rotor, a fundamental result of rotor dynamics emerges from the above analysis, that the flapping response is approximately 90◦ out of phase with the applied cyclic pitch, i.e., θ1s gives β1c, and θ1c gives β1s . For blades freely articulated at the center of rotation, or teetering rotors, the response is lagged by exactly 90 ◦ in hover; for hingeless rotors. The phase delay is a result of the rotor being aerodynamically forced, through cyclic pitch, close to resonance, i.e., oneper-rev. [5].

A.27


A.3.2.3. A.

Thrust Vectoring

Hinge Moment due to Gyroscopic Forces The hinge moment due to gyroscopic forces can be defined as the

hinge moment generated when the TPP is tilted by the fuselage rotation rates p, q and r. To analyze this following setup will be considered: A blade element with a mass m normalized in respect to an infinitesimal length Δl and placed at a distance l from the origin of the Hub Frame (HF). This is shown onFigure ‎A.15. It should be noted that the length l is in this gyroscopic moment analysis used instead of the notation r, hereby not confusing the length with the yaw rate r. Also the HP is in this analysis defined as a Hub Frame (HF) with the hz axes pointing in the same direction as bz. The blade element gyroscopic behavior is analyzed for a teetering type rotor, however the effect on the gyroscopic moment due to a hinge offset is minimal which is shown later when simplifications are introduced. The fundamental equation in gyroscopic behavior is M  

dL , dt

(‎A.106)

where L is the angular momentum of the blade element in question and M is the gyroscopic moment. The angular momentum can be described as [79]

L  l   (v.m.l ) ,

(‎A.107)

L  l.v .m.l ,

(‎A.108)

where v is the linear non rotating velocity of the blade element, l’ is the absolute value of the vector l = | l’| and v⊥ is the velocity vector

A.28


perpendicular to l’ that is given from the rotations of the fuselage and shaft rotation rates. Our first goal is therefore to derive v⊥. Later in Figure ‎A.17, a depiction of v⊥ is shown as a function of the pitch q, but will in reality be a function of all the rotation rates the blade element experiences.

Figure ‎A.15 Sketch of a teetering rotor used to analyze the gyroscopic moment

I.

Deriving the Motion of a Blade Element Relative to Spatial Frame The rotation rates effecting the gyroscopic moment of the blade

element are the fuselage roll pbs, pitch qbs and yaw rbs rates which together also can describe the rotation rate of HF relative to the spatial frame (SF). It should here be noted that rotation of BF relative to SF is the same as the rotation of the HF relative SF. The set notation (pbs, qbs, rbs) will therefore be used. Furthermore the blade element, indexed as B, rotates relative to the HF, the rotation rate ΩBh will therefore also be included. The motion of the blade element will in the following be derived relative to SF, there will therefore be no translateral movements described in v⊥. The effect of these is included in the aerodynamic hinge moment; However the translateral acceleration and deacceleration of the rigid body will not be included in the rotary wing modeling due to its minimal effect. A.29


The approach is to find v⊥ as a function of the blade elements B rotation rates pbs, qbs and rbs+Ωbs relative to SF; here adding Ω to hz axes rotation relative to SF. This is done in three steps:

Figure ‎A.16 The position of the blade element B relative to HF

Figure ‎A.17 The position of the blade element B relative to SF and described in BF. The velocity is depicted only as a function of q

1. We first represent the position of the blade element on vector form, relative to HF l’Bh. A.30


2. Next the position is made relative to SF l’Bh; this will be done in terms of the fuselage rotation rates changes to the blade elements position over an infinitesimal time increment dt. Likewise will Ω also change the blade elements position, which will also be included? The term l’Bh will hereby be a dynamic description to the position of the blade element relative to SF and in terms of rotation rates. The flapping rate ̇ and higher derivatives of the rotation rates will not be included due to their minimal effect. 3. Last the cross product between the blade elements rotation rates relative to SF and l’Bh is calculated. This results in the blade element B velocity v⊥ perpendicular to l’Bh. Vectorizing l in the HF: The vector hl’Bh representing the blade element B position relative to HF will now be described in the HF. We have h

 lBh

 c(  ).c( )   l   c(  ).s ( )  ,     s (  ) 

(‎A.109)

This projection is shown in Figure ‎A.16. In this figure it is important to notice that the hx axes is pointing in negative direction thus towards ψ=0o. The projection hxl, hyl and hzl are therefore negative terms. Description of the Position Vector Relative to SF: The description of the position vector relative to SF has to be described in HF. This is done by first finding the rotation matrix that maps HF onto SF. In the rigid body section A.1 the rotation matrix R that maps body frame BF coordinates onto SF coordinates was found. Since the rotations of BF and HF relative to SF are equal R will be reused. ‎A.1 c c Rbs ( )  c s    s

s s c  c s s s s  c c s c

c s c  s s  c s c  s c    c c

(‎A.110)

A.31


The matrix R has to be expressed in terms of the rotation rates phs, qhs, rhs and Ωhs. Consider Figure ‎A.17 depicting a 2 dimensional illustration of the blade element relative to SF. When pitching the helicopter at given rate q it will change the Euler angel θ over a infinitive small time period dt. This will hereby change the angle between HF and SF, however due to the inertia of the rotating blade; the flapping can’t change its attitude momentarily. The flapping will therefore, be seen from the longitudinal direction, follow the SF when given a pitch q. The tangential velocity will therefore point in the direction as shown in Figure ‎A.17. It can be seen from the figure that the blade elements movement described in HF does not include the Euler angles but will only be described in terms of the rotation rates. The rotation angels of the BF relative to SF as described in the BF is therefore written phsdt, qhsdt and (hrhs+hΩBh)dt and will replaced ϕ, θ and ψ respectively. Having the angles represented as rotation rates integrated over a infinitive small time period, gives us the opportunity for simplifying small value terms of R. Simplifications including the angle pdt will be presented, but will also be true for the angles qdt, rdt and β. 1. Small angle approximations leaves sin(pdt) = pdt, cos(pdt) = 1. 2. Since 1 >> pdt terms including a combination of the small angle pdt and one leaves 1 + pdt = 1. 3. Terms including multiplicative of the small elements dt and β are neglected. I.e. dt2 = 0 and βdt = 0. This assumption is valid because the multiplicative of the small elements are not multiplied by the larger Ω term, and therefore results in a relative small value in all the terms where it is present.

A.32


4. Since Ωdt >> rdt, the rotation rate in yaw seen from the blade element will be minimally effected by rdt compared to Ω. Ie. rdt = 0. The transformation can now be expressed

h ' l Bs



' R.h l Bh

 1  l. dt    qdt

dt 1 pdt

qdt   pdt  ,  1 

(‎A.111)

Keeping the simplifications hl’Bh can be expressed

h ' l Bs

II.

  c( )  s ( )dt    l. c( )dt  s ( )  . c( )qdt  s ( ) pdt   

(‎A.112)

Finding the Tangential Velocity Here find the crossproduct between hl’Bs, and the blade elements

rotation rates relative to SF. The laws of physics dictate that the tangential velocity v⊥ of a rotation vector Ω is v⊥ =Ω × l’ where l’ is the position where the velocity v⊥ is activated. Applying the same simplifications as earlier, the blade element velocity due to SF rotations can be described,

h

  p     q v  lhs       r  hs h



0  0    ,  Bh 

 qc( )qdt  s ( ) pdt       c( )dt  s ( )  l   c( )  s ( )dt   pc( )qdt  s ( ) pdt    ,    p c( )dt  s ( )  q c( )  s ( )dt  

(‎A.113)

(‎A.114)

To obtain a less complicated expression more simplifications can be made. Many of the terms are based only on small elements which are added to terms based on the larger element Ω. Therefore all terms including the small element dt, Ω or Euler angles, without being multiplied by the

