A SIX-DEGREE-OF-FREEDOM LAUNCH VEHICLE SIMULATOR FOR...

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A SIX-DEGREE-OF-FREEDOM LAUNCH VEHICLE SIMULATOR FOR RANGE SAFETY ANALYSIS

By

SHARATH CHANDRA PRODDUTURI

A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2007

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©2007 Sharath Chandra Prodduturi

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To my parents.

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ACKNOWLEDGMENTS

I would like to express my sincere gratitude to my supervisory committee chair (Dr.

Norman G. Fitz-Coy) for his continuous guidance, support, and help. I am really thankful to

him. I would also like to express my gratitude to my supervisory committee members (Dr.

Warren E. Dixon and Dr. Gloria J. Wiens) for their support and guidance.

I would like to express my gratitude to my parents for all their moral and financial support,

without which this task could not have been accomplished. I would be nowhere without them. I

would like to acknowledge my sisters (Shirisha and Swetha) for their help and support

throughout my life.

I would like to thank my friends and colleagues from AMAS (Frederick Leve, Shawn

Allgeier, Sharan Asundi, Takashi Hiramatsu, Jaime José Bestard, Andrew Tatsch, Andrew

Waldrum, Ai-Ai Cojuangco, Dante Buckley, Nick Martinson, Josue Munoz, Jessica Bronson and

Gustavo Roman) for their advice, help and support.

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TABLE OF CONTENTS page

ACKNOWLEDGMENTS ...............................................................................................................4

LIST OF FIGURES .........................................................................................................................7

ABSTRACT.....................................................................................................................................9

CHAPTER

1 INTRODUCTION AND BACKGROUND ...........................................................................11

2 EQUATIONS OF MOTION FORMULATION ....................................................................19

Coordinate Frames..................................................................................................................19 Kinematic Equation of Motion ...............................................................................................24 Dynamical Equations..............................................................................................................27 Generalized External Forces...................................................................................................30

External Forces................................................................................................................30 Thrust force ..............................................................................................................30 Aerodynamic forces (drag and lift) ..........................................................................32 Gravitational force....................................................................................................33

External Moments ...........................................................................................................34 Aerodynamic moments ............................................................................................34 Gravitational moment...............................................................................................35 Thrust moment .........................................................................................................36

3 DESCRIPTION OF MODELS USED ...................................................................................38

Gravity Model.........................................................................................................................38 Inertia Model ..........................................................................................................................49

Strap-on booster...............................................................................................................50 Cylindrical segment..................................................................................................50 Parabolic nose cone..................................................................................................52 Fins ...........................................................................................................................54

Liquid Engine ..................................................................................................................57 Solid Motor......................................................................................................................59 Payload ............................................................................................................................61

Drag Coefficient Model..........................................................................................................63 Center of Pressure Model .......................................................................................................64

Nose.................................................................................................................................66 Cylindrical Body .............................................................................................................67 Conical Shoulder .............................................................................................................67 Conical Boattail ...............................................................................................................68 Fins (Tail Section) ...........................................................................................................68

The WGS84 Ellipsoid Model .................................................................................................69

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4 SIMULATION RESULTS AND DISCUSSION...................................................................73

Simulation...............................................................................................................................73 Validation ...............................................................................................................................87

5 CONCLUSION AND FUTURE WORK ...............................................................................91

Conclusions.............................................................................................................................91 Future work.............................................................................................................................92

APPENDIX

A MATLAB FUNCTIONS AND SCRIPT................................................................................93

B SIMULATION CONFIGURATION....................................................................................116

LIST OF REFERENCES.............................................................................................................132

BIOGRAPHICAL SKETCH .......................................................................................................135

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LIST OF FIGURES

Figure page 1-1 Space-based range and range safety, today and future ......................................................13

2-1 Relative orientation of the various frames .........................................................................21

2-2 Euler angles and the relative orientation between the vehicle frame and the vehicle-centered horizontal frame ..................................................................................................24

2-3 Geometry of the launch vehicle and various position vectors ...........................................28

2-4 External forces acting on a launch vehicle during its flight...............................................31

3-1 Representation of a position vector in Cartesian and Spherical coordinates .....................42

3-2 Cylindrical segment of the strap-on booster ......................................................................51

3-3 Parabolic nose cone............................................................................................................53

3-4 Fin ......................................................................................................................................55

3-5 Liquid engine .....................................................................................................................58

3-6 Solid motor.........................................................................................................................60

3-7 Payload...............................................................................................................................62

3-8 Conical shoulder ................................................................................................................67

3-9 Conical Boattail .................................................................................................................68

3-10 Fin and Tail section............................................................................................................69

3-11 Geodetic Ellipsoid and Geodetic coordinates of an arbitrary point “P” ............................70

4-1 Various parameters of the launch vehicle as a function of time ........................................77

4-2 Velocity of the launch vehicle in the inertial frame...........................................................79

4-3 Position of the launch vehicle in the inertial frame ...........................................................80

4-4 Launch vehicle during the time of launch as seen from the J2000 inertial frame .............80

4-5 Moments of inertia of the launch vehicle about its instantaneous center of mass .............81

4-6 Moments of inertia of the strap-on booster about the instantaneous center of mass of the launch vehicle and about its instantaneous center of mass ..........................................82

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4-7 Moment of inertia of the first stage about the instantaneous center of mass of the launch vehicle and about its instantaneous center of mass ................................................84

4-8 Moment of inertia of the second stage about the instantaneous center of mass of the launch vehicle and about its instantaneous center of mass ................................................85

4-9 Moment of inertia of the third stage about the instantaneous center of mass of the launch vehicle and about its instantaneous center of mass ................................................86

4-10 The need for instrumental data or thrust vector in the vehicle frame ................................90

B-1 The DELTA II Launch vehicle geometry........................................................................116

B-2 Strap-on booster geometry...............................................................................................116

B-3 Elements of DELTA II Launch vehicle and Strap-on Booster ........................................120

B-4 Cylindrical shell ...............................................................................................................121

B-5 Propellant shell.................................................................................................................122

B-6 Parabolic nose cone..........................................................................................................123

B-7 Fins...................................................................................................................................124

B-8 First stage .........................................................................................................................125

B-9 Second stage.....................................................................................................................127

B-10 Third stage .......................................................................................................................128

B-11 Payload.............................................................................................................................130

B-12 Strap-on boosters around the Rocket ...............................................................................130

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Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the

Requirements for the Degree of Master of Science

A SIX-DEGREE-OF-FREEDOM LAUNCH VEHICLE SIMULATOR, FOR RANGE SAFETY ANALYSIS

By

Sharath Chandra Prodduturi

August 2007

Chair: Norman G. Fitz-Coy Major: Mechanical Engineering

Failure of a launch vehicle during its launch or flight might pose a hazard to the general

public. The United States Air Force Space Command (USAFSC) operates the United States

launch facilities and ensures safety to the general public, launch area and personnel, and foreign

land masses in case of such a failure. To ensure safety, USAFSC currently uses extensive

ground-based systems, which are expensive to maintain and operate and are limited to the

geographical area. To overcome these drawbacks, NASA proposed a concept called Space-

Based Telemetry and Range Safety (STARS) which uses space-based assets to ensure safety.

The STARS concept requires support tools in the form of simulation softwares that provide the

ability to quickly analyze new (or changes in) concept and ideas, an option not easily

accomplished with hardware only. Trajectory and link margin analysis tool is one of these

crucial support tools required by STARS.

My study focused on modeling the full dynamics of a launch vehicle and development of a

MATLAB based six-degree-of-freedom simulator for generating nominal and off-nominal

trajectories as part of the trajectory and link margin analysis. In my study, the J2000 coordinate

frame and the vehicle-centered horizontal frame were used as the reference frames to define the

position and orientation of a launch vehicle, respectively. Orientation and the kinematic

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equation of a launch vehicle are expressed in terms of quaternions instead of Euler angles, to

avoid intensive computations and singularities. The equations of motions of a launch vehicle are

developed by accounting for the variability in its mass and geometry. Various models are

developed for calculation of quantities such as gravity, inertia, center of pressure and drag

coefficient required for solving the equations of motion. The developed gravity model uses the

spherical harmonic representation of the gravitational potential to account for the variability in

Earth’s mass distribution and uses EGM96 (360 X 360) spherical harmonic coefficients and

WGS84 Earth ellipsoid model. The gravity model is singularity-free and numerically efficient.

A novel way of calculating the variable mass/inertial properties of a launch vehicle was

developed. This inertia model is a simple and approximate model and considers general

geometries to develop the inertia characteristics of a launch vehicle. The drag coefficient model

from the Missile Datcom database is used in this research. The kinematic equations, dynamic

equations, gravity, inertia, center of pressure, drag coefficient and other models are implemented

in MATLAB to form a six-degree-of-freedom launch vehicle simulator. The results and

discussions of a simulation performed using the developed simulator are presented in this thesis.

A validation of the developed simulator was attempted with flight data available from NASA

Kennedy Space Center; however, critical data needed for the validation could not be provided

due to ITAR restrictions.

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CHAPTER 1 INTRODUCTION AND BACKGROUND

Ensuring safe, reliable and affordable access to space is the fundamental goal of the U.S.

range safety program [23]. The Public Law 60 established the national range system based on

two primary concerns/factors: location and public safety. Thus, Range Safety, in the context of

national range activities, is rooted in PL 60 [14].

Range is defined to be the volume through which the launch vehicle must pass in order to

reach its destination from the launch point, and its projection on earth (in case of a space vehicle,

the destination can be outer space or a location on earth) [26]. A range includes space, facilities,

equipment and systems necessary for testing and monitoring launches, landing and recovery

operations of launch vehicles and other technical and scientific programs and activities [18].

The United States launch facilities are divided into Eastern and Western Ranges. The Air

Force Space Command operates the launch facilities of the United States. The 30th and the 45th

space wings manage and operate the Western Range and Eastern Range respectively. The

Eastern Range comprises of Cape Canaveral Air Station and its owned or leased facilities and

encompasses the Atlantic Ocean, including all surrounding land, sea, and air space within the

reach of any launch vehicle extending eastward into the Indian and Pacific Oceans. The Western

Range comprises of Vandenberg Air Force Base (VAFB) and its owned or leased facilities and

encompasses the Pacific Ocean, including all surrounding land, sea, and air space within the

reach of any launch vehicle extending westward through the Pacific and Indian Oceans[14].

The Eastern and Western Ranges, using a Range Safety Program provide safety to the

public by ensuring that the risk to the general public from launch and flight of launch vehicles

and payloads is no greater than that imposed by the over flight of conventional aircraft. Apart

from public protection, the national range safety includes launch area safety, launch complex

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safety, and the protection of national resources [14]. The objective of the Range Safety Program

as stated in “Eastern and Western Range 127-1, Range Safety Requirements” [14] is “The

objective of the Range Safety Program is to ensure that the general public, launch area personnel,

foreign land masses, and launch area resources are provided an acceptable level of safety and

that all aspects of pre-launch and launch operations adhere to public laws and national needs.

The mutual goal of the Ranges and Range Users shall be to launch launch vehicles and payloads

safely and effectively with commitment to public safety” (14, p. 1-5).

Range safety personnel evaluate vehicle design, manufacture and installation prior to

launch and monitor vehicle and environmental conditions during countdown. Range safety

personnel also monitor the performance of launch vehicles in flight and are responsible for their

remote destruction/termination if it should be judged that they pose a hazard. For all vehicle

termination cases, propulsion is terminated and based on the vehicle type, stage of flight, and

other circumstances of failure, the method of termination might vary. Depending on factors like

geographic location and population, the vehicle may be destroyed to disperse the propellants

before surface impact, or it may be kept intact to minimize the debris footprint. The launch

vehicle is also equipped with a break-wire or lanyard pull to initiate a flight termination in case

of a premature stage separation [23].

Extensive ground-based systems are utilized by the current United States Eastern and

Western Ranges for real-time tracking, communications, and command and control of the launch

vehicles. These ground-based assets are very expensive to maintain and operate and are limited

to the geographical area [31]. Therefore the current range systems need to be upgraded or

replaced. According to Whiteman et al. [31], NASA Dryden Flight Research Center, “Future

spaceports will require new technologies to provide greater launch and landing opportunities,

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support simultaneous missions, and offer enhanced decision support models and simulation

capabilities. These ranges must also have lower costs and reduced complexity, while continuing

to provide unsurpassed safety to the public, flight crew, personnel, vehicles, and facilities.

Commercial and government space-based assets for tracking and communications offer many

attractive possibilities to help achieve these goals” (31, p. 2).

Figure 1-1 shows the current primary Eastern and Western Ranges instrumentation sites

(solid lines) and a possible future space-based configuration with fewer ground-based assets

(dashed lines). From Fig 1-1, it should be noted that the future space-based configuration might

still include some launch-head ground-based assets for visibility and rapid response times shortly

after liftoff [31].

Figure 1-1. Space-based range and range safety, today and future. Reprinted with permission

from D. E. Whiteman, L. M. Valencia, and J. C. Simpson, “Space-Based Range Safety and Future Space Range Applications,” NASA Dryden Flight Research

Center, Edwards, California. Rep. H-2616, NASA TM-2005-213662, 2005.

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Space-Based Telemetry and Range Safety (STARS)

Space-Based Telemetry and Range Safety (STARS) is a multifaceted and multi-center

project to determine the feasibility of using space-based assets, including the Tracking and Data

Relay Satellite System (TDRSS) and Global Positioning System (GPS), to reduce operational

costs and increase reliability. The STARS study was established by the National Aeronautics

and Space Administration (NASA) to demonstrate the capability of space-based assets to provide

communications for Range Safety (low-rate, ultra-high reliability metric tracking data, and flight

termination commands) and Range User (video, voice, and vehicle telemetry) [31]. To support

the envisioned future space range, new and improved systems with Range Safety and Ranger

User capabilities are under testing and development. A brief description of the planned and

completed phases of the STARS project is given below [31], [30], [10], [21].

Phase 1

• Developed and tested a new S-band Range Safety system.

• During June-July 2003, seven test flights were performed on a F-15B aircraft at Dryden Flight Research Center using a Range User system representative of those on the current launch vehicles.

• Successfully demonstrated the basic ability of the STARS to establish and maintain satellite links with TDRSS and the GPS.

Phase 2

• The objective is to increase the Range User data rates by an order of magnitude by enhancing the S-band Range Safety system and a new telemetry system which utilizes a Ku-band phased-array antenna.

• TDRSS is the space-based communication link (i.e., TDRSS provides the tracking and data acquisition services between the launch vehicle/low earth orbiting spacecraft and NASA/customer control and data processing facilities [22]).

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Phase 3

• Phase 3 uses a small, lightweight hardware compatible with a fully operational system and demonstrates the ability to maintain a Ka-band TDRSS communications link during a hypersonic flight.

• Develop smaller, lighter version of the Range Safety Unit for the Range Safety system in fiscal year 2006.

• TDRSS is the space-based communication link.

• Test flights are planned for late fiscal year 2007.

• Space Based Range Safety system will be complete by the completion of Phase 3 development.

Phase 4

• Develop Ka-band transmitter (NASA) and phased array antenna (AFRL) for Range User system in fiscal year 2006-2007.

• Perform flight test on aircraft (Flight Demo 3a) to test performance of Glenn Research Center’s (GRC) Ka-band active phased array antenna in fiscal year 2007.

• Perform flight test of Ka-band system on F-15B in fiscal year 2008.

• Re-fly phase 3 Range Safety Unit design with enhancements.

Certification Phase

• Perform Certification of Range Safety and Range Users systems in fiscal year 2009–2011.

The STARS program was renamed to Space Based Range Demonstration and Certification

(SBRDC) program [20]. From the available information on the World Wide Web/internet,

Phases 1, 2 and 3 are completed and the current status of the STARS/SBRDC program is as

stated in Phase 4 above [19].

The STARS concept requires support tools in the form of simulation softwares which

provide the ability to quickly analyze new (or changes in) concepts and ideas, an option not

easily accomplished with hardware only. Trajectory and link margin analysis tool is one of these

crucial support tools required by STARS. The trajectory “portion” of the trajectory and link

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margin analysis involves generating trajectories (and orientation) of a launch vehicle. The link

margin “portion” involves calculating the telemetry link margin during the flight of a launch

vehicle. Link margin is defined as the difference in dB, between the magnitude of the received

signal at the receiver input and the receiver sensitivity (i.e., the minimum level of signal required

for reliable operation). The higher the link margin, the more reliable the communications link

[24]. The trajectory and orientation of the launch vehicle calculated using the trajectory

“portion” and dynamic parameters such as vehicle antenna patterns, locations of ground stations

and others are taken into account in order to compute the link margin. Trajectory and link

margin analysis is frequently required to ensure adequate link closure for range safety [15], [8].

Trajectory and link margin analysis involves simulating the launch vehicle for various

failure scenarios and checking if the command uplink can be closed with sufficient margin under

the worst possible conditions and from any intended ground site(s). The worst possible failure

scenario includes trajectories that result due to total loss of control of the launch vehicle. These

trajectories might include a sudden heading change to a populated area or may consist of series

of tumbles [8].

The Space Systems Group (University of Florida) and UCF collaborated to develop a

MATLAB based tool for trajectory and link margin analysis. The Space Systems Group is

responsible for modeling the dynamics of a launch vehicle while UCF is responsible for the

communications link model. The thrust of this thesis is to develop a MATLAB based launch

vehicle model/simulator which is capable of simulating a launch vehicle in flight.

Nominal launch vehicle trajectory simulation models have been done by many researchers.

Researchers have also attempted to simulate the off-nominal trajectory of launch vehicles; e.g.,

Chen et al. [8] has suggested a three-step algorithm to estimate the deviation of the launch

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vehicle from the nominal trajectory. In their approach, the sudden accelerations are treated as

the artificial maneuver controls, focusing on kinematics instead of dynamics. This research

intends to model the full dynamics of the launch vehicle.

In Chapter 2, the equations of motions of the launch vehicle are developed. The

definitions of the various coordinate frames used in the development of the simulator and the

transformations between them are discussed in detail. Following the above, the development of

the kinematic and dynamic equations of motion of the launch vehicle is presented. Finally, the

various external forces and external moments (acting on the launch vehicle) to be included in the

external force and moment terms in the equations of motion of the launch vehicle are discussed.

In Chapter 3, the models used in the development of the simulator are presented. The

development of the gravity, inertia, drag coefficient, center of pressure and WGS84 ellipsoid

models is presented in detail in this chapter. The gravity model computes the acceleration due to

gravity of Earth at a point of interest using its ECEF coordinates; the inertia model computes the

mass properties of a launch vehicle as a function of time; the drag coefficient model computes

the coefficient of drag for the launch vehicle as a function of position and velocity; the center of

pressure model computes the center of pressure for a specific geometry of the launch vehicle; the

WGS84 ellipsoid model defines a reference Earth ellipsoid and is used to compute the altitude of

a point of interest using its ECEF coordinates.

In Chapter 4, the simulation results are presented and discussed. Simulation results of a

DELTA II launch vehicle model for a fictitious thrust profile (constant axial thrust) are

discussed. Following the above, the details of the attempt to validate the developed simulator

with flight data available from NASA Kennedy Space Center were presented. It is shown that

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the simulator cannot be validated due to the lack of availability of critical data (an ITAR1 issue).

Finally, in Chapter 5, the conclusions of this research and the possible future work are discussed.

1 ITAR – International Traffic in Arms Regulations

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CHAPTER 2 EQUATIONS OF MOTION FORMULATION

This chapter discusses the equations of motions (i.e., the dynamic and kinematic equations)

of a launch vehicle. First the background is presented and then the derivations of the equations

of motions of an expendable launch vehicle are presented. Finally the generalized forces acting

on a launch vehicle during its flight are discussed.

The following assumptions are made in this research [9].

• The launch vehicle (with the strap-on boosters) is assumed to be rigid.

• The center of mass of the launch vehicle lies on the longitudinal axis.

• The longitudinal axis is the principal axis of inertia.

Coordinate Frames

In order to derive the equations of motion of a launch vehicle that describe its position and

orientation as a function of time, various coordinate frames are considered. These frames are

discussed below.

Inertial frame (XiYiZi): For studying the launch vehicle motion in the vicinity of Earth and at

an interplanetary level, the J2000 frame is considered as an inertial frame. This frame has the

origin at the Earth’s center of mass; its positive Z-axis points towards the Earth’s North Pole and

coincides with the Earth’s rotational axis. The positive X-axis lies in the Earth’s equatorial plane

and points towards the vernal equinox in J2000 epoch. The Y-axis lies in the equatorial plane

and completes a right-handed Cartesian frame [9], [28].

Rotating geocentric frame (XgYgZg): This frame rotates with the rotating Earth. This frame

has its positive Z-axis pointed towards the Earth’s North Pole and coincides with the Earth’s

rotational axis. The positive X-axis lies in the equatorial plane, crossing the upper branch of the

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Greenwich meridian. The Y-axis lies in the equatorial plane and completes a right-handed

Cartesian frame [9].

Vehicle-centered horizontal frame (XvYvZv): The orientation of the launch vehicle and its

velocity vector relative to the Earth’s surface can be described using this frame. The origin of

this frame coincides with the initial center of mass of the launch vehicle. The orientation of the

frame remains fixed through out the flight of the launch vehicle. The XY plane of this frame

coincides with the initial local horizontal plane (the local horizontal plane is the plane normal to

the radius vector from the center of mass of the Earth to the center of mass of the launch

vehicle). The positive X-axis points north and lies along the north-south direction. The positive

Y-axis points east and lies along the east-west direction. The Zv-axis is along the radius vector

from the center of Earth and is positive downwards [9].

Vehicle frame (XrYrZr): The origin of this frame coincides with the initial center of mass of the

launch vehicle. The X-axis coincides with the longitudinal axis of the launch vehicle and is

positive forwards (i.e., towards the nose of the launch vehicle). The Y-axis and Z-axis lie along

the other two principal axes of inertia of the vehicle such that they complete a right-handed

Cartesian frame [9].

Relative Orientations

Figure 2-1 shows the relative orientations of the various frames. The details of the relative

orientations and transformations between the above described frames are given below.

Inertial frame/Rotating geocentric frame [9]: The Earth and therefore the rotating geocentric

frame rotate about the Z-axis of the inertial frame with an angular velocity of Earth ( eω ). Thus,

the relative orientation of these frames is determined by a rotation about the Z-axis

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Figure 2-1. Relative orientation of the various frames

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through an angle that is equal to the angle between the Xi-axis and Xg-axis. This angle is equal

to the Greenwich hour angle of the vernal equinox HG. If both frames coincide at t = t0, the angle

HG at any time is given in Eq. 2-1.

( )0eGH t tω= − (2-1)

Since the inertial frame in our case is the J2000 frame, the term ( )0t t− is equal to the time

elapsed in seconds from January 1, 2000, 12:00 UTC until the time “t” of interest. The

transformation between the frames is given in Eq. 2-2. The vectors G E and I E in Eq. 2-3

represent an arbitrary vector E coordinatized in the rotating geocentric frame and the inertial

frame respectively. The transformation matrix is given in Eq. 2-3. G I

GIE C E= (2-2)

cos sin 0sin cos 0

0 0 1

G G

GI G G

H HC H H

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

= − (2-3)

Rotating geocentric frame/Vehicle-centered horizontal frame [9]: The relative orientation of

these two frames can be determined by means of two successive rotations. The rotating

geocentric frame (XgYgZg frame) is first rotated about its Z-axis (i.e., Zg axis) by an angle λ , the

geographic longitude of the launch vehicle. This new frame is then rotated about its new Y-axis

by an angle 2π φ⎛ ⎞⎜ ⎟⎝ ⎠

− + where φ is the geocentric latitude of the launch vehicle. The resulting

frame has the same orientation as the vehicle-centered horizontal frame. The transformation

between the frames is given in Eq. 2-4. The vectors V E and G E in Eq. 2-4 represent an

arbitrary vector E coordinatized in the vehicle-centered horizontal frame and the rotating

geocentric frame respectively. The transformation matrix is given in Eq. 2-5.

