final report
TRANSCRIPT
6 DOF Simulation
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DHRUBA
Dual(Ae & Av)/09/08
A4717209008
6 DOF Simulation
Table of Contents
1. Development of the 6DOF Nonlinear Model in MATLAB/Simulink
1.2Coordinate System
1.2.1 Euler Method
1.3 The 6DOF Equations of Motion System
1.4 Aerodynamics Module
1.5 Thrust Force Module
1.6 Gravity Module
List of Figures
Figure 1
Figure 2 Dhruba/Coefficients
Figure 2.1 dhruba/Coefficients/Body rate damping
Figure 2.1.1. dhruba/Coefficients/Body rate damping/Coefficient w. r. t r
Figure2.2 dhruba/Coefficients/Coefficient w r t actuatorsFigure 2.2.1 dhruba/Coefficients/Coefficient w r t actuators/Rudder/Angle
Conversion
Figure 2.2.2 dhruba/Coefficients/Coefficient w r t actuators/Coeff w r t elaevatorFigure 2.2.3 dhruba/Coefficients/Coefficient w r t actuators/Coefficient w rt
aileron
Figure 2.3 dhruba/Coefficients/Coefficient wrt to alpha beta
Figure: 3 Dhruba/gravity model
Figure:4 Dhruba/Flight Parameters
Figure: 5 Dhruba/Servo Command
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Acknowledgement
I express my sincere gratitude to, my mentor and guide, Prof M. S. Prasad for his able guidance,
continuous support and cooperation throughout my research, without which the present work
would not have been possible.
I would also like to thank all my friends for the constant support and help in the successful
completion of my project.
And above all, to the Almighty God, whose never ceaseing love and for the continued
guidance and protection.
Thank you to all mentioned above.
SIGNATURE:
(STUDENT.)
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1 Development of the 6DOF Nonlinear Model in MATLAB/Simulink
In this chapter, the development of a generic 6DOF nonlinear model is presented. This 6DOF nonlinear model is constructed using the stability derivatives of aircraft. This 6DOF model is built based on the MATLAB/Simulink platform. The structure of the 6DOF model can be broken in several modules as illustrated in Figure 1. Discussions in this chapter are associated with each individual module. Appendix A shows the details of the 6DOF model’s construction layer by layer.
Figure 1
1.1 Coordinate System
The Earth-Centered Earth-Fixed (ECEF) coordinate system is used in constructing the 6DOF nonlinear model. The origin of the ECEF coordinate system is defined at the center of the Earth. This ECEF frame is rotating about the Earth-Centered Inertial (ECI) reference frame. The representation of the ECEF frame from the ECI frame is simplified by considering only a constant rotation about the Z-axis. The body of interest, the aircraft, is assumed to be rigid and is represented in the ECEF frame. The ECEF coordinate system is implemented in the “6DOF ECEF (Euler)” block from the Aerospace Blockset in MATLAB/Simulink as shown in Appendix A.
1.2Aircraft Attitude Representations In aircraft simulations, there is a need for a mathematical technique to describe the position and orientation of the body of interest in both inertial and non-inertial coordinate systems. Various methods have been developed to determine the aircraft’s attitude representation. These include the well-known Euler angles the Euler-Axis rotation parameters, the direction cosines
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1.2.1 Euler Method
The Euler method is a well-known approach used for aircraft attitude representations and computations. The Euler method is defined based on the definition of Euler angle representations which can be found in many texts such as References. Using the Euler method, a vector transformation from the Earth-fixed axis to the Body-fixed axis and vice versa can be obtained through the transformation matrix shown in Eq.1 . The notations used in this equation are defined as follows: Sx = Sin(x) and Cx = Cos(x). The subscript “b” refers to the Body-fixed frames, and the subscript “f” refers to the Earth-fixed frame.
Equation 1
Since the Euler angles are functions of time, they need to be updated at each time step throughout the simulation process. The Euler kinematic equations are introduced for this purpose. Using the numerical integration method, the equations can be solved to compute the rate of change of the Euler angles in each given time step. The Euler kinematic equations are
shhown below.
Equation 2
Note that when the pitch angle (θ) is equal to ±90 degrees, the kinematic equations cannot be defined and thus cannot be solved. This is known as the gimbal lock singularity problem.
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Equation 5
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1.3 The 6DOF Equations of Motion System
The core of the 6DOF nonlinear model are the EOM used in this simulation. In this 6DOF model, the standard six degrees of freedom nonlinear differential equations for a conventional fixed wing aircraft are used. These 6DOF nonlinear differential equations are written as
Forces Equations:
Moment Equations:
From a mathematical viewpoint, Eq.4 and 5 and form a set of six nonlinear differential equations with six unknown variables (u, v, w, p, q, r). Each variable is presented in different equations and interacts with the others. Therefore, the equations cannot be solved individually. The total solution for the system can be obtained only by applying numerical integration to all equations for each given time step.