A.33


larger of the rotation rates, will be neglected. These assumptions leave us with: h

   s ( )   2 c( )dt  q   2 v  l    c( )   s ( )dt  p ,   c( ) pdt  s ( ) p  c( )q  s ( )qdt   

(‎A.115)

Analyzing the perpendicular velocities hv⊥ in each of the HF directions hx, hy and hz, we have the following. The perpendicular velocity in the directions hx, hy includes the terms qβ, pβ is the tangential velocity caused by the fuselage rotations, in respect to the change of the flapping angle. Since Ω >> p, q, β this term is relative small and will not be included in the model. Ω2c(ψ)dt, Ω2s(ψ)dt results in the Coriolis forces acting to twist the rotor blade. These blade loads is the result of the blades obeying The Law of Conservation of Momentum, which states that the product of the momentum of inertia and the rate of spin is constant. As the blade flaps up, its center of gravity moves in toward the spin axis; the same happens for the down flapping blade. This reduces the moment of inertia thus each blade tries to speed up however in opposite direction around the azimuth. In the early helicopter designs this courses much tension on the blades, but has now been solved with a vertical lead-lag hinge so the blade freely can accelerate and de-accelerate. The lead lag hinges are important in the study of the rotor loads and vibration; but they have no effect on performance, stability and control [78]. This term will therefore not be included in the model. Ωs(ψ), Ωc(ψ)

describes the tangential velocity of the blade

around azimuth, which is an element of the hinge moment due to aerodynamic force described in (Appendix ‎A section

‎C) .The

A.34


perpendicular velocity in the directions sz is the foundation of the gyroscopic moment. The terms included are: s(ψ)p, c(ψ)q

are the direct tangential velocity of the blade

elements rotation in respect to the fuselages rotation rates p and q Ωc(ψ)pdt, Ωs(ψ)qdt are the sz axes component of the direction change, to the tangential velocity of the blade elements rotation in respect to , inflicted by the angels pdt and qdt. This component is not present in the s

x and sy axes due to the simplifications. Without the simplifications the

term would appear as Ωβqdt and Ωβpdt in the sx and sy respectively which would be relative small value compared to the terms other terms. III.

Deriving the Gyroscopic Hinge Moment From equations (‎A.106) and (‎A.108) we have that the increment

of hinge moment due to gyroscopic acceleration is M  l

h

d hv ml , dt

(‎A.116)

The velocity terms in the directions hx and hy are already accounted for. The term dhv⊥/dt will therefore be represented with the scalar value of the differentiated velocity in the hz direction. We have M gyro  a gyro .l.m.l ,  c(t ) pdt  s(t ) p  c(t )q  s(t )qdt , dt  l (c( ) p  c( ) p  s( )q  s( )q) ,

a gyro  l

 2l (c( ) p  s( )q) .

(‎A.117) (‎A.118) (‎A.119) (‎A.120)

Total hinge moment is R

M gyro  2  pc( )  qs ( ) ml 2 dl ,

(‎A.121)

0

 2I b ( pc( )  qs( )) ,

(‎A.122) A.35


The definition of the inertia Ib have been presented in the analysis of the teetering type rotor section ‎A.3.2.2. The moment is now decomposed into sine and cosine components

M gyro  ( M gyro,cosc( )  M gyro,sin c( )) ,

(‎A.123)

M gyro,cos  2 pIb ,

(‎A.124)

M gyro,sin  2qI b ,

(‎A.125)

where

It can be seen that both components is negative terms of the 90 o phase lagged angular velocity input. The phase lag is due to the same facts as explained in the analysis of the teetering type rotor section ‎A.3.2.2; as an example will the input of a pitch q decomposed in the longitudinal cosine direction vary periodically with the frequency Ω as the blade rotates around azimuth. The system is therefore being forced at it natural frequency and will response with a 90o phase lagged moment as explained in the analysis of the teetering rotor. The negative terms can also be explained by the phase lag. Again a pitch q will naturally have the maximum input velocity influence on the cosine direction when the blade is pointing in the cosine direction and minimum when pointing in sine direction. The blade will therefore move from a given maximum vertical velocity, in the cosine direction, to have its maximum flapping angle β as a function of q in the sine direction. Depending on the way of rotation this flapping will be either in a positive or negative direction from the given definition of β1s and β1c. Rotating in clockwise as in our case, the gyroscopic moments would be defined as in equations (‎A.124) and (‎A.125). In rotating counter-clockwise, Mgyro,cosine would change sign, since the hy projection of the blade element position shown in Figure ‎A.16 changes sign. This consideration has to be taken in to account, when comparing with literature such as [78]. A.36


B.

Hinge Moment due to Aerodynamic Forces In order to decrease the complexity of the resulting equations, the

moment equations are based on the following assumptions: 1. Aerodynamic forces are considered to act from the hinge to the tip. 2. Reverse flow region is ignored (Region of the rotor disc where generated lift is negative due to translational velocities, see Figure ‎A.19 3. Induced velocity, v1, is uniform and lies in a plane parallel to the rotor shaft. 4. The airfoil lift characteristics are linear and free of stall and compressibility effects. 5. Lift on a blade element is perpendicular to the plane spanned by the leading and trailing edge of the blade. 6. The blade motion consists of only coning and first harmonic flapping. 7. Small angle approximations are valid.

Figure ‎A.18 Aerodynamic moment effecting the hinged type rotor

According to [78], the use of these assumptions has been found to be justified for conventional helicopters flying within conventional flight

A.37


envelopes, therefore we will adopt them in to derivation of flapping dynamics. The increment of moment about the flapping hinge due to aerodynamic force (lift on a blade element) is

M A  L.r  ,

(‎A.126)

The lift on a blade element is derived from blade element theory in section ‎A.3.1.2 as: L 

1 acUT2 r , 2

(‎A.127)

1 acU T2 r .r  , 2

(‎A.128)

Therefore M A 

Integrating out from the hinge gives MA 

R

1 ac  U T2 r .dr  , 2 e

(‎A.129)

where  a is the slope of the lift curve  c is the cord length,  ρ is the density of air  UT is the tangential velocity which has been previously derived in the blade element section ‎A.3.1.2. UT  r   (r  e) ,

(‎A.130)

The derived equation does not consider the added velocity which a blade encounters due to the helicopter translational movements. Figure ‎A.8 shows the contribution to the UT due to helicopter forward velocity, where it can be seen that on the left side of the helicopter (advancing side), the A.38


velocity of air that a blade encounters will be greater than on the right side (retreating side). This will result in a hange of lift distribution from one equally over the azimuth, to one slightly higher on the right side of the helicopter (azimuth interval 0◦ < ψ < 180◦). The change in lift distribution will result in a upward TPP tilt approximately 90◦ later, dependent of the rotor type. This will create a negative longitudinal component of the thrust vector, i.e. the helicopter will try to roll away from the flight direction. This effect is called the dihedral effect. Therefore the new tangential velocity equation is UT   (r  r )  u sin( )  v cos( ) ,

(‎A.131)

where  u sin(ψ) describes the tangential velocity contribution, due to translatory movement of helicopter body in bx direction, with respect to azimuth.  v cos(ψ) describes the tangential velocity contribution, due to translatory movement of helicopter body in by direction, with respect to azimuth.  The local angle of attack, α, derived in ‎A.3.2.2   

Up UT

,

(‎A.132)

where θ is the geometrical pitch of the blade and can be written as (Fourier decomposition) :    

r 1  A1 cos( )  B1 sin( ) , R

(‎A.133)

A1 and B1 are the tilt of the swash plate in the lateral and longitudinal direction respectively (see Figure ‎A.20), and Up is a velocity perpendicular to UT and is defined as: U p  w  v1  r    (r   e)(q cos( )  p sin( )) ,

