Aircraft-Flight-Dynamics-Control-and-Simulation-Using-Matlab-and-Simulink-Singgih-Satrio-Wibowo-2007

Preface 1

Aircraft Flight Dynamics, Control and Simulation Using MATLAB and SIMULINK: Cases and

Algorithm Approach

Singgih Satrio Wibowo

P REFACE

This book is written for students and engineers interesting in flight

control design, analysis and implementation. This book is written

during preparation of Matlab and Simulink course in UNIKL-MIAT

(University of Kuala Lumpur-Malaysian Institute of Aviation

Technology) in third week of February 2007. Although this book is still

in preparation, I hope that this book will be useful for the readers.

I wish to express my great appreciation to Professor Said D. Jenie

for his support. I wish to acknowledge Mr. Kharil Anuar and Mr.

Shahrul Ahmad Shah of MIAT for their invitation to the author to give

Matlab course in MIAT during the period of 26 February to 2 March

2007. I also wish to acknowledge the support of my colleagues at

Institut Teknologi Bandung (ITB): Javensius Sembiring and Yazdi I.

Jenie, and also my friends at Badan Pengkajian dan Penerapan

Teknologi (BPPT): Dewi Hapsari, Dyah Jatiningrum and Nina Kartika.

No words can express the thanks I owe to my parents: Ibunda Sulasmi

and Ayahanda Satrolan, and my family for their continuous support

through out my life. Finally and the most importantly, I would like to

thank The Highest Sweetheart Allah Almighty, The Creator and The

Owner of the universe.

Kuala Lumpur, 25 February 2007

Singgih Satrio Wibowo

Contents 2

Aircraft Flight Dynamics, Control and Simulation Using MATLAB and SIMULINK: Cases and

Algorithm Approach

Singgih Satrio Wibowo

C ONTENTS

Preface (1)

Contents (2)

List Of Figures (5)

List of Tables (7)

1Aircraft Dynamics and Kinematics (9)

1.1Coordinate Systems and Transformation (10)

1.1.1Local Horizon Coordinate Reference System (10)

1.1.2Body Coordinate Reference System (10)

1.1.3Wind Coordinate System (12)

1.1.4Kinematics Equation (15)

1.1.5Direction Cosine Matrix (16)

1.1.6Quaternions (17)

1.2Aircraft equations of motion (21)

1.2.1Translational Motion (21)

1.2.2Angular Motion (23)

1.2.3Force and Moment due to Earth’s Gravity (25)

1.2.4Aerodynamic Forces and Moments (26)

1.2.5Linearization of Equations of Motion (27)

1.1Matlab and Simulink Tools for Flight Dynamics Simulation (30)

2Flight Control (31)

2.1Attitude and Altitude Control using Root Locus Anlysis (32)

2.2Optimal Path-Tracking Control for Autonomous Unmanned Helicopter Using

Linear Quadratic Regulator (33)

2.2.1Linearized Model (34)

2.2.2Modified Linearized Model (37)

Contents 3

Aircraft Flight Dynamics, Control and Simulation Using MATLAB and SIMULINK: Cases and

Algorithm Approach

Singgih Satrio Wibowo

2.2.3Path Generator (39)

2.2.4Path-Tracking Controller Design (43)

2.2.5Matlab and Simulink Implementation (46)

2.2.6Numerical Results (54)

2.2.7Analysis and Discussion of the Results (63)

2.3Coordinated Turn Using Linear Quadratic Regulator (65)

2.3.1State-Space Equations for an Airframe (65)

2.3.2Problem Definition (65)

2.3.3Matlab and Simulink Implementation (66)

2.3.4Results (69)

2.3.5Analysis and Discussion of the Results (70)

2.4Adaptive Control for Yaw Damper and Coordinated Turn (71)

2.4.1Yaw Damper and Coordinated Turn: Definition (71)

2.4.2Model Reference Adaptive System (71)

2.4.3State-Space Model of XX-100 Aircraft (72)

2.4.4Matlab and Simulink Implementation (72)

2.4.5Results (72)

2.4.6Discussion of The Results (73)

3Flight Simulation (74)

3.1Matlab and Simulink tool for simulation (75)

3.1.1Matlab command for simulation purpose (75)

3.1.2Simulink toolbox for simulation purpose (75)

3.2Virtual Reality, an advance tool for visualization (76)

3.2.1Introduction to Virtual Reality toolbox: a user guide (76)

3.2.2Virtual Reality for transport aircraft (88)

3.3Simulation of Aircraft Dynamics: a VirtueAir transport craft (89)

