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

Transcription

1 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 007 1

2 007 Sharath Chandra Prodduturi

3 To my parents. 3

4 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. 4

5 TABLE OF CONTENTS ACKNOWLEDGMENTS...4 LIST OF FIGURES...7 ABSTRACT...9 CHAPTER 1 INTRODUCTION AND BACKGROUND...11 EQUATIONS OF MOTION FORMULATION...19 page Coordinate Frames...19 Kinematic Equation of Motion...4 Dynamical Equations...7 Generalized External Forces...30 External Forces...30 Thrust force...30 Aerodynamic forces (drag and lift)...3 Gravitational force...33 External Moments...34 Aerodynamic moments...34 Gravitational moment...35 Thrust moment DESCRIPTION OF MODELS USED...38 Gravity Model...38 Inertia Model...49 Strap-on booster...50 Cylindrical segment...50 Parabolic nose cone...5 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

6 4 SIMULATION RESULTS AND DISCUSSION...73 Simulation...73 Validation CONCLUSION AND FUTURE WORK...91 Conclusions...91 Future work...9 APPENDIX A B MATLAB FUNCTIONS AND SCRIPT...93 SIMULATION CONFIGURATION LIST OF REFERENCES...13 BIOGRAPHICAL SKETCH

7 LIST OF FIGURES Figure page 1-1 Space-based range and range safety, today and future Relative orientation of the various frames Euler angles and the relative orientation between the vehicle frame and the vehiclecentered horizontal frame Geometry of the launch vehicle and various position vectors External forces acting on a launch vehicle during its flight Representation of a position vector in Cartesian and Spherical coordinates Cylindrical segment of the strap-on booster Parabolic nose cone Fin Liquid engine Solid motor Payload Conical shoulder Conical Boattail Fin and Tail section Geodetic Ellipsoid and Geodetic coordinates of an arbitrary point P Various parameters of the launch vehicle as a function of time Velocity of the launch vehicle in the inertial frame Position of the launch vehicle in the inertial frame Launch vehicle during the time of launch as seen from the J000 inertial frame Moments of inertia of the launch vehicle about its instantaneous center of mass 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...8 7

8 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 Moment of inertia of the second stage about the instantaneous center of mass of the launch vehicle and about its instantaneous center of mass Moment of inertia of the third stage about the instantaneous center of mass of the launch vehicle and about its instantaneous center of mass The need for instrumental data or thrust vector in the vehicle frame...90 B-1 The DELTA II Launch vehicle geometry B- Strap-on booster geometry B-3 Elements of DELTA II Launch vehicle and Strap-on Booster...10 B-4 Cylindrical shell...11 B-5 Propellant shell...1 B-6 Parabolic nose cone...13 B-7 Fins...14 B-8 First stage...15 B-9 Second stage...17 B-10 Third stage...18 B-11 Payload B-1 Strap-on boosters around the Rocket

9 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 Chair: Norman G. Fitz-Coy Major: Mechanical Engineering By Sharath Chandra Prodduturi August 007 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 J000 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 9

10 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. 10

11 CHAPTER 1 INTRODUCTION AND BACKGROUND Ensuring safe, reliable and affordable access to space is the fundamental goal of the U.S. range safety program [3]. 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) [6]. 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 30 th and the 45 th 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 11

12 safety, and the protection of national resources [14]. The objective of the Range Safety Program as stated in Eastern and Western Range 17-1, 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 [3]. 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, 1

13 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. ). 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-616, NASA TM ,

14 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], [1]. Phase 1 Developed and tested a new S-band Range Safety system. During June-July 003, 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 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 []). 14

15 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 006. TDRSS is the space-based communication link. Test flights are planned for late fiscal year 007. 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 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 007. Perform flight test of Ka-band system on F-15B in fiscal year 008. Re-fly phase 3 Range Safety Unit design with enhancements. Certification Phase Perform Certification of Range Safety and Range Users systems in fiscal year The STARS program was renamed to Space Based Range Demonstration and Certification (SBRDC) program [0]. From the available information on the World Wide Web/internet, Phases 1, 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 15

16 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 [4]. 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 16

17 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, 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 17

18 the simulator cannot be validated due to the lack of availability of critical data (an ITAR 1 issue). Finally, in Chapter 5, the conclusions of this research and the possible future work are discussed. 1 ITAR International Traffic in Arms Regulations 18

19 CHAPTER 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 (X i Y i Z i ): For studying the launch vehicle motion in the vicinity of Earth and at an interplanetary level, the J000 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 J000 epoch. The Y-axis lies in the equatorial plane and completes a right-handed Cartesian frame [9], [8]. Rotating geocentric frame (X g Y g Z g ): 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 19