V GVGE C E= (2-4)

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sin cos sin sin cossin cos 0

cos cos cos sin sinVGC

φ λ φ λ φλ λ

φ λ φ λ φ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

− −= −

− − − (2-5)

Vehicle-centered horizontal frame/Vehicle frame [9]: The relative orientation of these two

frames can be determined by means three successive rotations as shown in Fig. 2-2. The three

angles through which these three successive rotations are performed are called Euler angles. The

vehicle-centered horizontal frame is first rotated about its Z-axis (i.e., vZ -axis) by an angle ψ to

obtain a new frame 1 1 1v v vX Y Z . ψ is called the yaw angle, the angles between the vertical plane

through the longitudinal axis of the launch vehicle and the vX - axis. Then the new frame

1 1 1v v vX Y Z is rotated about its Y-axis (i.e., 1v

Y axis) by an angle θ to obtain another new frame

2 2 2v v vX Y Z . θ is called the pitch angle, the angle between the longitudinal axis of the launch

vehicle and the local horizontal plane. Finally, the newest frame, 2 2 2v v vX Y Z , is rotated about its

X-axis (i.e., 2vX -axis) by an angle ϕ to obtain the vehicle frame r r rX Y Z . ϕ is called the

bank angle, the angle between the rZ - axis and the vertical plane through the longitudinal axis

of the launch vehicle. The transformation between the frames is given in Eq.2-6. The vectors

R E and V E in Eq. 2-6 represent an arbitrary vector E coordinatized in the vehicle frame and

the vehicle-centered horizontal frame respectively. The transformation matrix is given in Eq. 2-

7. In Eq. 2-7, Cθ and Sθ are used to represent the cosine and sine of an angle θ .

R VRVE C E= (2-6)

RV

C C C S SC C S S S C C C S S S S C

S S C S C S C C S S C C

θ ψ θ ψ θϕ ψ ϕ θ ψ ϕ ψ ϕ θ ψ ϕ θϕ ψ ϕ θ ψ ϕ ψ ϕ θ ψ ϕ θ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

−− + +

+ − + (2-7)

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Figure 2-2. Euler angles and the relative orientation between the vehicle frame and the vehicle-

centered horizontal frame

Inertial frame/Vehicle frame [9]: The transformation from the inertial frame to the vehicle

frame can be obtained by successively applying the transformations GIC , VGC and RVC to the

inertial frame. The transformation between the frames is given in Eq. 2-8. The vectors R E and

I E in Eq. 2-8 represent an arbitrary vector E coordinatized in the vehicle frame and the inertial

frame respectively. The transformation matrix is given in Eq. 2-9.

R IRIE C E= (2-8)

RI RV VG GIC C C C= (2-9) Kinematic Equation of Motion

The rotational kinematic equation of motion relates the orientation and the angular velocity

of a launch vehicle. The derivation of the kinematic equation is presented below.

Let 1

2

3

ωω ω

ω

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

= be the angular velocity of the vehicle frame with respect to the vehicle-

centered horizontal frame expressed in the vehicle frame. Since the vehicle-centered horizontal

frame is an inertial frame, ω is the absolute angular velocity of the launch vehicle. Let ψ& , θ&

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and ϕ& be the Euler angle rates for the 3-2-1 Euler rotation sequence from the vehicle-centered

horizontal frame to the vehicle frame. The angular velocity ω of the launch vehicle can be

expressed in terms of the Euler rates as given in Eq. 2-10. The rotation matrices 1R VC −

and R VC − in Eq. 2-10 are given in Eqs. 2-11 and 2-12.

1

1

2

3

0 00 00 0

R V R VC Cω ϕω θω ψ

− −

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

+ +&

&

&

(2-10)

1

1 0 0 cos 0 sin0 cos sin 0 1 00 sin cos sin 0 cos

R VCϕ ϕ

ϕ ϕϕ ϕ ϕ ϕ

−

−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

(2-11)

1 0 0 cos 0 sin cos sin 00 cos sin 0 1 0 sin cos 00 sin cos sin 0 cos 0 0 1

R VCϕ ϕ ψ ψ

ϕ ϕ ψ ψϕ ϕ ϕ ϕ

−

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(2-12)

Equation 2-10 can be rewritten as Eq. 2-13 where the matrix X in Eq. 2-13 is given in Eq.

2-14. Equation 2-13 can be rewritten as Eq. 2-15. The matrix X in Eq. 2-14 is inverted and

substituted into Eq. 2-15 to obtain Eq. 2-16.

1

2

3

X

ϕωω θω ϕ

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

&

&

&

(2-13)

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

2 1

2 1

2 1

1,1 1, 2 1,3

2,1 2, 2 2,3

3,1 3, 2 3,3

R VR V R V

R VR V R V

R VR V R V

X

C C CC C CC C C

−− −

−− −

−− −

⎡ ⎤⎢ ⎥

= ⎢ ⎥⎢ ⎥⎣ ⎦

(2-14)

11

2

3

X

ϕ ωθ ωϕ ω

−

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

&

&

&

(2-15)

1

2

3

cos sin sin cos sin1 0 cos cos sin cos

cos0 sin cos

ϕ θ ϕ θ ϕ θ ωθ ϕ θ ϕ θ ω

θψ ϕ ϕ ω

⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

= −

&

&

&

(2-16)

26

Eq. 2-16 is the kinematic equation of motion of the launch vehicle. This Euler angle

representation of the relative orientation of the vehicle-centered horizontal frame and the vehicle

frame has the following disadvantages (i) singularity at 2πθ = and (ii) solving the kinematic

equation of motion Eq. 2-16 is computationally intensive as it involves trigonometric quantities.

To avoid these problems, quaternions are used to represent the relative orientation of the vehicle-

centered horizontal frame and the vehicle frame. The transformation matrix RVC can also be

expressed in terms of quaternions as shown in Eq. 2-17. The quantites 0 1 2 3and, ,q q q q in Eq.

2-17 are calculated using the expressions in Eqs. 2-18–2-21.

2 20 1 1 2 0 3 1 3 0 2

2 21 2 0 3 0 2 2 3 0 1

2 21 2 0 3 2 3 0 1 0 3

2 2 1 2 2 2 22 2 2 2 1 2 22 2 2 2 2 2 1

RV

q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q

C⎡ ⎤+ − + −⎢ ⎥− + − +⎢ ⎥⎢ ⎥+ − + −⎣ ⎦

= (2-17)

0 cos cos cos sin sin sin2 2 2 2 2 2

q ψ θ φ ψ θ φ+= (2-18)

1 cos cos sin sin sin cos2 2 2 2 2 2

q ψ θ φ ψ θ φ−= (2-19)

2 cos sin cos sin cos sin2 2 2 2 2 2

q ψ θ φ ψ θ φ+= (2-20)

3 sin cos cos cos sin sin2 2 2 2 2 2

q ψ θ φ ψ θ φ−= (2-21)

The kinematic equation of motion in terms of quaternion rates is given in Eq. 2-22. The

quantites 1 2 3and,ω ω ω in Eq. 2-22 are the components of the angular velocity vector, ω , of

the vehicle frame with respect to the vehicle-centered horizontal frame expressed in the vehicle

frame 1

2

3

i.e.,ω

ω ωω

⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

= . Since the vehicle-centered horizontal frame is an inertial frame, ω is

the absolute angular velocity of the launch vehicle.

27

1 2 3

1 3 2

2 3 1

3 2 1

0 0

1 1

2 2

3 3

001

020

q qq qq qq q

ω ω ωω ω ωω ω ωω ω ω

⎡ ⎤ ⎡ ⎤− − −⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥−⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

=

&

&

&

&

(2-22)

Dynamical Equations

In this section, the derivation of the dynamic equations of a launch vehicle is presented.

Figure 2-3 shows the geometry of the launch vehicle and the various position vectors considered

in the derivation of the dynamical equations. The instantaneous center of mass of the launch

vehicle is represented by “C” and the initial center of mass of the launch vehicle is represented

by “C1”. The position vector of the mass element “dm” relative to the origin of the inertial frame

(coordinatized in the inertial frame) is represented by IdmR . The position vector of the initial

center of mass of the launch vehicle “C1” relative to the origin of the inertial frame

(coordinatized in the inertial frame) is represented by 1

ICR . The position vector of the

instantaneous center of mass of the launch vehicle “C” with respect to the initial center of mass

of the launch vehicle “C1” (coordinatized in the vehicle frame) is represented by Rcr . The

position vector of the mass element “dm” with respect to the instantaneous center of mass of the

launch vehicle “C” (coordinatized in the vehicle frame) is represented by R r . From Fig. 2-3, the

position vector of the mass element “dm” can be written as expressed in Eq. 2-23. The matrix

RIC in Eq. 2-23 is the transformation matrix from the inertial frame If to the vehicle frame Rf

obtained from Eq. 2-9. The acceleration of the mass element “dm” is obtained by differentiating

Eq 2-23 twice with respect to time “t” as shown in Eq. 2-24. The resulting expression for the

acceleration of the mass element “dm” is given in Eq. 2-25.

28

Figure 2-3. Geometry of the launch vehicle and various position vectors

1I I T R R

RICdm crR R C r⎡ ⎤⎣ ⎦= + + (2-23)

( )1

2

2I I T R R

RICdm crdR R C rdt

⎡ ⎤⎣ ⎦= + +&& (2-24)

( )( )( ) ( )( )1

2R R R R RcI I T

RICdm R R R R R R R

c

c c

r

r r

r r rR R C

r r

ω

ω ω ω

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

+ + × + += +

× + + × × +

&& && & &&& &&

& (2-25)

Applying Newton’s second law, we obtain Eq. 2-26. The expression for IdmR&& from Eq.

2-25 is substituted into Eq. 2-26 and then integrated over the entire launch vehicle mass as

shown in Eq. 2-27. The resulting expression after integration is given in Eq. 2-28.

IdmdF R dm= && (2-26)

( )( )( ) ( )( )1

2R R R R R

I ext I I TRICdm R R R R R R RM M

c c

c c

r r

r r

r rF R dm R C dm

r r

ω

ω ω ω

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦

+ + × + += = +

× + + × × +∫ ∫

&& &&& &&& &&

& (2-27)

29

( ) ( )12I ext I T R R R R R R R R

RIC c c c cr r r rF M R C M M M Mω ω ω ω⎡ ⎤⎣ ⎦= + + × + × + × ×&& &&& & (2-28)

(Q 0 and 0M

r r rdm= = =∫& && for a rigid body)

Equation 2-28 is the translational equation of motion of a launch vehicle. The term extF

in Eq. 2-28 represents the resultant of the external forces acting on the launch vehicle. The

external forces acting on the launch vehicle during its flight are described in the next section.

The term cr and its time derivatives in Eq. 2-28 are obtained by computing the instantaneous

center of mass of the launch vehicle with respect to the initial center of mass of the launch

vehicle as a function of time using the mass properties of the launch vehicle, the mass/inertia

properties of a launch vehicle are discussed in the inertia model in Chapter 3. A brief description

of the procedure used to compute cr and its time derivatives in the simulator is presented in

Chapter 4. Taking moments of all the forces about the instantaneous center of mass “C”, we

obtain Eq. 2-29. Substituting the expression for IdmR&& from Eq. 2-25 into Eq. 2-26 and then

substituting the resultant expression for dF into Eq. 2-29, and then integrating the resultant

expression over the entire launch vehicle mass, we obtain Eq. 2-30.

cdM r dF= × (2-29)

( ) ( )( )R ext R R R R R R Rc

M MM r r dm r r dmω ω ω

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= × × + × × ×∫ ∫& (2-30)

The terms ( ) ( )( )andR R R R R R R

M M

r r dm r r dmω ω ω× × × × ×∫ ∫& in Eq. 2-30 are represented

in terms of moment of inertia tensor of the launch vehicle, I , as shown in Eqs. 2-31 and 2-32.

Substituting Eqs. (2-31) and (2-32) into Eq.2-30, we obtain Eq. 2-33.

( ) RR R R

MIr r dm ωω = •× ×∫ && (2-31)

30

( )( ) ( )R RR R R R

MIr r dm ω ωω ω = × •× × ×∫ (2-32)

( )R R RR extc I IM ω ω ω• + × •= & (2-33)

Equation 2-33 is the rotational equation of motion of a launch vehicle. The term extcM

in Eq. 2-33 represents the resultant moment (about the instantaneous center of mass of the launch

vehicle) of the external forces acting on the launch vehicle. The moments of the external forces

acting on the launch vehicle during its flight are described in the next section.

Generalized External Forces

To solve the translational and rotational equations of motions of the launch vehicle given

by Eqs. 2-28 and 2-33, the external forces and the moments (of the external forces) acting on the

launch vehicle need to be calculated. This section discusses the external forces and moments

(due to external forces) acting on a launch vehicle. First, the external forces acting on a launch

vehicle are discussed and then the moments due to the external forces acting on a launch vehicle

are discussed.

External Forces

Figure 2-4. depicts the external forces acting on a launch vehicle during its flight. The

external forces acting on the launch vehicle during its flight are (i) thrust force, (ii) aerodynamic

forces (lift and drag) and (iii) gravity (weight)

Thrust force

The thrust force acting on a launch vehicle is ( )e e a eT mV p p A= + −& where eV is the

exhaust speed of the gases relative to the launch vehicle; m& is the propellant mass flow rate; ep

is the pressure at nozzle exit; ap is the ambient pressure; eA is the nozzle exit (exhaust) area.

31

Figure 2-4. External forces acting on a launch vehicle during its flight

The thrust force of a launch vehicle is generally expressed in the vehicle frame as given in

Eq. 2-34. The simulator requires the thrust force to be coordinatized in the inertial frame, this

can be obtained by using the rotation matrix from the vehicle frame to the inertial frame from Eq.

2-9, ( )TIR RIC C= . The expression for the thrust force coordinatized in the inertial frame is

given in Eq. 2-35

RThrust

x

y

z

TF T

T

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

= (2-34)

I T RRIThrust ThrustF C F= (2-35)

For a launch vehicle composed of strap-on boosters, solid motors and liquid engines, the

thrust acting on a launch vehicle at any instant is equal to the vector sum of the thrusts provided

by all of the strap-on boosters, solid motors and liquid engines at that instant. The simulator

requires the user to input the thrust profile of a launch vehicle.

32

Aerodynamic forces (drag and lift)

The aerodynamic forces acting on a launch vehicle can be neglected at altitudes greater

than or equal to 600 km [29]. However, the aerodynamic forces acting on a launch vehicle

below 600 km cannot be neglected and the expressions for these forces are shown below.

Drag

The drag force acting on a launch vehicle is expressed as 12Drag DF C A v vρ=− where

DC is the drag coefficient; A is the cross-sectional area perpendicular to the flow; ρ is the

density of the medium, andv v are the speed and velocity of the launch vehicle relative to the

medium. The simulator requires the drag force to be coordinatized in the inertial frame. The

expression for the drag force coordinatized in the inertial frame is given in Eq. 2-36.

12

I IDrag DF C A v vρ=− (2-36)

For a launch vehicle composed of strap-on boosters and a main section (i.e., the section

consisting of the different stages of the launch vehicle and payload), the resultant drag force

acting on a launch vehicle is equal to the vector sum of the drag forces acting on the strap-on

boosters and main section of the launch vehicle. The density of the medium/air, ρ , depends on

the altitude of the launch vehicle. The altitude of the launch vehicle is computed using the

WGS84 ellipsoid model discussed in Chapter 3. The procedure to calculate the drag coefficient

is presented in the drag coefficient model in Chapter 3

Lift

For low angles of attack, the lift force can be neglected. However, for high angles of

attack, the lift force cannot be neglected. The lift force acting on a launch vehicle is expressed

as 21 ˆ2 LLiftF C A v vρ ⊥= where LC is the lift coefficient; A is the surface area of the lifting

33

surface; ρ is the density of the medium; v is the speed of the launch vehicle relative to the

medium and v̂⊥ is a unit vector normal to the velocity of the launch vehicle. It should be noted

that the lift coefficient, LC , is a function of the angle of attack. The simulator requires the lift

force to be coordinatized in the inertial frame. The expression for the lift force coordinatized in

inertial frame is given in Eq. 2-37.

21 ˆ2

ILLift

IF C A v vρ ⊥= (2-37)

For a launch vehicle composed of strap-on boosters and a main section (consisting of the

different stages of the launch vehicle and payload), the resultant lift force acting on a launch

vehicle is equal to the vector sum of lift forces acting on the strap-on boosters and main section

of the launch vehicle. The density of the medium/air, ρ , depends on the altitude of the launch

vehicle. The altitude of the launch vehicle is computed using the WGS84 ellipsoid model

discussed in Chapter 3. In the current simulator, the lift force acting on a launch vehicle is

neglected.

Gravitational force

The gravitational force acting on a launch vehicle is W Mg= where M is the mass of the

launch vehicle; g is the acceleration due to Earth’s gravitational field. The gravitational force

can be best expressed in the inertial frame, the gravitational force acts approximately in the

negative direction along the radius vector from the center of the earth to the center of mass of the

launch vehicle. The expression for the gravitational force coordinatized in the inertial frame is

given in Eq. 2-38.

Ig

x

y

z

WF W

W

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

= (2-38)

34

For a launch vehicle composed of strap-on boosters, solid motors, liquid engines and

payloads, the gravitational force acting on the launch vehicle is equal to the product of the

instantaneous mass of the launch vehicle (i.e., sum of instantaneous masses of all the elements of

the launch vehicle) and the acceleration due to gravity vector acting at the instantaneous center

of mass of the launch vehicle. The gravitational force is assumed to be acting at the

instantaneous center of mass of the launch vehicle. The procedure to calculate the acceleration

due to gravity vector is presented in the gravity model in Chapter 3. The instantaneous mass and

the instantaneous center of mass of a launch vehicle can be calculated using the mass properties

of the launch vehicle, the mass/inertia properties of a launch vehicle are discussed in the inertia

model in Chapter 3. A brief description of the procedure used to compute the instantaneous

mass and center of mass of a launch vehicle in the simulator is presented in Chapter 4.

The resultant external force acting on a launch vehicle is the vector sum of the all the

forces acting on the launch vehicle as shown in Eq. 2-39. Substituting the external forces from

Eqs. 2-35–2-38 into Eq. 2-39, we obtain the expression for the resultant external force acting on

the launch vehicle given in Eq. 2-40

I ext I I I IgDragThrust LiftF F F F F+= + + (2-39)

21 1 ˆ2 2

I ext T IRI D L

x xI

y y

z z

T WF C T C A v v C A v v W

T Wρ ρ ⊥

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= − + + (2-40)

External Moments

The moments due to the external forces (i) thrust, (ii) drag, (iii) lift and (iv) gravity acting

on a launch vehicle are discussed below.

Aerodynamic moments

Aerodynamic moments due to the separation of the center of pressure and center of mass

are typically non-zero. The moments due to the aerodynamic forces acting on a launch vehicle

35

can be neglected at altitudes greater than or equal to 600 km [29]. However, the aerodynamic

moments acting on a launch vehicle below 600 km cannot be neglected and the expressions for

these moments are shown below.

The resultant aerodynamic forces (i.e., lift and drag) acting on a launch vehicle are

assumed to be acting at the center of pressure of the launch vehicle. The moments due to drag

force and lift force about the instantaneous center of mass of the launch vehicle are given in Eqs.

2-41 and 2-42 respectively, where Pr is the position vector of the center of pressure of the

launch vehicle with respect to the instantaneous center of mass of the launch vehicle. The vector

Pr can be computed by computing the center of pressure and center of mass locations of a

launch vehicle. The procedure to compute the center of pressure of a launch vehicle is presented

in the center of pressure model in Chapter 3. The center of mass of the launch vehicle can be

calculated using the mass properties of the launch vehicle, the mass/inertia properties of a launch

vehicle are discussed in the inertia model in Chapter 3.

( )Drag DragC p

I IIM Fr= × (2-41)

( )Lift LiftC p

I IIM Fr= × (2-42)

In the current simulator, the lift force and its moment acting on the launch vehicle are

neglected.

Gravitational moment

The gravitational force acts at the instantaneous center of mass of the launch vehicle.

Therefore the gravitational moment about the same point (instantaneous center of mass of the

launch vehicle) is a zero vector. The expression for the gravitational moment is given in Eq. 2-

43.

( ) 0g CI M = (2-43)

36

Thrust moment

If thrust vectoring is considered, the moment due to thrust force of the launch vehicle

about the instantaneous center of mass of the launch vehicle will be non-zero and will be the

major factor affecting the attitude of the vehicle. If thrust force is considered to act always along

the longitudinal axis (i.e., no thrust vectoring), the moment due to thrust force of the launch

vehicle about the instantaneous center of mass of the launch vehicle will be zero. The expression

for the moment due to thrust force of the launch vehicle about the instantaneous center of mass

of the launch vehicle is given in Eq. 2-44, where inr is the position vector of the center of nozzle

of the ith thrusting element (from which the burnt fuel/gases exits from the launch vehicle) with

respect to the instantaneous center of mass of the launch vehicle. The position vectors inr of all

the thrusting elements can be calculated from the knowledge of the geometry of the launch

vehicle and the location of the center of mass of the launch vehicle. The center of mass of a

launch vehicle can be calculated using the mass properties of the launch vehicle, the mass/inertia

properties of a launch vehicle are discussed in the inertia model in Chapter 3. A brief description

of the procedure used to compute the position vectors inr in the simulator is presented in

Chapter 4.

( )Thrust ThrustC n ii

I I I

iM Fr= ×∑ (2-44)

Therefore the resultant external moment acting on a launch vehicle is the vector sum of all

the moments due to the external forces as shown in Eq. 2-45.

( ) ( ) ( )Drag Lift Thrust CC CI ext I I I

C M M MM + += (2-45)

Substituting Eq. 2-40 into Eq. 2-28, yields Eq. .2-46, which is the composite translational

equation of motion for a launch vehicle. Similarly, substituting Eqs. 2-9 and 2-45 into Eq. 2-33

yields Eq. 2-47, which is governing equation for the rotational motion of a launch vehicle.

37

( ) ( )

2

1

1 1 ˆ2 2

2

T IRI D L

I T R R R R R R R Rc c c cRIC

x xI

y y

z z

T WC T C A v v C A v v W

T W

M R C M r M r M r M r

ρ ρ

ω ω ω ω

⊥

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤⎣ ⎦

− + +

+ + × + × + × ×

=

&& &&& &

(2-46)

( ) ( ) ( ) ( )T R R RDrag RILift Thrust CC C

I I I CM M M I Iω ω ω⎡ ⎤⎣ ⎦+ + = • + × •& (2-47)

This chapter presented the development of the kinematic equation of motion and the

dynamic equations of motion of a launch vehicle. To solve the dynamic equations given in Eqs.

2-46 and 2-47, quantites such as gravity, inertia, center of pressure, drag coefficient and others

need to be calculated. Chapter 3 presents some of the models developed to calculate these

quantities.

38

CHAPTER 3 DESCRIPTION OF MODELS USED

In the previous chapter, the derivation of the equations of motion and the discussion of the

generalized forces were presented. In order to implement these in the MATLAB based

simulator, quantities such as inertia, gravity, drag coefficient and altitude need to be calculated as

time dependent functions. In this chapter the models employed to calculate the above quantities

in the simulator are described. The models described in this chapter are (i) gravity model (ii)

inertia model (iii) drag coefficient model (iv) center of pressure model and (v) WGS84 ellipsoid

model. The gravity model calculates the gravity vector at a point of interest. The inertia model

calculates the mass properties of a launch vehicle; the center of pressure model calculates the

location of center of pressure necessary for computing the aerodynamic moments about the

center of mass of a launch vehicle; the drag coefficient model calculates the drag coefficient

necessary for the computation of drag forces and moments; the WGS84 ellipsoid model

calculates the altitude of a launch vehicle. The altitude of a launch vehicle is required to

calculate the density of air and the Mach number of the launch vehicle which in turn are required

for the computation of the aerodynamic forces and the drag coefficient respectively. The details

of the models are given below.

Gravity Model

The gravity model presented below calculates the acceleration due to gravity acting at a

point of interest due to the gravitational field of the Earth. The acceleration vector obtained from

this model is coordinatized in the ECEF frame. The details of the model are presented below.

The Earth is not a spherically symmetric mass body. The Earth bulges at the equator as a

consequence of its rotation. The density of the Earth also varies from location to location. This

variability of the Earth’s mass is modeled using a spherical harmonic expansion of the

39

gravitational potential [17]. The spherical harmonic representation of the gravitational potential

function (V) is given in Eq. 3-1 [27].

( )( )max

2 01 sin cos sin

nn nm

n nm nmn m

GM aV P C m S mr r

α λ λ= =

⎛ ⎞⎛ ⎞⎜ ⎟+ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠= +∑ ∑ (3-1)

In Eq 3-1, ( )GMμ is the Earth’s gravitational constant; ,,r α λ are the distance

from the Earth’s center of mass to the point of interest, geocentric latitude and

geocentric/geodetic longitude of the point of interest, respectively; a is the semi-major axis of

the WGS84 ellipsoid (discussed later); ,n m are the degree and order of the spherical harmonic

function respectively; ,nm nmC S are the spherical harmonic coefficients of degree n and order

m ; mnP is the associated Legendre function of degree n and order m . The associated

Legendre function mnP is defined as follows [27].

