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>

6 DOF Simulation

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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