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ROBUST NONLINEAR ATTITUDE CONTROL WITH DISTURBANCE COMPENSATION By KEVIN J. WALCHKO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2003 Copyright 2003 By Kevin J Walchko ACKNOWLEDGEMENTS I would like to thank Dr. Paul Mason for his guidance of my education and research. He has always made every effort to aid me in my work and life. Paul has held many roles in my life such as mentor, academic advisor, friend, and best man at my wedding. He has become more than an advisor over the years, and will forever be part of my family. I will always be in his debt. I would like to thank Dr. John Schueller who acted as my cochair during this dissertation. He provided much advice and guidance in the bureaucracy of University of Florida. Schueller was always willing to help me with various paperwork and other such matters pertaining to fellowships, pay checks, and graduation. He is a good friend and I look forward to working with him in the future. I would also like to thank my family who sacrificed much at times so this could be possible. My wife Nina took great care of me and our new son Michael. She made sure I had plenty of time to do my work and never complained about anything. She is the source from which I draw all of my strength, ambition, and dedication. Finally, I would like to thank the horde, our pack of cats. They provided much amusement and stress relief through their wacky antics. I treasure each and every one of them, and will never forget those who have passed away. TABLE OF CONTENTS page A C K N O W L E D G E M E N T S .............................................................................................. iii L IST O F FIG U R E S ............. ...................... .... .......................... .. .............. viii LIST OF TABLES ......... ........................ ........ .......... ....... xi L IST O F SY M B O L S ..... .... .................................................. .. ....... .............. xiii A B ST R A C T ..............................................................................xv 1 IN TR O D U C TIO N .................. ...... ............................ .... ......... .. .............1 Form action Flying ................................... ..... .................. ................... 1 G general Problem s Controlling Single Satellites ........................................ ................2 External D isturbances ........................ .. .................... .. ....... .......... ...... . Param etric U uncertainty ................................................. .............................. 3 L literature R review ................. ................................ .......... ....... ................ .4 Fuel Slosh ........................................ .............. M odelling fuel slosh dynam ics .................................. ..................................... 4 C controlling fuel slosh.................. .......... ....................... ......... ... .............. Do we need to worry about fuel slosh?..........................................................9 Missions involving fuel slosh disturbances ............................................. 12 Therm ally Induced Vibrations: Solar Snap ................................ ..................... 12 Controlling therm al vibrations...................................................................... 13 M missions w which involved solar snap ..................................... ............... 15 Param etric U uncertainty ............................................... .............................. 16 Sum m ary and Overview of the Thesis ........................................ ..... ............... 17 2 CONTROLLING THE UNKNOWN: FUEL SLOSH ..............................................19 In tro d u ctio n ......................................................................... 19 M odelling Fuel Slosh D ynam ics ........................................... ........... ............... 20 F uel Slosh M models ......................... ...... .............. ..................... .... .. ..23 Sim ulation M odel ............................................................. 23 C controller M odel ................................................................. .. ..... 24 3 D YN AM ICS OF SOLAR ARRAY S ................................................. .....................26 In tro d u ctio n ........................................................................................................... 2 6 Therm ally Induced Vibrations ............................................................................. 26 H e a t F lu x ............................................................................................................... 2 7 Sum m ing F forces ............. ........ ............................................ ........ ..... .......2 8 Nonhomogeneous Boundary Conditions ................................... ............... 33 Separation of V ariables ............................ ........... .............. ... .. ............... 34 Solving the position function F(x) ....................................... ............... 35 Solving the tim e function G (t) ........................................ ...... ............... 38 Su m m ary of resu lts ..................................................... .. ........ ........ ..... ... ... 3 8 Therm ally Induced Dynamics .................................. .....................................39 M odal Equations .......................... ........ .... .. ..... .. ............39 D isturbance Torques ........................ ........................ ......... ........... 40 Thickness Tem perature Gradient ........................................ ...................... 42 C on clu sion ............................................................................44 4 SATELLITE ATTITUDE DYNAM ICS ........................................... .....................46 Reference Fram es ................................... .. .......... .. ............46 E arth C entered In ertial ........................................ ............................................4 6 O rb ital F ram e ...............................................................4 7 Body Fixed Frame ....................................... .......... ................47 Fun With Vectors and Rotating Coordinate Systems ...............................................47 Spacecraft Attitude ................................................ ..............49 Q u aternion s ..................................................... ................. 50 Rotations of Rigid Bodies in Space. ............................... .. ....................... 51 Spacecraft Equations of M otion ......................... .................... .. ............ ... 51 A attitude K inem atics ....................................... ................... ......... 52 Spacecraft D ynam ics ....................... .. ....................... .... .. ........... 52 Sim ple Spacecraft ................................. ... .. ........ ............ 52 Angular Velocity .................................... .. .. ..... .. ............56 Internal D disturbances ............. .... ......... .......................... ............ ........ ......56 Spacecraft with Reaction or Momentum Wheels ..............................................56 F u el S lo sh ...................................................................................................... 6 0 Environm ental D isturbances in Space ............................................. ............... 61 Gravity Gradient ................................. .. .. .. ...... .. ............62 S o la r S n a p ....................................................... ................ 6 2 C on clu sion ............................................................................62 5 ATTITUDE CONTROL OF SPACECRAFT .................................... ...............64 Satellite C control H ardw are .............................................................. .....................64 Control Com putational H ardw are ........................................ ...... ............... 64 C control A ctu ators ............... .......................................... ...... .... ......... ........ 65 Reaction and m om entum wheels .......................................... ............... 65 M agnetic torque rods ............................................... ............................ 66 G a s jets ................................................................................ 6 7 Proportional D erivative Control .......................................... ........................... 67 Sliding Mode Control Theory and Background .................................. ...............69 W hat is Sliding M ode? ........................................................... .........................69 T h e o ry .......................................................................... 6 9 Sim ple E xam ple .................. ................... .............. ........ .. ...... .... 72 Other Sliding Mode Designs for Spacecraft .................................... ...............73 Satellite Controller Design For This Work ..................................... ............... 75 Controller D erivation ......................... ........... ........ ............ 75 Fuel Slosh D isturbance ............................................... .............................. 77 Stability of C ontroller .................... .. ........................ ......... ........... 78 Saturation ..................... .......................80 Com paring The Tw o Types of Controller ........................................ .....................81 C o n c lu sio n ................................................................. 8 1 6 SIMULATION AND RESULTS ............................................................................83 Simulation Dynamics and Parameters ............................................. ............... 83 S a te llite ......................................................................... 8 3 Satellite D ynam ics ....................... .................. ................... .. ...... 85 F u e l T a n k ............................................................................................................... 8 6 Reaction Wheels ................................. ..... ........ ..............86 S o la r P a n e ls ..................................................................................................... 8 7 C o n tro l S y ste m ............................................................................................... 8 7 F finding an O ptim al P D G ain ........................................................... .....................88 F u e l S lo sh ............................................................................................................. 9 3 IA E ...............................................................................9 4 IT A E ..............................................................................9 5 S ettlin g T im e ................................................................................................... 9 5 C M ................. ....... ... ...................................9 6 Torque and M om entum Lim its ....................................................... 96 Qualitative Analysis of Results .......... ... ...... ............. .....98 S o la r S n a p ..............................................................................9 8 Perform ance .............................................................98 Qualitative Analysis of Results ....... .............................99 7 C O N C L U SIO N ....................................................... 101 A ATTITUDE REPRESENTATIONS AND ROTATION MATRICES ....................104 F ix ed A ngle R rotation s ......................................................................................... 104 E uler A ngles ........................................ 105 Q u atern io n s ...............................................................10 7 Q uaternion A lgebra ................................................ ............... 108 Rotations of Rigid Bodies in Space. ........................... ..............110 vi A attitude Errors in Q uaternions .............. ........................................... .............. 111 Sum m ary of Q uaternions ................................................. ....................... 112 B M O D A L E Q U A TIO N S .................................................................... ...................113 M option of M ultiDegree of Freedom System ................................. .................... 113 Dam opening ................................. ......................... ..... ..... ........ 115 Sim ple E xam ple .............................................................................................. 116 C SIM U L A T IO N C O D E ............................................................................................... 117 Simulation Code .................. ..... ................... 