20 Greenwich meridian. The Y-axis lies in the equatorial plane and completes a right-handed Cartesian frame [9]. Vehicle-centered horizontal frame (X v Y v Z v ): 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 Z v -axis is along the radius vector from the center of Earth and is positive downwards [9]. Vehicle frame (X r Y r Z r ): 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 -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 ( ω ). Thus, the relative orientation of these frames is determined by a rotation about the Z-axis e 0

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

22 through an angle that is equal to the angle between the X i -axis and X g -axis. This angle is equal to the Greenwich hour angle of the vernal equinox H G. If both frames coincide at t = t 0, the angle H G at any time is given in Eq. -1. HG ωe ( t t0 ) Since the inertial frame in our case is the J000 frame, the term ( ) = (-1) t t 0 is equal to the time elapsed in seconds from January 1, 000, 1:00 UTC until the time t of interest. The transformation between the frames is given in Eq. -. The vectors G E and I E in Eq. -3 represent an arbitrary vector E coordinatized in the rotating geocentric frame and the inertial frame respectively. The transformation matrix is given in Eq. -3. G E = C E (-) I GI cos HG sin HG 0 CGI = sin HG cos HG 0 (-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 (X g Y g Z g frame) is first rotated about its Z-axis (i.e., Z g 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 π + 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. -4. The vectors V E and G E in Eq. -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. -5. V G E= CVG E (-4)

23 sinφ cosλ sinφsin λ cosφ C VG = sin λ cosλ 0 (-5) cosφ cosλ cosφsin λ sinφ 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. -. 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., obtain a new frame Zv -axis) by an angle ψ to XvY 1 vz 1 v. ψ is called the yaw angle, the angles between the vertical plane 1 through the longitudinal axis of the launch vehicle and the X v - axis. Then the new frame XvY 1 vz 1 v is rotated about its Y-axis (i.e., Y 1 v 1 axis) by an angle θ to obtain another new frame Xv Y v Z v. 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, X-axis (i.e., Xv Y v Z v, is rotated about its X v -axis) by an angle ϕ to obtain the vehicle frame XYZ r r r. ϕ is called the bank angle, the angle between the Zr - axis and the vertical plane through the longitudinal axis of the launch vehicle. The transformation between the frames is given in Eq.-6. The vectors R E and V E in Eq. -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 In Eq. -7, Cθ and Sθ are used to represent the cosine and sine of an angle θ. R E= C E V RV Cθ Cψ CθSψ Sθ CRV = CϕSψ + SϕSθCψ CϕCψ + SϕSθSψ SϕCθ SϕSψ + CϕSθCψ SϕCψ + CϕSθSψ CϕCθ (-6) (-7) 3

24 Figure -. Euler angles and the relative orientation between the vehicle frame and the vehiclecentered 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 C GI, C VG and C RV to the inertial frame. The transformation between the frames is given in Eq. -8. The vectors R E and I E in Eq. -8 represent an arbitrary vector E coordinatized in the vehicle frame and the inertial frame respectively. The transformation matrix is given in Eq. -9. R E = C E (-8) I RI C = C C C (-9) RI RV VG GI 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. ω1 Let ω = ω be the angular velocity of the vehicle frame with respect to the vehiclecentered horizontal frame expressed in the vehicle frame. Since the vehicle-centered horizontal ω 3 frame is an inertial frame, ω is the absolute angular velocity of the launch vehicle. Let ψ&, & θ 4

25 and ϕ& be the Euler angle rates for the 3--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 The rotation matrices C RV 1 and C RV in Eq. -10 are given in Eqs. -11 and -1. ω & ϕ ω = 0 + C & RV 1 θ + C RV 0 (-10) ω ψ & cosϕ 0 sinϕ C R V 1 = 0 cosϕ sinϕ (-11) 0 sinϕ cosϕ sinϕ 0 cosϕ cosϕ 0 sinϕ cosψ sinψ 0 C R V = 0 cosϕ sinϕ sinψ cosψ 0 (-1) 0 sinϕ cosϕ sinϕ 0 cosϕ Equation -10 can be rewritten as Eq. -13 where the matrix X in Eq. -13 is given in Eq Equation -13 can be rewritten as Eq The matrix X in Eq. -14 is inverted and substituted into Eq. -15 to obtain Eq ω & ϕ 1 ω = X & θ ω & ϕ 3 ( 1,1) R V ( 1, ) R V ( 1,3) (,1) R V (, ) R V (,3) ( 3,1) ( 3, ) ( 3,3) C R V C 1 C X = C R V C 1 C C R V C R V1 C R V & ϕ ω1 & 1 θ = X ω & ϕ ω 3 & ϕ cosθ sinϕsinθ cosϕsinθ ω1 & 1 θ = 0 cosϕcosθ sinϕcosθ ω cosθ ψ& 0 sinϕ cosϕ ω 3 (-13) (-14) (-15) (-16) 5