(‎A.134) A.39


where  w is the velocity of the helicopter along the body z axis.  v1 is the induced velocity derived in section ‎A.3.1.1. r’β’ is the perpendicular velocity contribution from the blade flapping, mapped on to the vector in the same direction as Up (small angle approximated). Derivative of β can be calculated to

Figure ‎A.19 Tangential velocity experienced by a blade element, Helicopter [78]

 

  1c sin( )  1s cos( ) , 

(‎A.135)

  (1c sin( )  1s cos( )) ,

(‎A.136)

r    r  ( 1c sin( )  1s cos( )) ,

(‎A.137)

(r′+e)(qcos(ψ)−psin(ψ)) is the contribution to the perpendicular velocity from the pitch and roll rates of the helicopter. However, in derivation of equation (‎A.129), it was assumed that lift distribution across the blade is linear, see Figure ‎A.21, this is not the case, and a realistic approximation of lift distribution can be seen on the same figure. To compensate for this assumption and to simplify the A.40


complexity of the aerodynamic moment equation even more, following expression for MA will be used MA

1  ac 2

R e

 U T r .dr  , 2

(‎A.138)

0

Substituting (‎A.131), (‎A.132) and (‎A.134) in to (‎A.138) and performing the integration MA

1  ac 2

R e



0

( 

Up UT

)U T2 r .dr  ,

(‎A.139)

Figure ‎A.20 Tilts of the swash plate.

Figure ‎A.21 Theoretical and realistic lift distributions, illustrated for a full sized helicopter [78].

Since only constant and first harmonic flapping are to be studied, the use of following trigonometric identities is required A.41

APPENDIX A: Helicopter Dynamic Analysis sin( ) , 2 1  cos(2 ) sin( ) 2  , 2

sin( ) cos( ) 

1  cos(2 ) , 2 3 sin( )  sin(3 ) sin( )3  , 4 3 cos( )  cos(3 ) cos( )3  , 4 sin( )  sin(3 ) sin( ) cos( ) 2  , 4 cos( )  cos(3 ) cos( ) sin( ) 2  , 4 cos( ) 2 

(‎A.140) (‎A.141) (‎A.142) (‎A.143) (‎A.144) (‎A.145) (‎A.146)

where after higher order harmonics, such as sin(2ψ), cos(2 ψ), are disregarded, due to their contribution to flapping is small. The aerodynamic moment can therefore be written in terms of the constant, sine, and cosine terms [44] M A  M A,const  M A,cos cos( )  M A,sin sin( ) ,

(‎A.147)

where M A,const 

 1 1 u2 v2 e2 ac( R  e) 2  2 R 2     2 2 2 4 2 R 2 12 R 2  4 4 R e  u  ve   ue  v    1c    1s  o   2 6R  6R   6R  6R 6R 2  u ue  v ve         B1     A1 2  3R 6 R    3R 6 R 2 



e 1  qve we pu       v1  2 2 2 2 12 R  6 R  6 R 2  6 R  3R  w pue qv      2 2 3R 12 R  6 R 2 

(‎A.148)

A.42


M A,cos 

  ve 1 2v  v1v ac( R  e) 2  2 R 2     o  2 2 3R  2 2 R 2   3R

  e2 1 e  uvB1 u2 e e2             12 R 2 4 6 R  1s 4 2 R 2  8 2 R 2 6 R 12 R 2    

3v 2 8 2 R 2

M A,sin 



 e2 1  e 1 A1     2   4 6R 4  12R

 q  wv   2 2 R 2  

(‎A.149)

 ue 1 2u  v1u ac( R  e) 2  2 R 2      o 2 2 3R  2 2 R 2  3R

 e2  1 e  uvA1 v2 e e2            12 R 2 4 6 R  1s 4 2 R 2  8 2 R 2 6 R 12 R 2     e2 1  e 1   B1     2 2 2   4 6R 4 8 R  12R 3u 2

  p  wu   2 2 R 2  

(‎A.150)

For most rotors hinge offset ratio, e/R will be small enough that it can be eliminated from the terms where it is squared. Thus, the final terms of hinge moment due to aerodynamic forces are M A, const 

 1 1 u2 v2 e  ac( R  e) 2  2 R 2      o 2 2 2 2 2 4 6 R  4  R 4  R  u  ve   ue  v     1c    1s 2 6R   6R  6R 6R 2  u ue  v ve         B1     A1 2  3R 6 R    3R 6 R 2  e 1  qve we pu       v1  2 2 2 2 12 R  6 R  6 R 2  6 R  3R  w pue qv      2 2 3R 12 R  6 R 2 

M A,cos 

(‎A.151)

  ve 1 2v  ac( R  e) 2  2 R 2     o 2 2 3R    3R

 e  uvB1 u2 e 1        1s  2 2 2 2  6R  4 6R  4 R  8 R 

3v 2 8 R 2

2



1  v1v 1  wv   e A1    q  2 2 4   6R 4  2 R 2 2 R 2 

(‎A.152) A.43


M A,sin 

 ue 1 2u  ac( R  e) 2  2 R 2    o 2 2 3R   3R

 e  uvA1 v2 e 1          1c 2 2 2 2  6R  4 6R  4 R  8 R 

C.

3u 2 8 2 R 2



1  v1u 1  wu   e B1    p   2 2 4   6R 4  2 R 2 2 R 2 

(‎A.153)

Hinge Moment due to Inertia of the Blade The hinge moment due to inertia of the blade can be written as M I   Ib   .

(‎A.154)

Figure ‎A.22 Moment of inertia effecting the hinged type rotor

Second derivative of β can be calculated as 2 

2



 1c sin( )  1s cos( )  

  1c sin( )  1s cos( )     1c cos( )  1s sin( )  

  2 1c cos( )  1s sin( ) 

(‎A.155)

Thus the final equation is M I  -Ib    -Ib  2 1c cos( )  1s sin( )  ,

(‎A.156)

Rewriting the equation as components of constant sine and cosine terms A.44


M I  M I ,const  M I ,cos cos( )  M I ,sin sin( ) ,

(‎A.157)

M I,const  0 ,

(‎A.158)

M I,cos  Ib 2 1c ,

(‎A.159)

M I,sin   Ib 1s .

(‎A.160)

where

2

D.

Hinge Moment due to Centrifugal Force The hinge moment due to centrifugal force has been derived in

section ‎A.3.2.2 as  eM b   M cf   2  I b  g    eM b     1c cos( )  1s sin( )    2  I b  g  

(‎A.161)

Rewriting the equation as components of constant sine and cosine terms

E.

M cf  M cf ,const  M cf ,cos cos( )  M cf ,sin sin( ) ,

(‎A.162)

 eM b   o , M cf , const   2  I b  g  

(‎A.163)

 eM b   1c , M cf , cos   2  I b  g  

(‎A.164)

 eM b  M cf ,sin   2  I b   1s , g  

(‎A.165)

Hinge Moment due to Gravitational Force Hinge moment due to gravitational force on the blade, also

known as weight moment can be derived as

A.45


Figure ‎A.23 The gravity moment effecting the hinged type rotor

MW   sin(  ) 

R e

0

mgr dr  ,

(‎A.166)

However it is assumed that beta will be small enough that the weight moment can be regarded as constant about the hinge regardless of flapping, therefore MW  

R e

0

mgr dr  2

mgR 2  e  1   . 2  R   M b   M W ,const

F.

(‎A.167)

Hinge Moment due to Blade Elasticity Hinge moment due to elasticity of the blade, also known as

spring moment can be derived as

A.46


Figure ‎A.24 spring moment effecting the hinged type rotor M s   K

  K   1c cos( )  1s sin( ) 

,

(‎A.168)

Rewriting the equation as components of constant sine and cosine terms

A.3.2.4.