117 M a tla b ................................................................1 1 7 C Programming Language .....................................................................117 C++ Programming Language ........... .......................................118 cM athlib H leader File ........... .................................... .......... .. ............ 119 cRK 4 H leader File ................................... .. .......... .. ............126 REFEREN CES .................................. .. ... ........ .............. 127 B IO G R A PH IC A L SK E T C H ............................................ ........................................ 134 vii LIST OF FIGURES Figure age 11 Finite element analysis of a fuel tank inside of a satellite. ......................................6 12 PP T for E O 1 ............................................................................ 10 13 D iagram of a PP T ........... .................................. ... ................... 10 14 NEAR spacecraft orbiting the asteroid Eros. ............. ......................................... 11 15 Hubble Space Telescope diagram .............................................. ............... 13 21 Traditional fuel slosh pendulum model where M is the total fuel mass center, mp is the pendulum mass, ms is the stationary mass fraction, L is the pendulum arm length, and g is the local acceleration due to some external force. .....................22 22 Propagation of fuel slosh in an elliptical tank over time. These images demonstrate the difficulty in modelling the fluid movement with simple representations. ......22 23 Diagram of a single sloshing mode, with its mass attached to the fuel tank walls by springs and dampers aligned along the x, y, and z axes. .....................................23 24 Plot of the effect of multiple modes in the velocity of a simply supported beam. This is representative of the fuel slosh disturbance in flight data. .............................24 31 The Soviet satellite Sputnik which orbited the Earth in October 1957. ................27 32 Diagram of the penumbra and umbra regions which block a satellite from sunlight. ....................................................................................................... 2 8 33 Another view showing the path of a satellite and how its orbit does effect the heat flux it is exposed to. ..................... .................. ................... ........ 28 34 Cross section of solar panel with coordinate systems superimposed. Note that the S's in the figures represent 's and M's are moments. ........................................30 35 Free body diagram of a section of the beam of size dx where M(x,t) is a moment, V(x,t) is a shear force, f(x) is a force per unit length, and u(x,t) is the height of the m ass elem ent at position x at tim e t. ........................................ ............... 32 36 Plot of [349]. Intersections of the two lines are solutions of the characteristic equation ...........................................................................36 37 A cross section of the beam in which a heat flux is applied to one side. Notice the beam starts off at a uniform temperature. .................................. ............... 42 38 Plot of the temperature change of a beam. Notice that the temperature difference between the two sides remains constant eventually. ...........................................42 39 Hubble Space Telescope which shows the solar panels bent due to a constant tem perature gradient. ..................... .. .... ................ ......................... 43 310 Another picture of the Hubble Space Telescope showing deflection of the solar panels during maintenance by the space shuttle. ............ ................................... 44 41 The three frames of reference commonly used in spacecraft dynamics. ...............47 42 A reference frame in motion relative to an inertially fixed frame of reference. ....48 43 A rigid body satellite. ..................... .. .... ................ ......................... 53 44 A rigid body spacecraft with a single reaction wheel. ........................................57 45 A plot of common environmental disturbances a spacecraft is subjected too. ......60 51 Two types of reaction wheels produced by Ithaco. ............................................66 52 Torque rods produced by Ithaco. ........................................ ....................... 67 53 Spacecraft engine produced by Boeing. ..................................... ............... 68 54 Two possible solutions for the double integrator, where (a) is and (b) is These plots are typically referred to as phase portraits. They plot position and velocity in this example, but later (when we are doing controls with full state feed back) the portraits will represent error and error velocity. ............ .................................... 70 55 Switching between the two controllers results in driving the system to zero in a stable m an o r. .......................................................... ................ 7 1 56 Graphical representation of how effects the sliding surface's orientation in the phase plane. The dotted lines represent the sliding surface (s = 0). .............................72 61 EO1 satellite with solar panel deployed. ................................... ............... 83 62 Diagram of satellite's internal components. .................................. .................84 63 Composition of a typical simple solar panel. ................................. .................86 64 Cost function surface plot for the optimal gain of the PD controller. ...................89 65 ITAE results from the optimal PD search. ................................. .................90 66 IAE results from the optimal PD search. .................................... .................90 67 Settling time results from optimal PD search ....................................................... 91 68 Control momentum or fuel used searching for the optimal PD controller. ...........92 69 A depiction of the step maneuver showing the orientation of the satellite at start and end. The angular distance and axis of rotation are also shown for better understanding of the term s. .............................................................................. 94 610 Quaternion error and control effort of the PD controller rotating from 30 to 30. ...................................................................................................... 9 6 611 Quaternion error and control of the SM controller rotating from 30 to 30. .......97 612 Quaternion error and control of the SMCNF controller rotating from 30 to 30. ...................................................................................................... 9 7 613 Arcsecond error of the PD controller during solar snap. ............... ............ 99 614 Arcsecond error of the SM controller during solar snap. ...........................100 615 Arcsecond error of the SMCNF controller during solar snap. ........................100 Ai Body reference frame attached to a rigid body. The xaxis points out the front of the v eh icle ........................................................................ 10 5 Bl Simple multidegree of freedom system composed of mass and spring elements. .................................................................................................... 1 1 3 C1 UML diagram of Mathlib class which contains C++ objects for vectors, matricies, and quaternion used in this work. .............................. .... ....................... 119 LIST OF TABLES Table page 11 Spacecraft with thermally induced disturbances. ................................................15 31 Solar panel thermal material properties. ..................................... ............... 32 32 First four eigen modes of the beam ............................... ..................35 51 Comparison of common active attitude control hardware. ....................................65 52 Sum m ary of the switching law ........................................ ......................... 70 61 Technical Specifications for the EO1 spacecraft ............................................84 62 EO 1 fuel slosh param eters. ............................................ ............................ 86 63 E O 1 w heel specifications. .......................................................... .....................86 64 EO1 solar panel param eters. .............................................................................87 65 Initial and final euler angles where roll, pitch, and yaw are equal. ......................93 66 IAE results with values having units of radians ............................... ...............94 67 ITAE results with values having units of radianseconds2. ............ .................95 68 Settling time results with values having units of seconds. ....................................96 69 CM results with values having units of Nms. ................................. ...............96 610 Maximum wheel momentum achieved where units are in Nms. .......................98 611 Solar snap results of the three controllers. ...................................................99 Ai Properties of a rotation m atrix. ............ ...... ............................................105 A2 Comparison of the two major types of rotations, sequential and first and third axes ro tatio n ........................................................................ 1 0 6 A 3 Quatem ion Algebra Sum m ary ........................................ ......................... 109 xii LIST OF SYMBOLS Thermally Induced Vibrations Symbols a Absorptivity CT Coefficient of thermal expansion. n Natural frequency of mode n. P Density of a material. K Thermal diffusivity. A Cross sectional area of the solar panel. c Heat capacity. E Young's Modulus F(x) Position function from the separation of variables method. Fn Position function for mode n. G(t) Time function from the separation of variables method. h Thickness of the solar panel. k Thermal conductivity. MT(t) Thermal moment. qo Total heat flux on surface. qs Heat flux from sun. qe Earth emitted heat flux. qa Earth reflected heat flux. u(x,t) Displacement of solar panel. V(x,t) Shear force. Satellite Dynamics and Fuel Slosh Symbols OB/I Angular velocity of satellite system relative to inertial frame. oB/O Angular velocity of satellite system relative to orbital frame. 0O/1 Angular velocity of orbital frame relative to inertial frame. Satellite Dynamics and Fuel Slosh Symbols C Rotation matrix. c First moment of inertia. Cfn Damper coefficient for sloshing mode n. F External forces and torques. h Angular momentum. J Second moment of inertia. kfn Spring coefficient for sloshing mode n. L Length of pendulum. M Mass matrix of satellite system. mp Pendulum mass. ms Stationary mass fraction. p Linear momentum. q Quaternion The imaginary part of the quaternion, 13 q Rc Distance of satellite from center of Earth. T Kinetic energy of satellite system. V Composite linear and angular velocity of satellite system. Controls Symbols x Weighting term on the sliding surface. Kd Derivative controller gain. Kp Proportional controller gain. PD Proportional Derivative controller s Sliding surface. SM Sliding Mode controller. SMCNF Sliding Mode controller with a colored noise filter Td Gain ratio between the proportional and derivative gains. Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy ROBUST NONLINEAR ATTITUDE CONTROLWITH DISTURBANCE COMPENSATION By Kevin J Walchko May 2003 Chair: Paul Mason Cochair: John Schueller Major Department: Mechanical and Aerospace Engineering Attitude control of small spacecraft is a particularly important component for many missions in the space program: Hubble Space Telescope for observing the cosmos, GPS satellites for navigation, SeaWiFS for studying phytoplankton concentrations in the ocean, etc. Typically designers use proportional derivative control because it is simple to understand and implement. However this method lacks robustness in the presence of disturbances and uncertainties. Thus to improve the fidelity of this simulation, two disturbances were included, fuel slosh and solar snap. Fuel slosh is the unwanted movement of fuel inside of a fuel tank. The fuel slosh model used for the satellite represents each sloshing mode as a massspringdamper. The mass represents the wave of fuel that propagates across the tank, the damper represents the baffling that hinders the movement, and the spring represents the force imparted to the spacecraft when the wave impacts the tank wall. This formulation makes the incorporation of multiple modes of interest simple, which is an advance over the typical one sloshing mode, pendulum model. Thermally induced vibrations, or solar snap, occur as a satellite transitions from the daytonight or nighttoday side of a planet. During this transition, there is a sudden change in the amount of heat flux to the solar panels and vibrations occur. Few authors have looked at the effects of solar snap. The disturbance dynamics were based on the work by Earl Thorten. The simulated effects compared favorably with real flight data taken from satellites that have encountered solar snap. A robust sliding mode controller was developed and compared to a more traditional proportional derivative controller. The controllers were evaluated in the presence of fuel slosh and solar snap. The optimized baseline proportional derivative controller used in this work showed little effort was needed to obtain better performance using sliding mode. In addition, a colored noise filter was developed to compensate for the fuel sloshing disturbance and incorporated into the sliding mode controller for greater performance increase at the expense of requiring a little more control effort. CHAPTER 1 INTRODUCTION This chapter provides the motivation and a literature review for this work. Additional literature will be presented in subsequent chapters as appropriate. Formation Flying As scientists aim for higher goals of discovery, satellites prove to be an increasingly important tool. Bigger, more powerful satellites can search the cosmos more easily for distant planets, map the Earth, or even try to discover the origins of the universe itself. However, when a satellite reaches a certain size it effectively can no longer be launched into space. The only answer to bigger, more powerful satellites is either construction in space or formation flying. Construction is space is difficult and costly, but perhaps one day when the international space station (or its successor) is finished this will be a viable solution. Formation flying is the only near term solution. Thus, the idea of not one big satellite but rather many little satellites, each acting as an individual element of a much larger sensor array, is actively being pursued by National Aeronautical and Space Administration (NASA) and other agencies. These missions greatly improve the sensing capabilities available to NASA, while keeping the associated cost and risk down. One of the main problems with formations and individual spacecraft in orbit is the amount of fuel consumed. Small differences in altitude between satellites will result in different velocities and orbits. This will require some satellites to expend more fuel in order to continually maintain formation. In addition, if they are in different inclinations, they are in different orbital planes (planes that cut through the center of the Earth) and thus will naturally move farther apart or closer together depending on where they are in the orbit. Therefore fuel is required for formation maintenance, which inevitably leaves fuel tanks partially filled. Partially filled tanks can lead to fuel slosh which can degrade the pointing accuracy and therefore affect the science mission. Fuel slosh compensation is not an easy task and unfortunately is often neglected. Satellites do not typically change their orbits unsupervised, but with formations there will have to be some level of autonomy where the satellites must make corrections to their orbits. Satellites will be required to change orbit to close gaps in formations or to avoid collisions from orbital debris. Gaps in the formation will occur as satellites run out of fuel or have failures. The control solution must be robust to account for uncertainty, produce stable motion, be easy to understand, and easy to implement. There has been much research on satellite formation control [18]; however these works assume satellites with robust individual control systems. General Problems Controlling Single Satellites To make a formation more stable, we must improve individual satellite control. This section will provide a brief overview of some of the problems with satellite control. Later greater detail will be presented on two specific problems: fuel slosh and solar snap. External Disturbances NASA engineers must contend with many types of disturbances when controlling a spacecraft. Low altitude satellites must fight gravitational and aerodynamic torque which can deteriorate their attitude accuracy and over time degrade their orbit. These gravitational torques can be present on large satellites where gravity has more of an effect on one end of a satellite than the other. In deep space or geosynchronous orbits, solar based disturbance becomes the dominate environmental disturbance since there is no friction or dampening. Also in these orbits, the gravity effects become negligible compared to effects produced by the sun. One of the major solar disturbances is solar snap which is thermally induced vibrations. Vibrations due to thermal changes have been known for a long time, but here on Earth there are few situations where these vibrations are of any interest. Another disturbance due to the sun is called radiation torque. Photons emitted from the sun slam into the satellite (a force) which produces a torque about the center of mass of the craft. This radiation torque grows and wanes with solar flare activity, but none the less pushes and prods satellites until they slip behind the dark side of a planet or moon. Parametric Uncertainty Control systems are typically designed using mathematical models of the system to be controlled. Thus there is a direct correlation between the performance of a standard classical controller and the accuracy of the mathematical model of the plant. Unfortunately in satellite control, the inertial matrix is only known within 5% of its in orbit value. This model inaccuracy must be addressed in order to achieve the higher level of performance expected from the missions. Thus engineers are currently examining new control schemes which help to combat this problem and improve performance in the presence of modelling error. Another problem that effects the inertia matrix is fuel slosh and fuel expenditure. Fuel can compose 40% of the total mass of a satellite prior to launch [9]. During orbit insertion, some of this fuel will be burned to perform various Av maneuvers to achieve a desired orbit. However, some of the fuel will be kept so that later orbital corrections can be made during the life span of the spacecraft. This remaining fuel poses a problem as it moves around inside of the satellite and thus changes the inertial matrix on orbit. Literature Review This section will provide a review of spacecraft that have had problems with fuel slosh and solar snap. The controllers developed to compensate for these disturbances will also be discussed. An analytical description of these disturbances will be provided in a later chapter. Fuel Slosh Modelling fuel slosh dynamics Accurate analysis of the fuel requires the use of the NaiverStokes equations, which are extremely computationally intensive. Since most satellites have a limited amount of processor power assigned to control, this is not practical for controller implementation. Therefore the most used models are the simplest ones such as the pendulum, massspring dampener, and moving fuel tank walls [11]. For small motions, the objective is to reproduce the sloshing mode resonance frequencies. Typically a mass is attached to the wall of the fuel tank by a spring. The movement of this mass accounts for the free surface wave motion and the influence on the spacecraft's dynamics is represented by the force the spring exerts on the wall of the fuel tank. Dampeners are commonly introduced to represent the viscous dampening of the fuel. There is typically one set of massspring dampeners for each sloshing mode that is being represented. Swirl effects of the fuel can also be included as reaction wheels with torsional springs attached. These models however are only accurate in representing the sloshing modes when small linear or angular motions are performed. For accuracy during larger motions, better models must be utilized. However these models are not ideal for control purposes. The typical pendulum model used for fuel slosh is derived from work done with rockets. Typically in this scenario, the rocket's main engine is firing and produces a local acceleration that pools the fuel at one end of the tank. The amount of fuel that remains pinned to this end of the tank is the stationary fuel mass fraction. The rest of the fuel is able to oscillate about the center line of the fuel tank in a pendulumlike motion. These oscillations are typically small since the fuel has only a small momentum compared to the forces produced by the local gravity. Unfortunately this model is not as useful in the situation of satellite attitude control, since there is no external force produced by a thruster that is constantly firing throughout the life of the vehicle. Thus a model is needed in which the fuel has more freedom of movement, can exhibit a larger range of movement, and produce accurate results. The work developed by Agrawal [10] for use on the INTELSAT IV, dual spin spacecraft, is a boundary layer model of the fuel for a spinning spacecraft. The author reached the conclusion that the analytical model gives the most accurate prediction of the fuel dynamics. The finite element model consisted of 81 nodes which represented a spherical fuel tank, shown in Figure 11. This fuel tank had a radius of 0.42 m, was 1.31 m from center of mass of the craft, and spun at 30 rpm. The solution for the fluid motion is obtained by solving three equations: an inviscous fluid problem, a boundary layer problem, and a viscous correction problem. The inviscid and viscous correction solutions are obtained by using finite element methods, while the boundary layer problem is solved analytically. The numerical results showed two slosh modes which could be accurately  J Il.: . \ L Figure 11. Finite element analysis of a fuel tank inside of a satellite. modeled by the pendulum. However there were also first azimuth and elevation modes, lower modes to be inertial modes and higher modes to be a combination of inertial and slosh modes. These other modes would not be accurately modeled by a pendulum. Controlling fuel slosh Fuel slosh is an example of underactuated control. The objective is to control both the rigid body dynamics of the spacecraft and the fluid dynamics of the fuel only having access to effectors that act directly on the spacecraft. The fuel's dynamics are controlled through the coupling of the two. One of the main problems with controlling this type of disturbance is that one can not measure the position, orientation, etc. of fuel in a satellite. One can only measure the effects the fuel slosh on the total system. The states and parameters of the fuel must be estimated, and a controller must try to account for the fuel's influence. Therefore it is difficult to control what can not be measured directly. Thus several types of passive methods are used to control slosh such as baffles [12], slosh absorbers [13] to dissipate energy through sloshing, and large fuel tanks are broken up into smaller ones. However these steps add complexity and weight to a satellite's design. The added complexity increases the construction time, the cost, and the chances of mechanical failure of the satellite. The added weight also increases the cost of launching the satellite since it costs approximately $10,000 per pound to put things into space. Fuel slosh directly affects the performance of the attitude control system, in both pointing and tracking maneuvers. One of the simplest ways to handle fuel slosh is to identify the sloshing frequencies and eliminate the influence on the feedback with a notch filter. Unfortunately this denies the satellite the ability to operate in certain frequency ranges since the filter attenuates the slosh frequencies in the feedback. Hence, normal feedback without sloshing effects will also be attenuated. Also as fuel is expended and the sloshing frequency changes, the filter would need to be updated to notch out the new sloshing frequencies again. Cho and McClaroch [14] developed a controller which utilized a common 2D pendulum model. Their equations only dealt with motion in a plane, but "a significant extension of what has been done previously, since it simultaneously controls both the rigid vehicle motion and the fuel slosh dynamics." One of the major problems they set out to solve, which others were unsuccessful in solving, was the transverse motion of the craft. Their method utilized a Lyapunov function approach, and was able to control the system. They utilized additional control effectors to counter this movement (i.e., gas jet thrusters) while also controlling the pitch and suppressing the slosh dynamics. Kuang and Leung [15] use a simple model of a liquidfilled spacecraft subject to external disturbances, and control it with a state feedback H, controller. The spacecraft's elliptical fuel tank is assumed to be completely filled with an ideal liquid in uniform vortex motion. Because of these assumptions, the slug of liquid can be represented by a finite number of variables. These variables can then be solved using Helmholtz equations. For the simulations, the liquid slug is located at the center of mass of the vehicle and surrounded by a viscous layer. An H. attitude controller was then developed using the theory of Hardyinfinity optimal control theory. The controller was capable of attenuating the vortices in the fuel tank. The pointing accuracy remained constant regardless of small changes in the viscosity. The authors plan to next look at partially filled fuel tanks, which