26 Eq. -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 π θ = and (ii) solving the kinematic equation of motion Eq. -16 is computationally intensive as it involves trigonometric quantities. To avoid these problems, quaternions are used to represent the relative orientation of the vehiclecentered horizontal frame and the vehicle frame. The transformation matrix C RV can also be q, q, q q in Eq. expressed in terms of quaternions as shown in Eq The quantites 0 1 and 3-17 are calculated using the expressions in Eqs q0 + q1 1 qq 1 + qq 0 3 qq 1 3 qq 0 CRV = qq 1 qq 0 3 q0 + q 1 qq 3+ qq 0 1 qq 1 + qq 0 3 qq 3 qq 0 1 q0 + q3 1 (-17) ψ θ φ ψ θ φ q0 = cos cos cos + sin sin sin (-18) ψ θ φ ψ θ φ q1 = cos cos sin sin sin cos (-19) ψ θ φ ψ θ φ q = cos sin cos + sin cos sin (-0) ψ θ φ ψ θ φ q3 = sin cos cos cos sin sin (-1) The kinematic equation of motion in terms of quaternion rates is given in Eq. -. The quantites ω1, ω and ω3 in Eq. - 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 ω1 i.e., ω = ω. Since the vehicle-centered horizontal frame is an inertial frame, ω is ω 3 the absolute angular velocity of the launch vehicle. 6

27 q& 0 0 ω1 ω ω3 0 q 1 1 ω1 0 ω q 3 ω & 1 = q& ω ω3 0 ω q 1 q& 3 ω3 ω ω1 0 q3 Dynamical Equations q (-) In this section, the derivation of the dynamic equations of a launch vehicle is presented. Figure -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 C 1. The position vector of the mass element dm relative to the origin of the inertial frame (coordinatized in the inertial frame) is represented by I R dm. The position vector of the initial center of mass of the launch vehicle C 1 relative to the origin of the inertial frame (coordinatized in the inertial frame) is represented by I R. The position vector of the C1 instantaneous center of mass of the launch vehicle C with respect to the initial center of mass of the launch vehicle C 1 (coordinatized in the vehicle frame) is represented by R r. 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. -3, the position vector of the mass element dm can be written as expressed in Eq. -3. The matrix C RI in Eq. -3 is the transformation matrix from the inertial frame f to the vehicle frame f I R obtained from Eq. -9. The acceleration of the mass element dm is obtained by differentiating Eq -3 twice with respect to time t as shown in Eq. -4. The resulting expression for the acceleration of the mass element dm is given in Eq. -5. c 7

28 Figure -3. Geometry of the launch vehicle and various position vectors R = R + C + r (-3) I I T R R dm C1 RI rc I d I T R R dm = C + 1 RI rc + ( ) R R R R R && r c + && r + ( ω ( r& c + r& )) + RI R R R R R R R & ω ( rc+ r) + ω ( ω ( rc+ r) ) R&& R C r (-4) dt I I T R&& dm = R&& C + C 1 (-5) Applying Newton s second law, we obtain Eq. -6. The expression for I R&& dm from Eq. -5 is substituted into Eq. -6 and then integrated over the entire launch vehicle mass as shown in Eq. -7. The resulting expression after integration is given in Eq. -8. I df Rdmdm = && (-6) I ext I I T dm C1 RI M M && r ( ω ( r& c r& )) ( rc r) ( rc r) R R R R R && r c F = R&& dm= R&& + C dm R R R R R R R & ω + + ω ( ω + ) (-7) 8

29 ( ω & ) ω ω ( ω ) F = M R&& + C M && + M + M & + M (-8) (Q r& = && r = 0and rdm= 0for a rigid body) I ext I T R R R R R R R R C1 RI rc rc rc rc M Equation -8 is the translational equation of motion of a launch vehicle. The term in Eq. -8 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 r c and its time derivatives in Eq. -8 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 r c 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. -9. Substituting the expression for I R&& dm from Eq. -5 into Eq. -6 and then substituting the resultant expression for df into Eq. -9, and then integrating the resultant expression over the entire launch vehicle mass, we obtain Eq dm = r df (-9) c ( ) M = r ( & ω r ) dm + r ω ( ω r ) dm (-30) R r R & ω R r dm and R r R ω R ω R r dm in Eq. -30 are represented R c ext R R R R R R R M M ( ) The terms ( ) ( ) M M in terms of moment of inertia tensor of the launch vehicle, I, as shown in Eqs. -31 and -3. Substituting Eqs. (-31) and (-3) into Eq.-30, we obtain Eq ext F M R ( ω ) = I r & r dm & ω (-31) R R R 9