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Equation 4
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For the force equations, the inputs for the system are the three major forces applied to the aircraft. These are the aerodynamic forces, thrust forces, and gravity forces. These forces are nonlinear and time variant. The components of each force are broken down in detail and are discussed in Section , Section and Section
The inputs for the moment equation are the three moment terms applied to the three aircraft Body-fixed axes. These moments are generated from the aerodynamic and thrust forces with respect to the aircraft’s center of gravity (C.G.). As the gravity forces are defined at the C.G. location, no moment is introduced by the gravity forces. The assumption is made that the thrust line passes through the C.G. Therefore, the terms that contribute to the moment equations are only associated with the aerodynamic forces. These aerodynamic moment terms are broken down one-by-one and are presented in Section. The variables P, Q, R from the 6DOF equations are applied to the Euler kinematic equations Eq. 6-8 to update the euler terms. The updated euler terms are then used for vector coordinate transformations. The 6DOF nonlinear equations are implemented in the “6DOF ECEF (Euler)” block from the Aerospace Blockset in MATLAB/Simulink as shown in Appendix A.
1.4 Aerodynamics Module As previously mentioned, the aerodynamics forces are one of the major forces applied to the aircraft, and these forces create aerodynamic moments that contribute to the moment equations. The main purpose of this aerodynamics module is to estimate the values of the aerodynamic forces and moments in the aircraft body frame. A high level block diagram for this module can be simplified as seen in Figure
Figure 2
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The aerodynamic forces are composed of three forces, which are the lift, drag, and sideforce. These forces generate moments with respect to the center of gravity about the X, Y, Z-axis and are described as the rolling moment, pitching moment, and yawing moment. The component build-up method is used to generate the forces and moments. The total forces and moments
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that act on the aircraft are simply the summation of the forces and moments contributed by each component. The aerodynamic forces and moments are first described as dimensionless coefficients, which are associated with the stability and control derivatives given in the modeling results. Equations above are used to implement the aerodynamic force and moment coefficients. Note that when modeling the aircraft in this study, only one set of derivative values at the trim speed. The six aerodynamic force and moment equations are implemented in the subsystem “Aerodynamics Coefficients” as shown on Appendix . This subsystem is broken down into three different subsystems They are:
1) Coefficient w.r.t. Body Rate,
2) Coefficient w.r.t. Alpha and Beta Angles, and
3) Coefficient w.r.t. Deflection Angles for each Control Surface,
which creates the inputs for equations shown below. Starting with the first vector, they are the drag coefficient, sideforce coefficient, lift coefficient, rolling moment coefficient, pitching moment coefficient, and yawing moment coefficient equations, respectively. They are illustrated on pages through in Appendix . The final dimensional values of the forces and moments are then computed from the dimensionless coefficients using the following equations.
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The force components shown in Eqs, ,are defined in the stability axis. To make them useable within Eq., which is defined in the Body-fixed axis, a coordinate frame transformation is needed to transfer the stability axis forces to Body-fixed axis forces using the equations expressedbelow.
The implementation of force and moment equations is within the “Aerodynamic Forces and Moments” block from the Aerospace Blockset in MATLAB/Simulink and is shown on page in Appendix A .
1.5 Thrust Force Module
Thrust force is assumed to be constant at trim condition. The implementation of Thrust force in MATLAB/Simulink and is shown on page in Appendix A .
1.6 Gravity Module Gravity is the last major force that is applied to the aircraft system. In most simple simulation models, gravity is always assumed to be constant regardless of the aircraft position and altitude with respect to the Earth. The WGS84 Gravity Model block implements the mathematical representation of the geocentric equipotential ellipsoid of the World Geodetic System (WGS84). The block output is the Earth's gravity at a specific location.
The WGS84 gravity calculations are based on the assumption of a geocentric equipotential ellipsoid of revolution. Since the gravity potential is assumed to be the same everywhere on the ellipsoid, there must be a specific theoretical gravity potential that can be uniquely determined from the four independent constants defining the ellipsoid.
Use of the WGS84 Taylor Series model should be limited to low geodetic heights. It is sufficient near the surface when submicrogal precision is not necessary. At medium and high geodetic heights, it is less accurate.
Use of the WGS84 Close Approximation model should be limited to a geodetic height of 20,000.0 m (approximately 65,620.0 feet). Below this height, it gives results with submicrogal precision.