M s  M s,const  M s,cos cos( )  M s,sin sin( ) ,

(‎A.169)

M s,const   Ko ,

(‎A.170)

M s, cos  K1c ,

(‎A.171)

M s, sin  K1s .

(‎A.172)

Control Rotor Flapping

M i  M a  M cf  M gyro ,

(‎A.173)

to find the flapping angle, the moments are equated to zero M a  M cf  M gyro  M i  0 .

(‎A.174)

Inserting the moments respective equations give

A.47

APPENDIX A: Helicopter Dynamic Analysis R

0

R R rl ( , r )dr   mr 2 ( )dr   mr 2  2 (  ) 0

R

0

  2mr ( p sin( )  q cos( ))dr  0 2

(‎A.175)

0

The expression for gyroscopic, centrifugal and inertial moments has a common part R

0

mr 2 dr ,

(‎A.176)

and this is in fact the inertial momentum of the control rotor ICR. Using this information we can rearrange the moments into





R I CR    2   2 ( p sin( )  q cos( ))   rl ( , r )dr , 0

(‎A.177)

which reveals an expression where only the aerodynamical moment need to be integrated along the length of the control rotor. Now we define the function describing lift of each blade element l(ψ,r) as a function of azimuth and radial station. This function is based on blade element theory which was used to describe lift and flapping of the main rotor disk l ( , r ) 

1 acV 2 , 2

(‎A.178)

The velocity term V is a sum of two perpendicular velocity components UT and UP. Where V is attack is described by

(

√

and the angle of

). Before we continue some

assumptions must be made. The tangential velocity component UT ≫ UP and this makes the assumption that V ≈ UT valid. Using small angle approximations we approximate

. A reminder from lift

equations and main rotor flapping equations is the angle of attack _ is the difference between blade pitch θ and the inflow angle ,

.

A.48


 U  V 2  V 2      U T2   P   U T2  U T U P , UT  

(‎A.179)

substituting (‎A.179) into (‎A.178), and inserting this into (‎A.177) yields      2 ( p sin( )  q cos( ))  2

RCR , P

ac



2 I CR

UT2  UTU P rdr

(‎A.180)

0

Figure ‎A.25 Control rotor paddles

Here the velocity components UT and UP are normalized with respect to the angular velocity ΩR. The radial station r is replaced by ̅ 4 1 acRCR ,P  2       2 ( p sin( )  q cos( ))   U T   U T U P  r dr 2

2 I CR

0





(‎A.181)

Now we can factor out the lock number from the integral. The lock number is an important non-dimensional coefficient describing the ratio of aerodynamic to inertia forces. γ is defined as  

4 acRCR ,P

I CR

   2   2 ( p sin( )  q cos( )) 

,

(‎A.182)

1

  2  U T   U T U P  r dr , 20

(‎A.183)

Until now the aerodynamic moment has been integrated from the shaft to the tip of the control rotor. As explained in the introduction the control rotor does not have blade reaching from the shaft to the tip, but A.49


rather small paddles on the outer tip of the control rotor, as can be seen in Figure ‎A.25. To get the correct aerodynamic moment we take the integral from the entire rotor range ( ̅

) and subtract the aerodynamic

moment of a rotor which only spans from the shaft to where the paddles start ( ̅

)

   2   2 ( p sin( )  q cos( ))    2  U T   U T U P  r dr  1

2 0



RCR RCR , P



0

U T2   U T U  r dr P  

(‎A.184)

The moments in (‎A.184) are calculated in [49], it can be presented as below

A.50


   2   2 ( p sin( )  q cos( ))   

8 

 sin( t ) 

4  sin( t ) RCR B 4  sp RCR , P 

 

4  cos(t ) RCR A   cos(t )  4  sp 8 R CR , P      sin( t ) sin( t ) R 4  2 cos(t )  p CR      4 8     RCR , P    

   cos(t ) cos(t ) R 4 CR    4 8   R CR , P  

 

 2 sin( t )  q      

4  sin( t ) CR ,1s cos(t ) RCR   4  CR ,1s 8 RCR 2 ,P   4  cos(t ) CR ,1c sin( t ) RCR      sin( t )    4  CR ,1c 8 RCR 2 ,P  



cos(t ) 

   sin( t ) sin( t ) R 4 CR    4 8  RCR , P  

 2 cos(t )      CR ,1s    

   cos(t ) cos(t ) R 4 CR    4 8  RCR , P   

 

v R3 v1  1 4CR 2  3RCR , P 3RCR , P 

   

 2 sin( t )      CR ,1c    

(‎A.185)

It can be seen that the last term in (‎A.185) is constant. This term will only have an effect on the coning angle which is not of interest here and can be neglected.

A.51

APPENDIX B: Complementary Filter derivation

B APPENDIX B: Complementary Filter derivation B.1. Orientation from gyroscope A three-axis gyroscope will measure the angular rate about the x, y and z axes of the senor frame, termed ωx, ωy and ωz respectively. If these parameters (in rads-1) are arranged into the vector Sω defined by equation (‎B.1), the quaternion derivative describing the rate of change of orientation of the earth frame relative to the sensor frame ES q can be calculated [80] as equation (‎B.2). S

  0  x  y  z  S E

q 

1S S ˆ E q  2

(‎B.1) (‎B.2)

The orientation of the earth frame relative to the sensor frame at time t, ES q ,t , can be computed by numerically integrating the quaternion derivative

S E  ,t

q

as described by equations (‎B.3) and (‎B.4) provided that

initial conditions are known. In these equations, St is the angular rate measured at time t, Δt is the sampling period and ES qêst ,t 1 is the previous estimate of orientation. The sub-script ω indicates that the quaternion is calculated from angular rates [80]. 1S S ˆ E qest ,t 1  t , 2

(‎B.3)

q  ES qêst ,t 1  ES q ,t t ,

(‎B.4)

S E

q ,t 

S E  ,t

B.2. Orientation from accelerometer and magnetometer In the context of an orientation estimation algorithm, it will initially be assumed that an accelerometer will measure only gravity and a magnetometer will measure only the earth’s magnetic field. If the direction of an earth’s field is known in the earth frame, a measurement of the field’s B.1


direction within the sensor frame will allow an orientation of the sensor frame relative to the earth frame to be calculated. However, for any given measurement there will not be a unique sensor orientation solution, instead there will infinite solutions represented by all those orientations achieved by the rotation the true orientation around an axis parallel with the field. A quaternion representation requires a single solution to be found. This may be achieved through the formulation of an optimization problem where an orientation of the sensor ES qˆ , is found as that which aligns a predefined reference direction of the field in the earth frame E dˆ , with the measured field in the sensor frame S S ; thus solving (‎B.5) where equation (‎B.6) defines the objective function [80].

 qˆ, dˆ, sˆ, S E

min f

f



S 4 E qˆ

S E

E

S

(‎B.5)



qˆ, E dˆ , S sˆ  ES qˆ * E dˆ ES q S sˆ ,

(‎B.6)

Gradient descent algorithm is one of the simplest to both implement and compute. Equation (‎B.7) describes the gradient descent algorithm for n iterations resulting in an orientation estimation of S E

guess’ orientation

S E

qˆ n 1 based on an ‘initial

qˆ 0 and a variable step-size μ. Equation (‎B.8) computes

an error direction on the solution surface defined by the objective function, f, and its Jacobian, J [81]. S E

qk 1  ES qk  

f

 f 

f

 

qˆ , E dˆ , S sˆ , k  0,1, 2...n , S qˆ , E dˆ , S sˆ

S E E

 qˆ , dˆ, sˆ  J  qˆ , dˆ  f  qˆ , dˆ, sˆ, S E

E

k

S

T

S E

E

k

S E

E

k

S

(‎B.7) (‎B.8)