30 ( ω ( ω R R )) = ( I ) R R R R r r dm ω ω (-3) M R ext R R R ( ) M c = I & ω + ω I ω (-33) Equation -33 is the rotational equation of motion of a launch vehicle. The term ext M c in Eq. -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. -8 and -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 -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 & e e a e where e The thrust force acting on a launch vehicle is = + ( ) T mv p p A V is the exhaust speed of the gases relative to the launch vehicle; m& is the propellant mass flow rate; p e is the pressure at nozzle exit; pa is the ambient pressure; Ae is the nozzle exit (exhaust) area. 30

31 Figure -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 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. T -9, CIR ( CRI ) given in Eq. -35 =. The expression for the thrust force coordinatized in the inertial frame is R T x FThrust = Ty (-34) T F = C F (-35) z I T R Thrust RI Thrust 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. 31

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 [9]. 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 F Drag 1 = CD Aρvv where C D is the drag coefficient; A is the cross-sectional area perpendicular to the flow; ρ is the density of the medium, v and 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 I 1 I FDrag = CD Aρv v (-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 F Lift 1 = ˆ where C L is the lift coefficient; A is the surface area of the lifting CAv ρ v L 3

33 surface; ρ is the density of the medium; v is the speed of the launch vehicle relative to the medium and ˆv is a unit vector normal to the velocity of the launch vehicle. It should be noted that the lift coefficient, C L, 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 I 1 I FLift = CAv L ρ vˆ (-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 I W x Fg = Wy (-38) W z 33

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 Substituting the external forces from Eqs into Eq. -39, we obtain the expression for the resultant external force acting on the launch vehicle given in Eq. -40 F = F + F + F F (-39) I ext I I I I Thrust Drag g + Lift T x W x I ext T 1 I 1 I F = CRI Ty CD Aρv v+ CL Aρv vˆ + Wy (-40) T z W z 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 34

35 can be neglected at altitudes greater than or equal to 600 km [9]. 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. -41 and -4 respectively, where r P 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 r P 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. I I neglected. ( MDrag ) ( M Lift ) C C I I = F (-41) I r p p I Lift Drag = F (-4) r In the current simulator, the lift force and its moment acting on the launch vehicle are 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 I ( g ) 0 M = (-43) C 35

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. -44, where r n i is the position vector of the center of nozzle of the i th 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 the thrusting elements can be calculated from the knowledge of the geometry of the launch r n i of all 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 Chapter 4. ( M ) I I I Thrust = C r F n Thrust i i i r n i in the simulator is presented in (-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 ( ) ( ) ( ) M = + + (-45) I ext I I I C MDrag M C Lift M C Thrust C Substituting Eq. -40 into Eq. -8, yields Eq..-46, which is the composite translational equation of motion for a launch vehicle. Similarly, substituting Eqs. -9 and -45 into Eq. -33 yields Eq. -47, which is governing equation for the rotational motion of a launch vehicle. 36

37 T x W x 1 1 I y ˆ y T z W z T I CRI T CD Aρv v+ CL Aρv v + W I T R R R R R R R R M R&& C + C ( ) ( ) 1 RI M && r c+ M ω r& c + M & ω r c + M ω ω r c T R R R ( MDrag ) + ( M ) + ( M ) = C RI I ω + ω ( I ω) = (-46) I I I & C Lift C Thrust (-47) C 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. -46 and -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. 37

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 38

39 gravitational potential [17]. The spherical harmonic representation of the gravitational potential function (V) is given in Eq. 3-1 [7]. n n n GM a V = 1 + P sin C cos m + S sin m r n= m= 0 r In Eq 3-1, μ ( ) is the Earth s gravitational constant;,, max m n nm nm (3-1) GM ( α )( λ λ ) from the Earth s center of mass to the point of interest, geocentric latitude and r α λ are the distance geocentric/geodetic longitude of the point of interest, respectively; a is the semi-major axis of the WGS84 ellipsoid (discussed later); nm, are the degree and order of the spherical harmonic function respectively; C, S are the spherical harmonic coefficients of degree n and order m ; nm nm m P n is the associated Legendre function of degree n and order m. The associated Legendre function m P n is defined as follows [7]. m Pn ( sinα ) = ( cosα ) Pn ( sin ) m d d m ( sinα ) α = Legendre polynomial m P n ( sinα ) 1 d = n n! d sin n ( α ) n ( sin α 1) A gravitational model is defined by the set of constants μ, a and the spherical harmonic coefficients C, S The gravitational model used in this research is the WGS84 EGM96 nm nm 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,, m C S and the associated Legendre functions, P ( sin ) nm nm n n α. The (unnormalized) spherical harmonic coefficients, C, S, tend to very small values and the associated m Legendre functions, P ( sin ) n nm nm α, tend to very large values as the degree increases. These problems can be circumvented by normalizing the associated Legendre function and the 39