is a more difficult problem. Thus far we have discussed more complex models to model liquids and said that in order to provide the level of control that a designer wants, one must use a complex model. An alternate example is Grundelius and Bernhardsson [16], who developed a minimum time optimal controller for moving open rectangular fluid containers down an assembly line. Proper control of the containers is important. If the accelerations applied to the containers are too high, then the contents of the containers may spill out of the open top. This is obviously a loss to the company in material and results in a more expensive process. Also, if the top of the container is to be eventually sealed, the contents may spill on to an area where glue may be applied, thus making for a bad seal. This could result in customers disliking the product because of "cheap" design. When the speed of the system began to exceed two meters per second, fluid began to slosh out of the containers and the controller was not able to achieve the desired result. The authors incorporated a minimum energy cost function into the design. Using only a simple linear model for the position of the surface of the liquid and the container, a controller that used a quadratic penalty on the slosh and a quadratic penalty on the terminal state was able to produce the desired results without a complex hydrodynamical model. Do we need to worry about fuel slosh? Fuel slosh is associated with liquid fuel which is used by thrusters. These types of thrusters are efficient and can provide a significant amount of control authority. Slosh forces can be eliminated via the use of alternate actuators. Let us take a look at some of the various types of control actuators available to satellites. Wisniewski and Blanke [17] developed a three axis magnotorquer controller for satellites subject to a gravity gradient while in a polar orbit about the Earth. Magnotorquers are an attractive form of control for small inexpensive satellites since they use the Earth's magnetic field to control the orientation of the spacecraft. However magnotorquers produce very small torques, and their principle of operation is nonlinear and difficult to use. They are difficult due to the fact that control torques can only be generated perpendicular to the Earth's magnetic field. Also the size of the spacecraft might prohibit the use of magnotorquers since they produce low torque. Chen et al. [18] developed an optimal controller which utilizes both thrusters and magnotorquers to unload extra momentum that has accumulated and saturated the reaction wheels of a spacecraft. Their method separates the required torques for the thrusters and magnotorquers using a simple optimal algorithm. Simulations conducted by Chen showed a 20% fuel savings using this method. Liquid [19] apogee motors are common in geosynchronous spacecraft design. This results in liquid fuel constituting almost half of the vehicles' mass. In space, liquid fuel motion greatly influences the attitude stability and control. The level and nature of these influences are dependent on the geometry and location of the fuel tanks, the ratio of spacecraft mass to fuel mass, the fill level of the fuel tanks, and the physical properties of the fuel itself. They also depend on the flexible structure of the spacecraft and the behavior of the estimation and control schemes that are present. Unfortunately, accurate prediction of the fuel dynamics is a difficult problem due to the complexity of the hydrodynamical equations of motion. LEASAT, which was launched in 1984, experienced attitude instability during the preapogee injection phase which immediately followed the activation of despin control. The instability was the result of the interaction between the lateral sloshing modes and the attitude control. Pulsed plasma thrusters (PPT) are becoming a more popular control solution for small spacecraft [20]. This fact is due to the small electrical consumption and the use of a solid teflon fuel. The fuel is attractive because it is stable, compact, and does not provide complications due to sloshing liquids. EO1 provided a successful demonstration of an attitude control system using these PPT's shown in Figure 12 and Figure 13. PPT's are capable of very high pulse width modulation frequencies (PPT's are only capable of on off or bangbang type of control). Although PPT's have some very favorable properties, they still have the liability of using a nonrenewable fuel and are only capable of operating F Figure 12. PPT for EO1. Figure 13. Diagram of a PPT. Figure 14. NEAR spacecraft orbiting the asteroid Eros. on small spacecraft. Another technology that shows promise for satellites ranging from 100 grams to 500 kg is MEMS engines [21]. MEMS (microelectricalmechanical systems) is an exciting new technology field which currently is beginning to make its way out of the R&D labs and into commercial products. MEMS are microscopic mechanicalelectrical systems that are developed using a combination of micromachining and standard integrated circuit design. Some of the common MEMS components on the market are accelerometers, gyros, and GPS receivers. Here MEMS rocket engines are developed for space applications. The MEMS engines are the size of a human hair and have a thrusttoweight ratio hundreds of times greater than the space shuttle. Although there are many good alternatives to liquid thrusters for small spacecraft, liquid fueled thrusters still provide the best source of high thrust capability for both small and large spacecraft. Therefore fuel slosh will always be present in these types of systems and thus affect high precession pointing and tracking accuracy. Missions involving fuel slosh disturbances We now understand why fuel slosh exists. Next let us look at missions where slosh has been observed. The Near Earth Asteroid Rendezvous (NEAR) spacecraft was sent to orbit and observe the Eros asteroid on December 23, 1998. The spacecraft, shown in Figure 14, had to fly by the asteroid following an unsuccessful firing of its main engine a few days earlier. A subsequent successful firing of the engine put NEAR on course to rendezvous with Eros to begin its planned year long orbital mission starting in midFebruary 2000. The reason the main engine firing did not work is the on board control system terminated the bum when sensors registered higher than expected lateral accelerations. These accelerations were later determined to be caused by the spacecraft's fuel. The NEAR's mission had to be delayed for over a year because the control system was incapable of handling the coupled dynamics. Situations like NEAR do not just happen around asteroids, but with the advent of the International Space Station similar problems could arise. As resupply ships travel to and from the station, similar problems could occur and a ship could crash into the station. This scenario is similar to the accident that occurred with MIR and one of its resupply vessels. Although that incident was not caused by fuel slosh, it still does represent a potential problem. Thermally Induced Vibrations: Solar Snap Thermally induced vibration or solar snap occurs when dynamics are excited by a rapid temperature change. This type of vibration has been known for some time but rarely Figure 15. Hubble Space Telescope diagram. appears in applications on Earth. However in the harsh, irradiated vacuum of space, without the presence of an atmosphere to dampen the vibrations, this disturbance can adversely affect the performance of high precision instruments and equipment. This section, like the proceeding one, will present missions that where adversely affected by solar snap and control systems that were designed to compensate for it. Controlling thermal vibrations The Hubble Space Telescope (HST), shown in Figure 15, was launched on 24 April 1990 with the hopes of making numerous new scientific discoveries. The HST was a 13 ton, freeflying spacecraft with a pointing precision requirement of 0.007 arcsec over a 24 hour period (which is the most stringent requirement ever imposed). During its initial checkout period, a pointing "jitter" was discover. The jitter was produced by a thermally induced vibrations imitating from the solar panel array. Later in December 1993 the solar arrays were replaced by the Space Shuttle Endeavour in an attempt to reduce the jitter. The solar panels exhibited an endtoend bending oscillation when the spacecraft transitioned between sunlight and shadow. They also had a sideways oscillation on the day side of the Earth. At its worst, the 20 ft. solar arrays would deflect as much as 3 ft. This would have the effect of rendering any long time exposures of over 25 min. pointless. The original controller was a digital PID controller with an FIR filter in the rate path to attenuate high frequency, mainbody bending modes. This design was incapable of handling the induced disturbances from the solar arrays, which caused a peak pointing jitter of 0.1 arcsec. ([22], p. 583) Wie and Liu [25] redesigned the attitude control system using an H. controller. This controller was designed to attenuate the two major modes from the solar panels which operated at 0.12 and 0.66 Hz. Through a trialanderror process, they were able to tune the weights of the controller and properly achieve the mission requirements. Singhose et al. [23], while working at M.I.T., designed a method to minimize vibrations in flexible spacecraft which utilized command profiles. The commands are designed by the following: 1. Formulate the equations of motion for a system subject to a sequence of impulses. 2. The equations are solved for the sequence of impulses that results in the smallest amount of residual vibrations in the system. 3. The sequence is convolved with the desired input to generate the command sequence which is issued to the real system. The command sequence is a pulse width modulation (i.e., onoff sequence) of commands for the spacecraft's thrusters. Later at Georgia Tech., Ooten and Singhose [24] extended the work to slidingmode control. The drawback for this work is, although flexible space craft are mentioned, all simulations involve only a massspringdamper system. Also, solving the equations of motion (step 2) for a nonlinear, flexible satellite would be very difficult. Missions which involved solar snap Originally spacecraft were very simple, but as technology progressed into the 1960s more appendages were being put on. These structures offered new challenges in controlling flexible structures and vibrations. Below is a table of spacecraft that had their performance compromised by thermal vibrations. These satellites do not include military and Soviet spacecraft. Table 11: Spacecraft with thermally induced disturbances. Spacecraft Dale Description Ulysses 1990 Spinstabilized solar probe experiences unexpected nutation due to 1991 thermally induced vibrations of berylliumcopper axial boom Upper Atmosphere 1991 "Thermal snap" at orbital night/day transition attributed to rapid Research Satellite heating of large single solar array Hubble Space Tele 1990 Jitter phenomena attributed to thermally induced vibrations of scope FRUSAtype solar arrays LANDSAT4/5 1980s Thermal elastic shock experienced during orbital night/day transition due to disturbance of large single solar array. Communications Tech 1978 Three axis stabilized satellite experienced "thermal elastic shock" nology Satellite during orbital night/day transition Voyager 1977 Low frequency oscillations of PRA booms during LEO operations attributed to "thermal flutter" Apollo 15 1971 Thermally induced vibrations of twin BiSTEM booms filmed during 64th lunar orbit by astronauts. Explorer 45 1971 Unexpected nutation of spin stabilized satellite attributed to thermal bending of four radial booms. NRL 161, 163, and 164 1969 Thermally induced vibrations of STEM booms used for gravitygra dient stabilization leads to large satellite motions. OGOIV 1967 NASA Orbiting Geophysical Observatory Satellite STEM booms experience strong thermally induced vibrations during orbital night/ day transition. OV10 1966 US Air Force gravitygradient satellite using STEM booms equipped with tip masses experiences thermal flutter and complete inversion in orientation. Table 11: Continued Spacecraft Date Description Explorer XX 1964 NASA spinstabilized satellite experiences spin decay due to interac tion between STEM boom thermal bending and solar radiation pres sure. 196483D and 1963 1964/ Dynamic thermal bending of STEM booms on APL gravitygradient 22A 1963 satellites. Aloutte 1 1962 Canadian spinstabilized satellite experiences spinrate decay due to thermal structural response of STEM booms similar to Explorer XX. a. Information contained in this table comes from Thorten [26] on page 345. "In spite of the practical importance of thermally induced vibrations to the successful operation of spacecraft, recurrent vibration problems from the 1960s to the present day suggest that the phenomena is either underemphasized or not well understood in spacecraft design" (Thorten [26], p. 344) Parametric Uncertainty A very simple way to model the effects of fuel and thermal vibrations is to represent