Equations (‎B.7) and (‎B.8) describe the general form of the algorithm applicable to a field predefined in any direction. However, if the reference direction of the field is defined to only have components within one or two of the principle axis of the earth coordinate frame then the B.2


equations simplify. An appropriate convention would be to assume that the direction of gravity defines the vertical, z axis as shown in equation (‎B.10). E Substituting gˆ and normalized accelerometer measurement S aˆ for E dˆ and

S

sˆ respectively, yields the simplified objective function and Jacobian

defined by equations (‎B.12) and (‎B.13). S E

qˆ  q1 E

S

fg

J



q2

g  0 0 0 1 ,



aˆ  0 ax

ay

(‎B.9) (‎B.10)



az ,

(‎B.11)

 2(q2 q4  q1q3 )  a x  qˆ k , aˆ   2(q1q2  q3 q4 )  a y  , 2(0.5  q22  q32 )  a x 

(‎B.12)

 2q3 qˆ   2q2  0

(‎B.13)

S E

S



 

S g E

q4  ’

q3

2q4

 2q1

2q1  4q2

2q4  4q3

2q2  2q3  , 0 

The earth’s magnetic field can be considered to have components in one horizontal axis and the vertical axis [82]. This can be represented by equation

(‎B.14).

measurement

S

E

Substituting

mˆ for

E

dˆ and

S

bˆ

and

normalized

magnetometer

sˆ respectively, yields the simplified

objective function and Jacobian defined equations (‎B.16) and (‎B.17). b  0 bx

Sˆ

S



0 bz  ,

mˆ  0 mx m y

(‎B.14)



mz ,

 2bx (0.5  q32 q 24 )  2bz (q2 q4  q1q3 )  mx    f b ES qˆ , Ebˆ, S mˆ   2bx (q2 q3  q1q4 )  2bz (q1q2  q3 q4 )  m y  , 2bx (q1q3  q2 q4 )  2bz (0.5  q22  q32 )  mz   





(‎B.15) (‎B.16)

B.3


J



S g E

 2bz q3   ˆ qˆ , b,   2bx q4  2bz q2  2bx q3 E



2bz q4

 4bx q3  2bz q1

2bx q3  2bz q1 2bx q4  4bz q2

2bx q2  2bz q4 2bx q1  4bz q3

 4bx q4  2bz q2   2bx q1  2bz q3  ,  2bx q2

(‎B.17)

As has already been discussed, the measurement of gravity or the earth’s magnetic field alone will not provide a unique orientation of the sensor. To do so, the measurements and reference directions of both fields may be combined as described by equations (‎B.18) and (‎B.19). Whereas the solution surface created by the objective functions in equations (‎B.12) and (‎B.16) have a global minimum defined by a line, the solution surface define by equation (‎B.18) has a minimum define by a single point, provided that bx  0 . f



S g ,b E







 f g ES qˆ , S aˆ  S ˆ qˆ , aˆ , b, mˆ   , S Eˆ S  f g ,b E qˆ , b, mˆ  S

E







 

 J T g ES qˆ  J g ,b ES qˆ , Ebˆ   T S Eˆ ,  J g ,b E qˆ , b 







(‎B.18) (‎B.19)

A conventional approach to optimization would require multiple iterations of equation (‎B.7) to be computed for each new orientation and corresponding senor measurements. However, it is acceptable to compute one iteration per time sample provided that the convergence rate of the estimated orientation governed by μt is equal or greater than the rate of change of physical orientation. Equation (‎B.20) calculates the estimated orientation orientation

S E S E

q,t computed at time t based on a previous estimate of

qêst ,t 1 and

measurements S aˆt and

the objective function error f defined by sensor S

mˆ t sampled

at time t. The form of f is chosen

according to the sensors in use, as shown in equation (‎B.21). The subscript

 indicates that the quaternion is calculated using the gradient descent algorithm

B.4

APPENDIX B: Complementary Filter derivation S E

q,t  ES qêst ,t 1  t

f f

(‎B.20)

,

T  J g  ES qêst ,t 1  f g  ES qêst ,t 1 , S aˆt  f   T S E S E S  J g ,b E qêst ,t 1 , bˆ f g ,b E qêst ,t 1 , bˆ, mˆ t



 

 , 



(‎B.21)

An appropriate value of μt is that which ensures the convergence rate of

S E

q,t is

limited to the physical orientation rate as this avoids overshooting

due an unnecessarily large step size. Therefore μt can be calculated as equation (‎B.22) where Δt is the sampling period,

q

S E  ,t

is the rate of change

of orientation measured by gyroscopes and α is an augmentation of μ to account for noise in accelerometer and magnetometer measurements [80]. t   ES q ,t t ,   1 ,

(‎B.22)

B.3. Filter fusion algorithm S E

An estimated orientation orientation calculations,

S E  ,t

q

and (‎B.20). The fusion of

qest ,t , is obtained out of the fusion of the

and ES q,t ; calculated using equations (‎B.4)

qˆ

S E  ,t

and

S E  ,t

q is described by equation (‎B.23)

where t and 1   t  are weights applied to each orientation calculation [80]. S E

qest ,t   t ES q,t 1   t ES q ,t , 0   t  1 ,

An optimal value of S E

(‎B.23)

can be defined as the weighted divergence of S

q is equal to the weighted convergence of E q . This is represented by

equation (‎B.24) where divergence rate of

S E

q

t t the

convergence rate of

S E

q and  is the

expressed as the magnitude of a derivative of

quaternion corresponding to the error of gyroscope measurement. Equation (‎B.24) can be rearranged to define  t as equation [80].

1   t    t t

t

(‎B.24)

B.5


t 

 t  t

(‎B.25)

Equations (‎B.23) and (‎B.25) ensure the optimal fusion of ES q ,t and S E ,t

q

assuming that the convergence rate of

S E

q governed by α is greater

than or equal the rate of orientation change. Therefore α has no upper bound. If α is assumed to be very large then  t , defined by equation (‎B.22), also becomes large and the orientation filter equations simplify. A large value of  t used in equation (‎B.20) means that ES qêst ,t 1 has small effect and the equation (‎B.20) can be written as. S E

q,t   t

f f ,

(‎B.26)

The definition of  t in equation (‎B.25) also simplifies as the β term in the denominator can be neglected and the equation can be rewritten as equation (‎B.27). It is possible from equation (‎B.27) to also assume that

t  0 . t 

 t t ,

(‎B.27)

Substituting equations (‎B.5), (‎B.26) and (‎B.27) into equation (‎B.23) yields equation (‎B.28). Important to note that in equation (‎B.28),  t substituted by zero as equation (‎B.26) S E

qest ,t 

 t  f   t  1  0 ES qêst ,t 1  ES q ,t t  ,  t  f 

Equation (‎B.28) can be simplified to equation (‎B.29) where

(‎B.28) S E

qest ,t

the estimated rate of change of orientation is defined by equation (‎B.30) S  and E qˆ,t is the direction of the error of ES qest ,t defined by equation (‎B.31).

B.6

APPENDIX B: Complementary Filter derivation S E

qest ,t  ES qêst ,t 1  ES qest ,t t ,

(‎B.29)

qest ,t  ES q ,t   ES qˆ,t ,

(‎B.30)

f qˆ,t  f ,

(‎B.31)

S E

S E

It can be seen from equations (‎B.28) through (‎B.31) that the filter calculates the orientation

S E

qest by numerically integrating the estimated

S orientation rate E q est . The filter computes E q est as the change of orientation

S

rates measured by the gyroscopes, E q , with the error magnitude of the S

gyroscope measurement, β, removed in the direction of the estimated error, S E

qˆ , computed from accelerometer and magnetometer measurements.