Appendix A (90)

Contents 4

Aircraft Flight Dynamics, Control and Simulation Using MATLAB and SIMULINK: Cases and

Algorithm Approach

Singgih Satrio Wibowo

Appendix B (93)

References (99)

List Of Figures 5

Aircraft Flight Dynamics, Control and Simulation Using MATLAB and SIMULINK: Cases and

Algorithm Approach

Singgih Satrio Wibowo

L IST O F F IGURES

Figure 1-1 Local horizon coordinate system (10)

Figure 1-2 Body-coordinate system (11)

Figure 1-3 Aircraft attitude with respect to local horizon frame: Euler angles (12)

Figure 1-4 Wind-axes system and its relation to Body axes (13)

Figure 1-5 Aerodynamic lift and drag (14)

Figure 2-1 A small-scale unmanned helicopter, Yamaha R-50 (33)

Figure 2-2 Dimension of the Yamaha R-50 Helicopter (34)

Figure 2-3 The complete state-space form of R-50 dynamics (35)

Figure 2-4 Trajectory for example 1, circular (40)

Figure 2-5 Velocity profile for example 1 (41)

Figure 2-6 Trajectory for example 2, rectangular (41)

Figure 2-7 Velocity profile for example 2 (42)

Figure 2-8 Trajectory for example 3, spiral (42)

Figure 2-9 Velocity profile for example 3 (43)

Figure 2-10 Path tracking controller model (49)

Figure 2-11 Path generator block (49)

Figure 2-12 Earth to inertial velocity transform block (50)

Figure 2-13 Optimal controller block (50)

Figure 2-14 Yamaha R50 dynamics model block (50)

Figure 2-15 Body to inertial transform block (51)

Figure 2-16 Inertial to Earth transform block (51)

Figure 2-17 Write to file block (51)

Figure 2-18 Flight trajectory geometry (55)

Figure 2-19 Trajectory history (55)

Figure 2-20 Velocity history (56)

Figure 2-21 Control input history (56)

Figure 2-22 Attitude history (57)

Figure 2-23 Trajectory error history (57)

Figure 2-24 Flight trajectory geometry (58)

Figure 2-25 Trajectory history (58)

Figure 2-26 Velocity history (59)

Figure 2-27 Control input history (59)

Figure 2-28 Attitude history (60)

Figure 2-29 Trajectory error history (60)

Figure 2-30 Flight trajectory geometry (61)

Figure 2-31 Trajectory history (61)

List Of Figures 6

Aircraft Flight Dynamics, Control and Simulation Using MATLAB and SIMULINK: Cases and

Algorithm Approach

Singgih Satrio Wibowo

Figure 2-32 Velocity history (62)

Figure 2-33 Control input history (62)

Figure 2-34 Attitude history (63)

Figure 2-35 Trajectory error history (63)

Figure 2-36 A Body Coordinate Frame for an Aircraft [16] (65)

Figure 2-37 Simulink diagram of coordinated turn (67)

Figure 2-38 Write to file block (67)

Figure 2-39 Attitude history (69)

Figure 2-40 Tracking error history (70)

Figure 2-41 Control input history (70)

Figure 2-42 Block diagramfor Turn Coordinator system (71)

Figure 2-43 Block diagram for Model Reference Adaptive System (72)

Figure 3-1 The 3D AutoCAD model of XW aircraft (77)

Figure 3-2 The 3D AutoCAD model of lake and hill (78)

Figure 3-3 The V-Realm Builder window (78)

Figure 3-4 The 3D studio model of XW craft after imported into the V-Realm Builder (79)

Figure 3-5 The 3D studio model of XW craft after a background is added (79)

Figure 3-6 Adding four ‘Transform’ (80)

Figure 3-7 Renaming the four ‘Transform’ and moving the ‘Wise’ (80)

Figure 3-8 Adding a dynamic observer (80)

Figure 3-9 Edit rotation (orientation) of the observer (81)

Figure 3-10 Edit position of the observer (81)

Figure 3-11 Edit description of the observer (82)

Figure 3-12 An example of an observer (82)

Figure 3-13 An example of an observer, Right Front Observer (82)

Figure 3-14 Final results of the Virtual World (83)

Figure 3-15 A new SIMULINK model with VR Sink (83)

Figure 3-16 Parameter window of VR Sink (84)

Figure 3-17 Parameter window of VR Sink after loading “wise8craftVR.wrl” (84)

Figure 3-18 The VR visualization window of WiSE-8 craft (85)