( )sinmnP α = ( )

( )( )cos sin

sin

mmnm

d Pd

α αα

⎡ ⎤⎣ ⎦

( )sinnP α = Legendre polynomial ( )

( )21 sin 12 ! sin

n

n nnn

dd

αα

−=

A gravitational model is defined by the set of constants μ , a and the spherical harmonic

coefficients ,nm nmC S The gravitational model used in this research is the WGS84–EGM96

model. The spherical harmonic representation of the gravitational potential function in Eq. 3-1

has numerical computation problems in the form of the (unnormalized) spherical harmonic

coefficients, ,nm nmC S and the associated Legendre functions, ( )sinmnP α . The (unnormalized)

spherical harmonic coefficients, ,nm nmC S , tend to very small values and the associated

Legendre functions, ( )sinmnP α , tend to very large values as the degree increases. These

problems can be circumvented by normalizing the associated Legendre function and the

40

spherical harmonic coefficients. In general, this normalization is achieved by multiplying the

spherical harmonic coefficients (and dividing the associated Legendre function) by a scale factor

which depends on the degree and order of the function. Denoting the normalized quantities by

an overbar, the normalized spherical harmonic coefficients and the normalized associated

Legendre polynomial are defined in Eqs. 3-2 and 3-3 respectively [27].

( )( ) ( )

12!

! 2 1nmnm

nmnm

Cn mCSn m n kS

⎡ ⎤⎡ ⎤ + ⎡ ⎤= ⎢ ⎥⎢ ⎥ ⎢ ⎥− +⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦

, for 0, 1;0, 2

m km k= =≠ =

(3-2)

( ) ( ) ( )( ) ( )

122 1

sin sin!

!m m

n nn m n k

P Pn m

α α⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

− +=

+, for

0, 1;0, 2

m km k= =≠ =

(3-3)

In Eqs. 3-2 and 3-3, ,nm nmC S are the normalized spherical harmonic coefficients; mnP

is the normalized associated Legendre function. The advantages of the normalization of the

associated Legendre polynomial and spherical harmonic coefficients can be retained by rewriting

the spherical harmonic representation of the gravitational potential function (V) in Eq. 3-1 in

terms of normalized spherical harmonic coefficients and normalized associated Legendre

function as given in Eq. 3-4 [27].

( )( )max

2 01 sin cos sin

nn nm

n nm nmn m

GM aV P C m S mr r

α λ λ= =

⎛ ⎞⎛ ⎞⎜ ⎟+ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠= +∑ ∑ (3-4)

The above “normalized” spherical harmonic representation of the gravitational potential

Eq. 3-4 is used to compute the acceleration due to gravity. The acceleration due to gravity can

be computed by taking the gradient of the gravitational potential V as given in Eq. 3-5. The

gradient of the potential V is shown in Eq. 3-6. It should be noted that the ( ), ,x y z in Eq. 3-6

represents the Cartesian coordinates in the ECEF frame.

g V= ∇ (3-5)

41

ˆˆ ˆV V VV i j kx y z

∂ ∂ ∂∇ = + +

∂ ∂ ∂ (3-6)

Let , ,x y zg g g be defined as xVgx

∂=∂

, yVgy

∂=∂

and zVgz

∂=∂

. Since V is a function of

( ), ,r α λ , , andx y zg g g must be expressed in terms of the partial derivatives of V with respect to

, andr α λ as in Eqs. 3-7–3-9.

xV V r V Vgx r x x x

α λα λ

=∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(3-7)

yV V r V Vgy r y y y

α λα λ

=⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂

= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (3-8)

zV V r V Vgz r z z z

α λα λ

=∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(3-9)

The expressions for the terms , andV V Vr α λ

∂ ∂ ∂∂ ∂ ∂

in Eqs. 3-7–3-9 are given in Eqs. 3-10-3-

13, where the expression for the term ( )sinmnP α

α∂ ⎡ ⎤⎣ ⎦∂

in Eq. 3-11 is given in Eq. 3-12. The

term 1C in Eq. 3-11 is defined as ( ) ( )( )

12

1

! 2 1!

n m n kC

n m⎡ ⎤− +

= ⎢ ⎥+⎢ ⎥⎣ ⎦

.

( ) ( )( )max

22 0

11 sin cos sin

nm

n nm nmn

n n

n m

n aV GM P C m S mr r r

α λ λ= =

−⎛ ⎞+∂

+⎜ ⎟⎜ ⎟∂ ⎝ ⎠= + ∑ ∑ (3-10)

( ) ( )max

12 0

sin cos sinnn n

mn nm nm

n m

V GM a P C m S mr r

C α λ λα α= =

⎛ ⎞∂ ∂⎛ ⎞ ⎡ ⎤= +⎜ ⎟⎜ ⎟ ⎣ ⎦⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠∑ ∑ (3-11)

( ) ( ) ( )1sin sin tan sinm m mn n nPP P mα α α α

α+∂ ⎡ ⎤ = −⎣ ⎦∂

(3-12)

( )( )max

2 0sin sin cos

nn nm

n nm nmn m

V GM a P mC m mS mr r

α λ λλ = =

⎛ ⎞∂ ⎛ ⎞= − +⎜ ⎟⎜ ⎟⎜ ⎟∂ ⎝ ⎠⎝ ⎠∑ ∑ (3-13)

To calculate all other partial derivatives in Eqs. 3-7, 3-8 and 3-9, the spherical coordinates

( ), ,r α λ must be represented in terms of the Cartesian coordinates ( ), ,x y z. To do this, let us

consider the geometry shown in Fig. 3-1.

42

Figure 3-1. Representation of a position vector in Cartesian and Spherical coordinates

From Fig. 3-1, the Cartesian position vector, ( ), ,R x y z can be represented in terms of

spherical coordinates, , ,r α λ as given in Eqs. 3-14-3-16. The partial derivatives of the spherical

coordinates , ,r α λ with respect to Cartesian coordinates ( ), ,x y zare calculated using Eqs. 3-

14–3-16 and are expressed in Eqs. 3-17-3-19.

( )2 2 2r x y z= + + (3-14)

1tanyx

λ −= ⎛ ⎞⎜ ⎟⎝ ⎠

(3-15)

( )1

2 2tan

z

x yα −=

+

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

(3-16)

( )( ) 2 22 2 2, ,r x zx y

x r x x x yx y r

α λ∂ ∂ − ∂−

∂ ∂ ∂ ++= = = (3-17)

( )( ) 2 22 2 2, ,r y zy x

y r y y x yx y r

α λ∂ ∂ − ∂∂ ∂ ∂ ++= = = (3-18)

43

( )2 2

2, 0,x yr z

z r z r zα λ+∂ ∂ ∂

∂ ∂ ∂= = = (3-19)

Substituting Eqs. 3-10–3-19 into Eqs. 3-7, 3-8 and 3-9, we obtain the X, Y and Z

components of the acceleration vector in the ECEF frame. The computation of the acceleration

vector using the above procedure becomes cumbersome at or near poles (i.e., at 90α = ± ° )

because Eq. 3-12 contains tanα which becomes infinite when α approaches 90± o .

Furthermore, Eqs. 3-17 and 3-18 also become infinite as and/orx y approach zero.

A change of coordinates is carried out which avoids the above difficulty yet retains similar

recursive and orthogonality properties. The coordinate change and the derivation of the

acceleration vector in terms of these new coordinates (as a result of the coordinate change) is

presented below [25].

Let the new coordinates be and, , ur s t where each coordinate is defined as

( )1

2 2 2 2r x y z= + + , , ,x y zs t ur r r

= = = (i.e., r is the scalar magnitude of the vector R , ,, us t

are the three components of the unit vector R̂ ). From the above definition of the new

coordinates, it can be seen that 2 2 2 1s t u+ + = and s

R r tu

⎧ ⎫⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭

. To obtain the acceleration vector

g in terms of these new coordinates, the gravitational potential V in Eq. 3-4 should be first

expressed in terms of these new coordinates. In place of the normalized associated Legendre

polynomial ( ).mnP , a new polynomial ( ).m

nA is defined as given in Eq. 3-20. In place of

sin and cosm mλ λ in the gravitational potential V expression Eq. 3-4, define ( ),mR s t and

44

( ),mI s t as given in Eqs. 3-21–3-22. Using the definitions in Eqs. 3-20–3-22, the gravitational

potential V in Eq. 3-4 can be expressed in terms of these definitions as given in Eq. 3-23.

( ) ( ) ( )( ) ( )2

122 1 1

12

!! !

n mnm

n n n mu un m n k dA

n m n du

+

+ −⎡ ⎤− +

= ⎢ ⎥+⎢ ⎥⎣ ⎦

(3-20)

( ) ( ), cos cos mmR s t mλ α= = Real part of ( )ms it+ (3-21)

( ) ( ), sin cos mmI s t mλ α= = Imaginary part of ( )ms it+ (3-22)

( ) ( ) ( )( )max

2 01 , ,

nn nm

n nm m nm mn m

uGM aV A C R s t S I s t

r r= =

+⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∑ ∑ (3-23)

The above spherical harmonic representation of the gravitational potential Eq. 3-23 is now

expressed in terms of the new coordinates and, , ur s t and is used to compute the acceleration

vector. The acceleration vector can be computed by taking the gradient of the gravitational

potential V as given in Eq. 3-24. The gradient of the potential V is defined in Eq. 3-25. It

should be noted that the ( ), ,x y z in Eq. 3-25 represents the Cartesian coordinates in the ECEF

frame.

g V= ∇ (3-24)

ˆˆ ˆV V VV i j kx y z

∂ ∂ ∂∇ = + +

∂ ∂ ∂ (3-25)

Let , ,x y zg g g be defined as xVgx

∂=∂

, yVgy

∂=∂

and zVgz

∂=∂

. Since, V is a function of

( ), , ,r s t u , , andx y zg g g must be expressed in terms of the partial derivatives of V with respect

to , , andr s t u as given in Eqs. 3-26–3-28.

xV V r V s V t V ugx r x s x t x u x

= +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(3-26)

yV V r V s V t V ugy r y s y t y u y

= +⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (3-27)

zV V r V s V t V ugz r z s z t z u z= +

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (3-28)

45

Calculating and substituting the quantities , , , , , ,r r r s s sx y z x y z∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂

, , ,t t tx y z∂ ∂ ∂∂ ∂ ∂

, andu u ux y z∂ ∂ ∂∂ ∂ ∂

in Eqs. 3-26–3-28 and then substituting the resulting expressions of Eqs. 3-26–

3-28 into Eqs. 3-24 and 3-25, we obtain the expression for the acceleration vector g as given in

Eq. 3-29.

1 1 1 ˆˆ ˆ ˆV V V V V V Vg V R i j kr s t u r s r t r u

s t ur r r

∂ ∂ ∂ ∂ ∂ ∂ ∂⎡ ⎤= ∇ − + + +⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦= − − (3-29)

The acceleration vector can be calculated by substituting the expressions for the quantites

, , andV V V Vr s t u

∂ ∂ ∂ ∂∂ ∂ ∂ ∂ into Eq. 3-29. Before computing and substituting the quantities

, , andV V V Vr s t u

∂ ∂ ∂ ∂∂ ∂ ∂ ∂ into Eq. 3-29, the definitions in Eqs. 3-30–3-35 are used to express the

gravitational potential V in a compact form.

ar

ρ = (3-30)

0GM

rρ = (3-31)

1 0ρ ρ ρ= (3-32)

( ) ( ) ( ), , ,m m mn n m n mD s t C R s t S I s t= + (3-33)

( ) ( ) ( )1 1, , ,m m mn n nm mE s t C R s t S I s t− −= + (3-34)

( ) ( ) ( )1 1, , ,m m mn n nm mF s t S R s t C I s t− −= − (3-35)

Using Eqs. 3-30–3-32, the recursion equations in Eqs. 3-36–3-37 are formed. Using Eqs.

3-31, 3-33 and 3-36, the spherical harmonic representation of the gravitational potential V in Eq.

3-23 can be compactly represented as given in Eq. 3-38.

1 for 1n n nρ ρ ρ −= > (3-36)

1n n

r aρ ρ += , 1

1nn

nr aρ ρ +

∂ +⎡ ⎤= − ⎢ ⎥∂ ⎣ ⎦ (3-37)

46

( ) ( )max

02 0

,m mn n

n n

nn m

V A u D s tρ ρ= =

+= ∑ ∑ (3-38)

The spherical harmonic representation of the gravitational potential V in Eq. 3-38 is used

with Eq. 3-29 to compute the acceleration vector g . The recursion relationships (Eqs. 3-39 and

3-40) and partials for ( ),mR s t and ( ),mI s t with respect to ands t (Eqs. 3-41 and 3-42) are

useful in computing the acceleration vector g .

( ) ( ) ( )1 1, , ,m m mR s t sR s t tI s t− −= − (3-39)

( ) ( ) ( )1 1, , ,m m mI s t sI s t tR s t− −= + (3-40)

( ) ( ) ( )1

, ,,m m

mR s t I s t

mR s ts t −

∂ ∂= =

∂ ∂ (3-41)

( ) ( ) ( )1

, ,,m m

mI s t R s t

mI s ts t −

∂ ∂= − =

∂ ∂ (3-42)

Define 1 2 3 4, , anda a a a as given in Eqs. 3-43–3-46. From Eqs. 3-29 and 3-43–3-46, the

acceleration vector g can be expressed in terms of 1 2 3 4, , anda a a a as given in Eq. 3-47. Using

Eqs. 3-30–3-42, the quantities , , andV V V Vr s t u

∂ ∂ ∂ ∂∂ ∂ ∂ ∂

are computed and substituted into Eqs. 3-

43–3-47 to obtain the expressions for 1 2 3 4, , anda a a a as given in Eqs. 3-48–3-51.

11 Var s∂

=∂

(3-43)

21 Var t∂

=∂

(3-44)

31 Var u∂

=∂

(3-45)

4V V V Var s t u

s t ur r r

−∂ ∂ ∂ ∂

=∂ ∂ ∂ ∂

− − (3-46)

4 1 2 3ˆˆ ˆ ˆg a R a i a j a k= + + + (3-47)

( ) ( )max

11

2 0,m m

n n

n nn

n ma A u mE s t

aρ +

= == ∑ ∑ (3-48)

( ) ( )max

21

2 0,m m

n n

n nn

n ma A u mF s t

aρ +

= == ∑ ∑ (3-49)

47

( ) ( )max

13

1

2 0,m m

n n

n nn

n ma A u D s t

aρ ++

= == ∑ ∑ (3-50)

( ) ( ) ( )max

04 1 2 3

1

2 0

1,m m

n n

n nn

n m

na A u D s t sa a a

r at uρρ +

= =

+= − − − − −∑ ∑ (3-51)

The expressions for 1 2 3 4, , anda a a a from Eqs. 3-48–3-51 are substituted into Eq. 3-47 to

obtain the acceleration vector g . The acceleration vector obtained by the above procedure is

coordinatized in ECEF frame. The above procedure for calculating the acceleration vector g

Eqs. 3-20–3-51, however, has a numerical computation problem in the form of the polynomial

( )mnA u , which is addressed below. From Eq. 3-20, the polynomial ( )m

nA u is defined as

( ) ( ) ( )( ) ( )2

122 1 1

12

!! !

n mnm

n n n mu un m n k dA

n m n du

+

+ −⎡ ⎤− +

= ⎢ ⎥+⎢ ⎥⎣ ⎦

(3-52)

From Eq. 3-52, it can be seen that the computation of ( )mnA u for high values of degree

n and order m becomes cumbersome specifically due to the presence of terms ( )!n m+ and

( )2 1n m

n

n m uddu

+

+ − . This problem can be circumvented by using the recursion relationships for the

polynomial ( )mnA u given in Eqs. 3-53–3-56 [16], [6].

( ) ( )( ) ( ) ( ) ( )1

1 20 0 021 2

1 2 12 1 2 1 12 3n n n

nu u n n u n uA A An n− −

⎡ ⎤+⎡ ⎤⎢ ⎥= + − − −⎡ ⎤⎣ ⎦ ⎢ ⎥⎢ ⎥−⎣ ⎦⎣ ⎦

(3-53)

( ) ( )1

1

122 1

2n n

n n

nu uA An−

−

+⎡ ⎤= ⎢ ⎥⎣ ⎦ (3-54)

( ) ( ) ( )( ) ( )

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

1

2

12

12

2

12 11

2 11

1

m m

n n

m

n

u m u uA An m n m

n m n mu uAn m n m

+

+

⎡ ⎤= + +⎢ ⎥+ + −⎣ ⎦

⎡ ⎤+ + − −− ⎢ ⎥+ + −⎣ ⎦

(3-55)

( ) 0,mnA u m n= > (3-56)

48

It should be noted that the recursion relationships Eqs. 3-53–3-55 cannot be used if the

values of the polynomial ( ) for at least 2 3, 0mnA u n m n≤ ≤ ≤ ≤ are not provided. Therefore the

values of polynomials ( ) , 2 3, 0mnA u n m n≤ ≤ ≤ ≤ need to be calculated manually. Once the

values of ( ) , 2 3, 0mnA u n m n≤ ≤ ≤ ≤ are calculated, they can be used along with the recursion

relationships Eqs. 3-53–3-56 to generate ( ) , 3, 0mnA u n m n> ≤ ≤ by following the procedure

below [6].

1. Equation 3-53 is used to generate ( )0 , 3n u nA >

2. Equation 3-54 is used to generate the diagonal elements ( ) , 3n

n u nA >

3. Equations 3-55 and 3-56 are used to generate all the remaining elements.

The recursion relationships Eqs. 3-53–3-56 have been proved to be the most numerically

stable of all the recursion relationships [16], [6]. The polynomials ( ) , 2 360,mnA u n≤ ≤

0 m n≤ ≤ generated using the recursion relationships in Eqs. 3-53–3-56 have a maximum

percentage error of about 0.007983. This maximum percentage error was determined by

computing the polynomial ( ) , 2 360, 0mnA u n m n≤ ≤ ≤ ≤ (i) using the recursion relationships

Eqs. 3-53–3-56 and (ii) using a MATLAB inbuilt function ( )mnN u related to ( )m

nA u by the

expression ( )( )

( )2 2

, where1 21

mn m

mnu uk N

uA =

−

1 for 02 for 0

k mk m= == ≠

and then taking the

difference between the corresponding values of ( )mnA u obtained by the above two methods and

dividing it by the corresponding value of ( )mnA u obtained by using the second method (i.e.,

using ( )mnN u ). It should be noted that the maximum percentage error was computed for

49

0.99 0.99u− ≤ ≤ because ( )mnA u cannot be calculated at 1u = ± using the second method (i.e.,

using ( )mnN u ). It can be concluded that the gravity model described above i.e., Eqs. 3-20–3-56

is singularity-free and numerically efficient.

Inertia Model

The inertia model presented here is used to calculate the mass properties of the launch

vehicle (i.e., mass and mass moment of inertia of the launch vehicle). The mass moment of

inertia of the launch vehicle is calculated about a Cartesian axes through the instantaneous center

of mass of the launch vehicle. A simple but reasonable inertia model was developed and is being

used in this research. The details of the developed inertia model are presented below.

The launch vehicle is assumed to have (i) solid strap-on boosters, (ii) solid motors, (iii)

liquid motors and (iv) a payload. To develop the inertia characteristics, general geometries were

considered. The simulator requires the inertia tensor of the launch vehicle to be computed about

the instantaneous centroidal axes of the launch vehicle. Therefore the inertia tensors of the

individual “elements” (solid strap-on boosters, solid motors, liquid motors and payload) of the

launch vehicle need to be computed about the instantaneous centroidal axes of the launch

vehicle. This is calculated by first computing the inertia tensor of the individual elements about

their own centroidal axes and then using the parallel axis theorem, the inertia tensors of the

individual elements about the instantaneous centroidal axes of the launch vehicle are computed.

The sum of the instantaneous inertia tensors of the individual elements about the instantaneous

centroidal axes of the launch vehicle is the instantaneous inertia tensor of the entire launch

vehicle about its instantaneous centroidal axes. These details are presented in the following

subsections.

50

Strap-on booster

The solid strap-on booster is divided into following segments (i) a cylindrical segment, (ii)

a parabolic nose cone and (iii) four fins. To calculate the inertia tensor of a strap-on booster

about the instantaneous centroidal axes of the launch vehicle, the inertia tensors of the individual

segments (of the strap-on booster) about their own centroidal axes are calculated and then using

the parallel axis theorem, the inertia tensors of the individual segments about the instantaneous

centroidal axes of the strap-on booster are calculated. The sum of these inertia tensors of the

individual segments about the instantaneous centroidal axes of the strap-on booster is the

instantaneous inertia tensor of the entire strap-on booster about its instantaneous centroidal axes.

This instantaneous inertia tensor of the strap-on booster is used with the parallel axis theorem to

calculate the instantaneous inertia tensor of the strap-on booster about the instantaneous

centroidal axes of the launch vehicle. The inertia properties of the segments of the strap-on

booster are presented below.

Cylindrical segment

The cylindrical segment of the strap-on booster depicted in Fig. 3-2 is assumed to be made

of a cylindrical shell structure and solid propellant (the solid propellant is assumed to be

distributed as a cylindrical shell). The cylindrical shell structure of outer radius, osR , inner

radius, isR and height, cL , is modeled as a solid cylinder of radius, osR and height, cL , from

which another cylinder of radius, isR and height, cL , is removed (the bases and the centroidal

axes of both cylinders coincide). The propellant shell is modeled similar to the cylindrical shell

structure with a time varying radius of the inner cylinder as the propellant burns. Therefore the

propellant shell of outer radius, opR , inner radius, ipR and height, cL , is modeled as a solid

cylinder of radius, opR and height, cL , from which another cylinder of time varying radius, ipR

51

and height, cL , is removed (the bases and the centroidal axes of both cylinders coincide). It is

assumed that as the propellant burns, the radius of the inner cylinder, ipR , increases. When all

the propellant is burnt, opipR R= . The densities of the materials of the cylindrical shell structure

and the solid propellant are assumed to be ands pρ ρ respectively.

The mass of the cylindrical segment, cM , is the sum of masses of the cylindrical shell

structure and cylindrical propellant shell as shown in Eq. 3-57. The mass moment of inertias,

and,C C CXX YY ZZI I I of the cylindrical segment about its centroidal axis is the sum of mass

moment of inertias of the cylindrical shell structure and the propellant shell about the centroidal

axis of the cylindrical segment as shown in Eqs. 3-58, 3-59 and 3-60, respectively.

Figure 3-2. Cylindrical segment of the strap-on booster

( ) ( )2 2 2 2c s c os p c opis ipM L R R L R Rπρ πρ= − + − (3-57)

52

( ) ( )4 4 4 41 12 2C s c os p c opis ipXXI L R R L R Rπρ πρ= − + − (3-58)

( ) ( )( ) ( )

2 2 2 2 2 2

2 2 2 2 2 2

1 13 312 12

1 13 312 12

C s c os os c s c cis isYY

p c op op c p c cip ip

I L R R L L R R L

L R R L L R R L

πρ πρ

πρ πρ

+= + − +

+ − + (3-59)

( ) ( )( ) ( )

2 2 2 2 2 2

2 2 2 2 2 2

1 13 312 12

1 13 312 12

C s c os os c s c cis isZZ

p c op op c p c cip ip

I L R R L L R R L

L R R L L R R L

πρ πρ

πρ πρ

+= + − +

+ − + (3-60)

Therefore the inertia tensor of the assumed homogeneous cylindrical segment about its

centroidal axis is

0 00 00 0

C

C

C

XX

YY

ZZ

II I

I

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

. The instantaneous inertia tensor of the cylindrical

segment about the instantaneous centroidal axes of the solid booster, CI , is computed using the

parallel axis theorem as shown in Eq.3-61.

CM 1T Tc C C C CI I r r r r⎡ ⎤= ⎣ ⎦+ − (3-61)

The vector, Cr , in Eq. 3-61 is the position vector of the instantaneous center of mass of

the solid strap-on booster with respect to the center of mass of the cylindrical segment and 1 is

the 3 X 3 identity matrix.

Parabolic nose cone

The nose cone of the solid strap-on booster depicted in Fig. 3-3 is modeled as a solid

parabolic nose cone of radius, NCR and length, NCL . The density of material of the nose cone is

assumed to be NCρ .The details of the model are presented below.

The mass of the parabolic nose cone, NCM , is given by the expression as shown in Eq.

3-62. The mass moment of inertias, and,NC NC NCXX YY ZZI I I , of the parabolic nose cone about

53

its centroidal axis are given by the expressions as shown in Eqs. 3-63, 3-64 and 3-65

respectively.