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1.7 Atmosphere Module
During the simulation process, updated atmospheric data are frequently required to give feedback to some of the modules in order to provide the necessary computational information. For this reason, an atmosphere module is required to provide the latest atmospheric data for the current altitude.
The ISA Atmosphere Model block implements the mathematical representation of the international standard atmosphere values for ambient temperature, pressure, density, and speed of sound for the input geopotential altitude.
The use of this “ISA Atmosphere” model is shown on page in Appendix A.
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APPENDIX A
6 DOF Non Linear Model developed using Matlab/Simulink
Figure 1
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thrust
25000
gravity model
PLANT BUSGravity Force
Servo Command
Pilot CommandActuator
Scope3
Scope2
Scope1Scope
Pilot command
Pilot command
Flight Parameters
PLANT BUS
Aerodynamic angles
Total Air Speed
Dynamic Pressure1
Mach
Temperature
Pressure
Constant1
[6; 0; 0]
Constant
[3x1]
Coefficients
Flight Parameter Bus
PLANT BUS
Actuator
coefficient out
AerodynamicForces and Moments
Coefstab
qbar
CGCPVb
Fbody
Mbody
Add
6DoF (Euler Angles)
Fxyz
(lbf)
Mxyz
(lbf-ft)
Ve (ft/s)
Xe (ft)
(rad)
DCMbe
Vb (ft/s)
(rad/s)
d
/dt
Ab (ft/s
2)
BodyEuler Angles
FixedMass
<signal3>
<signal5>
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Dhruba/Coefficients
Figure 2
coefficient out
1
Product
Constant
.0478Coefficient wrt to alpha beta
Flight Parameters Coeff wrt alpha beta
Coefficient w r t actuators
Actuator Coefficient w r t actuators
Body rate damping
Plant Bus
Flight Parameters
Body Rate Damping
Add
Actuator3
PLANT BUS2
Flight Parameter Bus1
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Body rate damping
Figure 2.1 dhruba/Coefficients/Body rate damping
Body Rate Damping
1
Product2
Product1
Product
Gain2
-K-
Gain1
-K-
Gain
-K-
Div ide
Coeffic ient w.r t q
Coeffic ient w. r. t r
Coeffic ient w. r. t p
Add
Flight Parameters
2
Plant Bus
1 <signal6>
<signal2>
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Coefficient w. r. t r
Figure 2.1.1. dhruba/Coefficients/Body rate damping/Coefficient w. r. t r
Coeff w.r.t r
1
Cyr
0
Constant
0
Cnr
-.325
Clr
.20
Coefficient w.r t q
Figure 2.1.2. dhruba/Coefficients/Body rate damping/Coefficient w.r t q
Coeff w.r.t q
1
Constant
0
Cmq
-20.8
Clq
6.58
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Coefficient w r t actuators
Figure2.2 dhruba/Coefficients/Coefficient w r t actuators
Coefficient w r t actuators
1
Rudder
Actuator Coeff w.r.t rudder
Coefficient w rt aileron
Actuator Coeff w.r.t aeleron
Coeff w r t elaevator
Actuator Coeff w.r.t elevator
Actuator1
<signal1>
<signal2>
<signal3>
Angle Conversion
Figure 2.2.1 dhruba/Coefficients/Coefficient w r t actuators/Rudder/Angle Conversion
Out11
Unit Conversion
slopeIn11
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Coeff w r t elaevator
Figure 2.2.2 dhruba/Coefficients/Coefficient w r t actuators/Coeff w r t elaevator
Coeff w.r.t elevator
1
Product2
Product1
Product
Constant
0
Cmde
-1.2
Clde
.3
Cdde
0
Angle Conversion
deg radActuator
1
Coefficient w rt aileron
Figure 2.2.3 dhruba/Coefficients/Coefficient w r t actuators/Coefficient w rt aileron
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Coeff w.r.t aeleron
1
Product5
Product4
Product3Cyda
.0000
Constant
0
Cnda
.003
Clda
.014
Angle Conversion
deg radActuator1
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Coefficient wrt to alpha beta
Figure 2.3 dhruba/Coefficients/Coefficient wrt to alpha beta
Coeff wrt alpha beta
1
coefficient w r t zero alpha
Coeff w.r.t alpha
Product2
Product1
Product
Gain
-K-
DivideDerivative
du/dt
Coefficient w.r t beta
Coefficient w.r t alphadot
Coefficient alpha
Add1
Flight Parameters1<signal1>
<signal1>
<signal2><signal2>
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Coefficient alpha
Figure 2.3.1 dhruba/Coefficients/Coefficient wrt to alpha beta/Coefficient alpha
Coeff w.r.t alpha
1
Constant
0
Cma
-1.6
Cla
5.5
Cda
.47