Figure 3-19 The VR parameter after VRML Tree editing (86)

Figure 3-20 The VR Sink after VR parameter editing (87)

Figure 3-21 The VR Transform subsystem (88)

List of Tables 7

Aircraft Flight Dynamics, Control and Simulation Using MATLAB and SIMULINK: Cases and

Algorithm Approach

Singgih Satrio Wibowo

L IST OF T ABLES

Table 1 Physical Parameter of The Yamaha R-50 (34)

Table 2 Parameter values of matrix A (35)

Table 3 Parameter values of matrix B (37)

List of Tables 8

Aircraft Flight Dynamics, Control and Simulation Using MATLAB and SIMULINK: Cases and

Algorithm Approach

Singgih Satrio Wibowo

Aircraft Dynamics and

Kinematics

9

Aircraft Flight Dynamics, Control and Simulation Using MATLAB and SIMULINK: Cases and

Algorithm Approach

Singgih Satrio Wibowo

1A IRCRAFT D YNAMICS AND K INEMATICS

Nature of Aircraft dynamics and kinematics in three-dimensional (3D)

space can be described by a set of Equations of Motion (EOM), which

contains six degrees of freedom: three translational modes and three

rotational modes. In the equations, it needs to define the forces and

moments acting on the vehicle since it is the factors responsible for

the motion. Therefore, the modeling of the forces and moments is a

must. The mathematical model of forces and moments include the

aerodynamic, propulsion system and gravity. These models will be

discussed in detail in this chapter.

In this chapter, first we briefly overview the coordinate systems that used as the reference frame for the description of aircraft motion.

Then, a complete nonlinear model of the aircraft motion will be

discussed briefly.

Aircraft Dynamics and Kinematics

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Aircraft Flight Dynamics, Control and

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Using MATLAB and SIMULINK: Cases and Algorithm Approach Singgih Satrio Wibowo 1.1 C OORDINATE S YSTEMS AND T RANSFORMATION

A number of coordinate systems will employed here to be use as a reference for the motion of the aircraft in three-dimensional space, ? Local horizon-coordinate system

? Body-coordinate system

? Wind-coordinate system

1.1.1 L OCAL H ORIZON C OORDINATE R EFERENCE S YSTEM

The local horizon coordinate system is also called the tangent-plane; it is a Cartesian coordinate system. Its origin is located on pre-selected point of interest and its h x , h y , h z axes align with the north, east and down direction respectively as shown in Figure 1-1.

F IGURE 1-1 L OCAL HORIZON COORDINATE SYSTEM

For simulation purpose, the local horizon local will be used as reference (inertial) frame. It is correct since the most of aircraft is flying in low altitude and range relative to the earth surface.

1.1.2 B ODY C OORDINATE R EFERENCE S YSTEM

e X Z

e Y

Aircraft Dynamics and Kinematics

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Aircraft Flight Dynamics, Control and

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Using MATLAB and SIMULINK: Cases and Algorithm Approach Singgih Satrio Wibowo The body coordinate system is a special coordinate system which represents the aircraft body. Its origin is attached to the aircraft center of gravity, see Figure 1-2. The positive b x axis lies along the symmetrical axis of the aircraft in the forward direction, its positive b y axis is perpendicular to the symmetrical axis of the aircraft to the right direction, and the positive b z is perpendicular to the b b ox y plane making the right hand orientation.

F IGURE 1-2 B ODY -COORDINATE SYSTEM

The transformation of body axes to the local horizon frame is carried out using Euler angle orientation procedures. The orientation of the body axes system to the local horizon axes system is expressed by Euler angles as shown in Figure 1-3.

b y b b x

Aircraft Dynamics and Kinematics 12

Aircraft Flight Dynamics, Control and Simulation

Using MATLAB and SIMULINK: Cases and Algorithm Approach

Singgih Satrio Wibowo

F IGURE 1-3 A IRCRAFT ATTITUDE WITH RESPECT TO LOCAL HORIZON FRAME :

E ULER ANGLES

The transformation of local horizon coordinate system to body coordinate system can be expressed as [2]

cos cos cos sin sin sin sin cos cos sin sin sin sin cos cos sin cos cos sin cos sin sin cos sin sin sin cos cos cos h b C θψ

θψ

θ

?θψ?ψ

?θψ?ψ?θ?θψ?ψ

?θψ?ψ

?θ--++-??

??=??????