Figure 3-3. Parabolic nose cone

212NC NC NC NCM R Lπρ= (3-62)

213NCXX NC NCI M R= (3-63)

2 21 16 3NCYY NC NC NCI M R L⎛ ⎞

⎜ ⎟⎝ ⎠

= + (3-64)

2 21 16 3NCZZ NC NC NCI M R L⎛ ⎞

⎜ ⎟⎝ ⎠

= + (3-65)

Therefore the inertia tensor of the parabolic nose cone about its centroidal axis is

0 00 00 0

NC

NC

NC

XX

YY

ZZ

II I

I

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

. The instantaneous inertia tensor of the parabolic nose cone about

the instantaneous centroidal axes of the solid booster, NCI , is computed using the parallel axis

theorem as shown in Eq. 3-66.

NCM 1T TNC NC NC NC NCI I r r r r⎡ ⎤= ⎣ ⎦+ − (3-66)

54

The vector, NCr , in Eq. 3-66 is the position vector of the instantaneous center of mass of

the solid strap-on booster with respect to the center of mass of the parabolic nose cone.

Fins

The geometry of the fin is modeled by removing two triangular slabs 2f and 3f from a

rectangular slab 1f as shown in Fig. 3-4. Each strap-on booster is assumed to have four fins of

thickness ft towards the aft of the solid strap-on booster. The masses of the three slabs

1 2 3, andf f f are given by the expressions as shown in Eqs. 3-67, 3-68 and 3-69. The mass

moment of inertias of the slabs 1 2 3, andf f f about their own centroidal axes are given in Eqs.

3-70–3-78. The expressions for 1 2andL L in Eqs. 3-70–3-80 are given by 1 T RL X C= + and

2 T T RL X C C= + − . The X- and Y- components of the location of center of mass of the fin,

andCM CMX Y , are given by the expressions as shown in Eqs. 3-79 and 3-80 respectively. The

mass of the fin, fM ,is obtained by subtracting the mass of the slabs 2f and 3f from the mass

of the slab 1f as shown in Eq. 3-81. The moment of inertia of the fin about its centroidal axes is

obtained by subtracting the moment of inertias of slabs 2f and 3f from the corresponding

moment of inertias of the slab 1f as shown in Eqs 3-82, 3-83 and 3-84 respectively.

1 1f f fM L t Sρ= (3-67)

2

12 Tf f fM X t Sρ= (3-68)

3 212f f fM L t Sρ= (3-69)

( ) ( )1 11

221

12 2 cmXX f f ffSI M S t M Y⎛ ⎞

⎜ ⎟⎝ ⎠

= + + − (3-70)

55

Figure 3-4. Fin

( ) ( ) 1

1 11

22

11

12 2 cmYY f f ffLI M L t M X⎛ ⎞

⎜ ⎟⎝ ⎠

= + + − (3-71)

( ) ( ) 11 11

222

11

12 2 2cm cmZZ f ffL SI M L S M X Y

⎛ ⎞⎛ ⎞⎛ ⎞⎜ ⎟+ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠= + + − − (3-72)

( )2 22

22 21 1 2

18 12 3 cmXX f f ffI M S t M S Y⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= + + − (3-73)

( )2 22

22 21 1 1

18 12 3 cmYY T Tf f ffI M X t M X X⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= + + − (3-74)

( ) ( )2 22

2 22 21 1 2

18 3 3cm cmZZ T Tf ffI M S X M X X S Y

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠= + + − + − (3-75)

( )3 33

22 21 1 1

18 12 3 cmXX f f ffI M S t M S Y⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= + + − (3-76)

( ) 23 33

22 22 1

1 1 118 12 3 cmYY f f ff

I M L t M L L X⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠= + + − (3-77)

( ) ( ) 23 33

2 22 2

2 11 1 1

18 3 3cm cmZZ f ffI M S L M S Y L L X

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟−⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠= + + − + − (3-78)

2 21 2 1 2

1 2

1 13 32

T

CMT

L X L L LX

L X L

⎛ ⎞⎜ ⎟⎝ ⎠

− − −=

− − (3-79)

56

1 2

1 2

2 13 3

2T

CMT

L S X S L SY

L X L

− −=

− − (3-80)

1 2 3f f f fM M M M= − − (3-81)

( ) ( ) ( )1 2 3fXX XX XX XXf f f

I I I I− −= (3-82)

( ) ( ) ( )1 2 3fYY YY YY YYf f f

I I I I− −= (3-83)

( ) ( ) ( )1 2 3fZZ ZZ ZZ ZZf f f

I I I I− −= (3-84)

Therefore the inertia tensor of the fin about its centroidal axis is

0 0

0 0

0 0

f

f

f

XX

YY

ZZ

I

I I

I

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

.

The instantaneous inertia tensors of the four fins about the instantaneous centroidal axes of the

solid booster, 1 2 3 4

and, ,f f f fI I I I , are computed using the parallel axis theorem as shown in

Eqs. 3-85–3-88.

1 1 1 1 1fM 1T Tf f f f fI I r r r r⎡ ⎤= ⎢ ⎥⎣ ⎦

+ − (3-85)


+ − (3-86)


+ − (3-87)


+ − (3-88)

The vectors, 1 2 3 4, , andf f f fr r r r , in Eqs. 3-85–3-88 are the position vectors of the

instantaneous center of mass of the solid strap-on booster with respect to the center of mass of

the four fins respectively.

The instantaneous inertia tensor of the entire solid strap-on booster about its

instantaneous centroidal axes, SI ; is obtained by summing the inertia tensors of the individual

segments of the strap-on booster (i.e., cylindrical segment, parabolic nose cone and fins) about

the instantaneous centroidal axis of the strap-on booster as shown in Eq. 3-89. The instantaneous

57

mass of the solid booster is obtained by summing the masses of the individual segments of the

strap-on boosters as shown in Eq. 3-90.

1 2 3 4S C NC f f f fI I I I I I I= + + + + + (3-89)

4S C NC fM M M M= + + (3-90)

As mentioned at the beginning of this mass moment of inertia section, the inertia tensor

of the solid strap-on booster needs to be computed about the instantaneous centroidal axes of the

launch vehicle. Therefore the instantaneous inertia tensor of the entire solid strap-on booster

about the instantaneous centroidal axes of the launch vehicle , ( )C SI , is computed using the

parallel axis theorem as shown in Eq. 3-91.

( ) SM 1T TC S S S S SS

I I r r r r⎡ ⎤= ⎣ ⎦+ − (3-91)

The vector , Sr , in Eq. 3-91 is the position vector of the instantaneous center of mass of

the launch vehicle with respect to the instantaneous center of mass of the solid strap-on booster.

Liquid Engine

The liquid engine depicted in Fig. 3-5 is assumed to be made of a cylindrical shell structure

and liquid propellant. The cylindrical shell structure of outer radius, osR , inner radius, isR and

height, L , is modeled as a solid cylinder of radius, osR , and height, L , from which another

cylinder of radius, isR and height, L , is removed (the bases and the centroidal axes of both

cylinders coincide). The liquid propellant is modeled as a solid cylinder of radius, pR and

height, L , whose density (and mass) decreases with time (as the propellant burns). The densities

of the materials of the cylindrical shell structure and the liquid propellant are assumed to be

ands pρ ρ respectively.

58

The mass of the liquid engine, LM , is the sum of masses of the cylindrical shell structure

and propellant cylinder as shown in Eq. 3-92. The mass moment of inertias,

and,L L LXX YY ZZI I I , of the liquid engine about its centroidal axis is the sum of mass moment of

inertias of the cylindrical shell structure and the propellant cylinder about the centroidal axis of

the liquid engine as shown in Eqs. 3-93, 3-94 and 3-95 respectively.

Figure 3-5. Liquid engine

( )2 2 2s os p pisLM L R R LRπρ πρ= − + (3-92)

( )4 4 41 12 2L s os p pisXXI L R R LRπρ πρ= − + (3-93)

( ) ( )( )

2 2 2 2 2 2

2 2 2

1 13 312 12

1 312

C s os os s is isYY

p p p

I LR R L LR R L

L R R L

πρ πρ

πρ

+= + − +

+ (3-94)

59

( ) ( )( )

2 2 2 2 2 2

2 2 2

1 13 312 12

1 312

C s os os s is isZZ

p p p

I LR R L LR R L

L R R L

πρ πρ

πρ

+= + − +

+ (3-95)

Therefore the inertia tensor of the liquid engine about its centroidal axis is

0 00 00 0

L

L

L

XX

YY

ZZ

II I

I

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

. As mentioned at the beginning of this inertia model section, the

inertia tensor of the liquid engine needs to be computed about the instantaneous centroidal axes

of the launch vehicle. Therefore the instantaneous inertia tensor of the liquid engine about the

instantaneous centroidal axes of the launch vehicle, LI , is computed using the parallel axis

theorem as shown in Eq.3-96.

LM 1T TL L L L LI I r r r r⎡ ⎤= ⎣ ⎦+ − (3-96)

The vector, Lr , in Eq. 3-96 is the position vector of the instantaneous center of mass of

the launch vehicle with respect to the center of mass of the liquid engine.

Solid Motor

The solid motor model is similar to the cylindrical segment model of the solid strap-on

booster model discussed above in the strap-on booster section. However, the details of model

are presented again for the sake of completeness. The solid motor depicted in Fig. 3-6 is

assumed to be made of a cylindrical shell structure and solid propellant (the solid propellant is

assumed to be distributed as a cylindrical shell). The cylindrical shell structure of outer radius,

osR , inner radius, isR and height, L , is modeled as a solid cylinder of radius, osR , and height,

L , from which another cylinder of radius, isR and height, L , is removed (the bases and the

centroidal axes of both cylinders coincide). The propellant shell is modeled similar to the

60

cylindrical shell structure with a time varying radius of the inner cylinder as the propellant burns.

Therefore the propellant shell of outer radius, opR , inner radius, ipR and height, L , is modeled

as a solid cylinder of radius, opR , and height, cL , from which another cylinder of time varying

radius, ipR and height, cL , is removed (the bases and the centroidal axes of both cylinders

coincide). It is assumed that as the propellant burns, the radius of the inner cylinder, ipR ,

increases. When all the propellant is burnt, opipR R= . The densities of the materials of the

cylindrical shell structure and the solid propellant are assumed to be ands pρ ρ respectively.

( ) ( )2 2 2 2s os p opis ipSMM L R R L R Rπρ πρ= − + − (3-97)

( ) ( )4 4 4 41 12 2SM s os p opis ipXXI L R R L R Rπρ πρ= − + − (3-98)

Figure 3-6. Solid motor

61

( ) ( )( ) ( )

2 2 2 2 2 2

2 2 2 2 2 2

1 13 312 12

1 13 312 12

SM s os os s is isYY

p op op p ip ip

I LR R L LR R L

L R R L L R R L

πρ πρ

πρ πρ

+= + − +

+ − + (3-99)

( ) ( )( ) ( )

2 2 2 2 2 2

2 2 2 2 2 2

1 13 312 12

1 13 312 12

SM s os os s is isZZ

p op op p ip ip

I LR R L LR R L

L R R L L R R L

πρ πρ

πρ πρ

+= + − +

+ − + (3-100)

The mass of the solid motor, SMM , is the sum of masses of the cylindrical shell structure

and cylindrical propellant shell as shown in Eq. 3-97. The mass moment of inertias,

and,SM SM SMXX YY ZZI I I , of the solid motor about its centroidal axis is the sum of mass moment

of inertias of the cylindrical shell structure and the propellant shell about the centroidal axis of

the solid motor as shown in Eqs. 3-98, 3-99 and 3-100 respectively.

Therefore the inertia tensor of the solid motor about its centroidal axis is

0 00 00 0

SM

SM

SM

XX

YY

ZZ

II I

I

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

. The instantaneous inertia tensor of the solid motor about the

instantaneous centroidal axes of the launch vehicle, SMI , is computed using the parallel axis

theorem as shown in Eq.3-101.

CM 1T TSM SM SM SM SMI I r r r r⎡ ⎤= ⎣ ⎦+ − (3-101)

The vector, SMr , in Eq. 3-101 is the position vector of the instantaneous center of mass of

the launch vehicle with respect to the instantaneous center of mass of the solid motor.

Payload

The payload model presented below is similar to the parabolic nose cone model of the

strap-on booster model discussed above in the strap-on booster section. However, the details of

model are presented again for the sake of completeness. The payload depicted in Fig. 3-7 is

62

modeled as a solid parabolic nose cone of radius, PR and length, PL . The density of material of

the nose cone is assumed to be Pρ .The details of the model are presented below.

Figure 3-7. Payload

The mass of the parabolic nose cone, PM , is given by the expression as shown in Eq. 3-

102. The mass moment of inertias, and,P P PXX YY ZZI I I , of the parabolic nose cone about its

centroidal axis are given by the expressions as shown in Eqs. 3-103, 3-104 and 3-105

respectively.

212P P P PM R Lπρ= (3-102)

213PXX P PI M R= (3-103)

2 21 16 3PYY P P PI M R L⎛ ⎞

⎜ ⎟⎝ ⎠

= + (3-104)

2 21 16 3PZZ P P PI M R L⎛ ⎞

⎜ ⎟⎝ ⎠

= + (3-105)

63

Therefore the inertia tensor of the parabolic nose cone about its centroidal axis is

0 00 00 0

P

P

P

XX

YY

ZZ

II I

I

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

. The instantaneous inertia tensor of the payload about the

instantaneous centroidal axes of the launch vehicle, PI , is computed using the parallel axis

theorem as shown in Eq. 3-106.

PM 1T TP P P P PI I r r r r⎡ ⎤= ⎣ ⎦+ − (3-106)

The vector, Pr , in Eq. 3-106 is the position vector of the instantaneous center of mass of

the solid strap-on booster with respect to the center of mass of the parabolic nose cone.

Drag Coefficient Model

The drag force acting on the launch vehicle depends on the drag coefficient, cross-sectional

area perpendicular to the flow, density of the medium and the velocity of the launch vehicle

relative to the medium. The drag force acting on a launch vehicle can be expressed as

12Drag DF C A v vρ= , where DC is the drag coefficient; A is the cross-sectional area

perpendicular to the flow; ρ is the density of the medium; andv v are the speed and velocity of

the launch vehicle relative to the medium.

The simulator requires the drag force to be coordinatized in the inertial frame. The

expression for the drag force coordinatized in inertial frame is given as 12

i iDrag DF C A v vρ= .

In this research, the drag coefficient is calculated using a drag coefficient model obtained from

the Missile Datcom database [13]. The description of the drag coefficient model is given below.

The total drag coefficient acting on the launch vehicle can be expressed as the sum of

body zero-lift wave drag coefficient, body base drag coefficient and body skin friction drag

64

coefficient i.e., ( ) ( ) ( ) ( ), ,o o o oD D DBody Body Wave Base Body FrictionC CD C C= + + , where ( )oD Body

C is

the body zero-lift drag coefficient or simply the drag coefficient; ( ) ,oD Body WaveC is the body zero-

lift wave drag coefficient; ( )oD BaseC is the body base drag coefficient and ( ) ,oD Body Friction

C is the

body skin friction drag coefficient. The expressions for the body zero-lift wave drag coefficient,

body base drag coefficient and body skin friction drag coefficient are given below.

( )

1.69

12,

1.83 0.51.59 tanoD Body Wave N

C LMd

−

⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥⎛ ⎞ ⎪ ⎪

⎢ ⎥⎨ ⎬⎜ ⎟⎝ ⎠ ⎢ ⎥⎪ ⎪

⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

= + , for M>1 (Based on Bonney reference,

1tan− in radians)

( ) ,

0.25oD Base Coast

CM

= , if M>1 and ( ) ( )2

,0.12 0.13

oD Base CoastMC += , if M<1

( ) ,Ref

0.251o

eD Base Powered

ACS M

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

= − , if M>1

( ) ( )2,

Ref1 0.12 0.13

oe

D Base Powered

AC MS

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= − + , if M<1

( )0.2

,0.053

oD Body Friction

L MCd qL

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠= (Based on Jerger reference, turbulent boundary

layer, q in psf, L in feet) where NL is the nose length; d is the rocket diameter; L is the rocket body length; eA is the

nozzle exit area; RefS is the reference area; q is the dynamic pressure and M is the mach

number.

Center of Pressure Model

The aerodynamic forces acting on the launch vehicle are assumed to be concentrated at the

center of pressure of the launch vehicle. In order to compute the moments of the aerodynamic

forces (i.e., drag and lift) about the instantaneous center of mass, the center of pressure needs to

be calculated. The center of pressure for a launch vehicle can be computed using Barrowman

65

equations [2]. The assumptions considered in the calculation of the center of pressure using

Barrowman equations are given below [3].

• The angle of attack is very near zero. • The flow is steady state and subsonic. • Flow over launch vehicle is potential flow. • Point of the nose is sharp. • Fins are thin plates with no cant. • The launch vehicle is a rigid body. • The launch vehicle is axially symmetric.

The first two assumptions severely restrict the applicability of the Barrowman equations

for calculation of the center of pressure for a real-time launch vehicle. Despite these restrictions,

the Barrowman equations are used for the calculation of the center of pressure of the launch

vehicle as it gives a rough estimate of the location of the center of pressure of the launch vehicle.

However, to be more conservative, the location of the center of pressure can be chosen so that

the launch vehicle is statically stable throughout its flight. For static stability, the center of

pressure must be located aft of the center of mass and the static margin must be at least one

caliber (i.e., distance between the center of pressure and the center of mass of the launch vehicle

is equal to the largest diameter of the launch vehicle) [1]. It should be noted that a modern full

scale launch vehicles do not rely on aerodynamics for stability. Active control systems like

thrust vector control provides stability and control to the modern full scale launch vehicles [5].

Specifically, the above assumption of one caliber static margin was made in order to compute the

aerodynamic moments acting on the launch vehicle. However, the procedure to compute the

center of pressure of a launch vehicle using Barrowman equations is presented below for the

sake of completeness.

The procedure for calculating the center of pressure of a launch vehicle using Barrowman

equations involves dividing the vehicle into separate sections, analyzing each section separately,

66

analyzing the interference effects between sections and then recombining the results of the

separate analyses to obtain the final answer i.e., the normal force coefficients and center of

pressure locations of the separate sections with respect to a common point are calculated and

then the center of pressure of the launch vehicle is calculated by using the formula in Eq. 3-107

[2], [4]. The term cpx in Eq. 3-107 is the location of the center of pressure of the launch vehicle

with respect to an arbitrary point “O”; ( )N iC is the normal force coefficient of the ith section of

the launch vehicle; ix is the location of the center of pressure of the ith section of the launch

vehicle with respect to the same arbitrary point “O”.

( )( )

iN ii

cpN i

i

C xx

C=∑∑

(3-107)

A launch vehicle is assumed to be composed of the following sections (i) nose, (ii)

cylindrical body, (iii) conical shoulder, (iv) conical boattail and (v) fins (tail section). The

normal force coefficients and locations of the center of pressure for the above sections (i)–(v) are

given below.

Nose

The normal force coefficient for all shapes of the nose is always equal, ( )noseNC = 2.

However, the location of the center of pressure is different for different shapes of the nose. The

locations of the center of pressure measured from fore of the nose for various shapes are given

below.

(i) conical nose: 23cp Lx = ; (ii) ogive nose: 0.466cp Lx = and (iii) parabolic nose: 1

2cp Lx =

67

Cylindrical Body

The normal force coefficient for a cylindrical body at low angles of attack is zero, i.e.,

( )cylindrical_bodyNC = 0. The location of center of pressure for a cylindrical body at low angle of

attack is undefined or not necessary.

Conical Shoulder

The normal force coefficient for a conical shoulder is given in Eq. 3-108. The term d jn

Eq. 3-108 is the diameter of the nose of the launch vehicle. The location of the center of

pressure of the conical shoulder as shown in Fig. 3-8 is given in Eq. 3-109.

( )2 2

2 1conical_shoulder

2Nd dCd d

⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

= − (3-108)

1

22

1

2

11

31

cp

ddLxdd

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

−= +

−

(3-109)

Figure 3-8. Conical shoulder

68

Conical Boattail

The normal force coefficient for a conical boattail is given in Eq. 3-110. The term d in

Eq. 3-110 is the diameter of the nose of the launch vehicle. The location of the center of

pressure of the conical shoulder as shown in Fig. 3-9 is given in Eq. 3-111.

( )2 2

2 1conical_boattail 2N

d dCd d

⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

= − (3-110)

1

22

1

2

11

31

cp

ddLxdd

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

−= +

−

(3-111)

Figure 3-9. Conical Boattail

Fins (Tail Section)

The normal force coefficient for a fin geometry as shown in Fig. 3-10 is given in Eq. 3-

112. The term R in Eq. 3-112 is the radius of the tail section of the launch vehicle; d is the

diameter of the nose of the launch vehicle; n is the number of fins and interK is the fin

interference factor to be considered in order to account for the interference between the fins and

the tail section (i.e., launch vehicle body section to which the fins are attached) and is calculated

69

using Eq. 3-113. The location of the center of pressure of the tail section of the launch vehicle

(i.e., the section of the launch vehicle body to which the fins are attached) as shown in Fig. 3-10

is independent of the number of fins and is given by Eq. 3-114 (where all the fins attached to the

tail section are of same size and shape).

( )

2

inter 2fin

4

21 1N

sndK

La b

C

⎛ ⎞⎜ ⎟⎝ ⎠=⎛ ⎞+ + ⎜ ⎟+⎝ ⎠

(3-112)

interK = 1 Rs R

++

for 3 or 4 fins per stage (3-113)

= 0.51 Rs R

++

for 6 fins per stage

( )( ) ( ) ( )

2 13 6

R TR R Tcp R T

R T R T

C CX C Cx C CC C C C

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

+= + + −

+ + (3-114)

Figure 3-10. Fin and Tail section

The WGS84 Ellipsoid Model

WGS (World Geodetic System) defines a reference model for the earth, for use in

geodesy and navigation. WGS84 is used as the reference ellipsoid in this research. WGS84 was

70

last revised in 2004 and will be valid up to about 2010 [11]. The model presented below

computes the geodetic coordinates (i.e., geodetic latitude, longitude and altitude) from the Earth

centered Earth fixed (ECEF) coordinates. The geodetic coordinates and the ECEF coordinates of

an arbitrary point “P” are depicted in Fig. 3-11. The computed altitude is used to calculate the

density of air at that altitude. The calculated density of air in turn is necessary to compute the

aerodynamic forces acting on the launch vehicle. The details of the model are given below [12].

The geodetic coordinates ( ), , hλ φ are defined as follows.

• Geodetic latitude, λ , is the angle between the ellipsoidal normal “N” and the equatorial plane.

• Longitude, φ , is the angle measured in the equatorial plane between the prime meridian (i.e., the X-axis of the ECEF frame) and the projection of the point “P” onto the equatorial plane.

• Altitude, h , is the distance between the surface of the ellipsoid and the point “P” measured along the ellipsoidal normal.

Figure 3-11. Geodetic Ellipsoid and Geodetic coordinates of an arbitrary point “P”

The WGS84 ellipsoid is defined by the following parameters.

Semi-major axis length of the ellipsoid, a = 6378137.0 meters

Semi-minor axis length of the ellipsoid, b = 6356752.3142 meters

71

Flatness of the ellipsoid, a bfa−

= = 0.0034

Eccentricity of the ellipsoid, ( )2e f f= − = 0.0818

The relationship between ECEF coordinates ( ), ,x y z and geodetic coordinates ( ), , hλ φ of an

arbitrary point of interest “P” can be expressed by Eqs. 3-115–3-117.

( )cos cosx N h λ φ= + (3-115)

( )cos siny N h λ φ= + (3-116)

( )21 sinz N e h λ⎡ ⎤= − +⎣ ⎦ (3-117)

Given the ECEF coordinates of an arbitrary point “P”, geodetic coordinates of that point can be

computed using the following algorithm.

1. From Eqs. 3-115 and 3-116, the longitude of the point “P” can be calculated using Eq.3-

118.

1tan yx

φ − ⎛ ⎞= ⎜ ⎟⎝ ⎠

(3-118)

2. Initialize h = 0 and N a= and define 2 2p x y= + .

3. Iterate the following expressions until λ converges as given below.

( )2sin

1z

N e hλ =

− + (3-119)

21 sintan z e N

pλλ − ⎛ ⎞+

= ⎜ ⎟⎝ ⎠

(3-120)

( )2 21 sinaN

eλ

λ=

− (3-121)

cosph Nλ

= − (3-122)

Using the above algorithm (i.e., Eqs. 3-119–3-122), geodetic coordinates of an arbitrary

point “P” can be computed from the ECEF coordinates of “P”.

This chapter presented the models developed for calculating quantities such as gravity,

inertia, mass, center of pressure and drag coefficient. These models along with the equations of

motion of the launch vehicle developed in Chapter 2 were implemented in MATLAB to form a

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six-degree-of-freedom launch vehicle simulator. The results and discussions of a simulation

performed by this simulator are presented in the next chapter.