B.4. Magnetic distortion compensation Measurements of the magnetic field of earth are distorted by the presence of ferromagnetic materials in the vicinity of the magnetometer. The effect of distortions of magnetic on an orientation sensor performance have shown that major errors may be introduced by different sources including metal furniture, metal structures within a buildings construction and electrical appliances [83]. Interference sources can be fixed in the sensor frame, through calibration [84]. Interference sources in the earth frame, named soft iron, causes errors in the direction measurement of the earth's magnetic field. Errors of declination in the horizontal plane relative to the surface of earth can be corrected with an additional heading reference from magnetometer. Errors of inclination in the vertical plane relative to the surface of earth may be corrected with gravitational direction as the accelerometer. The measured direction of the earth's magnetic field in the earth frame at time t,

E

hˆt , can be computed as the normalized magnetometer

B.7


measurement, S mˆ t , rotated by the orientation of the sensor provided by the filter when estimated , ES qêst ,t 1 ; as described by equation (‎B.32). The effect of inclination errors of the measured direction earth's magnetic field, E hˆt , can be corrected if the filter's reference direction of the earth's magnetic field, E bˆt , is of the same inclination. This will be achieved by computing E

bˆt as

E

hˆt normalized to get only x and z axes components in the earth

frame; as described below. E





* hˆt  0 hx hy hz  ES qêst ,t 1 S mˆ t ES qêst ,t 1 , E

bˆt  0 

hx  hy 2

2

0 hz  , 

(‎B.32) (‎B.33)

Magnetic distortions compensating in this way ensures that disturbances of magnetic are limited to only affect the estimated heading component. The approach can neglect need for the earth’s magnetic field reference direction to be predefined; a potential disadvantage of other orientation filter designs [61]. Figure ‎4.11 illustrate representation of the complete filter implementation for a MARG sensor array. B.5. Filter gains The filter gain β represents all mean zero gyroscope measurement errors, expressed as the quantity of a quaternion derivative. Most sources of error include: signal aliasing, sensor noise, quantization errors, calibration errors, sensor miss-alignment, and sensor axis nonorthogonally and frequency response characteristics. It is convenient to define β using the angular quantity   , where ~ represents the estimated mean zero gyroscope measurement error of each axis. By using the relationship described by equation (‎B.2), β may be defined by equation (‎B.34) where qˆ is any unit quaternion.

B.8






1 qˆ  0 ~ 2

3~ ~ ~    4 ,

(‎B.34)

B.9

APPENDIX C: Calibration Procedure of Accelerometer

C APPENDIX C: Calibration Procedure of Accelerometer C.1. Acquisition of a set of points A simple test rig (see Figure ‎C.1) consisting of an adjustable platform, a cube mounted with a triaxial accelerometer and a V-block was built to perform these forty eight positions [85]. Set of measurements with various attitudes are needed, so the parameters in equation (‎4.1) can be estimated. In the case of an accelerometer, the magnitude of the local force (or acceleration) must be precisely known; it equals the magnitude of the Earth gravity force (1g). The calibration procedure must be stop when the measurement points cover most of the equation surface (‎4.1) [86]. The minimum procedure that is describes a closed solid by rotating the sensor in the all references coordinate system xyz to get enough measure a set of points.

Figure ‎C.1 Experiment platform setup

C.1


C.2.

2D projection of measurements The calibration can be facilitated by a display of measured

acceleration magnitude in spherical coordinates r(t) ,ϕ(t) and θ(t) (see Figure ‎C.2). This transformation requires an estimation of parameters that is obtained from the first measured points [87]: r (t )  ax (t )2  a y (t )2  az (t )2 ,

(‎C.1)

 a (t ) 2  a (t ) 2 x y  (t )  tan  2 a z (t )  

(‎C.2)

1

  ,  

 a y (t ) 2    a (t ) 2  ,  x 

 (t )  tan 1

(‎C.3)

Each measurement t is represented by a point with coordinates φ(t) and θ(t). In Figure ‎C.3, we can see areas without measurements, the projection of data points help us to get a uniform set of measurements z

θ

r

φ

y

x

Figure ‎C.2

Sphere coordinates

C.2


Figure ‎C.3 Measurement data of acceleration in 2d projection

C.3. Estimation of parameters 

To estimate the matrix α and the vector  in equation (‎4.1), we can minimize the following error function [88]: E ( p)   (e p (t ))2 , t

(‎C.4)

with:  e p (t )  a

2

     (a(t )   )T ( 1) 2 (a(t )   ) ,

(‎C.5)

where  a

2

 ax (t )2  a y (t )2  az (t )2  1g ,

p  [ x ,  y ,  z , S xy , S xz , S yz ,  x ,  y ,  z ]T

(‎C.6)

, the parameters vector to be estimated

using LMA (Levenberg–Marquardt Algorithm)

C.4. Levenberg-Marquardt Algorithm for nonlinear least squares Levenberg-Marquardt Algorithm is a standard technique used to finding solution of nonlinear problems. Least squares problems occur when fitting a function to a set of data points by making errors squares between C.3


the function and the data points equal to minimum. When function is not linear in the parameters, we say, this problem is nonlinear least square. Nonlinear least squares methods take an iterative improvement to find value of parameters in order to decrease the sum of the errors squares between the measured data points and the function. The LevenbergMarquardt fitting method is actually a combination between two minimization methods: the Gauss-Newton method and the gradient descent method. The Levenberg-Marquardt method is acting like Gauss-Newton method when the function parameters are close from optimal value and acting like gradient-descent method when the function parameters are far away from optimal value. yˆ (t; p) of

In fitting a function

a vector of n parameters p and an

independent variable t to a set of m data points (ti , yi ) , it is convenient and customary minimizing the sum of the weighted residuals (or weighted squares of the errors) between the curve-fit function yˆ (ti ; p) and the measured data y(ti ) . Chi-squared error criterion, this label is calling goodness-of-fit measure of scalar-valued [86].  y (t )  yˆ (ti ; p))   ( p)    i  wi  i 1  m

2

2

,

(‎C.7)

 2 ( p)  ( y  yˆ ( p))T W ( y  yˆ ( p)) ,

(‎C.8)

 2 ( p)  yTWy  2 yTWyˆ  yˆ TWyˆ ,

(‎C.9)

The value

wi

is an error in measurement y(ti ) . The weighting

matrix W is diagonal with

. If the function ̂ is nonlinear in

parameters p that is mean will minimize χ2 with respect to the parameters must be carried out iteratively. The aim of each iteration is reduces χ2 by finding a perturbation h to the parameters p.

C.4


The

Levenberg-Marquardt

method

adaptively

varies

the

parameter updates between the Gauss-Newton update and the gradient descent update [86],

J

T



WJ  I h  J TW ( y  yˆ ) ,

̂

where m × n Jacobian matrix

(‎C.10) and small values of the

algorithmic parameter λ result in a Gauss-Newton update and large values of λ result in a gradient descent update. The parameter λ is initialized to be large. If iteration happens to result in a worse approximation, λ is increased. As the solution approaches the minimum, λ is decreased, the Levenberg-Marquardt method approaches the Gauss-Newton method, and the solution typically converges rapidly to the local minimum [85]. Marquardt’s suggested update relationship [86],

J

T



WJ   diag ( J TWJ ) h  J TW ( y  yˆ ) ,

(‎C.11)

The algorithm adjusts λ according to whether χ2 is increasing or decreasing as follows: 1.

Given an initial guess for the set of fitted parameters p

2.

Compute χ2(p)

3.

Choose a value for λ, for instance λ = 0:001

4.