(1-1)

The above formula is very useful for determining the orientation of

the aircraft with respect to the earth surface. This matrix is an orthogonal class of matrix, meaning that its inverse can be obtained by

transposing the matrix above as 1

T

b h h

h b b C C C -????==????.

1.1.3 W IND C OORDINATE S YSTEM

Wind coordinate system represents the aircraft velocity vector. This frame defines the flight path of the aircraft. The term ‘wind’ used here is relative wind flowing through the aircraft body as the aircraft fly in the air [2].

h x

h

Aircraft Dynamics and Kinematics

13

Aircraft Flight Dynamics, Control and

Simulation

Using MATLAB and SIMULINK: Cases and Algorithm Approach Singgih Satrio Wibowo Its origin is attached to the center of gravity while its axes define the direction and the orientation of flight path. The positive w x axis coincides to the aircraft velocity vector V . The w z axis lies on the symmetrical plane of the aircraft, perpendicular to the w x axis and positive downward. And the last, positive w y axis is perpendicular to the w w ox z plane obeying the right-hand orientation. These axes definition are shown in Figure 1-4.

F IGURE 1-4 W IND -AXES SYSTEM AND ITS RELATION TO B ODY AXES

Wind axes system can be transformed to the body axes system using the following matrix of transformation,

cos cos -cos sin -sin sin cos 0sin cos -sin sin cos w b C αβαβαββαβαβα????=?????? (1-2)

This equation is useful for transforming the aerodynamic lift and drag forces to body axes system. As can be seen in Figure 1-4, the

b y w

Aircraft Dynamics and Kinematics

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Aircraft Flight Dynamics, Control and

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Using MATLAB and SIMULINK: Cases and Algorithm Approach Singgih Satrio Wibowo aerodynamic lift vector is along the negative w z axis while the aerodynamic drag is along the negative w x axis. Since the equations of motion are derived in body axes system, it needs to express all forces and moments which acting on the aircraft in the body axes. Therefore the aerodynamic lift and drag vectors should be transformed from wind axes to the body axes.

F IGURE 1-5 A ERODYNAMIC LIFT AND DRAG

Using Equation (1-2), Aerodynamic lift and drag can be transformed to body axes system by the following relation

cos cos -cos sin -sin sin cos 00sin cos -sin sin cos X Y Z A A A F D F L F αβαβαββαβαβα??-??????????=????????????-?????? (1-3)

Similarly, after dividing Equation (1-3) by 212

T V S ρ, the aerodynamic coefficients can be expressed as

b

y w

Aircraft Dynamics and Kinematics

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Aircraft Flight Dynamics, Control and

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Using MATLAB and SIMULINK: Cases and Algorithm Approach Singgih Satrio Wibowo

cos cos -cos sin -sin sin cos 00sin cos -sin sin cos X D Y Z L C C C C C αβαβαββαβαβα-????????????=

????????????-?????? (1-4)

Equation (1-4) will be used for transforming aerodynamic lift and drag coefficients to body axes aerodynamic coefficients X C , Y C , and Z C .

The translational velocity can also be transformed into the body axes system as follows:

cos cos -cos sin -sin sin cos 00sin cos -sin sin cos 0cos cos sin sin cos T T T T U V V W V V V αβαβαββαβαβααββαβ????????????=

??????????????????????=?????? (1-5)

in which the total velocity T V

is defined as T V =of attack α, and angle of sideslip β can be derived from equation (2-9) as follows:

arctan arcsin T W U V V αβ??= ?????= ??? (1-6)

Equation (2-10) will also be used in the simulation for calculating angle of attack and sideslip angle from body axes velocity.

1.1.4 K INEMATICS E QUATION

Aircraft Dynamics and Kinematics

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Aircraft Flight Dynamics, Control and

Simulation

Using MATLAB and SIMULINK: Cases and Algorithm Approach Singgih Satrio Wibowo Kinematics equation shows the relation of Euler angles and angular velocity []T

b P Q R =ω. The physical definition of Euler angles can be seen in Figure 1-3. The kinematics equations are listed as follows: sin tan cos tan cos sin sin cos cos cos P Q R Q R Q R ?

?θ?θθ

????ψθθ

=++=-=+ (1-7)

The above equation can be rewritten in the form of matrix as 1sin tan cos tan 0

cos sin sin cos 0cos cos P Q R ??θ?θθ??ψ??θθ?

?????????????=-???????????????????

? (1-8)

Equations (2-2) and (2-3) are used to obtained the Euler angles from the angular velocity P , Q , and R . But the above equations have disadvantage, i.e. can be singular for θ = ± 90 degrees. It motivates to use another way that can avoid the singularity. This can be done using quaternion which will be discussed in the next section.