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CHAPTER 4 SIMULATION RESULTS AND DISCUSSION

The dynamic equations (translational and rotational), kinematic equations, gravity,

inertia, drag coefficient, center of pressure and WGS84 ellipsoid models developed in Chapters 2

and 3 were implemented in MATLAB, these (along with the other MATLAB scripts/functions)

collectively form the six-degree-of-freedom launch vehicle simulator. Using the developed

simulator, a DELTA II launch vehicle model was simulated for a fictitious constant axial thrust

profile. This chapter presents the results and discussions of that simulation. Following the

results and discussions, the details of the attempt to reconstruct the thrust profile of an ATLAS

IIA launch vehicle from the flight data provided by NASA Kennedy Space Center are presented.

This reconstructed thrust profile is required for the validation of the simulator. It was shown that

the simulator cannot be validated due to the lack of availability of sufficient real-time data

necessary to reconstruct the thrust profile as required by the simulator.

Simulation

In this section, the results and discussions of a DELTA II launch vehicle model simulation

are presented. The various parameters used for this simulation are listed below.

1. Launch vehicle

• DELTA II [7]

o Four solid strap-on boosters o First stage solid motor o Second stage liquid engine o Third stage liquid engine o Payload enclosed in a parabolic nose cone

2. Thrust: Constant magnitude and acting along the longitudinal axis of the launch vehicle. The

thrust profile for this simulation can be seen in Fig. 4-1 (Tx vs time).

3. Place of launch: Kennedy Space Center, Merrit Island, Florida.

4. Date of launch: 15 June 2004.

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5. Time of launch: 1640:30 UTC.

The developed simulator requires the following data to be placed in individual text files

(a) thrust profile of the main section of the launch vehicle (the section consisting of solid motors,

liquid engines and payloads), (b) thrust profiles of the individual strap-on boosters, (c) the

instantaneous mass of the entire launch vehicle, (d) the position of the instantaneous center of

mass of the launch vehicle with respect to its initial center of mass and its time derivatives, (e)

instantaneous inertia tensor of the launch vehicle about its instantaneous center of mass and (f)

the positions of the nozzles of all the thrusting elements (solid motors, liquid engines and strap-

on boosters) with respect to the instantaneous center of mass of the launch vehicle. The

procedure used to populate the text files stated in (a)–(f) for this simulation is explained in brief

in the following (1)–(4) steps.

(1) The assumed fictitious thrust profile was used to populate the text files stated in (a) and (b).

(2) The text files populated in the step (1) i.e., the thrust profiles text files were used with Eqs.

4-1 and 4-2 to compute the instantaneous mass of the launch vehicle and populate the text file

stated in (c).

ii oi

sptt

dm T g Idt

⎛ ⎞⎜ ⎟⎝ ⎠

= (4-1)

( )ii

t t ti t t

dmm m tdt ⎛ ⎞

⎜ ⎟⎝ ⎠

−Δ −Δ

⎛ ⎞⎜ ⎟⎝ ⎠

= −Δ (4-2)

The term it

T in Eq. 4-1 represents the magnitude of the thrust provided by the ith thrusting

element at time t, it

T is calculated from the text files stated in (a) and (b); it

dmdt

⎛ ⎞⎜ ⎟⎝ ⎠

in Eqs. 4-1

and 4-2 is the rate of change of mass of the ith thrusting element at time t; the term ispI in Eq. 4-

1 is the specific impulse of the ith thrusting element; tim in Eq. 4-2 is the mass of the ith thrusting

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element at time t; ( )it tm−Δ

in Eq. 4-2 is the mass of the ith thrusting element at time ( )t t−Δ ; og in

Eq. 4-2 is the acceleration due to gravity at the surface of the Earth. (3) The launch vehicle

instantaneous mass text file populated in step (2) is used along with the inertia model discussed

in Chapter 3 and DELTA II launch vehicle geometry (obtained from DELTA II payload planners

guide [7]) to compute the instantaneous mass, the initial and the instantaneous center of mass and

the instantaneous inertia tensor of the launch vehicle and populate the text files stated in (d) and

(e). (4) The text file stated in (d) i.e., the instantaneous center of mass position text file

populated in step (3) is used with the DELTA II launch vehicle geometry (obtained from

DELTA II payload planners guide [7]) to compute positions of the nozzles of all the thrusting

elements with respect to the instantaneous center of mass of the launch vehicle and populate the

text file stated in (f). (For detailed DELTA II launch vehicle parameters and calculations, please

refer to APPENDIX B.) The simulation results and discussions are presented below.

Figure 4-1 shows the following parameters of the launch vehicle as a function of time.

• Ms is the instantaneous mass of the booster.

• M1 is the instantaneous mass of the first stage.

• M2 is the instantaneous mass of the second stage.

• M3 is the instantaneous mass of the third stage.

• rc is the location of the instantaneous center of mass of the launch vehicle with respect to its initial center of mass.

• Tx is the magnitude of the thrust acting on the launch vehicle as a function of time and is user defined.

The strap-on booster provides thrust to the launch vehicle by burning and expelling its

fuel/propellant. Therefore the instantaneous mass of the strap-on booster “Ms” decreases as a

function of time which can be seen in Fig 4-1. The mass of the strap-on booster after 64th second

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is zero because the strap-on booster is jettisoned at the 64th second, and therefore its mass

contribution to the launch vehicle is zero. Similar kind of arguments as the above explains the

instantaneous mass plots of the first, second and third stages of the launch vehicle. The plot of rc

in Fig. 4-1 is the X-component of the position vector of the instantaneous center of mass with

respect to the initial center of mass in the vehicle frame. The vehicle frame was described in

Chapter 2 and its origin coincides with the initial center of mass of the launch vehicle. The X-

axis of the vehicle frame coincides with the longitudinal axis of the launch vehicle and is positive

forwards (i.e., towards the nose of the launch vehicle). As the fuel is burnt and expelled from the

bottom stages/strap-on boosters, the instantaneous center of mass of the launch vehicle shifts up

(i.e., towards the nose of the launch vehicle) compared to the instantaneous center of mass of the

launch vehicle at the previous instant because relatively more mass is concentrated towards the

top than towards the bottom when compared to the previous instant in time. Therefore rc

increases with time as fuel/propellant is burnt and expelled. The Y and Z components of the

position vector of the instantaneous center of mass with respect to the initial center of mass are

zero always because the center of mass of the launch vehicle lies always on its longitudinal axis.

The Tx in Fig. 4-1 shows the X component of the thrust vector acting on the launch vehicle as a

function of time. The time history of the thrust vector is called the thrust profile. Generally,

thrust profiles are (i) coordinatized in the vehicle frame and (ii) designed based on the trajectory

of mission (i.e., user defined). In this present case, constant axial thrust is considered for all the

stages and strap-on boosters of the launch vehicle. Therefore the X component is equal to the

magnitude of the thrust vector (since thrust is axial, Y and Z components of the thrust vector in

the vehicle frame are zero), and the magnitude of the thrust remains constant between two

successive jettisons as seen in Fig. 4-1. Specifically, between the lift-off and the 64th second,

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Figure 4-1. Various parameters of the launch vehicle as a function of time

2673736 N of thrust is provided by the four strap-on boosters and the first stage liquid engine;

between the 64th second and 269th second, 889644 N of thrust is provided by the first stage liquid

engine; between the 269th second and the 700th second, 43657 N of thrust is provided by the

second stage liquid engine and between the 700th second and the 787th second, 66367 N of thrust

is provided by the third stage solid motor. After the 787th second, no thrust is provided to the

launch vehicle i.e., Tx = 0.

Figure 4-2 shows the x, y, z components and magnitude of the velocity vector of the

launch vehicle in the J2000 inertial frame as a function of time. From the velocity magnitude

plot, it can be seen that the magnitude of velocity of the launch vehicle increases slowly and after

the 64th second from the time of launch, the magnitude decreases by a small amount and then

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increases until the 289th second from the time of launch and then suddenly decreases, a similar

decrease and increase in the velocity magnitude can be observed again at the 880th second from

the time of launch. This response can be analyzed as follows. The strap-on boosters and the first

stage of the launch vehicle provide 2673736 N of thrust to the launch vehicle from the beginning

of the launch until the 64th second at which the strap-on boosters are jettisoned. During this

time, the fuel/propellant of the strap-on boosters and the first stage of the launch vehicle is burnt

and expelled which decreases the mass of the launch vehicle. Since, the thrusts of the strap-on

boosters and the first stage of the launch vehicle are constant, it can be seen that the same

magnitude of the thrust (2673736 N) is propelling a launch vehicle of decreasing mass and

therefore it results in accelerating the launch vehicle and hence the increase in the magnitude of

velocity of the launch vehicle is seen. At the 64th second, the strap-on boosters are jettisoned

while the first stage of the launch vehicle continues to provide a thrust of 889644 N until the

289th second at which the first stage is jettisoned. Just before the 64th second, both the strap-on

booster and the first stage of the launch vehicle provide a total thrust of 2673736 N to the launch

vehicle and just after the 64th second (and until the 289th second), only the first stage provides a

thrust of 889644 N. Therefore it can be seen that after the 64th second, lesser thrust (when

compared to the thrust just before the 64th second) is propelling the launch vehicle of almost the

same mass (since, the mass of the jettisoned boosters frames are very small compared to the

mass of the entire launch vehicle) and therefore it results in decelerating the launch vehicle and

hence the slight decrease in the magnitude of the velocity can be seen after the 64th second. The

magnitude of velocity of the launch vehicle keeps on decreasing until a point where the constant

thrust of 889644 N from the first stage of the launch vehicle is sufficient to accelerate the launch

vehicle of decreasing mass and therefore the magnitude of velocity increases until the 289th

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second. At 289th second, the first stage is jettisoned and the magnitude of velocity decreases

again and then the cycle repeats. This response can be explained by means of similar arguments

as stated above.

Figure 4-2. Velocity of the launch vehicle in the inertial frame

Figure 4-3 shows the x, y, z components of the position vector of the launch vehicle and

the position of the launch vehicle in 3D in the J2000 inertial frame (discussed in Chapter 2) as a

function of time. At the time of launch, the launch vehicle points in the +XI, -YI, +ZI direction as

seen from the J2000 inertial frame which is depicted in Fig. 4-4. Since the thrust is assumed to

be constant and axial, the thrust force always acts along the longitudinal axis of the launch

vehicle, and since the longitudinal axis of the launch vehicle is almost parallel to a vector from

the center of Earth to the center of launch vehicle, the thrust force continuously accelerates the

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Figure 4-3. Position of the launch vehicle in the inertial frame

Figure 4-4. Launch vehicle during the time of launch as seen from the J2000 inertial frame

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launch vehicle approximately in the +XI, -YI, +ZI direction and therefore the launch vehicle

position changes accordingly (i.e., position becomes more positive in X direction, more negative

in Y direction and more positive in the Z direction) which can be seen in Fig. 4-3. The 3D plot

in Fig. 4-3 shows the launch vehicle starting at point “A” and reaching point “B” after 1000

seconds from the time of launch.. The rates at which the X-, Y- and Z-components of the

position vector change in Fig. 4-3 are dictated by the corresponding X-, Y- and Z-components of

the velocity vector in Fig 4-2.

Figure 4-5 shows the moments of inertia of the launch vehicle about its instantaneous

center of mass as a function of time. xx yy zz, andI I I are the (instantaneous) moments of inertia

of the launch vehicle about the X, Y, and Z- axis respectively of a coordinate frame whose

orientation is the same as that of the vehicle frame and passes through the instantaneous center of

mass of the launch vehicle. Since the launch vehicle is symmetric about the X-axis,

Figure 4-5. Moments of inertia of the launch vehicle about its instantaneous center of mass

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yy zzandI I are equal. The change in the inertia of the launch vehicle is due to the change in (i)

mass of the launch vehicle and (ii) location of the instantaneous center of mass of the launch

vehicle. These changes in turn are due to the consumption of propellant and jettisoning of the

consumed stages of the launch vehicle.

Figure 4-6 shows the moments of inertia of a strap-on booster of the launch vehicle about

(i) a centroidal axis passing through the instantaneous center of mass of the launch vehicle and

(ii) a centroidal axis passing through its instantaneous center of mass as a function of time.

Figure 4-6. Moments of inertia of the strap-on booster about the instantaneous center of mass of

the launch vehicle and about its instantaneous center of mass

( ) ( ) ( )xx yy zzs ss, andI I I are the (instantaneous) moments of inertia of the strap-on booster about

the X, Y, and Z- axis respectively of a coordinate frame whose orientation is the same as that of

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the vehicle frame and passes through its instantaneous center of mass. ( ) ( )c c c c sx x y ys,I I

( )c cz z sand I are the (instantaneous) moments of inertia of the strap-on booster about the X, Y,

and Z- axis respectively of a coordinate frame whose orientation is the same as that of the

vehicle frame and passes through the instantaneous center of mass of the launch vehicle. The

change in the inertia of the strap-on booster is due to the changes in (i) mass of the strap-on

booster and (ii) locations of the instantaneous center of mass of both the launch vehicle and the

strap-on booster. These changes in turn are due to the consumption of propellant and jettisoning

of the consumed stages of the launch vehicle.

Figure 4-7 shows the moment of inertia of the first stage of the launch vehicle about (i) a

centroidal axis passing through the instantaneous center of mass of the launch vehicle and (ii) a

centroidal axis passing through its instantaneous center of mass as a function of time.

( ) ( )xx yy1 1,I I ( )zz 1and I are the (instantaneous) moments of inertia of the first stage about the

X, Y, and Z- axis respectively of a coordinate frame whose orientation is the same as that of the

vehicle frame and passes through its instantaneous center of mass. ( ) ( )c c c c 1x x y y1,I I ( )c cz z 1

and I

are the (instantaneous) moments of inertia of the first stage about the X, Y, and Z- axis

respectively of a coordinate frame whose orientation is the same as that of the vehicle frame and

passes through the instantaneous center of mass of the launch vehicle. From Fig. 4-7, it can be

seen that there is a slight increase (as opposed to decrease) in ( )c c 1y yI ( )( )c cz z 1and I around

64th second. This is due to jettisoning of the strap-on boosters. When the strap-on boosters are

jettisoned, there is a sudden increase in the distance between the instantaneous center of mass of

the first stage and the instantaneous center of mass of the launch vehicle. Therefore the increase

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in the inertia of the first stage about the center of mass of the launch vehicle (computed using

parallel-axis theorem)is greater than the decrease in the inertia of the first stage due to the

consumption of its propellant which results in a slight increase in ( )c c 1y yI ( )( )c cz z 1and I when

the strap-on boosters are jettisoned. After this slight increase in inertia of the first stage, its

Figure 4-7. Moment of inertia of the first stage about the instantaneous center of mass of the

launch vehicle and about its instantaneous center of mass

inertia decreases again until it is jettisoned. The change in the inertia of the first stage is due to

the changes in (i) mass of the first stage and (ii) locations of the instantaneous center of mass of

both the launch vehicle and the first stage. These changes in turn are due to the consumption of

propellant and jettisoning of the consumed stages of the launch vehicle.

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Figure 4-8 shows the moments of inertia of the second stage of the launch vehicle about (i)

a centroidal axis passing through the instantaneous center of mass of the launch vehicle and (ii) a


Figure 4-8. Moment of inertia of the second stage about the instantaneous center of mass of the


( )xx 2I ( )yy 2, I ( )zz 2and I are the (instantaneous) moments of inertia of the second stage about


the vehicle frame and passes through its instantaneous center of mass. ( ) ( )c c c c 2x x y y2,I I

( )c cz z 2and I are the (instantaneous) moments of inertia of the second stage about the X, Y, and

Z- axis respectively of a coordinate frame whose orientation is the same as that of the vehicle

frame and passes through the instantaneous center of mass of the launch vehicle. The change in

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the inertia of the second stage is due to the changes in (i) mass of both the second stage and (ii)

locations of the instantaneous center of mass of both the launch vehicle and the second stage.

These changes in turn are due to the consumption of propellant and jettisoning of the consumed

stages of the launch vehicle.

Figure 4-9 shows the moments of inertia of the third stage of the launch vehicle about (i) a

centroidal axis passing through the instantaneous center of mass of the launch vehicle and (ii) a


Figure 4-9. Moment of inertia of the third stage about the instantaneous center of mass of the


( )xx 3I ( )yy 3, I ( )zz 3and I are the (instantaneous) moments of inertia of the third stage about


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the vehicle frame and passes through its instantaneous center of mass. ( ) ( )c c c c 3x x y y3,I I

( )c cz z 3and I are the (instantaneous) moments of inertia of the third stage about the X, Y, and Z-

axis respectively of a coordinate frame whose orientation is the same as that of the vehicle frame

and passes through the instantaneous center of mass of the launch vehicle. The change in the

inertia of the third stage is due to the changes in (i) mass of both the third stage and (ii) locations

of the instantaneous center of mass of both the launch vehicle and the third stage. These changes

in turn are due to the consumption of propellant and jettisoning of the consumed stages of the

launch vehicle.

Validation

Validation of the simulator requires comparing the simulator output to the real-time flight

data. This can be done by providing real-time input of a launch vehicle for a particular mission

(date, place and time of launch and thrust profile data coordinatized in vehicle frame) to the

simulator and comparing the output of the simulator (attitude and position vectors) to the real-

time flight data (attitude and position vectors) of the same launch vehicle for the same mission.

As the validation requires all real-time input and flight data and since the real-time thrust

profile of a launch vehicle is not available to public (an ITAR issue), one of the alternatives is to

extract the approximate thrust profile from (i) place, date and time of launch, (ii) position,

velocity, and acceleration vectors of the launch vehicle as a function of time, (iii) attitude of the

launch vehicle as a function of time and (iv) launch vehicle configuration and ascent profile

Assuming that all the above data is available, approximate thrust profile of the launch vehicle

can be calculated by following the procedure presented below.

The translational equation of motion of a launch vehicle discussed in Chapter 2 is given

in Eq. 4-3. In Eq. 4-3, I a , is the acceleration vector of the launch vehicle coordinatized in the

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inertial frame. The forces and, ,I I I IgDragThrust LiftF F F F in Eq. 4-3 can be computed using Eqs.

2-37–2-39 developed in Chapter 2

I ext I I I I IgDragThrust Lift m aF F F F F+ == + + (4-3)

For simplicity, the frame of reference notation is dropped i.e., a instead of I a . Also, the

notation tx is used to represent a vector quantity x at a time “t”. Using Eq. 4-3 and the above

notation, the expression for tThrustF is given in Eq. 4-4. The relation between thrust and rate of

change of mass is given in Eq. 4-5.

t t ttt tgDragThrust Liftm aF F F F= − − − (4-4)

ot

t

tThrust

sp

dmdt

g IF

⎛ ⎞⎜ ⎟⎝ ⎠= (4-5)

In Eq. 4-3, og is the acceleration due to gravity at the surface of the Earth and tspI is the specific

impulse. The algorithm (steps 1–5) presented below is used to compute the thrust profile.

1. Calculate the mass at time “t” using 11

t tt

dmm m dtdt−

−

⎛ ⎞= − ⎜ ⎟⎝ ⎠

2. Calculate the thrust vector at time “t” using t t ttt tgDragThrust Liftm aF F F F= − − −

3. Calculate the rate of change of mass at time “t” using o ttThrustt

spdm g Idt

F⎛ ⎞ =⎜ ⎟⎝ ⎠

4. Repeat steps 1,2 and 3 to calculate the thrust vectors at time intervals “t+1”, “t+2”, ……,until the final time interval “t+n” (i.e., step 1 to calculate the mass at time “t+1” and step 2 to calculate the thrust vector at time “t+1” and step 3 to calculate the rate of change of mass at time “t+1”).

5. The thrust vectors at time intervals t = 0, 1, 2, ……, “t+n” represents the approximate thrust profile of the launch vehicle (the launch vehicle’s configuration and ascent profile play a critical role in the computation of thrust profile).

The thrust profile obtained by the above procedure is coordinatized in the inertial frame, but, the

simulator requires the input thrust profile to be coordinatized in the vehicle frame. The attitude

89

of the launch vehicle can be used to transform the thrust profile from the inertial frame to the

vehicle frame using the transformation matrix from the inertial frame to the vehicle frame RIC

(Eqs. 2-8 and 2-9) discussed in Chapter 2.

Validation Attempt

“Partial” data of an ATLAS- IIA launch vehicle for a real-time mission is available. An

attempt was made to validate the simulator as described at the beginning of this (validation)

section using the available data. The attempt proved to be futile; the details of the attempt are

given below.

The following real-time flight data of an ATLAS-IIA launch vehicle available for the

validation of the simulator (i) place, date and time of launch, (ii) position and velocity vectors of

the launch vehicle as a function of time and (iii) launch vehicle configuration and ascent profile.

The real-time attitude of the ATLAS-IIA launch vehicle is not available. As mentioned

previously, the simulator requires the thrust profile to be coordinatized in the vehicle frame.

Since the real-time attitude of the launch vehicle is unavailable, it is not possible to coordinatize

the thrust profile in the vehicle frame as required by the simulator. Therefore the simulator

cannot be validated with the available data.

Therefore in addition to the available data, instrumented flight data (i.e., accelerations and

gyro on board the vehicle) or real-time thrust profile of the launch vehicle is necessary to

validate the simulator. Figure 4-10 shows the graphical depiction of the need for instrumental

data or thrust vector in the vehicle frame.

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Figure 4-10. The need for instrumental data or thrust vector in the vehicle frame

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CHAPTER 5 CONCLUSION AND FUTURE WORK

Conclusions

This research presents the development of a six-degree-of-freedom launch vehicle

simulator which along with the communications link tool being developed by UCF forms the

trajectory and link margin analysis tool. This trajectory and link margin analysis tool is one of

the crucial support tools for STARS which provide the ability to quickly analyze new (or

changes in) concepts and ideas, an option not easily accomplished with hardware only.

Chapter 1 provided an overview of the United States range safety, STARS concept and the

trajectory and link margin analysis which motivated the development of a six-degree-of-freedom

launch vehicle simulator. Chapter 2 modeled the full dynamics of an expendable launch vehicle

by developing its kinematic and the dynamic equations. It was shown that the kinematic

equation expressed in terms of Euler angles had problems in the form of a singularity and

numerical computation. These problems were circumvented by expressing the kinematic

equation in terms of quaternions. Following the development of the kinematic equation, the

dynamic equations of an expendable launch vehicle were developed by accounting for the

variability in its mass and geometry.

Chapter 3 developed the models for the calculation of quantities such as gravity, inertia,

center of pressure and drag coefficient required for solving the equations of motion presented in

Chapter 2. The gravity model utilized the spherical harmonic representation of the gravitational

potential to account for the variability in Earth’s mass distribution and used EGM96 (360 X 360)

spherical harmonic coefficients and WGS84 Earth ellipsoid model. The gravity model was

shown to be singularity-free and numerically efficient. A novel way of calculating the variable

mass/inertial properties of a launch vehicle was developed. This inertia model was a simple and

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approximate model and considered general geometries to develop the inertia characteristics of a

launch vehicle. The drag coefficient model used in this research was obtained from the Missile

Datcom database.

The models developed in Chapters 2 and 3 were implemented in MATLAB to form a six-

degree-of-freedom launch vehicle simulator. The results and discussions of a simulation

performed using this simulator were presented in Chapter 4. These results with the support of

the discussions were proven to be logically true. However, it was shown that the simulator

cannot be validated as the critical data needed for the validation could not be provided due to

ITAR restrictions. This inability to validate the developed simulator prevented the completion of

the development of the trajectory and link margin analysis tool required by the STARS as a

support tool.

Future work

The future work would first include validating the simulator. Once validated, the simulator

developed as a result of this research can be used as a basis/framework for the development of a

future high-fidelity six-degree-of-freedom simulator. The future work/modifications to the

current simulator would include the following.

The aerodynamic model used to compute quantities such as the coefficient of drag,

coefficient of lift, center of pressure and other aerodynamic coefficients will be modified. The

inertia model will be modified to handle non-symmetric configurations of the launch vehicles.

The developed six-degree-of-freedom simulator will be interfaced with the MATLAB based

telemetry link margin analysis tool which is being developed by UCF to form the trajectory and

link margin analysis tool. A GUI will be developed for the trajectory and link margin analysis

tool and will be finally interfaced with Satellite Tool Kit (STK).