them as external disturbances (i.e., deterministic noise). Depending on the mass fuel ratio, fuel slosh affects the mass distribution of the vehicle which affects the inertia matrix. Since control systems are model based, this will directly affect the level of control. If a control system could be designed that provided generic control, or model independent control, then the effects of these disturbances could be negated. This idea of generic control was the subject of work done by Walchko and Mason [27] utilizing fuzzy logic. The idea was that fuzzy logic is rule based and not model based control. Therefore fuzzy logic would be capable of providing better control in the presence of uncertainty. Although several different configurations of fuzzy logic were developed (PD, PID, and sliding mode) none produced the desired results when there was a significant amount of uncertainty. In fact it was shown that the two best were sliding mode and fuzzy sliding mode. However the nonfuzzy sliding mode had the added advantage that stability could be shown for the controller, whereas fuzzy logic lacks any general method to prove stability Petroff et al. [28] attempted to address this problem of fuzzy logic stability. They looked at simple linear single input single output systems and were able to develop a stability analysis. However, this method did not naturally extend to multiinput multi output systems and definitely not to nonlinear systems. Thus although fuzzy logic appeared to be attractive as a model independent control architecture, it was extremely weak with respect to stability analysis. This fatal flaw ultimately disqualified it for our applications. Summary and Overview of the Thesis What is needed are controllers that are easy to adapt from one satellite to another but still retain their robustness and efficiency. Problems due to simple control systems being implemented on satellites are numerous. This usually results in a redesign of the control system which is costly and a waste of time and resources. Now couple these problems with the difficulties of formation flying and things get very risky. This work will primarily focus on fuel slosh and solar snap. Both of these phenomena are interesting in that they have rarely been observed. Rather, engineers have examined the data from satellites that have experienced unexplained problems and deduced these were the causes. Neither of these problems is generally looked at when a control system is developed, but rather they are dealt with only when they appear. This attitude proves to be a problem when applied to formation control. The possibility of one satellite going unstable due to these phenomena then introduces the possibility of that satellite affecting the entire formation, thus endangering the mission. Chapter 2 will provide a more in depth discussion of fuel slosh. Various models will be described that are used to model fuel slosh for simulation purposes and models that are used in controllers. Chapter 3 will cover the interactions of solar panels with satellite dynamics. Thermally induced vibrations are described by the solution to their partial differential equations and modal equations. Chapter 4 will cover satellite attitude dynamics and control. This chapter will introduce the reader to the fundamental dynamics of satellites. Chapter 5 will introduce the basics of sliding mode control and other attitude control algorithms and control hardware. The specific controller for this work will also be derived and discussed. Chapter 6 will provide an overview of the simulations. All parameters, equations, and assumptions will be presented here. Also all results will be discussed in this chapter. Chapter 8 will provide conclusion to the work done. CHAPTER 2 CONTROLLING THE UNKNOWN: FUEL SLOSH The introduction covered much of the problems associated with fuel slosh and highlighted missions where this phenomena occurred. This chapter will expand on the introduction by focusing more on the fuel slosh disturbance. Two models were developed, one for use in the simulation to provide the disturbance torques on the satellite and another model to help the controller account for the fuel slosh disturbances. Introduction The sloshing behavior of liquids and its modeling has been studied in many different areas: fuel slosh in aerospace applications, movement of fluid filled containers in industrial/manufacturing applications, earth quakes, vehicle and ship dynamics. Unfortunately the choice of a model is not a simple one. Most analytic models that try to accurately represent the dynamics are three dimensional partial differential equations. These models are heavily dependant on boundary conditions and computationally expensive to solve. Thus they are typically not used in controller design. However, the nonlinear dynamics that appear from these complex models are important when designers wish to move large amounts of fluid rapidly. The aerospace industry has looked at controlling the effects of fuel movement in aircraft fuel tanks for years. Most of the ideas (i.e., baffled fuel tanks, breaking one big tank up into many smaller ones, etc.) can be traced back to standard aircraft design. The interaction of the fuel and the spacecraft is more complex in threeaxis stabilized systems than spinstabilized systems [29]. This is due to the fact that this system has constraints that reduce the number of degrees of freedom, simplify the model structure, and is numerically easier to compute. However, spacecraft will typically operate in several modes of operation over their life span and thus it is important to be able to control fuel slosh in any mode of operation. Modelling Fuel Slosh Dynamics Models for fuel slosh have not changed much since Abramson started in 1961 [9]. Models over the years have been: (single and multi) massspringdamper, pendulum, liquid slug, and CFD/FEA models. Typically, the controller uses a model to account for the fuel present inside of the spacecraft. Unfortunately, the models listed are too simple or based on a static representation of the fuel and do not attempt to modify the parameters of the model during run time. The coordinate system for the fuel is typically the body coordinate system [29]. This has the result of making the spacecraft's motions appear as a bodyforce field in the NaiverStokes equations. The boundary conditions at the wall also become stationary with only the free surface remaining transient while the viscous forces provide a dampening of the liquid. The fuel motion is influenced by four classes of forces: gravity, inertia, viscous, and capillary forces. The early work with slosh dynamics and spacecraft centered around liquid filled rockets and their stability [29]. This topic is still of interest and the assumptions [9] for this problem typically are: SThe fuel has small displacements, velocities, and slopes of the liquidfree surface. * The fuel tank is rigid. * The fuel is assumed to be a nonviscous liquid. * The fuel is also modelled as an incompressible, homogenous fluid. Propellant slosh is the induced motion of liquid due to acceleration of the container. This motion along with its reactive forces can deteriorate the pointing performance. This is especially true for systems with large fuel/weight ratio. Due to the nonlinear dynamics of sloshing, it is not always possible to compensate for its affects using a standard attitude controller. Typically a pendulum system, shown in Figure 21, is used to model a single sloshing mode. Since slosh dynamics typically display several modes, several pendulums should be included in the model of the system. Thus the more dominate modes are modelled, the more realistic the simulation. A problem with this type of model is determining the values of the pendulum lengths, masses, springs, and dampeners. One way to do this is to look at actual flight data where you can determine the modal frequencies. Assuming the amount of mass in the slosh mode, the value of the spring constant can be solved. The damper value is picked so that the system's oscillations die out after an appropriate amount of time. The main problem with liquid slosh dynamics is estimating the hydrodynamic pressure distribution, forces, and moments. One reason that this is so difficult is the dynamic boundary conditions at the free surface varies with time in a manor not known a priori, as depicted in Figure 22. Hydrodynamic pressure in many rigid tanks is composed of two parts. The first part is the fluid that moves with the tank in unison. The second component is the sloshing at the free surface. This component is typically modelled as a L Figure 21. Traditional fuel slosh pendulum model where M is the total fuel mass center, mp is the pendulum mass, ms is the stationary mass fraction, L is the pendulum arm length, and g is the local acceleration due to some exter nal force. Figure 22. Propagation of fuel slosh images demonstrate the difficulty in simple representations. in an elliptical tank over time. These modelling the fluid movement with massspringdamper or pendulum. Suggested values for the pendulum shown in Figure 2 1 are given by ELSayad et al. [30] as L = coth 1.84h) mp 1.84 \ RI) [21] where L is the length of the pendulum, R is the radius of the tank, and mp is the mass at the end of the pendulum. ~k~E~q~, totally tanh( 1.84 Fuel Slosh Models Two different models for the fuel will be utilized in this work. The first model will be included in the satellite dynamics and provide the proper disturbance torques on the system. The second model will be computationally simpler, but still contain the important aspects of the fuel's disturbance on the system. Simulation Model The model utilized is based on the massspringdamper models by Sidi [9] and Hughes [31] shown in Figure 23. Each sloshing mode will be composed of a mass, a spring, and a damper. The mass represents the mass of fuel in a specific mode. The spring represents the force exerted on the tank wall by a sloshing mode. The damper represents the baffling in the fuel tank that dissipates the sloshing movement. The actual dynamics of this disturbance will be derived later when the dynamics of the satellite are covered. This is due to the fact that both the dynamics of the spacecraft and fuel are tightly coupled and it is easier to present them all at once. Figure 23. Diagram of a single sloshing mode, with its mass attached to the fuel tank walls by springs and dampers aligned along the x, y, and z axes. S107 Mi 0de 1 6 10 100 o FrPieqcy (Hz) Figure 24. Plot of the effect of multiple modes in the velocity of a sim ply supported beam. This is representative of the fuel slosh disturbance in flight data. Controller Model The model used in the controller needs to be capable of representing several sloshing modes, but can not be computationally expensive. Figure 24 provides a frequency domain visualization of the proposed model. The model represents the sum of the slosh modes as a band pass filter. This will allow us to incorporate the effects of the multiple sloshing modes without having to specifically identify their individual characteristics. A simple band pass filter is the combination of a low pass and high pass filter in series. The standard transfer function for this filter is as follows: as 1 as [22] as+ lbs+ 1 abs2 + (a+b)s+ 1 1 1 a = b = [23] freqhighpass freqlowpass where the transfer function on the left is a highpass filter and the one on the right is a low pass filter. The variables a and b are the inverse of the high and low pass cutoff frequen cies. Putting [22] into state space form yields ab ab + u [24] x 1 0 x y = o0 x [25] The interesting thing to note about 25 is the output given by the model is velocity and not position. This is due to the derivative term (i.e., s in the numerator) in 23. This is a reasonable characteristic, since the fuel slosh effects show up in the rate terms and not the position terms. CHAPTER 3 DYNAMICS OF SOLAR ARRAYS As discussed in the introduction, thermally induced vibrations are a source of problems for satellite attitude control. This chapter will provide the theoretical development of thermally induce vibrations in long, thin beams. These results will finally be used to generate the solar snap disturbance in the simulations. Introduction Research on thermally induced vibrations began back in the mid1950's, before Sputnik orbited the Earth in October 1957. However it was not until the 1960's that applications for this new idea appeared. Initially spacecraft appendages (i.e., solar panels, communication arrays, booms, etc.) where thought to be simple systems like those used by Sputnik (Figure 31), but slowly it became apparent that they were much more complicated. Strange and unexplained spacecraft behavior was now being attributed to this new nonlinear behavior being observed. Thermally Induced Vibrations First, this section will derive and solve the partial differential equations associated with solar snap. Then differential equations will be obtained from the PDEs which are utilized in the simulations. The derivation that follows will parallel work done by Thorten [26] and Rietz [67]. However, there was great difficulty in actually