Coefficient w r t zero alpha
Figure 2.3.2. dhruba/Coefficients/Coefficient wrt to alpha beta/coefficient w r t zero alpha
Coeff w.r.t alpha
1
Constant
0
Cm0
-.1
Cl0
.5
Cd0
.042
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Coefficient w.r t alphadot
Figure 2.3.3 dhruba/Coefficients/Coefficient wrt to alpha beta/Coefficient w.r t alphadot
dot
Coeff w.r.t alphadot
1
Constant
0
Cma
-9.0
Cladot
.006
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Coefficient w.r t beta
Figure 2.3.4. dhruba/Coefficients/Coefficient wrt to alpha beta/Coefficient w.r t beta
Coeff w.r.t beta
1
Cyb
-.85
Constant
0
Cnb
-.20
Clb
-.10
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Dhruba/gravity modelFigure: 3
Gravity Force
1
boeing 747
636600
WGS84 Gravity Model
WGS84
(Taylor Series) l h (m) g (m/s
2)
Reshape
U( : )
Product1
Product
Matrix
Multiply
Math
Function
uT
Flat Earth to LLA
X e
href
l
h
Constant
0
PLANT BUS
1
<signal2>
<signal4>
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Dhruba/Flight Parameters
Figure:4
Pressure
6
Temperature
5
Mach
4
Dynamic Pressure1
3
Total Air Speed
2
Aerodynamic angles
1
Selector
Scope3
Mach Number
V
aMach
Incidence, Sideslip,& Airspeed
V b
V
ISA Atmosphere Model
h (m)
T (K)
a (m/s)
P (Pa) (kg/m
3)
ISA
Dynamic Pressure
V
q
1/2
V2
PLANT BUS
1
<signal2>
<signal5>
Dhruba/Servo Command
Figure: 5
Actuator
1
Transfer Fcn2
20
s+20
Transfer Fcn1
20
s+20
Transfer Fcn
20
s+20
Pilot Command
1
<signal1>
<signal2>
<signal3>
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Appendix B
Result for simulation:
For our simulation we have considered boeing 747. We require stability derivatives ,cg location ,cp location aerodynanmic center , weight, inertia tensor, chord, span ,aspect ratio. We got these values from r.c nelson’s flight stability and control. We consider trim condition, where the flight is at an altitude of 20000 feet and a constant thrust 0f 13000 lbs. We are giving pilot command of elevator deflection of 2 degree for 2 seconds and analyse the response of aircraft which is shown by the graphs below.
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Future Work
The USAF digital DATCOM program should be used as another modeling method to potentially improve the models used in this research.
The 6DOF model can be interfaced with other existing autopilot systems to develop a hardware-in-the-loop simulation platform.
References
1. Center of Remote Sensing of Ice Sheets Research Institute, Lawrence, KS, URL: http://www.cresis.ku.edu [cited 19 September 2008].
2. Bernstein, L., Bosch, P., et. al., “Climate Change 2007: Synthesis Report. Summary for Policy Markers,” Intergovernmental Panel on Climate Change [online publication], URL: http://www.ipcc.ch/pdf/assessment-report/ar4/syr/ar4_syr_spm.pdf [cited 20 August 2008].
3. Donovan, W. R., “The Design of an Uninhabited Air Vehicle for Remote Sensing in the Cryosphere,” Master’s Thesis, Department of Aerospace Engineering, The University of Kansas, Lawrence, KS, 18 December 2007.
4. Donovan, W. R., “CReSIS UAV Critical Design Review: The Meridian,” Technical Report 123, Center for Remote Sensing of Ice Sheets, The University of Kansas, Lawrence, KS, 25 June 2007.
5. Underwood, S., “Performance and Emission Characteristics of an Aircraft Turbo diesel Engine using JET-A Fuel,” Master’s Thesis, Department of Aerospace Engineering, The University of Kansas, Lawrence, KS, 05 May 2008.
6.Anderson, J. D., and Wendt, J. F., Computational Fluid Dynamics: An Introduction, 2nd ed., Springer-Velag, New York, 1996.
7. Anderson, J. D., Computational Fluid Dynamics: The Basics with Applications, McGraw Hill Inc., 1995.
8. Phillips, W. F., Mechanics of Flight, John Wiley & Sons, Hoboken, NJ, 2004.
9. Pamadi, B. N., Performance, Stability, Dynamics, and Control of Airplanes, 2nd edition, AIAA Education Series, Reston, VA, 2004.
10. Roskam, J., Aircraft Flight Dynamics and Automatic Flight Controls (Part I), DAR Corporation, Lawrence, KS, 2003.
11,MATLAB/Simulink, Software Package, Version R2012-a, The MathWorks Inc., Natick, MA.
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