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APPENDIX A MATLAB FUNCTIONS AND SCRIPT

% % A Six-Degree-of-Freedom Launch Vehicle % % % Simulator for Range Safety Analysis % % % Author : Sharath Chandra Prodduturi, Graduate Student, SSG % University of Florida. % Date : 06-23-2007 close all clear all clc % Constants % Angular velocity of Earth omega_earth = (2*pi)/(24*60*60); % Standard Gravitational Parameter for Earth mu_earth = 398600.4418*10^9; input('enter the year of launch (20XX) = '); input ('enter the month of launch (1 - 12) = '); input('enter the day of launch (1 - 31) = '); input('enter the hour of launch (1 - 24) = '); input('enter the minute of launch (1 - 60) = '); input('enter the second of launch (1 - 60) = '); % Function for checking the input dates and times % If the input data isnt valid, the program terminates. [year,month,day,hour,min,sec,check] = timecheck(year1,month1,day1,hour1,... min1,sec1); if isempty(check) | (check == 0) error('Error in the Input. Execute the file again...!!') else % Function for Calculating the time elapsed (in sec) from JAN 1 2000 to the %input date [total_days,t0] = time(year,month,day,hour,min,sec); end disp('Enter the Launch site') disp('1 for Kennedy Space Center, Merrit Island, Florida') disp(['2 for Vandenberg Air Force Base, Space Command 30th Space Wing',... ', California']) disp('3 for Virginia Space Flight Center, Wallops Island, Virginia') disp('4 for Edwards Air Force Base, California') disp('5 for Poker Flat Research Range, Alaska') disp('6 for Alaska Spaceport, Kodiak Island, Alaska') disp('7 for Mojave Civilian Aerospace Test Center, California') disp('8 for White Sands Space Space Harbor, Las Cruces, New Mexico') input('enter the launch site = '); % Initial Euler Angles input('Enter the initial Euler angles [th_1; th_2; th_3] = '); eu_in = [0; 90; 0];

94

eu_in = pi/180 * eu_in; % Function for extracting the Latitude and Longitude of the selected Launch Site [long,lat] = place(place_1); load vehicle.m vehicle_parameters = vehicle; %ODE45, Outputs of the function are x,y,z components of position,velocity of the initial Center of mass of the %rocket in the Inertial frame and Euler angles % % Initial conditions for the ODE solver initial = initial_conditions(lat, long, eu_in, omega_earth, t0); options = []; % odeset('RelTol',3e-2,'AbsTol',3e-5); [t,y] = ode113('position', [0:1:1000], initial, options, t0, omega_earth,... lat, long, mu_earth, vehicle_parameters); format long % plots of position of the rocket in the Inertial frame (Non-rotating % geocentric equatorial reference frame) figure(1) subplot(2,2,1); plot(t,y(:,1)) xlabel('t'); ylabel('x'); subplot(2,2,2); plot(t,y(:,2)) xlabel('t'); ylabel('y'); subplot(2,2,3); plot(t,y(:,3)) xlabel('t'); ylabel('z'); subplot(2,2,4); plot3(y(:,1),y(:,2),y(:,3)) xlabel('x'); ylabel('y'); zlabel('z'); % plots of velocity of the rocket in the Inertial frame (Non-rotating % geocentric equatorial reference frame) figure(2) subplot(2,2,1); plot(t,y(:,4)) xlabel('t'); ylabel('velocity_x'); subplot(2,2,2); plot(t,y(:,5)) xlabel('t'); ylabel('velocity_y'); subplot(2,2,3);

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plot(t,y(:,6)) xlabel('t'); ylabel('velocity_z'); function [year,month,day,hour,min,sec,check] = timecheck(year1,month1,day1,hour1,min1,sec1); % Function for checking the input dates and times % If the input data isnt valid then the program terminates. if ((year1 < 2000) | (year1 > 2099) | (month1 > 12) | (day1 > 31) | (day1 < 0) | (hour1 > 24) | (hour1 < 0)| (min1 > 60) | (min1 < 0)| (sec1 > 60) | (sec1 < 0)) year = 0; month = 0; day = 0; hour = 0; min = 0; sec =0; elseif ((rem((year1-2000),4) == 0) & (month1 == 2) & (day1 > 29)) year = 0; month = 0; day = 0; hour = 0; min = 0; sec =0; elseif ((rem((year1-2000),4) ~= 0) & (month1 == 2) & (day1 > 28)) year = 0; month = 0; day = 0; hour = 0; min = 0; sec =0; elseif ((month1 == 4 | 6 | 9 | 11) & (day1 > 30)) year = 0; month = 0; day = 0; hour = 0; min = 0; sec =0; else year = year1; month = month1; day = day1; hour = hour1; min = min1; sec = sec1; end if ((year == 0) | (month == 0) | (day == 0)) check = 0; else check = 1; end function [total_days,t0] = time(year,month,day,hour,min,sec) %Function for Calculating the time elapsed (in sec) from JAN 1 2000 to the %input date days = 0; y1 = (year-2000)/4; y = (year-2000)+(ceil(y1))/365; y2 = rem((year-2000),4); for k=1:month-1 switch k case {1,3,5,7,8,10,12} days = days + 31; case {4,6,9,11} days = days + 30; case 2 switch y2 case 0 days = days + 29; otherwise days = days + 28;

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end end end total_days = y*365 + days + (day-1); % Time in seconds from JAN 1 2000 00:00:00.00 to the input time given. t0 = total_days*86400 + hour*3600 + min*60 + sec; % Time in seconds from JAN 1 2000 12:00:00.00 to the input time given. t0 = t0 - 43200; function [long,lat] = place(place_1) % % Function for extracting the Latitude and Longitude of the selected Launch Site % If the input data isnt valid, the program terminates. % % calling sequence: % [long,lat] = place(place_1) % Define Variables: % place_1 -- Launch Site % long -- Longitude of the Launch Site % lat -- Latitude of the Launch Site switch place_1 case 1 long = -81.0; lat = 28.5; case 2 long = -120.35; lat = 34.4; case 3 long = -75.5; lat = 37.8; case 4; long = -118; lat = 35; case 5 long = -147.8; lat = 64.9; case 6 long = -146; lat = 67.5; case 7 long = -118.2; lat = 35; case 8 long = -106.8; lat = 32.3; case 9 disp('You have not entered a proper choice') disp('You can select the Launch site of your choice') long = input('Enter the geographic longitude of the Launch site = '); lat = input('Enter the geocentric latitude of the Launch site = '); otherwise error('Error in the Input. Execute the file again...!!')

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end long = long*pi/180; % in rad lat = lat*pi/180; % in rad % [long,lat] = place(place); % Ref : http://www.spacetoday.org/Rockets/Spaceports/LaunchSites.html function initial = initial_conditions(lat, long, eu_in, omega_earth, t0) % % This function calculates the initial conditions of the launch vehicle. % These initial conditions are required by the ODE solver. The initial % conditions produced by this function are: % (i) Initial position of the launch vehicle in the Inertial frame % (ii) Initial velocity of the launch vehicle in the Inertial frame % (iii) Initial quaternion parameters (i.e., q0, q1, q2 and q3) % Initial position of the rocket in the inertial frame e = 0.0818; a = 6378137.0; N = a/sqrt(1-(e*sin(long))^2); % x_g, y_g, z_g are the components of initial postion of the launch vehicle % in the rotating geocentric frame x_g = N*cos(lat)*cos(long); y_g = N*cos(lat)*sin(long); z_g = N*(1-e^2)*sin(lat); % Initial angle between X_I and X_G H_G = omega_earth*t0; % Initial transformation matrix from E_I to E_G C_GI = [cos(H_G) sin(H_G) 0; -sin(H_G) cos(H_G) 0; 0 0 1]; % P_I is the initial position of the launch vehicle in the inertial frame P_I = C_GI'*[x_g; y_g; z_g]; % V_I is the initial velocity of the launch vehicle in the Inertial frame w_e = [0; 0 ; omega_earth]; V_I = cross(w_e, P_I); % Initial Quaternion Parameters q0 = cos(eu_in(3)/2)*cos(eu_in(2)/2)*cos(eu_in(1)/2) + ... sin(eu_in(3)/2)*sin(eu_in(2)/2)*sin(eu_in(1)/2); q1 = cos(eu_in(3)/2)*cos(eu_in(2)/2)*sin(eu_in(1)/2) - ... sin(eu_in(3)/2)*sin(eu_in(2)/2)*cos(eu_in(1)/2); q2 = cos(eu_in(3)/2)*sin(eu_in(2)/2)*cos(eu_in(1)/2) + ... sin(eu_in(3)/2)*cos(eu_in(2)/2)*sin(eu_in(1)/2); q3 = sin(eu_in(3)/2)*cos(eu_in(2)/2)*cos(eu_in(1)/2) - ... cos(eu_in(3)/2)*sin(eu_in(2)/2)*sin(eu_in(1)/2); %Initial conditions for the ODE solver

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initial = [P_I(1), P_I(2), P_I(3), V_I(1), V_I(2), V_I(3), q0, q1, q2,... q3,0,0,0]; function ydot = position(t, y, initial, t0, omega_earth, lat, long, ... mu_earth, vehicle_parameters) % % ODE45, Outputs of the function are x,y,z components of position, velocity % vectors of the initial Center of mass of the rocket in the Inertial % frame and Euler angles ydot = zeros(size(y)); % ------------------------------------------------------------------------- % Co-ordinate frame Transformations % Initial angle between X_I(Y_I) and X_G(Y_G) H_G = omega_earth*t0; % Instantaneous angle between X_I(Y_I) and X_G(Y_G) H_G_instant = omega_earth*(t + t0); % Instantaneous Transformation matrix from E_I to E_G C_GI_instant = [cos(H_G_instant) sin(H_G_instant) 0; -sin(H_G_instant) cos(H_G_instant) 0; 0 0 1]; % Initial Transformation matrix from E_I to E_G C_GI = [cos(H_G) sin(H_G) 0; -sin(H_G) cos(H_G) 0; 0 0 1]; % Transformation matrix from E_G to E_V C_VG = [-sin(lat)*cos(long) -sin(lat)*sin(long) cos(lat); -sin(long) cos(long) 0; -cos(lat)*cos(long) -cos(lat)*sin(long) -sin(lat)]; % Transformation matrix from E_V to E_R C_RV = [2*y(7)^2+2*y(8)^2-1, 2*y(8)*y(9)+2*y(7)*y(10),... 2*y(8)*y(10)-2*y(7)*y(9); 2*y(8)*y(9)-2*y(7)*y(10), 2*y(7)^2+2*y(9)^2-1,... 2*y(9)*y(10)+2*y(7)*y(8); 2*y(8)*y(10)+2*y(7)*y(9), 2*y(9)*y(10)-2*y(7)*y(8),... 2*y(7)^2+2*y(10)^2-1]; % Transformation matrix from E_I to E_R C_RI = C_RV*C_VG*C_GI; % ------------------------------------------------------------------------- % This function generates a flag which is used in calculation of the mass % properties, forces and moments. flag = flag_generator(t); % -------------------------------------------------------------------------

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% Mass properties % Instantaneous Center of mass and Instantaneous mass calculations [r_c, v_c, a_c, m_r] = cm_function(t, flag); % Instantaneous Mass Moment of Inertia Tensor/Matrix Calculations [I_xx I_xy I_xz I_yy I_yz I_zz] = Inertia_function(t, flag); % ------------------------------------------------------------------------- % Forces and Moments % Thrust [Thrust_Force Thrust_Moment] = Thrust_function(t, flag, C_RI, ... vehicle_parameters); % Aerodynamic forces [Drag_Force Drag_Moment Lift_Force Lift_Moment] = ... AeroForces_function(t, y, C_GI, C_RI, flag, vehicle_parameters); % Gravity force [g_Force, g_Moment] = Gravity_function(y(1:3), C_GI_instant, m_r); % ------------------------------------------------------------------------- Moment_External = Thrust_Moment + Drag_Moment + Lift_Moment + g_Moment; % Rotational Equation of motion ydot(11) = (Moment_External(1) - (I_zz - I_yy)*y(12)*y(13))/I_xx; ydot(12) = (Moment_External(2) - (I_xx - I_zz)*y(11)*y(13))/I_yy; ydot(13) = (Moment_External(3) - (I_yy - I_xx)*y(11)*y(12))/I_zz; omega_r = [y(11); y(12); y(13)]; omega_dot_r = [ydot(11); ydot(12); ydot(13)]; % ------------------------------------------------------------------------- % Relation between the quaternion rates and the angular velocity M = 0.5*[0 -y(11) -y(12) -y(13); y(11) 0 y(13) -y(12); y(12) -y(13) 0 y(11); y(13) y(12) -y(11) 0] * [y(7); y(8); y(9); y(10)]; ydot(7) = M(1); ydot(8) = M(2); ydot(9) = M(3); ydot(10) = M(4); % ------------------------------------------------------------------------- % Instantaneous Center of mass of the Rocket in the Vehicle frame R_c = r_c; % Instantaneous Velocity of the Center of mass of the Rocket in the Vehicle % frame dR_c = v_c;

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% Instantaneous Acceleration of the Center of mass of the Rocket in the % Vehicle frame d2R_c = a_c; % ------------------------------------------------------------------------- % y(1),y(2),y(3) are the x,y,z components of the position of the Initial % center of mass of the rocket in the Inertial frame % y(4),y(5),y(6) are the x,y,z components of the velocity of the Initial % center of mass of the rocket in the Inertial frame % ydot(4),ydot(5),ydot(6) are the x,y,z components of the acceleration of % the Initial center of mass of the rocket in the Inertial frame ydot(1) = y(4); % y(1) = R_c1(1) ydot(2) = y(5); % y(2) = R_c1(2) ydot(3) = y(6); % y(3) = R_c1(3) % Translational Equation of motion Force_External = Thrust_Force + Drag_Force + Lift_Force + g_Force; YDOT = (1/m_r)*(Force_External - C_RI'*(d2R_c + 2*cross(omega_r,dR_c)... + cross(omega_dot_r,R_c) + cross(omega_r,(cross(omega_r,R_c))))); ydot(4) = YDOT(1); ydot(5) = YDOT(2); ydot(6) = YDOT(3); % ------------------------------------------------------------------------- function flag = flag_generator(t) % % This function generates a flag based on the thrust profile on the Launch % Vehicle This function checks if the thrust is zero during the immediate % past instant or immediate future instant in order to avoid interpolation % load Thrust_data/Thrust.txt load Thrust_data/Nozzle.txt % Checking if the thrust is zero during the immediate past instant or % immediate future instant in order to avoid interpolation [tmi, te, tpi] = numerical_rounding(t, 1); t_x_check1 = interp1(Thrust(:,1), Thrust(:,2), tpi); t_y_check1 = interp1(Thrust(:,1), Thrust(:,3), tpi); t_z_check1 = interp1(Thrust(:,1), Thrust(:,4), tpi); t_x_check2 = interp1(Thrust(:,1), Thrust(:,2), te); t_y_check2 = interp1(Thrust(:,1), Thrust(:,3), te); t_z_check2 = interp1(Thrust(:,1), Thrust(:,4), te); if (norm([t_x_check1; t_y_check1; t_z_check1]) == 0) flag = 1; elseif (norm([t_x_check2; t_y_check2; t_z_check2]) == 0) flag = 2;

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else flag = 0; end function [xmi, xe, xpi] = numerical_rounding(x, n) % % This function rounds the input x to the specified n decimal points. n >=1 % The outputs of this function are the rounded x and two numerical values % which are greater and lesser than the rounded x by (10)^(-n) % if (n >= 1) x1 = x - fix(x); x1 = x1*(10)^n; x1 = fix(x1); x1 = x1*(10)^(-n); xe = fix(x) + x1; xmi = xe - (10)^(-n); xpi = xe + (10)^(-n); else error('n should be atleast 1') end function [r_c, v_c, a_c, m_r] = cm_function(t, flag) % % This function calculates/outputs the following: % (a) Position Vector % (b) Velocity Vector % (c) Acceleration Vector % of the Instantaneous Center of Mass of the Launch Vehicle % w.r.t the Initial Center of Mass of the the Launch Vehicle and % (d) Instantaneous Mass of the Launch Vehicle % % The input to this function is the time 't' of interest and the flag % generated by the flag_generator. % % This function requires the following data in text files in a folder % named 'Mass_properties' (without quotes) : % % (1) Time and the components of the Position, Velocity and Acceleration % vectors of the Instantaneous Center of Mass of the Launch Vehicle % w.r.t the Initial Center of Mass of the Launch Vehicle in a text % file named 'CM_data' (without quotes) in the following format: % % t1 Px(t1) Py(t1) Pz(t1) Vx(t1) Vy(t1) Vz(t1) Ax(t1) Ay(t1) Az(t1) % t2 Px(t2) Py(t2) Pz(t2) Vx(t2) Vy(t2) Vz(t2) Ax(t2) Ay(t2) Az(t2) % . . . . . . . . . . % . . . . . . . . . . % % where: % Px, Py and Pz are the X-, Y-, Z- components of the Position Vector % of the Instantaneous Center of Mass of the Launch Vehicle w.r.t. the % Initial Center of Mass respectively % % vx, Vy and Vz are the X-, Y-, Z- components of the Velocity Vector % of the Instantaneous Center of Mass of the Launch Vehicle w.r.t. the

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% Initial Center of Mass respectively % Ax, Ay and Az are the X-, Y-, Z- components of the Acceleration Vector % of the Instantaneous Center of Mass of the Launch Vehicle w.r.t. the % Initial Center of Mass respectively % % (2) Time and Instantaneous Center of Mass of the Launch Vehicle in a % text file named 'Mass_data' (without quotes) in the following format: % % t1 M(t1) % t2 M(t2) % . . % . . % load Mass_properties/CM_data.txt load Mass_properties/Mass_data.txt [tmi, te, tpi] = numerical_rounding(t, 1); switch flag case 0 time = t; case 1 time = tpi; case 2 time = te; otherwise error(['Invalid value of "flag" in cm_function, Please check', ... ' the value of "flag" generated by flag_generator']) end Px = interp1(CM_data(:,1), CM_data(:,2), time); Py = interp1(CM_data(:,1), CM_data(:,3), time); Pz = interp1(CM_data(:,1), CM_data(:,4), time); Vx = interp1(CM_data(:,1), CM_data(:,5), time); Vy = interp1(CM_data(:,1), CM_data(:,6), time); Vz = interp1(CM_data(:,1), CM_data(:,7), time); Ax = interp1(CM_data(:,1), CM_data(:,8), time); Ay = interp1(CM_data(:,1), CM_data(:,9), time); Az = interp1(CM_data(:,1), CM_data(:,10), time); m_r = interp1(Mass_data(:,1), Mass_data(:,2), time); r_c = [Px; Py; Pz]; v_c = [Vx; Vy; Vz]; a_c = [Ax; Ay; Az]; function [I_xx I_xy I_xz I_yy I_yz I_zz] = Inertia_function(t, flag) % % This function calculates/outputs the components of the Inertia tensor % of the launch vehicle. The input to this function is the time 't' of % interest and the flag generated by the flag_generator. % % This function requires the following data in a text file named % 'Inertia_data' (without quotes) placed in a folder named % 'Mass_properties' (without quotes) : % % (i) Time and the components of the Inertia tensor of the launch vehicle

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% about the instantaneous center of the launch vehicle in the % following format: % % t1 I_xx(t1) I_xy(t1) I_xz(t1) I_yy(t1) I_yz(t1) I_zz(t1) % t2 I_xx(t2) I_xy(t2) I_xz(t2) I_yy(t2) I_yz(t2) I_zz(t2) % . . . . . . . % . . . . . . . % load Mass_properties/Inertia_data.txt [tmi, te, tpi] = numerical_rounding(t, 1); switch flag case 0 time = t; case 1 time = tpi; case 2 time = te; otherwise error(['Invalid value of "flag" in Inertia_function, Please', ... ' check the value of "flag" generated by flag_generator']) end I_xx = interp1(Inertia_data(:,1), Inertia_data(:,2), time); I_xy = interp1(Inertia_data(:,1), Inertia_data(:,3), time); I_xz = interp1(Inertia_data(:,1), Inertia_data(:,4), time); I_yy = interp1(Inertia_data(:,1), Inertia_data(:,5), time); I_yz = interp1(Inertia_data(:,1), Inertia_data(:,6), time); I_zz = interp1(Inertia_data(:,1), Inertia_data(:,7), time); function [Thrust_Force Thrust_Moment] = Thrust_function(t, flag, C_RI, ... vehicle_parameters) % % This function calculates the Total Thrust & Thrust Moment (about the % Instantaneous Center of Mass of the Launch Vehicle) of the Launch Vehicle % (including Boosters). The output is cordinatized in the vehicle frame % % The input parameters (to be passed) to this function are the time 't' % of interest, number of Boosters, the flag generated by the % flag_generator, the transformation matrix C_RI and the vehicle_parameters % array % % This function requires the following data in text files in a folder % named 'Thrust_data' (without quotes) : % % (1) Time and the Thrust vector (co-ordinatized in the Vehicle frame) % of the Launch Vehicle in a text file named 'Thrust.txt' (without % quotes) in the following format: % % t1 Thrust_x(t1) Thrust_y(t1) Thrust_z(t1) % t2 Thrust_x(t2) Thrust_y(t2) Thrust_z(t2) % . . . . % . . . . % % (2) Time and the (Position) vector from the instantaneous center of mass % of the launch vehicle (co-ordinatized in the Vehicle frame) to the

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% nozzle of the launch vehicle through which the consumed fuel / gases % are expelled in a text file named 'Nozzle.txt' (without quotes) in % the following format: % % t1 Nozzle_x(t1) Nozzle_y(t1) Nozzle_z(t1) % t2 Nozzle_x(t2) Nozzle_y(t2) Nozzle_z(t2) % . . . . % . . . . % % (3) Time and Thrust vector (co-ordinatized in the Booster / % Vehicle frame) of the i'th booster in a text file named % 'Thrust_bi.txt' (without quotes) where 'i' in '_bi' represents the % i'th booster (i = 1,2,...,n) in the following format. (For example, % 1st booster data should be in a text file named 'Thrust_b1.txt' % (without quotes), 2nd booster data in 'Thrust_b2.txt' (without % quotes) and so on..) % % t1 Thrust_x_bi(t1) Thrust_y_bi(t1) Thrust_z_bi(t1) % t2 Thrust_x_bi(t2) Thrust_y_bi(t2) Thrust_z_bi(t2) % . . . . % . . . . % % (4) Time and the (Position) vector from the instantaneous center of mass % of the launch vehicle (co-ordinatized in the vehicle frame) nozzle of % the i'th booster through which the consumed fuel / gases are expelled % in a file named 'Nozzle_bi.txt' (without quotes) where 'i' in '_bi' % represents the i'th booster (i = 1,2,...,n) in the following format. % (For example, 1st booster data should be in a text file named % 'Nozzle_b1.txt' (without quotes), 2nd booster data in 'Nozzle_b2.txt' % (without quotes) and so on..) % % t1 Nozzle_x_bi(t1) Nozzle_y_bi(t1) Nozzle_z_bi(t1) % t2 Nozzle_x_bi(t2) Nozzle_y_bi(t2) Nozzle_z_bi(t2) % . . . . % . . . . % % Initialize to zero vector(s) Thrust_Force = [0;0;0]; Thrust_Moment = [0;0;0]; % Miscellaneous Parameters Calculations [L_vehicle L_strap D_vehicle D_strap A_e_vehicle A_e_strap Ln_vehicle ... Ln_strap Ref_A_vehicle Ref_A_strap boosters] = ... misc_calculations_function(t, flag, vehicle_parameters); % Boosters Thrust Data for i = 1:boosters dummy = int2str(i); load (strcat('Thrust_data/','Thrust_b',dummy,'.txt')) load (strcat('Thrust_data/','Nozzle_b',dummy,'.txt')) eval(sprintf('t_x = interp1(Thrust_b%d(:,1), Thrust_b%d(:,2), t);',... i, i)); eval(sprintf('t_y = interp1(Thrust_b%d(:,1), Thrust_b%d(:,3), t);',... i, i)); eval(sprintf('t_z = interp1(Thrust_b%d(:,1), Thrust_b%d(:,4), t);',... i, i));

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eval(sprintf('n_x = interp1(Nozzle_b%d(:,1), Nozzle_b%d(:,2), t);',... i, i)); eval(sprintf('n_y = interp1(Nozzle_b%d(:,1), Nozzle_b%d(:,3), t);',... i, i)); eval(sprintf('n_z = interp1(Nozzle_b%d(:,1), Nozzle_b%d(:,4), t);',... i, i)); eval(sprintf('thrust_b%d_x = t_x;', i)); eval(sprintf('thrust_b%d_y = t_y;', i)); eval(sprintf('thrust_b%d_z = t_z;', i)); eval(sprintf('nozzle_b%d_x = n_x;', i)); eval(sprintf('nozzle_b%d_y = n_y;', i)); eval(sprintf('nozzle_b%d_z = n_z;', i)); end for i = 1:boosters eval(sprintf(['Thrust_Force = Thrust_Force + [thrust_b%d_x; '... 'thrust_b%d_y; thrust_b%d_z];'],i,i,i)); eval(sprintf(['Thrust_Moment = Thrust_Moment + '... 'cross(([nozzle_b%d_x; nozzle_b%d_y; nozzle_b%d_z])'... ' ,[thrust_b%d_x; thrust_b%d_y; thrust_b%d_z]);']... ,i,i,i,i,i,i)); end % Launch Vehicle Data load Thrust_data/Thrust.txt load Thrust_data/Nozzle.txt switch flag case {1, 2} t_x = 0; t_y = 0; t_z = 0; n_x = 0; n_y = 0; n_z = 0; case 0 t_x = interp1(Thrust(:,1), Thrust(:,2), t); t_y = interp1(Thrust(:,1), Thrust(:,3), t); t_z = interp1(Thrust(:,1), Thrust(:,4), t); n_x = interp1(Nozzle(:,1), Nozzle(:,2), t); n_y = interp1(Nozzle(:,1), Nozzle(:,3), t); n_z = interp1(Nozzle(:,1), Nozzle(:,4), t); otherwise error(['Invalid value of "flag" in Thrust_function, Please', ... ' check the value of "flag" generated by flag_generator']) end Thrust_Force = Thrust_Force + [t_x;t_y;t_z]; Thrust_Moment = Thrust_Moment + cross(([n_x; n_y; n_z]),... [t_x;t_y;t_z]); % Co-ordinatizing the thrust force in the inertial frame Thrust_Force = transpose(C_RI)*Thrust_Force; function [L_vehicle L_strap D_vehicle D_strap A_e_vehicle A_e_strap ...