following their work. Fortunately, all key equations were independently derived. The results in this work were also similar to Figure 31. The Soviet satellite Sputnik which orbited the Earth in Octo ber 1957. derivations performed on cantilever beams excited by sources other than thermal vibrations by Greenburg [32], Kreyszig [33], Meirovitch [34], and Zill and Cullen [35]. Heat Flux Thermal vibrations are excited by a sudden change in heat flux to the object. In this work, heat flux changes greatly during the daytonight and nighttoday transitions. Here the satellite passes into and out of the Earth's shadow. The Earth's shadow is actually composed of two regions shown in Figure 32. The penumbra is a region of partial shadow and the umbra is a region of full shadow. When a satellite is in low Earth orbit (LEO), the penumbra is small and the satellite essentially transitions from the day side to the night side (i.e., umbra). This sudden change from day tonight or nighttoday results in larger thermal vibrations. This is due to the sudden change in heat flux, much like turning on or off a light switch. The amount of heat flux (q) seen by a satellite is composed of solar flux from the sun (qs), Earth emitted radiation flux1 (q,), and Earth's reflected heat flux (qa) from the sun. solar vector S1 1 1 1 1 rr :i I Figure 32. Diagram of the penumbra and Figure 33. Another view showing the umbra regions which block a satellite path of a satellite and how its orbit does from sunlight, effect the heat flux it is exposed to. This amount changes with the satellite's position in orbit and its orientation relative to the solar flux vector (see Figure 33). q = qs+qe +a [31] q, = 1350acos(p) [32] where as is the surface absorptivity for solar radiation and Vp is the angle between the sur face normal and the solar flux vector. Summing Forces The solar panel is modelled as an EulerBernoulli cantilever beam subjected to a thermal moment. The beam, a cross section is shown in Figure 34, will have the following assumptions: * Plane cross sections before bending will remain planes after bending. 1. The radiation referred to here is black body radiation. * Lateral displacements in v and w are due to bending only. Deformations from shearing and changes in transverse beam dimensions are negligibly small. * The beam is a EulerBernoulli cantilever beam. * The system is uncoupled, meaning that changes in beam orientation do not effect the amount of heat flux entering the system. * The temperature gradient is only one dimensional (i.e., through the thickness of the beam). The moments on the system are defined as Mz = foydA [33] A MT = fEaTydA [34] A Linear strain displacement is as follows: 2 2 Ou duo v W [3 x x dx ax 2 2[35 Hooke's uniaxial law is ox = EExEaAT [36] Then substituting [35] into [36] yields: 2 2 du o Ov 8 w x = E y2 EaAT [37] x = dx 9x2 x2 Now substituting [37] into [33] produces S2 2 2 2 du o v O wL V E W S = y z EaTydA = El EM [38] z dx Tx2 yx2 9x2 yz9x2 T where Sxz Mz z x Mx Figure 34. Cross section of solar panel with coordinate systems superim posed. Note that the S's in the figures represent o's and M's are moments. I= fy2dA and Iy = fy2dA [39] A A Next, summing the forces in they direction in Figure 35 results in Fy = f(x) V(x, t) + V(x, t) + V(x, t)dx + mu(x, t) = 0 [310] Note that from here on out, the terms in parenthesis will be dropped unless it is felt that they are necessary for clarity. 2 8 u aV pA dx + dx = fdx [311] Oat ax 2 a u OV pA + =f [312] at2 Ox Summing the moments about the center of mass of the beam yields: M= M+ dx+ Vdx+ V+ dx dx M = 0 [313] Sx 2 2\ x M+V = 0 [314] ax V [315] ax Substituting [38] into [35] and ignoring the middle term (since we are only looking at motion in the u direction and not the w direction). al__E + M) = V [316] ax\ ax2 T) = Now taking [312] and substituting [316] gives us: 2 2 d ( ( a u f pA + EI + = [317] dt2 Ox[Ox Ox2 T)) 2 4 2 du u MTr pA +EIf + = [318] dt2 aX4 X2 This equation can be further simplified by assuming that no uniform load (f(x)) on the solar panel exists and realizing the thermal moment is not a function of x, thus its second derivative is zero. f= 0 MT = 0 34 = E[319] pA Incorporating these, [318] becomes utt + (34Uxxxx = 0 [320] where a subscript ofx or t refers to differentiation with respect to that variable. The bound ary conditions of the system are as follows: u(x, t) = 0 [321] u (O, t) = 0 [322] f(x) i iI Figure 35. Free body diagram of a section of the beam of size dx where M(x,t) is a moment, V(x,t) is a shear force, f(x) is a force per unit length, and u(x,t) is the height of the mass element at position x at time t. Mz(L) = EluxxMT = 0 V(L, t) = Eluxxx + M = 0 Table 31: Solar panel thermal material properties. Variable Description Units k thermal conductivity W mK q heat flux W m2K p density kg m3 c heat capacity j kgK K thermal diffusivity m2 s a coefficient of thermal expansion 1 K [323] [324] Nonhomogeneous Boundary Conditions Unfortunately the boundary conditions for the system are nonhomogeneous and in order to solve the equations this must be corrected. Getting rid of the nonhomogeneous boundary conditions will be accomplished by changing the dependent variable. u(x, t) = v(x, t) + p(x) [325] Substituting [325] into [323] yields: Mr Ux = + xx [326] +x xx El The second derivative of u must be equal to zero to make the system homogenous, thus p must negate the right hand side of the equation. Mxx [327] Pxx Now we must solve for Vp(x), thus integrating [327] twice results in an equation with two constants that can be solved by looking at the initial conditions. = MT2 + C X + C2 [328] 2EI p(0) = = 0= C2 x(0) = 0 = C1 [329] MT (x) = 2 [330] V (x) = 2Ex Thus [325] can be rewritten as MTX2 u(x, t) = v(x, t) [331] 2El Our new beam equation and boundary conditions using [331] are shown below. Notice how this is now a simpler homogeneous equation to solve. vtt + 34vxxxx = 0 [332] v(0, t) = 0 [333] Vx(O, t) = 0 [334] vxx(L, t) = 0 [335] xxx(L, t) = 0 [336] Separation of Variables Now [332] with its homogeneous boundary conditions can be solved via separation of variables. We will assume that the function v(x, t) can be broken up into two separate functions. Each of these two functions are dependent on only one variable, either position (x) or time (t). v(x, t) = F(x)G(t) [337] Substituting [337] into [332] gives us an equation which can be separated. FxxxxG + 4FGt = 0 [338] G Fxx S xxxx = (02 [339] G [4F p4 [340] pA Since each side of the equation is dependent on a single variable, it must be equal to a constant. We will choose this constant to be co2, which is squared to simplify the derivation. Each of the two sides can now be solved, which gives us one differential equation in x and one differential equation in t. Gtt + 2G = 0 [341] Fxxxx (024F = 0 [342] Solving the position function F(x) We will begin by solving the F(x) equation first. Assuming that the solution to this differential equation is an exponential, the following solution is found. F(x) = B cosh(pnx) + B2sinh(p 3x) + B3cos(nx) + B4sin(3nx) [343] F(0) = B +B3 = 0 [344] Fx(O) = B2 +B 4 0 [345] Fxx(L) = B(cosh(P3nL) + cos(P3nL)) +B2(sinh (P3L) + sin([3nL)) = 0 [346] Fxxx(L) = B(sinh( 3L) sin([3nL)) +B2(cosh(P3nL) + cos(P3nL)) = 0 [347] Combining the last two equations into a matrix give us: cosh(3PL) + cos(3nL) sinh(PnL) + sin(InL) 0 [348] sinh(PnL) sin(PnL) cosh (3L) + cos (iL) B The determinate of this matrix gives us the characteristic equation from which we can solve for the values of P3,L that result in zero. The characteristic equation is shown below and a plot (shown in Figure 36) which can be used to graphically solve the equation. cosh (PL) cos (PL) + 1 = 0 [349] Table 32: First four eigen modes of the beam n 1 2 3 4 [3nL 1.875104 4.694091 7.854757 10.995541 It is now possible to solve the equations in the matrix as a ratio B2/B1. Thus it is possible to solve for 3 of the constants in terms of the fourth. F,(x) = cosh(3px) cos(3px)+ ,(sinh(Px) sin(3,x)) [350] (P3,L)2 4 pA r 2 B2 sin(3L) sinh(3,L) SK n a [351] n L2 p E" El n B1 cos(PL)+ cosh(P3,L) This represents the n'' spatial solution or mode shape. In order to calculate the complete response of the beam, a summation over all modes must be done. 00 F(x) = aFn(x) [352] n= 1 The coefficient an is found by the orthogonality property of the modes. This coefficient is what defines the amplitude of the vibrations while F,(x) only defines the shape of the vibrating modes. Now because it is assumed that the eigenvectors form an orthonormal basis, the dot product of any one eigenvector with another is zero. For example, if we take the above equation and dot both sides with the first eigenvector only the first eigenvector 0.4  0.2  0.4 0 Z 4 6 8 10 12 14 16 18 20 Figure 36. Plot of [349]. Intersections of the two lines are solutions of the characteristic equation. and its coefficient will remain. We are now free to assume that the solution to the differen tial equation is a form of the first eigenvector and solve for the unknown coefficient. Mt(0) x2 S2EI F,(x)dx an = L2 [353] foF2(x)dx where the thermal moment (Mt(t)) comes from Johnson and Thorten which is similar to Reitz [66] shown below respectively. Eawh2 MT(t) = h2AT [354] 1n272Kt 48EIqoc J4 0 h2 MT(t) =  4 e [355] The AT in Thorten's equation is the temperature difference through the thickness of the solar panel. An equation for AT, assuming the solar panel can be modelled as a thin plate, will be derived later in the chapter. The temperature difference equation is dependent on the physical shape of the solar panel, while the thermal moment equation is more general. A quick numerical example, to calculate the coefficients for the first mode, set P3,L to 1.875104 and L to a position on the beam (w). (1.875104)2 El 3L 1.875104 01 = 2 L w sin(1.875104) sinh(1.875104) cos(1.875104)+ cosh(1.875104) Then solve for the coefficient an and finally for F(x). Although the derivations show that the summation symbol goes from zero to infinity, in actuality you would only sum over a finite number of modes. Solving the time function G(t) The solution of the time part of the separation of variables method is a little complicated. The change of the dependent variable is what complicates finding the result. Our solution for the time variable has the form shown below. Gtt(t) + 02G(t) = MTt [357] The solution of [357] will be in the form shown below, where the term Gp is the particular solution which can be determined if we have an equation for the thermal moment. The general form of the solution is shown below. G(t) = Asin(nt) + Bcos(ont) + Gp [358] Summary of results We have taken a long, winding road in order to solve the equations of motion for the beam. The final solution of the beam, which describes the motion in one direction, will need to be numerically solved is shown below. M,(t)x2 u(x, t) = v(x, t) + p(x) = F(x)G(t) [359] 2El Eawh2 M(t) = wh2AT [360] F(x) = a,[cosh(Px) cos(P3x)+ c,(sinh(P x) sin(P3,x))] [361] (IPL)2 ELI InL sin (3nL) sinh(P3,L) Sn P = cos(3 +cosh(3nL [362] L2 N P L cos(PnL) + cosh(PnL) G(t) = A sin(ct) +Bcos(cmt) + Gp [363] 2Mt(0) x2 an 2Ej X [364] F F(x) dx Again, summation occurs only over the number of modes of interest. Thermally Induced Dynamics Modal Equations The preceding derivation results in the path of the beam at some point x over time. However for the simulation modal equations are developed which are ordinary differential equations (ODE). This section follows the ideas in McConnell [36] and Thompson [37] and will cover the formulation of the modal equations. Further explanation of the theory pertaining to modal equations can be found in Appendix B or Tongue [71]. Each mode of the system will be a second order equation as shown below. qn + C,q, + Coq,, = Q, [365] The subscript n denotes that the equation is for the n'1' mode. The q's are the generalized coordinates of the mode and Q is the generalized force applied to that mode. Note that in this formulation, the generalized force for the second mode will only effect the second mode and have no effect on the first mode or any other mode. This idea is due to orthogo nality and as in the previous derivation, the modes are assumed to be orthogonal and do not interact with each other. The relationship between our physical parameter of distance x and the generalized coordinate q is as follows: x = Fq, [366] n where F, is our position function1 for mode n from the previous derivation, note that again it's dependence on x is not shown for simplicity. L Q = M P(x)Fndx [367] 0 where P(x) is some external distributed load per unit length. The mass, spring, and damp ener matrices are shown below: L M, = fpAF2dx [368] L 2 2 Kn = fEl 2 dx [369] C,n =a + 3,2 [370] The dampening matrix is assumed to be the same magnitude as the mass and spring coefficient matrices. Thus the dampening matrix is just a scaled version of the two where a and P3 are scalar weighting coefficients that are tuned until the desired amount of dampening is present in the system. The natural frequencies ((cn) are defined by [351] and are shown again below: (IPL)2 EI mcn L [371] n L2 N p Disturbance Torques In order to include the solar snap disturbance