Calculate h and evaluate χ2 (p+h)

5.

If χ2(p+h) ≥ χ2(p) increase λ by a factor and go to (3) and try an update again.

6.

If χ2(p+h) < χ2(p) decrease λ by a factor, accept the updated trial solution p ← p+h and go to (3) and try an update again. The reasoning of the method is that if the error is increasing, the

quadratic approximation in the Gauss-Newton method is not working well and we are likely not near a minimum, so λ should be increased in order to blend more towards steepest descent. On the other hand, if the error is C.5


decreasing, the approximation is working well, and we expect that we are getting closer to a minimum so λ is decreased to blend more towards Gauss-Newton. Levenberg-Marquardt’s algorithm has the disadvantage that if the value of damping factor, λ, is large, inverting (JTWJ + λ I) is not used at all. Marquardt [88] realized that each component of the gradient can be scaled according to the curvature so that there is larger movement along the directions where the gradient is smaller. Thereby avoiding slow convergence in the direction of small gradient. Marquardt utilized this by replacing the identity matrix, I in Levenberg-Marquardt’s original equations Equation (‎C.10) with the diagonal matrix consisting of the diagonal elements of JTWJ (the Hessian matrix), resulting in the Levenberg-Marquardt algorithm, Equation (‎C.11) which then includes an estimation of the local curvature information and uses this to move further in the directions in which the gradient is smaller. C.5. Numerical implementation This work including the enhancement of a rank-1 Jacobian update. In iteration i, the step h is evaluated by comparing

( )

(

) the step is accepted if the metric ρi [86] is greater than a user-specified value, ϵ4, i (h) 

 2 ( p )   2 ( p  h) , 2hT (i  J T W ( y  yˆ ( p)))

(‎C.12)

If in an iteration ρi(h) > ϵ4 then p + h is sufficiently better than p, p is replaced by p + h, and λ is reduced by a factor. Otherwise λ is increased by a factor, and the algorithm proceeds to the next iteration  Initialization and update of the L-M parameter, λ, and the parameters p [86] 1. [

is user-specified . [ ]]

(

̂( ));

C.6

APPENDIX C: Calibration Procedure of Accelerometer ( ) otherwise:

[

[ ((

[

[ ]]

2. (

̂( )))

]] ( ) (( (

( ) otherwise:

| ( [

[ ]]

3. [

( ) otherwise:

where

] ];

]] ( | (

[

]

is user-specified . ̂( )); )

( ))

)

[

( ( )| (

(

̂( )))

(

) ) ];

)

]

is user-specified . ̂( )); ( [ ) ( )| ( ) ];

) ]

is determines acceptance of a L-M step

 Computation and rank-1 update of the Jacobian, [

]

In the first iteration, in every 2n iterations, and in iterations where

(

)

( ),

the

Jacobian

(

ℝ

)

is

numerically

approximated using forward differences, J ij 

yˆ (ti ; p  p j )  yˆ (ti ; p) yˆ i  , p j p j

(‎C.13)

or central differences (default) J ij 

yˆ (ti ; p  p j )  yˆ (ti ; p  p j ) yˆ i  , p j 2 p j

(‎C.14)

where the j-th element of δpi is the only non-zero element and is set to (

| |) n all other iterations, the Jacobian is updated using the Broyden

rank-1 update formula, J J

( yˆ ( p  h)  yˆ ( p)  Jh)hT hT h

,

(‎C.15)

For problems with many parameters, a finite differences Jacobian is computationally expensive. Convergence can be achieved with fewer function evaluations if the Jacobian is re-computed using finite differences only occasionally. The rank-1 Jacobian updates equation (‎C.15) requires no additional function evaluations. C.7


 Convergence criteria Convergence is achieved when one of the following three criteria is satisfied, 1. Convergence in the gradient,

|

(

̂)|

2. Convergence in parameters,

|

|

; or

(

3. Convergence in where

,

and

)

;

.

are convergence tolerance for gradient, parameters and

Chi-square respectively. Otherwise, iterations terminate when the iteration count exceeds a pre-specified limit. To get data from sensor and solve equation (‎4.1) numerically for estimate nine parameters, Figure ‎4.18 illustrates the graphical user interface of the developed calibration software, interfaced with a Matlab.

C.8

APPENDIX D: Verification of the Linear and Non-Linear

D APPENDIX D: Verification of the Linear and Non-Linear Model As the non-linear model is linearized, it would be desirable to verify that the linear and the non-linear model acts in the same way to a given input. This will be done by applying both a positive and negative value on all of the four inputs one by one and comparing the outputs from the non-linear and the linear models with the expected output. When comparing the two models, only the trend of the states are taken into consideration since the value of the states cannot be compared due to the models being highly unstable and only the states directly affected by the input are compared. This is also the reason why only the first two seconds are considered. The input used is a step given to the system at time 0. The trend of the states will be checked using both positive and negative inputs. The verification setup, with no input, is shown in Figure ‎D.1. First the case is considered where no inputs are present. The expected outcome of this is that nothing should happen to the states in the linear model, whereas the states of the non-linear model will develop as expected that v will become negative. This is due to the tail rotor which gives a negative v when counteracting the drag on the main rotor. This translatory motion has the effect that p becomes positive. When no input is given, nothing should happen to the linear model as it is in its operating point. Figure ‎D.2 and Figure ‎D.3 show how the non-linear and the linear model act when not getting an input. As expected nothing happens to the linear model when it is not getting an input. The non-linear acts as expected on the variables v and p. That u becomes negative is caused be cross-couplings in the system. This translatory velocity causes p to become positive. D.1


Figure ‎D.1 Illustration of the set-up used to verify the linear model

Figure ‎D.2 Plot of the rotational velocity of the helicopter model with no input

D.2


Figure ‎D.3 Plot of the translatory velocity of the helicopter model with no input

D.1. Verification Using Lateral Input D.1.1. Positive Input The models will now be tested with positive step on the lateral input ulat. The expected outcome of this is that the translatory velocity along the by axis and the rotational velocity about bx both become positive. Figure ‎D.4 and Figure ‎D.5 shows that p and v becomes positive as expected when giving positive lateral input. D.1.2. Negative Input The models will now be tested with a negative step on the lateral input ulat. The expected outcome of this is that the translatory velocity along the by axis and the rotational velocity about bx both become negative. Figure ‎D.6 and Figure ‎D.7 shows p and v becomes negative as expected when giving positive lateral input.

D.3


D.2. Verification Using Longitudinal Input D.2.1. Positive Input The models will now be tested with positive step on the longitudinal input ulong. The expected outcome of this is that the translatory velocity along the bx axis becomes negative and the rotational velocity about by becomes positive. Figure ‎D.8 and Figure ‎D.9 shows that the states q and u developes as expected. D.2.2. Negative Input The models will now be tested with a negative step on the longitudinal input ulong. The expected outcome of this is that the translatory velocity along the bx axis becomes positive and the rotational velocity about by becomes negative. Figure ‎D.10 and Figure ‎D.11 shows that the states q and u developes as expected.