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Ln_vehicle Ln_strap Ref_A_vehicle Ref_A_strap N_s]... = misc_calculations_function(t, flag, vehicle_parameters) % % This function calculates/outputs various miscellaneous parameters: % 1. Length of the Launch Vehicle (and Strap-On Boosters) % 2. Diameter of the Launch Vehicle (and Strap-On Boosters) % 3. Nozzle exit area of the Launch vehicle (and Strap-On Boosters) % 4. Nose length of the Launch Vehicle (and Strap-On Boosters) % 5. Reference area of the Launch Vehicle (and Strap-On Boosters) % 6. Position vector of the Initial Center of Mass of the Launch Vehicle % w.r.t the nose of the Launch Vehicle expressed in the Vehicle Frame % (Body Co-ordinate Frame) % % The input parameters (to be passed) to this function are the time 't' % of interest, the flag generated by the flag_generator and the % vehicle_parameters array % % This function requires the following data in a text file named % 'Length_data' (without quotes) placed in a folder named % 'Mass_properties' (without quotes) : % % (i) Time and the length of the Launch Vehicle (excluding the Strap-On % Boosters) in the following format: % % t1 L(t1) % t2 L(t2) % . . % . . % % Length of the Launch Vehicle load Mass_properties/Length_data.txt [tmi, te, tpi] = numerical_rounding(t, 1); switch flag case 0 L_vehicle = interp1(Length_data(:,1), Length_data(:,2), t); case 1 L_vehicle = interp1(Length_data(:,1), Length_data(:,2), tpi); case 2 L_vehicle = interp1(Length_data(:,1), Length_data(:,2), te); otherwise error(['Invalid value of "flag" in misc_caclculations_', ... 'function, Please check the value of "flag" generated'... ' by flag_generator']) end % Length of the Strap-on Booster(s) L_strap = vehicle_parameters(1); % Diameter of the Launch Vehicle D_vehicle = vehicle_parameters(2); % Diameter of the Strap-on Booster(s) D_strap = vehicle_parameters(3);

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% Nozzle exit area of the Launch Vehicle A_e_vehicle = vehicle_parameters(4); % Nozzle exit area of the Strap-on Booster(s) A_e_strap = vehicle_parameters(5); % Nose length of the Launch vehicle Ln_vehicle = vehicle_parameters(6); % Nose length of the Strap-on Booster(s) Ln_strap = vehicle_parameters(7); % Reference area of the Launch Vehicle Ref_A_vehicle = (pi/4)*D_vehicle^2; % Reference area of the Strap-On Booster(s) Ref_A_strap = (pi/4)*D_strap^2; % Number of boosters N_s = vehicle_parameters(8); function [Drag_Force Drag_Moment Lift_Force Lift_Moment] = ... AeroForces_function(t, y, C_GI, C_RI, flag, vehicle_parameters) % % This function calculates/outputs the Atmospheric Forces and Moments: % 1. Drag Force (coordinatized in the inertial frame) % 2. Drag Moment (coordinatized in the vehicle frame) % 3. Lift Force (coordinatized in the inertial frame) % 4. Lift Moment (coordinatized in the vehicle frame) % % The input parameters (to be passed) to this function are the time 't' % of interest, the flag generated by the flag_generator and the vehicle % parameter array % % This function requires the following data in text files in a folder % named 'Mass_properties' (without quotes) : % % (1) Position Vector of the Instantaneous Center of Mass of the ith % Strap-on Booster w.r.t the Instantaneous Center of Mass of the % entire Launch Vehicle expressed in the Vehicle Frame in a text file % named 'cm_bi.txt' (without quotes) where 'i' in '_bi' represents % the i'th booster (i = 1,2,...,n) in the following format. % (For example, 1st booster data should be in a text file named % 'cm_b1.txt' (without quotes), 2nd booster data in 'cm_b2.txt' % (without quotes) and so on..) position_I = y(1:3); velocity_I = y(4:6); % Position vector of the Launch vehicle in the ECEF frame position_ECEF = C_GI*position_I; % Velocity vector of the Launch Vehicle in the Vehicle Reference frame velocity_R = C_RI*velocity_I; % Miscellaneous Parameters Calculations

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[L_vehicle L_strap D_vehicle D_strap A_e_vehicle A_e_strap Ln_vehicle ... Ln_strap Ref_A_vehicle Ref_A_strap boosters] = ... misc_calculations_function(t, flag, vehicle_parameters); % Density of air and Co-efficient of Drag Calculation if t == 0 if boosters ~= 0 C_D_strap = 0; end C_D_main = 0; rho_air = 1.225; else [geod_lat, geod_long, h] = wgs84(position_ECEF); if h < 600000 rho_air = AtmDens2(h/1000); if boosters ~= 0 C_D_strap = C_Drag(rho_air, h, velocity_I, Ln_strap, ... D_strap/2, A_e_strap, Ref_A_strap, L_strap); end C_D_main = C_Drag(rho_air, h, velocity_I, Ln_vehicle, ... D_vehicle/2, A_e_vehicle, Ref_A_vehicle, L_vehicle); else rho_air = 0; if boosters ~= 0 C_D_strap = 0; end C_D_main = 0; end end [tmi, te, tpi] = numerical_rounding(t, 1); switch flag case 0 time = t; case 1 time = tpi; case 2 time = te; otherwise error(['Invalid value of "flag" in AeroForces_function, ', ... 'Please check the value of "flag" generated by ', ... 'flag_generator']) end % ------------------------------------------------------------------------- % Drag % Forces % Drag force acting on the Launch Vehicle (excluding boosters) Drag_Force_main = 0.5*C_D_main*rho_air*Ref_A_vehicle*... norm(velocity_R)*velocity_R; % Drag force acting on the boosters Drag_Force_strap = 0.5*C_D_strap*rho_air*Ref_A_strap*... norm(velocity_R)*velocity_R;

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% Total Drag force acting on the Launch Vehicle (i.e., including boosters) Drag_Force = Drag_Force_main + boosters*Drag_Force_strap; % Moments % Moment due to the Drag force acting on the Launch Vehicle about the % instantaneous center of mass of the launch vehicle including the % strap-on boosters. (The drag force is assumed to be acting at the center % of pressure of the launch vehicle Drag_Moment = cross([-D_vehicle;0;0], Drag_Force); % Co-ordinatize the Drag Force in the Inertial Frame Drag_Force = transpose(C_RI)*Drag_Force; % ------------------------------------------------------------------------- % Lift % Forces % Lift force acting on the Launch Vehicle (excluding boosters) Lift_Force_main = [0;0;0]; % Lift force acting on the boosters Lift_Force_strap = [0;0;0]; % Total Lift force acting on the Launch Vehicle (i.e., including boosters) Lift_Force = Lift_Force_main + boosters*Lift_Force_strap; % Moments % Moment due to the Lift force acting on the Launch Vehicle about the % instantaneous center of mass of the launch vehicle including the % strap-on boosters. (The Lift force is assumed to be acting at the center % of pressure of the launch vehicle) Lift_Moment = cross([-D_vehicle;0;0], Lift_Force); % Co-ordinatize the Lift Force in the Inertial Frame Lift_Force = transpose(C_RI)*Lift_Force; % ------------------------------------------------------------------------- function [geod_lat, geod_long, alt] = wgs84(R_ECEF) % This function calculates the Geodetic Latitude, Longitude and the % Altitude from the rectangular ECEF coordinates. % % Form: [geod_lat, geod_long, alt] = wgs84(R_ECEF) % % Input to this function is the Position Vector of the point of interest in % the ECEF frame. % % Outputs of this function are: % 1. geod_lat = Geodetic Latitude of the Point of Interest. % 2. geod_long = Longitude of the Point of Interest. % 3. alt = Altitude (is the distance along the ellipsoidal normal, away from % the interior of the ellipsoid, between the surface of the ellipsoid and % the point of interest)

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% Reference: The Global Positioning System & Inertial Navigation by Jay % A.Farell & Matthew Barth. x = R_ECEF(1); y = R_ECEF(2); z = R_ECEF(3); % WGS-84 Ellipsoid parameters a = 6378137.0; % Semimajor axis length of the Ellipsoid (in meters) b = 6356752.3142; % Semiminor axis length of the Ellipsoid (in meters) f = (a-b)/a; % Flatness of the Ellipsoid e = sqrt(f*(2-f)); % Eccentricity of the Ellipsoid p = sqrt(x^2 + y^2); h = 0; % Initialization N = a; % Initialization lambda = 0; % Initialization epsilon = 1; % Initialization % Iteration while epsilon > 1e-8 lambda_old = lambda; sine_lambda = z/(N*(1-e^2) + h); lambda = atan((z + e^2*N*sine_lambda)/p); N = a/sqrt(1 -(e*sin(lambda))^2); h = (p/cos(lambda)) - N; epsilon = lambda - lambda_old; end geod_lat = lambda; geod_long = atan2(y,x); alt = h; function [g_Force, g_Moment] = Gravity_function(R_I, C_GI, Mass) % % This function calculates/outputs the Gravity Force Vector and the Gravity % Force Moment (about the Instantaneous Center of Mass of the Launch % Vehicle) acting on the Launch Vehicle due to the gravitational attraction % of the Earth. The Force and Moment vectors calculated in this function % are co-ordinatized in the Inertial frame. % % The input parameters (to be passed) to this function are the position of % the Launch Vehicle co-ordinatized in the Inertial frame, the % (instantaneous) transformation matrix from the Earth Centered Inertial % frame to the ECEF frame (C_GI) and the instantaneous Mass of the % Launch Vehicle. % Position vector of the Launch Vehicle (co-ordinatized in the ECEF frame) R_ECEF = C_GI*R_I; % Acceleration due to gravity acting on the Launch Vehicle (co-ordinatized % in the ECEF Frame) g_ECEF = gravity(R_ECEF); % Acceleration due to gravity acting on the Launch Vehicle (co-ordinatized % in the Inertial Frame) g_I = transpose(C_GI)*g_ECEF;

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% Gravity Force acting on the Launch Vehicle (co-ordinatized in the Inertial % Frame) g_Force = Mass*g_I; % Gravity Force Moment (about the Instantaneous Center of Mass of the % Launch Vehicle) acting on the Launch Vehicle (co-ordinatized in the % Inertial Frame) g_Moment = [0;0;0]; function [g_e] = gravity(R_g) % This function computes the Gravity vector in the ECEF frame. This % function is based on WGS84-EGM96 Gravity model. % % The input to this function is the position vector of the point of % interest in the ECEF frame. (3 X 1 vector) load WGS84EGM96-normalized % The coefficients Cnm, Snm, Jn in the file WGS84EGM96 are Normalized % Gravitational coefficients. C = nC; S = nS; J = nJ; n_max = length(J); x = R_g(1); y = R_g(2); z = R_g(3); r = norm(R_g); R_g_unitvector = R_g/r; mu_earth = 398600.4418*10^9; % Standard Gravitational Parameter for Earth; a = 6378137.0; s = x/r; t = y/r; u = z/r; % ------------------------------------------------------------------------- % rho_n calculation rho_0 = mu_earth/r; rho = a/r; for i = 1:(1+n_max) if i == 1 rho_n(i) = rho*rho_0; else rho_n(i) = rho*rho_n(i-1); end end % -------------------------------------------------------------------------

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% Anm_bar calculations A0n_bar(2) = 1/2*5^(1/2)*(3*u^2-1); A0n_bar(3) = 1/2*7^(1/2)*u*(5*u^2-3); Anm_bar = zeros(n_max,n_max+1); Anm_bar(2,1) = 15^(1/2)*u; Anm_bar(2,2) = 1/2*15^(1/2); Anm_bar(3,1) = 1/4*42^(1/2)*(5*u^2-1); Anm_bar(3,2) = 5.1235*u; Anm_bar(3,3) = 2.0917; for n = 4:n_max A0n_bar(n) = (1/n)*(u*sqrt((2*n+1)*(2*n-1))*A0n_bar(n-1) - (n-1)*sqrt((2*n+1)/(2*n-3))*A0n_bar(n-2)); end for n = 4:n_max Anm_bar(n,n) = sqrt((2*n+1)/(2*n))*Anm_bar(n-1,n-1); end for n = 4:n_max for m = n-1:-1:1 Anm_bar(n,m) = 2*(m+1)*u*sqrt(1/((n+m+1)*(n-m)))*Anm_bar(n,m+1) + (u^2-1)*sqrt((n+m+2)*(n-m-1)/((n+m+1)*(n-m)))*Anm_bar(n,m+2); end end % ------------------------------------------------------------------------- % a1_bar calculation a1_bar = 0; for n = 2:n_max a1_bar_1 = 0; for m = 0:n if m == 0 a1_bar_2 = 0; else a1_bar_2 = Anm_bar(n,m)*m*Enm_bar(n,m,s,t,C,S,J); end a1_bar_1 = a1_bar_1 + a1_bar_2; end a1_bar = a1_bar + a1_bar_1*rho_n(n+1)/a; end % ------------------------------------------------------------------------- % a2_bar calculation a2_bar = 0; for n = 2:n_max a2_bar_1 = 0; for m = 0:n if m == 0 a2_bar_2 = 0;

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else a2_bar_2 = Anm_bar(n,m)*m*Fnm_bar(n,m,s,t,C,S,J); end a2_bar_1 = a2_bar_1 + a2_bar_2; end a2_bar = a2_bar + a2_bar_1*rho_n(n+1)/a; end % ------------------------------------------------------------------------- % a3_bar calculation a3_bar = 0; for n = 2:n_max a3_bar_1 = 0; for m = 0:n if m == 0 a3_bar_2 = A0n_bar(n)*Dnm_bar(n,m,s,t,C,S,J); else a3_bar_2 = Anm_bar(n,m+1)*Dnm_bar(n,m,s,t,C,S,J); end a3_bar_1 = a3_bar_1 + a3_bar_2; end a3_bar = a3_bar + a3_bar_1*rho_n(n+1)/a; end % ------------------------------------------------------------------------- % a4_bar calculation a4_bar = 0; for n = 2:n_max a4_bar_1 = 0; for m = 0:n if m == 0 a4_bar_2 = A0n_bar(n)*Dnm_bar(n,m,s,t,C,S,J); else a4_bar_2 = Anm_bar(n,m)*Dnm_bar(n,m,s,t,C,S,J); end a4_bar_1 = a4_bar_1 + a4_bar_2; end a4_bar = a4_bar + a4_bar_1*(n+1)*rho_n(n+1)/a; end a4_bar = -rho_0/r - a4_bar - s*a1_bar - t*a2_bar - u*a3_bar; % ------------------------------------------------------------------------- % Acceleration due to gravity in ECEF g_e = [a1_bar; a2_bar; a3_bar] + a4_bar*R_g_unitvector; end % ------------------------------------------------------------------------- % Dnm_bar subfunction function sol = Dnm_bar(n,m,s,t,C,S,J) rmst = r_m(m,s,t);

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imst = i_m(m,s,t); if m == 0 sol = J(n)*rmst; else sol = C(n,m)*rmst + S(n,m)*imst; end end % ------------------------------------------------------------------------- % Enm_bar subfunction function sol = Enm_bar(n,m,s,t,C,S,J) rm1st = r_m(m-1,s,t); im1st = i_m(m-1,s,t); if m == 0 sol = J(n)*rm1st; else sol = C(n,m)*rm1st + S(n,m)*im1st; end end % ------------------------------------------------------------------------- % Fnm_bar subfunction function sol = Fnm_bar(n,m,s,t,C,S,J) rm1st = r_m(m-1,s,t); im1st = i_m(m-1,s,t); if m == 0 sol = -J(n)*im1st; else sol = S(n,m)*rm1st - C(n,m)*im1st; end end % ------------------------------------------------------------------------- % r_m subfunction function sol = r_m(m,s,t) sol = real((s+i*t)^m); end % ------------------------------------------------------------------------- % i_m subfunction

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function sol = i_m(m,s,t) sol = imag((s+i*t)^m); end % -------------------------------------------------------------------------

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APPENDIX B SIMULATION CONFIGURATION

Figure B-1. The DELTA II Launch vehicle geometry

Figure B-2. Strap-on booster geometry

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Instantaneous Center of Mass of the Rocket

From the geometry of the rocket and strap-on boosters, instantaneous center of mass of the

rocket is calculated

Strap-on booster parameters

Lns = Length of the nose-cone Xbs = Length from the nose tip to fin root leading edge Lc = Length from the base of the nose-cone to the base of the strap-on booster Ys= Length from the base of the nose-cone to the top of the solid-motor Hs= Length of the solid-motor R3s= Outer radius of the cylindrical strap-on booster frame R2s= Inner radius of the cylindrical strap-on booster frame R1s = Inner radius of the solid-propellant shell CRs = Fin root chord XRs= Length from fin root leading edge to fin tip leading edge parallel to the strap-on booster body CTs= Fin tip chord Lfs = Length of fin mid chord line Ss= Span of one fin tfs = Thickness of the fin Nfs = Number of fins

1sρ = Density of the material of the strap-on

pρ = Density of the solid propellant Thrusts= Thrust of the solid motor Isp_s= Specific impulse of solid motor in sec Rocket Parameters Ln = Length of the nose-cone Ls = Length from the nose tip of the rocket to the nose tip of the strap-on booster L1 = Length of the first stage of the rocket L2 = Length of the second stage of the rocket L3 = Length of the third stage of the rocket R1 = Outer radius of the cylindrical rocket frame R2 = Inner radius of the cylindrical shell of the 1st stage rocket frame R3 = Inner radius of the cylindrical shell of the 2nd stage rocket frame R4 = Inner radius of the cylindrical shell of the 3rd stage rocket frame R5 = Inner radius of the solid propellant shell of the 3rd stage M1pi = Initial mass of the propellant of the first stage M2pi = Initial mass of the propellant of the second stage M4 = Mass of the payload (parabolic nose-cone) Thrust1 = Thrust of the 1st stage

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Isp_1 = Specific impulse of the 1st stage in sec Thrust2 = Thrust of the 2nd stage Isp_2 = Specific impulse of the 2nd stage in sec Thrust3 = Thrust of the 3rd stage Isp_3 = Specific impulse of the 3rd stage in sec

1ρ = Density of the material of the rocket frame Ns = Number of strap-on boosters Re = Rocket nozzle exit radius Instantaneous mass of the rocket Mr = M1 + M2 + M3 + M4 + NsMs Initial mass of the rocket = Mri = M1i + M2i + M3i + M4 + NsMsi Instantaneous center of mass of the rocket w.r.t intial center of mass

2 31 1 2 3 4 4ic

s s sc

rXM x M x M x M x N M Xr

M+ + + += −

where

icX = Initial Center of mass of the rocket 1 1 2 2 3 3 4 4 s si icsi i iic

riX M x M x M x M x N M X

M+ + + +=

Center of mass of first stage = x1 = – (L1/2 + L2 + L3 + Ln) Initial mass of the first stage M1i = M1_1 – M1_2 + M1pi Instantaneous mass of the first stage = M1 = M1i – t1 dm1p/dt M1_1 = 1ρ π R1

2 L1 M1_2 = 1ρ π R2

2 L1 Rate of change of mass of the first stage liquid engine = dm1p/dt = Thrust1/(Isp_1 g) g = Acceleration due to gravity on the surface of the Earth. t1 = time elapsed in sec from the moment the first stage was ignited Center of mass of second stage = x2 = – (L2/2 + L3 + Ln) Initial mass of the second stage = M2i = M2_1 – M2_2 + M2pi Instantaneous mass of the second stage = M2 = M2i – t2 dm2p/dt M2_1 = 1ρ π R1

2 L2 M2_2 = 1ρ π R3

2 L2 Rate of change of mass of the second stage liquid engine = dm2p/dt = Thrust2/(Isp_2 g) g = Acceleration due to gravity on the surface of the Earth. t2 = time elapsed in sec from the moment the second stage was ignited Center of mass of third stage = x3 = – (L3/2 + Ln) Initial mass of the third stage = M3i = M3_1 – M3_2 + M3_3 –M3_4 Instantaneous mass of the third stage = M3 = M3i – t3 dm3p/dt M3_1 = 1ρ π R1

2 L3

M3_2 = 1ρ π R42 L3

M3_3 = pρ π R42 L3

M3_4 = pρ π R52 L3

Rate of change of mass of the third stage solid motor = dm3p/dt = Thrust3/(Isp_3 g) g = Acceleration due to gravity on the surface of the Earth. t3 = time elapsed in sec from the moment the third stage was ignited

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Center of mass of nose cone/payload = x4 = – 2 Ln/3 Initial mass of the strap-on booster Msi = M1s – M2s +M3is + Ns Mfs + M5s Instantaneous mass of the strap-on booster Ms = M1s – M2s +M3ps + Ns Mfs + M5s Instantaneous center of mass of the strap-on booster

5 51 1 2 2 3 3 4 s ss s s s ps s sfs fss

s

M x M x M x N M x M xX

M− + + +

=

Initial center of mass of the strap-on booster 5 51 1 2 2 3 3 4 s ss s s s is s sfs fs

icssi

M x M x M x N M x M xX

M− + + +

=

x1s = – (Ls + Lns + Lc/2) M1s = 1sρ π R3s

2 Lc x2s = – (Ls + Lns + Lc/2) M2s = 1sρ π R2s

2 Lc x3s = – (Ls + Lns + Ys + Hs/2) M3is = M3s – M4s M3s = pρ π R2s

2 Hs

M4s = pρ π R1s2 Hs

M3ps = M3is – ts dms/dt Rate of change of mass of the solid motor = dms/dt = Thrusts/(Isp_s g) ts = time elapsed in sec from the moment the strap-on booster was ignited g = Acceleration due to gravity on the surface of the Earth. x4s = – Ls – (L1s

2- XRs2/3 – L2s (L1s – L2s/3))/(2 L1s – XRs – L2s)

Mass of fin Mfs= M1fs – M2fs – M3fs M1fs = 1sρ L1s tfs Ss/8 M2fs = 0.5 1sρ Ss tfs Ss/8 M3fs = 1sρ L1s tfs Ss/8 where L1s = XRs + CTs, L2s = XRs + CTs - CRs x5s = – Ls – 2 Lns/3 M5s = 500;

Mass Moment of Inertia Calculation

The Mass moment of inertia of the rocket (and booster) is calculated about the Cartesian

axes through the instantaneous center of mass of the rocket.

The major elements of the rocket are

1. First stage liquid motor

2. Second stage liquid motor

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3. Third stage solid motor

4. Payload

5. Solid strap-on boosters

Figure B-3. Elements of DELTA II Launch vehicle and Strap-on Booster

Strap-on booster: The booster is assumed to be made of

1. Cylindrical shell (Frame)

2. Propellant shell (the solid propellant is assumed to be distributed as a cylindrical shell)

3. Parabolic nose cone

4. Fins

Cylindrical shell: A cylindrical shell of outer radius R3s, inner radius R2s and height Lc is

basically a solid cylinder of radius R3s and height Lc from which another cylinder of radius R2s

and height Lc is removed (the bases and the centroidal axes of both cylinders coincide).