torques in a simulation, not all of the preceding steps need to be performed. This section will follow along the work of Johnson and Thorten [68]. Instead of solving [325], a differential equation is developed which 1. Thompson refers to F, as a normal mode and uses the variable %,, see page 438 in [37] draws on the material already presented. This solution is easier to understand and integrate into a computer simulation. The equations Johnson and Thorten give for the coupled satellite solar panel system using a generalized form of Lagranian equations is as follows: L Isat0 +fpA(Ro + x)u(x, t)dx = 0 [372] 0 An expression for the disturbance torque can be obtained by moving terms associated with movement of the solar panel to the right side of the equation.The disturbance torque (g,,) is a composite of two torques. The first torque (gqs) is generated from the quasistatic terms, or the nonhomogeneous boundary conditions. The second torque (g,) is derived from the vibrational aspects of the beam. gss = gqs + v [373] This can be seen by substituting the second derivative of [325] into the disturbance torque equation obtained from [372]. L gss = fpA(Ro + x)u(x, t)dx [374] 0 d2 2 u(x, t) = u(x, t) = (v(x, t) + p(x, t)) [375] L d2 pA2ha RL3 376] gqs = fpA(Ro+ x) dd + A [376] 0 L 2 N = pA(R,+x)2v(x, t)dx = pA [(Ro+x) (x)dx qn [377] 0 n = T T qsL 2k T(L,t) t Figure 37. A cross section of the beam in Figure 38. Plot of the temperature which a heat flux is applied to one side. change of a beam. Notice that the tem Notice the beam starts off at a uniform tem perature difference between the two perature. sides remains constant eventually. where is the shape function (which use to be F, but is changed to match Johnson and Thorten's work) and q, is the generalized coordinate for mode n, and AT is the tempera ture difference through the thickness (h) of the solar panel. Thickness Temperature Gradient The equation for the temperature in a thin plate of thickness h can be found in Thorten's book [26] on pg. 86 or Incropera and DeWitt [69]. A diagram of the solar panel cross section is shown in Figure 37, the equation for the temperature change through the thickness is given below. o 242Kt qh xKt l(y + 1 2 1 2 (1)" ( 1 7 T(Y, = + 6 cos n2 + 2 eh [378] k h2 2 h 2 6 2 h 2 The temperature difference through the thickness of the beam can be found by subtracting the temperature at the heated surface from the temperature at the insulated surface at time t. After subtracting the two equations and some algebra, the temperature gradient at time t, steady state at time oo, and its acceleration becomes: q(O,t) T(x,O) = T Sy Sq(L,t)=0 SL  ko 2 7U2 o n2 qfh 1 2 2 h AT(t) I+ 2 [379] Sn = 1, 3, 5, ...  qoh AT(oo) =q [380] 2k AT(t) = [381] dt2 Notice that Figure 38 and [380] show the temperature gradient of the solar panel will eventually reach a steady state. The two sides of the beam never simultaneously achieve the same temperature while there is a heat flux constantly applied. This means there is always a moment on the solar panels, and thus always a deflection of the solar panels. Examples of this are shown in Figure 39 and Figure 310 where the solar panels of Hubble are deflected due to the constant temperature gradient. Hubble's solar panels are more complex than what is represented in this work. Hubble's solar panels incorporate bi stems, solar blankets, and spreader bars which are very thin, light weigh structures that are highly susceptible to solar snap. Hubble's solar panels have been replaced several times Figure 39. Hubble Space Telescope which shows the solar panels bent due to a constant temperature gradient. Figure 310. Another picture of the Hubble Space Telescope showing deflection of the solar panels during maintenance by the space shuttle. over the years, and the current ones are smaller and stiffer than the original design. This is an attempt to reduce the influence of solar snap on Hubble. Even though there is a constant temperature gradient through the panels, this does not mean there is constant vibrations. The vibrations are driven by a change in the heat flux, and thus a change in the temperature gradient. These changes primarily occur during the daytonight and nighttoday transitions of the satellite around the Earth. Conclusion Thermally induced vibrations or solar snap is a disturbance which occurs when a spacecraft transitions into and out of the Earth's shadow. A temperature gradient occurs in the long thin solar panel which in turn creates a thermal moment. This moment causes the panel to vibrate until the transients die out. During this time frame, the scientific mission has to be temporarily halted and the spacecraft put into a stable orientation. 45 The theory and math behind this phenomenon is not a mystery and the aim of this chapter was to provide the reader with a complete derivation of this problem for a simple solar panel. Here the assumption was made that the temperature gradient only occurs though the thickness of the beam which reduced the problem to one dimension. CHAPTER 4 SATELLITE ATTITUDE DYNAMICS Satellite attitude control can be difficult due to the various types of disturbances and limited control capabilities of satellites. In order to develop a controller, a good understanding of the dynamics of the system is needed for proper simulations to be conducted. This chapter will give an overview of the dynamics of rigid bodies in rotating reference frames. Reference Frames Several different coordinate systems will be used to develop the equations of motion for a satellite. This section will cover the reference frames used in this work and are shown in Figure 41. Earth Centered Inertial The first reference frame is the Earth centered inertial (ECI) frame. This frame is a nonrotating frame relative to the fixed stars1. Another way to say this is, the ECI frame is an inertially fixed frame where Newton's laws are valid. This frame has its origin located at the center of the Earth with its zaxis along the Earth's mean axis of rotation. The yaxis and xaxis are pointing to some convenient set of stars. Note that the Earth is allowed to rotate in this fixed frame. 1. Actually the stars move, and thus our ECI frame moves with them. This movement is small enough to be ignored in most cases. Z {Inertial} Earh l X , $ Reaction " Wheel Z { XX Y {LVLH} {Body} Figure 41. The three frames of reference commonly used in spacecraft dynamics. Orbital Frame The orbital reference frame or local vertical, local horizontal (LVLH) frame is the reference frame from which all of our spacecraft's angle will be defined. The orbital frame's origin is located at the center of mass of the spacecraft. The zaxis points towards the center of the Earth. The yaxis is perpendicular to the orbital plane. The xaxis points in the general direction of travel. When the orbit is circular, then the velocity vector and the xaxis are colinear. Body Fixed Frame The body reference frame is located with its origin located at the center of mass of the spacecraft. The x, y, and z axis are located along the x, y, and z principle axis of the spacecraft. Fun With Vectors and Rotating Coordinate Systems This section will present some simple vector identities and properties which maybe useful to the reader. Although all of the important steps are shown in the following x f I \x derivations for spacecraft dynamics, at times some algebraic simplifications are not shown. a(b+c) = ab+ac axb = bxa (axb)c = a (bxc) ab = b'a ax(bxc) = b(a c)c(a b) ab = aTb Conventions used: * All vectors and matrices are bold face. * 1 and 0 represent the identity matrix and the zero matrix respectively. The dynamics for a spacecraft are written with respect to several frames of reference. Two important reference frames will be the ECI inertially fixed frame and a body fixed frame which is allowed to rotate in space. Differentiating a vector relative to a rotating reference frame is given by: 0A1 dAB a = d + oxAB [41] at dt Y p Figure 42. A reference frame in motion relative to an inertially fixed frame of reference. where (o is the relative angular rotation rate between the inertially fixed frame I and the rotating body frame B. Now for the point located in the rotating system at position r, its velocity is: dr1 dR, dr' dr's V + = R+ +coxr'B = R+v+coxr'B [42] S dt dt dt dt r = R + r' [43] where R is the velocity of the origin of the rotating system relative to the fixed system and vB is the velocity of the point in the rotating system. A similar derivation can be shown for the acceleration of the point which results in: a, = aB+R+2c xvB+c x(coxr')+ Ixr' [44] When dealing with rigid bodies, with the rotating frame representing a body fixed frame, the point does not move within this frame. This is due to the fact that is fixed in the body. Thus vB = a = 0, and the proceeding equations reduce to: v = R+ x r' ai = R+cox(coxr')+ Cxr' [45] where the velocity and acceleration are functions of the rotation rate of the rotating frame and its velocity and acceleration. Spacecraft Attitude Attitude refers to the orientation a spacecraft occupies in 3D space. Although there are several different ways to represent this (i.e., euler angles, Gibbs vector, fixed angles, Euler symmetric parameters, etc.), typically space applications utilize quaternions. This section does not attempt to provide the extensive understanding needed to employ quaternions but rather a simple introduction. Further information can be found in Wertz [37] or Crane and Duffy [42], or Appendix A. Quaternions Quaternions were invented by William Rowan Hamilton in 1843. Prior to his discovery, it was believed impossible that any algebra could violate the laws of commutativity for multiplication. His work introduced the idea of hypercomplex numbers. Here real numbers can be thought of as hypercomplex numbers with a rank of 1, ordinary complex numbers with a rank of 2, and quaternions with a rank of 4. Hamilton's crucial rule that made this possible: i2 = j2 = k2 = ijk = 1 [46] Hamilton supposedly developed this rule while on his way to a party. When he realized what the solution was, he took out his pocket knife and carved the answer into a wooden bridge. This rule would forever change mathematics as was known at the time. Now mathematicians could look at algebra where communitivity did not work. This is where Gibbs and others developed algebra of vector spaces, and quickly eclipsed Hamilton's work until recently. Quaternions, also known as Euler symmetric parameters, are more mathematically efficient ways to compute rotations of rigid and nonrigid body systems than traditional methods involving standard rotational matrices or Euler angles. Quaternions have the advantage of few trigonometric functions needed to compute attitude. Also, there exists a product rule for successive rotations that greatly simplifies the math, thus reducing processor computation time. Quaternions also hold the advantage of being able to interpolate between two quaternions (through a technique called spherical linear interpolation or slerp) without the danger of singularities, maintaining a constant velocity, and minimum distance travelled between points. The major disadvantage of quaternions is the lack of intuitive physical meaning. Most people would understand where a point was in cartesian space if they were given [1 2 3]. However, few would comprehend where a point was if given the quaternion [1 2 3 4]. Rotations of Rigid Bodies in Space. Quaternions are able to represent a rotation of a rigid body in space. To perform a rotation () of a rigid body about an arbitrary moving/fixed axis (e) in space, the quaternion representation of this operation is q = [q4] Norm(q) = 1 [47] q =e sin( q4 =cos( [48] e = [e e2 eT Norm(e) = 1 [49] Notice that only one sine and one cosine function call is needed to calculate a quaternion, where an euler rotation matrix would require three sine and three cosine function calls, one each for roll, pitch, and yaw. Since trigonometric function calls are computationally expensive, this is a great savings. Spacecraft Equations of Motion In this section, the kinematic and dynamic equations of motion for a spacecraft are presented. Attitude Kinematics A spacecraft's orientation in space is represented by a quaternion. The quaternion kinematic equations of motion are given by Wertz [38] as, 1 1 q = G2q = E(q)co [410] 2 2 where S= Ox and E(q)= 433x3+q [411] IT 00^ T q The terms with the x by them signify a skew matrix. Spacecraft Dynamics First the equations of motion will be developed for a satellite and then expanded to include other dynamics. Thus a lengthy derivation is needed to derive something as simple as Euler's equation. This, however, is necessary so that when reaction wheels and fuel slosh are included, understanding how these dynamics are incorporated will be easier. Simple Spacecraft To model a satellite, we starting off with a couple of definitions for the total mass of the spacecraft, first moment of inertia, and the second moment of inertia respectively. N N N m = mn c mnrn J mn(rTrnlrnr ) [412] n= 1 n= 1 n= 1 where mn is a point mass in the spacecraft, Nis the total number of points. These defini tions are in reference