D.3. Verification Using Collective Input D.3.1. Positive Input The models will now be tested with positive step on the collective input ucol. The expected outcome of this is first of all that the translatory velocity along the bz axis will become negative, the reason a positive input gives a negative movement is that bz is defined downwards on the helicopter. In the non-linear model the drag on the main rotor causes the model to move negatively along by axis, this is not the case for the linear model. However, when increasing the collective input, the drag is changed and the model will increase the thrust on the tail rotor and thereby move along the by axis in negative direction, in the non-linear model this

D.4


velocity will simply become "more negative". Figure ‎D.12 shows that the states. D.3.2. Negative Input The models will now be tested with a negative step on the collective input ucol. The expected outcome of this is that the translatory velocity along the bz axis increases in both models. In the linear model the translatory velocity along by becomes positive, there will also be an effect on the translatory velocity along by in the non-linear model, but as it already has a negative velocity along this axis the effect from the collective input might not be large enough to overcome the negative velocity already present Figure ‎D.13 shows that the states develop in the expected manner, the translatory velocity along by in the nonlinear model does not become positive but it does become "less negative". D.4. Verification Using Pedal Input D.4.1. Positive Input The models will now be tested with positive step on the pedal input uped. The expected outcome of this is first of all that the rotational velocity along the bz axis will become positive. In the linear model the increased force generated by the tail rotor causes the model to move negatively along by axis, in the non-linear model this velocity will simply be increased. D.4.2. Negative Input The models will now be tested with a negative step on the pedal input uped. The expected outcome of this is first of all that the rotational velocity along the bz axis will become negative. In the linear model the decreased force generated by the tail rotor causes the model to move

D.5


positively along by axis, in the non-linear model the negative velocity along b

y will simply be decreased. Figure ‎D.18 and Figure ‎D.19 shows that the states develop as

expected, it also show that the change of v is fairly small.

D.6


Figure ‎D.4 Plot of the rotational velocity of the helicopter model with positive lateral input

Figure ‎D.5 Plot of the translatory velocity of the helicopter model with positive lateral input

D.7


Figure ‎D.6 Plot of the rotational velocity of the helicopter model with negative lateral input

Figure ‎D.7 Plot of the translatory velocity of the helicopter model with negative lateral input

D.8


Figure ‎D.8 Plot of the rotational velocity of the helicopter model with positive longitudinal input

Figure ‎D.9 Plot of the translatory velocity of the helicopter model with positive longitudinal input

D.9


Figure ‎D.10 Plot of the rotational velocity of the helicopter model with negative longitudinal input

Figure ‎D.11 Plot of the translatory velocity of the helicopter model with negative longitudinal input

D.10


Figure ‎D.12 Plot of the translatory velocity of the helicopter model with positive collective input

Figure ‎D.13 Plot of the translatory velocity of the helicopter model with negative collective input

D.11




D.12


Figure ‎D.16 Plot of the translatory velocity of the helicopter model with posetive pedal input

Figure ‎D.17 Plot of the translatory velocity of the helicopter model with posetive pedal input

D.13


Figure ‎D.18 Plot of the translatory velocity of the helicopter model with negative pedal input

Figure ‎D.19 Plot of the translatory velocity of the helicopter model with negative pedal input

D.14

‫ملخص البحث‬ ‫تمتلك الطائرات المروحية سمة اقتران عالية مع نظام متعدد المدخالت متعدد‬ ‫المخرجات‪ ،‬مما يزيد من تعقيد النظام الدينامي للطائرة المروحية‪ .‬بالتالي سيكون نطام دينامي غير‬ ‫مستقر‪ .‬تعالج هذه االطروحة موضوع الديناميك والسيطرة للطائرة العمودية المسيرة خالل مناورة‬ ‫الحوم‪ ،‬وتقدم للقارئ مسار التطوير ابتداءاً من النمذجة وتحليل النظام مروراً بدمج اجهزة‬ ‫َ‬ ‫االستشعار وتقدير متجة الحالة الى توليف وحدة التحكم‪ .‬يتم توجيه التركير على تطبقين‪ :‬االول‪:‬‬ ‫جلب النظام من الحالة غير المستقرة مع متجة حاله ال يساوي الصفر الى حالة مستقرة مع متجة‬ ‫حالة يسوي الصفر‪ ،‬الثاني‪ :‬ايجاد الحاالت الحرجة التي ال تستطيع بها وحدة التحكم جلب النظام من‬ ‫الحالة غير المستقرة الى الحالة المستقرة والحفاظ عليها ‪.‬‬ ‫تم اختيار الطائرة (‪ )Trex500‬كمنصة اختبار في مشرع االطروحه هذا كما وجهزت الطائرة‬ ‫بأجهزة استشعار مع حاسوب داخلي‪ .‬تم بناء نموذج رياضي عام غير خطي لهذه المروحية‬ ‫بأستخدام طريقة المبادئ االساسية وتنفيذها في بيئة برمجة الماتالب‪-‬سيمولينك‪ .‬أستخدمت طرق‬ ‫عديدة وبعضها مبتكر لحساب عوامل النموذج الرياضي وتصميم وحدة التحكم االمثل المبنية على‬ ‫اساس طريقة وضع المعايير (اي جعل المعادلة معادلة ذات حدود) و التقريب الخطي في حالة‬ ‫الحوم المتزن‪ .‬جمعت مختلف معلومات المستشعرات بواسطة خوارزميات دمج المستشعرات‬ ‫المطورة المختلفة الى متجة القياسات‪ .‬استعملت طريقة مرشح كالمان الخطي للحركة الخطية‬ ‫والمرشح التكميلي للحركة الدورانية لحساب متجة القياسات والذي يمثل متجة الحالة المحسوب‬ ‫للنموذج‪.‬‬ ‫تم تعريف الوسيط بين النموذج الرياضي الغير خطي وبيئة الماتالب‪-‬سيميولينك والتي بموجبها تم‬ ‫تطوير وحدة التحكم وتنفيذ المحاكاة‪ .‬المروحية قادرة على االنتقال من حالة الحركة مع متجة حالة‬ ‫اليساوي الصفر الى مناورة الحوم مع متجة حالة يساوي الصفر والحفاظ على هذه الحالة ضمن‬ ‫بيئة المحاكاة‪.‬‬ ‫تم التحقق من صحة النماذج الرياضية في حالة الحوم‪ .‬النتائج المستحصلة تشير الى تطابق جيد مع‬ ‫مبادئ ديناميك طائرات الهليكوبتر‪ .‬أعطت نتائج خوارزميات المالحة ألجهزة االستشعار من‬ ‫سرعة خطية وسرع زاوية وزوايا اويلر الثقة لدمجها الحقا مع وحدة التحكم‪ .‬في الجزء التجريبي‬ ‫كنا قادرين على تحقيق رحلة ذاتية الحكم‪ ،‬حيث كانت المروحية قادرة على أداء طيران مستقل في‬

‫مناورة الحوم‪ .‬مما يشير الى أن جميع مراحل العمل بدءا من النمذجة وأنتهاءا بوحدة التحكم هي‬ ‫مراحل صحيحة رياضيا وبرمجيا‪ .‬تم تحديد الحاالت الدينامية الحرجة في النظام‪ ،‬الذي ال يمكن‬ ‫لوحدة التحكم أن تحافظ فيه على استقرارية النظام الدينامي‪ ،‬وذلك باستخدام برنامج المحاكاة الذي‬ ‫يمثل نموذجا رياضيا لطائرة هليكوبتر مع وحدة تحكم‪.‬‬

‫جمهورية العراق‬ ‫وزارة التعليم العالي والبحث العلمي‬ ‫الجامعة التكنولوجية‬ ‫قسم الهندسة الميكانيكية‬

‫دراسة االستقرارية الدينامية‬ ‫لطائرة مروحية مسيرة‬ ‫أطروحة مقدمة إلى‬ ‫قسم الهندسة الميكانيكية في الجامعة التكنولوجية وهي جزء من متطلبات نيل‬ ‫درجة الدكتوراه في علوم الهندسة الميكانيكية ‪ /‬ميكانيك تطبيقي‬

‫تقدم بها‬

‫عقيل علي وناس‬ ‫بإشراف‬ ‫أستاذ مساعد‬

‫أستاذ مساعد‬

‫الدكتور عماد ناطق عبد الوهاب‬

‫الدكتور محمد أدريس محسن‬

‫‪1026‬‬