Outer radius of the cylindrical shell = R3s Inner radius of the cylindrical shell = R2s Height of the cylindrical shell = Lc Density of the material of the frame = 1sρ

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M1s = 1sρ π R3s2 Lc

M2s = 1sρ π R2s2 Lc

Figure B-4. Cylindrical shell

Mass moments of inertias of the two cylinders about their center of masses are (Ixxs)1 = 0.5 M1s R3s

2 (Iyys)1 = (1/12) M1s (3 R3s

2 + Lc2)

(Izzs)1 = (1/12) M1s (3 R3s2 + Lc

2) (Ixxs)2 = 0.5 M2s R2s

2 (Iyys)2 = (1/12) M2s (3 R2s

2 + Lc2)

(Izzs)2 = (1/12) M2s (3 R2s2 + Lc

2) Mass moments of inertias of the two cylinders about the center of mass of the solid booster are (Ixcxcs)1 = (Ixxs)1 (Iycycs)1 = (Iyys)1 + M1s (-(Lc/2 + Lns) - Ls - Xics)2 (Izczcs)1 = (Izzs)1 + M1s (-(Lc/2 + Lns) - Ls - Xics)2 (Ixcxcs)2 = (Ixxs)2 (Iycycs)2 = (Iyys)2 + M2s (-(Lc/2 + Lns) - Ls - Xics)2 (Izczcs)2 = (Izzs)2 + M2s (-(Lc/2 + Lns) - Ls - Xics)2

Propellant shell: A cylindrical shell of outer radius R2s, inner radius R1s and height Hs is

basically a solid cylinder of radius R2s and height Hs from which another cylinder of radius R1s

and height Hs is removed (the bases and the centroidal axes of both cylinders coincide). It can be

seen that as the propellant burns, the radius of the inner cylinder R1s increases. When all the

propellant is burnt, R1s = R2s.

Outer radius of the propellant shell = R2s

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Initial inner radius of the propellant shell = R1si Height of the propellant shell = Hs Density of the solid propellant = pρ

Figure B-5. Propellant shell

M3s = pρ π R2s2 Hs

M4si = pρ π R1si2 Hs

Initial mass of the strap-on booster propellant = Mspi = M3s – M4si Instantaneous mass of the strap-on booster propellant = Msp = M3s – M4s where M4s = M4si + ts dms/dt Rate of change of mass of the solid motor = dms/dt = Thrusts / (Isp_s g) ts = time elapsed in sec from the moment the strap-on booster was ignited g = Acceleration due to gravity on the surface of the Earth.

Instantaneous inner radius of the propellant shell = R1s = 4s

s

MHpρ π

Mass moments of inertias of the two cylinders about their center of masses are (Ixxs)3 = 0.5 M3s R2s

2 (Iyys)3 = (1/12) M3s (3 R2s

2 + Hs2)

(Izzs)3 = (1/12) M3s (3 R2s2 + Hs

2) (Ixxs)4 = 0.5 M4s R1s

2 (Iyys)4 = (1/12) M4s (3 R21s

2 + Hs2)

(Izzs)4 = (1/12) M4s (3 R1s2 + Hs

2) Mass moments of inertias of the two cylinders about the center of mass of the solid booster are (Ixcxcs)3 = (Ixxs)3 (Iycycs)3 = (Iyys)3 + M3s (-(Hs/2 + Ys + Lns) - Ls - Xics)2 (Izczcs)3 = (Izzs)3 + M3s (-(Hs/2 + Ys + Lns) - Ls - Xics)2 (Ixcxcs)4 = (Ixxs)4 (Iycycs)4 = (Iyys)4 + M4s (-(Hs/2 + Ys + Lns) - Ls - Xics)2

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(Izczcs)4 = (Izzs)4 + M4s (-(Hs/2 + Ys + Lns) - Ls - Xics)2

Parabolic nose cone: The nose cone is assumed to be parabolic as shown in Fig. B-6.

Height of the nose cone = Lns Radius of the nose cone = R3s Mass of the nose cone = M5s Mass moments of inertia of the parabolic nose cone about its center of mass are (Ixxs)5 = (1/3) M5s R3s

2 (Iyys)5 = (1/6) M5s (R3s

2 + Lns2/3)

(Izzs)5 = (1/6) M5s (R3s2 + Lns

2/3) Mass moments of inertia of the parabolic nose cone about the center of mass of the solid booster are: (Ixcxcs)5 = (Ixxs)5 (Iycycs)5 = (Iyys)5 + M5s (-2 Lns/3 - Ls - Xics)2 (Izczcs)5 = (Izzs)5 + M5s (-2 Lns/3 - Ls - Xics)2

Figure B-6. Parabolic nose cone

Fins: Each strap-on booster is assumed to have four fins. The (geometry of the) fin is obtained

by removing two triangular slabs from a rectangular slab as shown in figure B-7.

M1fs = 1sρ L1s tfs Ss/8 M2fs = 0.5 1sρ Ss tfs Ss/8 M3fs = 1sρ L1s tfs Ss/8 where L1s = XRs + CTs L2s = XRs + CTs - CRs The center of mass of the fin is given by XCm_s = (L1s

2- XRs2/3 – L2s (L1s – L2s/3))/(2 L1s – XRs – L2s)

YCm_s = (L1s Ss- 2 XRs Ss/3 – L2s Ss/3)/(2 L1s – XRs – L2s) (Ixx)1fs = (1/12) M1fs (Ss

2 + tfs2)

(Iyy)1fs = (1/12) M1fs (L1s2 + tfs

2) (Izz)1fs = (1/12) M1fs (Ss

2 + L1s2)

(Ixx)2fs = M2fs (Ss2/18 + tfs

2/12)

124

(Iyy)2fs = M2fs ( XRs2/18 + tfs

2/12) (Izz)2fs = (1/18) M2fs (Ss

2 + L2s2)

(Ixx)3fs = M3fs (Ss2/18 + tfs

2/12) (Iyy)3fs = M3fs ( L2s

2/18 + tfs2/12)

(Izz)3fs = (1/18) M3fs (Ss2 + L2s

2)

Figure B-7. Fins

∴ The mass moments of inertia of the fin about its centroidal axes are given by (Ixx)fs = (Ixcxc)1fs - (Ixcxc)2fs - (Ixcxc)3fs (Iyy)fs = (Iycyc)1fs - (Iycyc)2fs - (Iycyc)3fs (Izz)fs = (Izczc)1fs - (Izczc)2fs - (Izczc)3fs where (Ixcxc)1fs = (Ixx)1fs + M1fs (Ss/2 – Ycm_s)2 (Iycyc)1fs = (Iyy)1fs + M1fs (L1s/2 – Xcm_s)2 (Izczc)1fs = (Izz)1fs + M1fs ((L1s/2 – Xcm_s)2 + (Ss/2 – Ycm_s)2) (Ixcxc)2fs = (Ixx)2fs + M2fs (2 Ss/3 – Ycm_s)2 (Iycyc)2fs = (Iyy)2fs + M2fs (XRs/3 – Xcm_s)2 (Izczc)2fs = (Izz)2fs + M2fs ((2 Ss/3 – Ycm_s)2 + (XRs/3 – Xcm_s)2) (Ixcxc)3fs = (Ixx)3fs + M3fs (Ss/3 – Ycm_s)2 (Iycyc)3fs = (Iyy)3fs + M3fs (L1s – L2s/3 – Xcm_s)2 (Izczc)3fs = (Izz)3fs + M3fs ((Ss/3 – Ycm_s)2 + (L1s – L2s/3 – Xcm_s)2) Mass of fin Mfs= M1fs – M2fs – M3fs ∴ (Ixcxcs)6 = (Ixcxcs)8 = (Ixx)fs + Mfs (YCm_s + R3s)2

(Iycycs)6 = (Iycycs)8 = (Iyy)fs + Mfs (-(XCm_s + Xbs) - Ls - Xics)2 (Izczcs)6 = (Izczcs)8 = (Izz)fs + Mfs ((YCm_s + R3s)2 + (-(XCm_s + Xbs) - Ls - Xics)2) (Ixcxcs)7 = (Ixcxcs)9 = (Ixx)fs + Mfs (YCm_s + R3s)2

(Iycycs)7 = (Iycycs)9 = (Izz)fs + Mfs ((YCm_s + R3s)2 + (-(XCm_s + Xbs) - Ls - Xics)2) (Izczcs)7 = (Izczcs)9 = (Iyy)fs + Mfs (-(XCm_s + Xbs) - Ls - Xics)2 ∴ The Mass moments of inertia of the booster about its centroidal axes are (Ixx)s = (Ixcxcs)1 - (Ixcxcs)2 + (Ixcxcs)3 - (Ixcxcs)4 + (Ixcxcs)5 + (Ixcxcs)6 + (Ixcxcs)7 + (Ixcxcs)8 + (Ixcxcs)9 (Iyy)s = (Iycycs)1 - (Iycycs)2 + (Iycycs)3 - (Iycycs)4 + (Iycycs)5 + (Iycycs)6 + (Iycycs)7 + (Iycycs)8

125

+ (Iycycs)9 (Izz)s = (Izczcs)1 - (Izczcs)2 + (Izczcs)3 - (Izczcs)4 + (Izczcs)5 + (Izczcs)6 + (Izczcs)7 + (Izczcs)8 + (Izczcs)9 First stage: The first stage is assumed to be made of a cylindrical shell (frame) and liquid

propellant. The liquid propellant is assumed to be distributed as a solid cylinder whose density

decreases with time (i.e., as the propellant burns).

Figure B-8. First stage

Cylindrical shell: A cylindrical shell of outer radius R1, inner radius R2 and height L1 is

basically a solid cylinder of radius R1 and height L1 from which another cylinder of radius R2 and

height L1 is removed (the bases and the centroidal axes of both cylinders coincide).

Outer radius of the cylindrical shell = R1 Inner radius of the cylindrical shell = R2 Height of the cylindrical shell = L1 Density of the material of the frame = 1ρ M1_1 = 1ρ π R1

2 L1 M1_2 = 1ρ π R2

2 L1 Mass moments of inertias of the two cylinders about their center of masses are (Ixx)1_1 = 0.5 M1_1 R1

2 (Iyy)1_1 = (1/12) M1_1 (3 R1

2 + L12)

(Izz)1_1 = (1/12) M1_1 (3 R12 + L1

2) (Ixx)1_2 = 0.5 M1_2 R2

2 (Iyy)1_2 = (1/12) M1_2 (3 R2

2 + L12)

(Izz)1_2 = (1/12) M1_2 (3 R22 + L1

2)

126

Propellant cylinder: The liquid propellant is assumed to be distributed as a solid cylinder

whose density decreases with time (i.e., as the propellant burns).

Radius of the cylinder = R2 Instantaneous mass of the unburnt first stage liquid propellant M1_3 = M1pi – t1 dm1p/dt Where M1pi = Initial mass of the first stage propellant t1 = time elapsed in sec from the moment the first stage is ignited dm1p/dt = Mass flow rate of the first stage propellant Rate of change of mass of the first stage liquid engine = dm1p/dt = Thrust1 / (Isp_1 g) g = Acceleration due to gravity on the surface of the Earth. (Ixx)1_3 = 0.5 M1_3 R2

2 (Iyy)1_3 = (1/12) M1_3 (3 R2

2 + L12)

(Izz)1_3 = (1/12) M1_3 (3 R22 + L1

2) ∴ The mass moments of inertia of the first stage about its centroidal axes are given by (Ixx)1 = (Ixx)1_1 - (Ixx)1_2 + (Ixx)1_3 (Iyy)1 = (Iyy)1_1 - (Iyy)1_2 + (Iyy)1_3 (Izz)1 = (Izz)1_1 - (Izz)1_2 + (Izz)1_3 Mass of the first stage =M1 = M1_1 – M1_2 + M1_3 ∴ The mass moments of inertia of the first stage about the instantaneous centroidal axes of the rocket are given by: (Ixcxc)1 = (Ixx)1 (Iycyc)1 = (Iyy)1 + M1 (L1/2 + L2 + L3 + Ln – Xic + rc)2 (Izczc)1 = (Izz)1 + M1 (L1/2 + L2 + L3 + Ln – Xic + rc)2

Second stage: The second stage is also assumed to be made of a cylindrical shell (frame) and

liquid propellant. The liquid propellant is assumed to be distributed as a solid cylinder whose

density decreases with time (i.e., as the propellant burns).





2 L2 M2_2 = 1ρ π R3

2 L2 Mass moments of inertias of the two cylinders about their center of masses are (Ixx)2_1 = 0.5 M2_1 R1

2 (Iyy)2_1 = (1/12) M2_1 (3 R1

2 + L22)

127

Figure B-9. Second stage

(Izz)2_1 = (1/12) M2_1 (3 R12 + L2

2) (Ixx)2_2 = 0.5 M2_2 R3

2 (Iyy)2_2 = (1/12) M2_2 (3 R3

2 + L22)

(Izz) 2_2 = (1/12) M2_2 (3 R32 + L2

2)

Propellant cylinder: The liquid propellant is assumed to be distributed as a solid cylinder

whose density decreases with time (i.e., as the propellant burns).

Radius of the cylinder = R3 Instantaneous mass of the unburnt second stage liquid propellant M2_3 = M2pi – t2 dm2p/dt Where M2pi = Initial mass of the second stage liquid propellant t2 = time elapsed in sec from the moment the second stage is ignited dm2p/dt = Mass flow rate of the second stage propellant Rate of change of mass of the first stage liquid engine = dm2p/dt = Thrust2/(Isp_2 g) g = Acceleration due to gravity on the surface of the Earth. (Ixx)2_3 = 0.5 M2_3 R3

2 (Iyy)2_3 = (1/12) M2_3 (3 R3

2 + L22)

(Izz)2_3 = (1/12) M2_3 (3 R32 + L2

2) ∴ The mass moments of inertia of the second stage about its centroidal axes are given by: (Ixx)2 = (Ixx)2_1 - (Ixx)2_2 + (Ixx)2_3 (Iyy)2 = (Iyy)2_1 - (Iyy)2_2 + (Iyy)2_3 (Izz)2 = (Izz)2_1 - (Izz)2_2 + (Izz)2_3 Mass of the second stage =M2 = M2_1 – M2_2 + M2_3 ∴ The mass moments of inertia of the second stage about the instantaneous centroidal axes of the rocket are given by: (Ixcxc)2 = (Ixx)2 (Iycyc)2 = (Iyy)2 + M2 (L2/2 + L3 + Ln – Xic + rc)2 (Izczc)2 = (Izz)2 + M2 (L2/2 + L3 + Ln – Xic + rc)2

128

Third stage: The third stage is assumed to be made of a cylindrical shell and cylindrical

propellant shell.




Figure B-10. Third stage


2 L3 M3_2 = 1ρ π R4

2 L3

Mass moments of inertias of the two cylinders about their center of masses are (Ixx)3_1 = 0.5 M3_1 R1

2 (Iyy)3_1 = (1/12) M3_1 (3 R1

2 + L32)

(Izz)3_1 = (1/12) M3_1 (3 R12 + L3

2) (Ixx)3_2 = 0.5 M3_2 R4

2 (Iyy)3_2 = (1/12) M3_2 (3 R4

2 + L32)

(Izz)3_2 = (1/12) M3_2 (3 R42 + L3

2)

Propellant shell: A cylindrical shell of outer radius R4, inner radius R5 and height L3 is



Outer radius of the propellant shell = R4

129

Initial inner radius of the propellant shell = R5si Height of the Propellant shell = L3 Density of the solid propellant = pρ

M3_3 = pρ π R42 L3

M3_4i = pρ π R5i2 L3

Initial mass of the third-stage solid propellant = M3pi = M3_3 – M3_4i Instantaneous mass of the third-stage propellant = M3p = M3_3 – M3_4 where M3_4 = M3_4i + t3 dm3/dt Rate of change of mass of the third stage solid motor = dm3/dt = Thrust3 / (Isp_3 g) t3 = time elapsed in sec from the moment the third stage was ignited g = Acceleration due to gravity on the surface of the Earth.

Instantaneous inner radius of the propellant shell = R5 = 3_4

3

MLpρ π

Mass moments of inertias of the two cylinders about their center of masses are (Ixx)3_3 = 0.5 M3_3 R4

2 (Iyy)3_3 = (1/12) M3_3 (3 R4

2 + L32)

(Izz)3_3 = (1/12) M3_3 (3 R42 + L3

2) (Ixx)3_4 = 0.5 M3_4 R5

2 (Iyy)3_4 = (1/12) M3_4 (3 R5

2 + L32)

(Izz)3_4 = (1/12) M3_4 (3 R52 + L3

2) ∴ The mass moments of inertia of the third stage about its centroidal axes are given by: (Ixx)3 = (Ixx)3_1 - (Ixx)3_2 + (Ixx)3_3 - (Ixx)3_4 (Iyy)3 = (Iyy)3_1 - (Iyy)3_2 + (Iyy)3_3 - (Iyy)3_4 (Izz)3 = (Izz)3_1 - (Izz)3_2 + (Izz)3_3 - (Izz)3_4 Mass of the third stage = M3 = M3_1 – M3_2 + M3_3 –M3_4 ∴ The mass moments of inertia of the third stage about the instantaneous centroidal axes of the rocket are given by: (Ixcxc)3 = (Ixx)3 (Iycyc)3 = (Iyy)3 + M3 (L3/2 + Ln – Xic + rc)2 (Izczc)3 = (Izz)3 + M3 (L3/2 + Ln – Xic + rc)2

Payload: The payload is assumed to be a parabolic (nose) cone

Mass of the nose-cone (pay load) = M4 Radius of the nose-cone = R1 Length of the nose-cone = Ln Mass moments of inertia of the nose-cone about its centroidal axes are given by (Ixx)p = M4 R1

2/3 (Iyy)p = (M4/6) (R1

2 + Ln2/3)

(Izz)p = (M4/6) (R12 + Ln

2/3) Mass moments of inertia of the nose-cone about the instantaneous centroidal axes of the rocket are given by: (Ixcxc)p = (Ixx)p (Iycyc)p = (Iyy)p + M4 (Ln/2 – Xic + rc)2

130

(Izczc)p = (Izz)p + M4 (Ln/2 – Xic + rc)2

Figure B-11. Payload

The mass moments of inertias of “Ns” boosters about the instantaneous centroidal axes of the rocket are given by: (Ixcxc)s = Ns ((Ixx)s + Ms (R3s+Lcr+R1)2) (Iycyc)s = Ns ((Iyy)s + Ms (Xics + Ls - Xic + rc)2) + Ms (R3s+Lcr+R1)2 Yres (Izczc)s = Ns ((Izz)s + Ms (Xics + Ls - Xic + rc)2) + Ms (R3s+Lcr+R1)2 Zres

Figure B-12. Strap-on boosters around the Rocket

where Yres = 1

2

0sin ( * )

sN

ii θ

−

=∑

Zres = 1

2

0cos ( * )

sN

ii θ

−

=∑

360

sNθ =

∴ The mass moments of inertia of the rocket about its instantaneous centroidal axes are given by: Ixx = (Ixcxc)1 + (Ixcxc)2 + (Ixcxc)3 + (Ixcxc)p + (Ixcxc)s Iyy = (Iycyc)1 + (Iycyc)2 + (Iycyc)3 + (Iycyc)p + (Iycyc)s Izz = (Izczc)1 + (Izczc)2 + (Izczc)3 + (Izczc)p + (Izczc)s

131

The products of inertias of the rocket are zero because of the symmetry (i.e., the Cartesian axes through the instantaneous center of mass is the "principal axes" of the rocket).

∴The inertia tensor = 0 0

0 00 0

xx

yy

zz

II I

I

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

132

LIST OF REFERENCES

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5. T. Benson, “Rocket Stability,” Cleveland, Ohio: Glenn Research Center, 2006. [Online]. Available: http://exploration.grc.nasa.gov/education/rocket/rktstab.html. [Accessed May 15, 2007].

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7. Boeing Company, DELTA II Payload Planners Guide. Huntington Beach, California: The Boeing Company, 1996.

8. J. C. Chen, C. C. Wang, D. Taggart, and E. Ditata, “Modeling an off-nominal launch vehicle trajectory for range safety link analysis,” in Proceedings of the 2003 IEEE Aerospace Conference, Vol. 7, March 2003, pp. 3419-3426.

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10. E. Denson, L. M. Valencia, R. Sakahara, J. C. Simpson, D. E. Whiteman, S. Bundick, D. Wampler, and R. B. Birr, “Space-based Telemetry and Range Safety Flight Demonstration #1 Final Report,” Kennedy Space Center, FL. Rep. KSC-YA-6400 Revision Basic, 2004.

11. European Cooperation for Space Standardization, “System engineering – coordinate systems,” Noordwijk, Netherlands: ESA Publication Division, 2007. [Online]. Available: www.ecss.nl/.../dispatch.cgi/home/showFile/100317/d20070507095706/No/ecss-e-10-09A-Draft1.1(3April2007).pdf. [Accessed June 03, 2007].

http://www.nar.org/pdf/launchsafe.pdf

http://www.rockets4schools.org/education/Calculating_Cp.pdf

http://www.apogeerockets.com/education/downloads/Barrowman_report.pdf

http://exploration.grc.nasa.gov/education/rocket/rktstab.html

http://www.gmat.unsw.edu.au/snap/publications/bethencourt_etal2005a.pdf

http://www.ecss.nl/.../dispatch.cgi/home/showFile/100317/d20070507095706/No/ecss-e-10-09A-Draft1.1(3April2007).pdf

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133

12. J. A. Farrell and M. Barth, The Global Positioning System & Inertial Navigation, 1st ed. McGraw-Hill Professional Publication, 1998.

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19. NASA, “F-15B SBRDC/ECANS,” Edwards, California: Dryden Flight Research Center, 2007. [Online]. Available: http://www1.dfrc.nasa.gov/Gallery/Photo/F-15B_SBRDC-ECANS/index.html. [Accessed June 10, 2007].

20. NASA, “Emerging Technology Space Based Telemetry and Range Safety 2006,” Merritt Island, Florida: Kennedy Space Center, 2006. [Online]. Available: http://kscsma.ksc.nasa.gov/Range_Safety/Annual_Report/2006/Printpages/ET-Space-Based-Telemetry-images.pdf. [Accessed June 10, 2007].

21. NASA, “Flight Demonstration #2 (Technical Interchange Meeting), Space Based Range Demonstration & Certification,” Kennedy Space Center, Florida. Rep. STARS FD#2, 2006.

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http://www.dbf.gatech.edu/performance.ppt

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http://www1.dfrc.nasa.gov/Gallery/Photo/F-15B_SBRDC-ECANS/index.html

http://www1.dfrc.nasa.gov/Gallery/Photo/F-15B_SBRDC-ECANS/index.html

http://kscsma.ksc.nasa.gov/Range_Safety/Annual_Report/2006/Printpages/ET-Space-Based-Telemetry-images.pdf

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25. S. Pines, “Uniform Representation of the Gravitational Potential and its Derivatives,” AIAA Journal, vol. 11, pp. 1508-1511, Nov. 1973.

26. L. C. Rabelo, J. Sepulveda, J. Compton, and R. Turner, “Simulation of range safety for the NASA space shuttle,” Aircraft Engineering & Aerospace Technology, vol. 78, no. 2, pp. 98–106, 2006.

27. G. Rosborough, “Spherical Harmonic Representation of the Gravity Field Potential,” Palo Alto, California: cdeagle, 2006. [Online]. Available: http://www.cdeagle.com/pdf/gravity.pdf. [Accessed May 15, 2007].

28. F. Takahashi, Y. Takahashi, T. Kondo, and Y. Koyama, Very Long Baseline Interferometer. Amsterdam: IOS Press, 2000.

29. J. R. Wertz and W. J. Larson., Space Mission Analysis and Design, 3rd ed. Microcosm Press, October 1999, pp. 208.

30. D. E. Whiteman, L. M. Valencia, and R. B. Birr, “Space-Based Telemetry and Range Safety Project Ku-Band and Ka-Band Phased Array Antenna,” NASA Dryden Flight Research Center, Edwards, California. Rep. H-2603, NASA TM-2005-212872, 2005.

31. D. E. Whiteman, L. M. Valencia, and J. C. Simpson, “Space-Based Range Safety and Future Space Range Applications,” NASA Dryden Flight Research Center, Edwards, California. Rep. H-2616, NASA TM-2005-213662, 2005.

http://www.cdeagle.com/pdf/gravity.pdf

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BIOGRAPHICAL SKETCH

Sharath Chandra Prodduturi was born in Hyderabad, India, in 1983. He graduated with a

bachelor’s degree in mechanical engineering from Osmania University, India, in 2004. In 2005,

he moved to Gainesville, Florida, to pursue his Master of Science degree in mechanical

engineering at the University of Florida.

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