1.1.5 D IRECTION C OSINE M ATRIX

Intersection angle i θ of any two vectors in three-dimensional (3D) space, denoted by 1r and 2r , can be found by the inner product relationship:

Aircraft Dynamics and Kinematics

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Aircraft Flight Dynamics, Control and

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Using MATLAB and SIMULINK: Cases and Algorithm Approach Singgih Satrio Wibowo 1212arccos i θ??=?

?????r r r r (1-9)

Using above idea, the transformation coordinate from local horizon axes system (,,h h h i j z ) to body axes system (,,b b b i j z ) can be cast into the matrix form [48]:

h b h b h b h b h b h b h b h b h b DCM c c c s s s s c c s s s s c c s c c s c s s c s s s c c c θψθψθ?θψ?ψ?θψ?ψ

?θ?θψ?ψ?θψ?ψ

?θ????=??????

-?

???=-+????+-??i i i j i z j i j j j z z i z j z z (1-10)

where symbol ()()sin s ?=? and ()()cos c ?=? are used for abbreviation. Equation (1-10) is identical to Equation (1-1). Therefore the term DCM will be used together with the transformation matrix h b C in the simulation.

1.1.6 Q UATERNION S

Quaternions were discovered by Sir William Hamilton in 1843. He used quaternion for extensions of vector algebras to satisfy the properties of division rings (roughly, quotients exist in the same domain as the operands). It has been widely discussed as interesting topic in algebra and for its amazing applicability in dynamics.

The following paragraphs discuss the application of Quaternion starting with its definition while more detail discussion will be presented in Appendix C. Quaternion is define as

Aircraft Dynamics and Kinematics 18

Aircraft Flight Dynamics, Control and Simulation

Using MATLAB and SIMULINK: Cases and Algorithm Approach

Singgih Satrio Wibowo

[]01230

12

31T

q q q q q q q q =?+?+?+?=q i j k

(1-11)

where 0q , 1q , 2q , 3q are reals, 1 is the multiplicative identity element, and i , j , k are symbolic elements having the properties:

21=-i , 21=-j , 21=-k

===ij k jk i ki j

=-=-=-ji k kj i ik j

(1-12)

The time-derivative of the quaternion can be expressed as follows:

[]001122333210230110

3201230

0102012b K q q R Q P q q R P Q K q q Q P R q q P Q R q q q q P q q q q Q K q q q q R q q q q K εεεε=--????????????-??????=-????

??-??????????---??????

-????

??????-??????=-??????-??????????---????=-q

ψq Q ωq

(1-13)

where ()

2222

01231q q q q ε=-+++ is an error coefficient.

Obviously, integrating equation (1-13) is much more efficient

than (1-3) because it does not involve computationally expensive trigonometric functions. This integration can be evaluated using the following relation:

Aircraft Dynamics and Kinematics 19

Aircraft Flight Dynamics, Control and

Simulation

Using MATLAB and SIMULINK: Cases and Algorithm Approach

Singgih Satrio Wibowo

()00t

t t dt =+?q q q

(1-14)

where ()t q denotes quaternion at time t and 0q is initial quaternion calculated from initial Euler angles using Eqn. (1-17).

The rotational transformation matrix can be directly found with

quaternion:

()

()()()()()

222201231203130222221203021323012222130223010312

222222h b C DCM

q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q =??

+--+-?

?

=-+--+????+-+--?

?

(1-15)

Euler angles can be determined from the quaternion by comparing Eqn. (2-15) to Eqn. (2-1) which yields

()()()230122220

31213021203222201

232arctan arcsin 22arctan q q q q q q q q q q q q q q q q q q q q ?θψ??+=?

?+--????=--????

+=?

?+--??

(1-16)

This quaternion can also be expressed in terms of Euler angles as [8]:

Aircraft Dynamics and

Kinematics

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Aircraft Flight Dynamics, Control and Simulation Using MATLAB and SIMULINK: Cases and

Algorithm Approach

Singgih Satrio Wibowo

0 1 2 3cos cos cos sin sin sin 222222

sin cos cos cos sin sin 222222

cos sin cos sin cos sin 222222

cos cos sin sin sin cos 222222

q q q q

?θψ?θψ

?θψ?θψ

?θψ?θψ

?θψ?θψ

??

=±+

?

??

??

=±-

?

??

??

=±+

?

??

??

=±-

?

??

(1-17)

The above equations will be used in the simulation which will be conducted in this book.

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