to Figure 43 (Adapted from Hughes [31] p. 43). The second moment of inertia (J) is typically just referred to as the moment of inertia. The moment of inertia has two important properties: symmetry and positive definite. *^, (m.) Ln Imn .0' (m,) Figure 43. A rigid body satellite. The momentum and velocity of each point in the spacecraft is N mnvn = fn + n vn = V+rn [413] m= 1 where R = vo, f, is the external forces acting on point, and fmn is the force exerted by point m on point n. The linear momentum (p) of the total spacecraft is N N N P = pn= mnvn n= v m +rn) = mv+c [414] n=1 n=l n=1 This equation could be further simplified by realizing that a spacecraft1 is a rigid body, and thus c = 0. Thus N p =f= Ifn [415] n= 1. Actually since spacecraft are composed of light weight, flexible materials, some authors model them as flexible structures. *Y,(m,) where N N SI fimn =0 [416] n=lm=l which comes from Newton's third law, fnn +fnm = 0. The angular momentum of the spacecraft is: N ho= rn xPn [417] n= Now differentiating this equation with respect to time, results in the following equation. N N ho= (rn xpn + rxpn)= [(vn v)x mn'n+ r xfn] [418] n= n=1 Realizing that vn x mnvn = 0 results in the following equation. ho = v xp +go [419] where go is the total external torque about the point O. Now if the equations are defined about the center of mass of a rigid body spacecraft, then c = 0, vo = vn, and the cross product with momentum in [419] disappears. The equations collapse to the more familiar ones for linear and angular momentum. p = mvy ho = go [420] Now using the previous equations involving rotating reference frames, we can write the linear and angular momentum equations for a spacecraft in a rotating body fixed frame. p = mvo +c+ xc [421] p = oxp+f [422] mxho v xp +go As before, these equations can be simplified using certain assumptions. However, deriving the equations without the assumptions makes including internal dynamics (i.e., fuel slosh and reaction/momentum wheels) easier. These equations put into a matrix form are as follows: p= p h V= [v]T F= f goT [424] p = MV M = ml cx [425] where Mis the mass matrix of the system. The kinetic energy Tof the system is given by: 1 T T = VMV [426] 2 T = mVyo + oWT x vo + TJom [427] The dynamics can be obtained from the kinetic energy by using the following: aT aT p = ho [428] av am daT aT = d aT +aT (a T [429] + x  =f t ) + mx a+v )x = go [429] dtK8v) K9v) dt\m) m v 9J These equations are called the quasiLagrangian equations and are shown in Hughes [31] and Meirovitch [39]. Meirovitch derives the equations using quasicoordinates, which is somewhat difficult to understand and very long. However, this method has been used by some authors such as Miller et al. [40] to model various flexible appendages on a satellite and moving submasses inside of a satellite. [423] Angular Velocity The spacecraft's orbit in this work is assumed to be circular. Since we are using an LVLH reference frame for our spacecraft, the angular velocity of the system must account for the orbital rate in addition to the maneuvers that the satellite is conducting. COB/I = OaiB/ + apol [430] col1 = Cn n = T [431] where B refers to the body frame, O refers to the orbital frame, I refers to the fixed inertial frame, a/10 refers to angular velocity of B relative to O, C is a rotation matrix, [t is the gravitational parameter of the Earth, and Rc is the distance the spacecraft is from the cen ter of the Earth. Internal Disturbances Various factors complicate the dynamics of a satellite's dynamics. Moving masses such as reaction wheels and sloshing fuel create additional linear and angular momentum that needs to be accounted for. This section will show how the above equations for a simple spacecraft need to be modified in order to account for these effects. Spacecraft with Reaction or Momentum Wheels Starting off with reaction wheels, we will introduce a method to account for any number of moving wheels in any configuration. However, since this work only deals with reaction wheels, we will not concern ourselves with momentum wheels which change their orientation and not their rate of spin. aV o2, Figure 44. A rigid body spacecraft with a single reaction wheel. Now we will add the contribution of the reaction or momentum wheels to the spacecraft dynamics. Looking at Figure 44 (adapted from Hughes [31] p. 66), we can define the following terms: m = msat + m c = csat +mb J = Jsat +Ia + m(bTb bbT) [432] where terms with the subscript sat refer to the satellite and terms with the subscript w refer to the wheels. The linear momentum is given by the following equations: Psat = msatv + cmx sat [433] p, = mv + m, x b [434] P = Psat +Pw = my + x c [435] where b is the vector from the point O to the wheel. The angular equations of motion are given by the following equations: hsat = sato+ csat xv [436] h, = IsCo. [437] h =hsat + ha [438] h = cxv + Jo+ ah, [439] where a is a vector that describes the direction of the wheel axis when dealing with a sin gle wheel. When there are multiple wheels, a becomes a matrix where the columns are the individual direction vectors for each wheel. For example, a system with n wheels would look like: a = [{al} {aw2} {awn [440] where awi = e2 e 3]T Norm(aw) = 1 [441] The wheel speed co, is a vector where each element is the speed of a specific wheel. Thus for a system with n wheel, the vector would be: O = [Owl ,w2 ... wnr [442] The equations of motion can be derived from kinetic energy for the system by using the quasiLagrangian. The kinetic energy for the spacecraft with wheel dynamics is: T= mvTv+ CTJ w+ I ~mwlC x m+IwmoTaTC [443] Putting these equations into a matrix form results in the following equations. S= [ph h, h V= [vo o o]T F= fg ga] [444] ml c x 0 S= MV M= c x J Ia [445] OT IaT I _,aT The equations of motion, like before, are as follows: p = coxp+f [446] ho = wx hovoxp+go [447] ha = ga [448] Although these equations look the same as the previous equations without wheel dynam ics, remember the equations for linear and angular momentum are different. The linear and angular momentum of the spacecraft and wheels are now coupled in these equations. Equations for spacecraft with wheel dynamics are often simpler than what are shown here. However, if the wheels are assumed to be located at the center of mass (which is physically impossible) then the equations are greatly simplified, = w x ho + g [449] since p = mvo and vo xp = vo x myv = 0. Further more, h = Jco+h since c = 0 and substituting this into [449] results in: Ji +h, = o x (Jw +h) + go [450] Jm = x Jo + x h hw + g [451] JC = ox Jw + o x h, ga + go [452] This final equation is the common equation found in Wie [22] and Sidi [9]. 60 Fuel Slosh Fuel slosh, as stated in a previous chapter, is the unwanted movement of fuel inside a spacecraft. Each sloshing mode is modelled as a mass, with the forces exerted on the spacecraft body represented by a spring, and the effects of baffling in the fuel tank 5.0 Geostationary Orbit 4.6 Cosmic Dust S CMCP= Gl m 4.2 CMC m Gravity Gradient E 0 ei.o* ILJ O _, Magnetic (Equator) 38 1.0 AmpTurn I rSpecular Reflection Diffuse Reflection Solar Radiation CMCP=& I rm 0 34 \ 3.0 Aerodynamic unspot Min. ym 2.6 L' ight / nspo Mox. M eight 2.2 10'7 105 10"3 10' TORQUE, [Nm] Figure 45. A plot of common environmental disturbances a space craft is subjected too. represented by a dampener. The dynamics of this are incorporated into our simple satellite model by the following equations which will represent one sloshing mode. p = myc x w+mpn [453] h = cxw+Jw+ mfb xn [454] pf = m(v b x o+ n) [455] where n is the position of the sloshing mode, mf is the mass of the fuel, and b is the vector from the center of mass of the spacecraft to the center of the fuel tank. These equations reflect how the linear, angular, and fuel slosh momentum change with the addition of fuel slosh's massspringdamper. The mass matrix now becomes: ml c x mf M = x J mb x [456] mf mjbx mf As with the wheel dynamics, the differential equations of motion for the linear and angular momentum remain the same. The new differential equation for the fuel slosh is: pf = mW x (v rd x w) cy kin [457] where the cf and kf are the dampening coefficient and spring constant for the sloshing mode. Environmental Disturbances in Space There are many environmental disturbances that plague a spacecraft in orbit. These disturbances constantly effect the performance of the spacecraft which necessitates the use of a control system that counter acts them. The disturbance torques produced by some of these are shown in Figure 45 (adapted from Hughes [31] p. 271). A further explanation of the important disturbances follows. Gravity Gradient Euler and d'Alemert in 1749 first pointed out how the Earth's gravitational field was not uniform. Later in 1780, Lagrange used these ideas to explain why the moon always had the same side facing the Earth. Since the gravitational field over a rigid body is not uniform, the center of mass is not the center of gravity. Thus there are torques about the center of mass which depend on the orientation of the spacecraft. The gravity gradient torque disturbance is found in [22], [31], and [38] as ggg 3m 23 lo [458] where ome is the orbital rate of the spacecraft, 03 is the third column in the orbital frame to body frame rotation matrix. When a satellite is in low Earth orbit, gravity gradient torques become the predominate environmental disturbance. Solar Snap In a previous chapter, modelling solar snap was discussed. The disturbance torque due to solar snap is a combination of the quasistatic displacement of the solar panel and its vibration. gss = gqs + [459] Conclusion This chapter showed the basic concepts of spacecraft and disturbances dynamics. These dynamics produce a system where the satellite body, wheels, and fuel are all coupled. The disturbances presented in this chapter were internal (fuel slosh) and external 63 (gravity gradient). A complete system can now be produced which has all of the dynamics of interest: satellite, wheel, fuel slosh, and thermally induced vibrations. CHAPTER 5 ATTITUDE CONTROL OF SPACECRAFT This chapter will introduce the reader to attitude control of a satellite and specifically to a variable structure method called sliding mode control. First however we will cover some of the typical hardware used in a satellite to control its attitude. Next the traditional proportional derivative controller and sliding mode controllers will be introduced. Later in this chapter we will compare and contrast the two controllers. Satellite Control Hardware This section will provide a brief overview of hardware used by a satellite to control attitude. Control Computational Hardware Satellite's are typically equipped with only the bare minimum in computational ability in an effort to reduce the cost of the vehicle. Unfortunately the complexity and performance of the control system is constrained only to the simpler algorithms. In fact some spacecraft, such as the lunar explorer, had no on board control systems. Instead it was remotely controlled from Earth, which was a very brittle solution at best. This was hailed as a great step forward in reducing the cost of space flight, but what happens if you lose the connect between ground control and the satellite? Any control algorithm used must be computationally inexpensive so it does not overburden the processor, and is typically designed to operate at 1 Hz. One of the simplest, easiest, and most used controller available is the venerable proportional derivative (PD) controller. This is one reason that the PD has flourished in spacecraft control. Control Actuators In order to overcome environmental disturbances, satellites are typically equipped with various types of actuators that can provide an attitude control effort. Some of these are active control systems which utilized electrical power or consumable resources to effect the spacecraft's attitude. While others are passive, and rely on conservation of energy to effect the satellite's motion. Passive methods of control are generally relegated to dampening out disturbances during flight. These methods do not produce the same levels of torque as their active counterparts, but are typically employed along with active controls in an effort to reduce the power consumption of the spacecraft. Some of the most common types of active control actuators will now be discussed. A quick comparison of their control torques is given below. Table 51: Comparison of common active attitude control hardware. Control Hardware Control Force Range (Nm) Magnetic Torque Rods 10E6 (geostationary) 2.5E3 (400 km) Reaction Wheels 0.052 Thrusters 0.130 Reaction and momentum wheels Reaction and momentum wheels are cylindrical masses that rotate about their center of mass driven by a motor. They are produced in a variety of types and sizes, some of which are shown in Figure 51. These devices are capable of changing or stabilizing the orientation of a spacecraft with respect to its axis of rotation through conservation of angular momentum. These are the best for delivering smooth continues control efforts. .i. un *, .FfP i. 5 . *Iwrv  **:r i, *ijwii \ IRONLESS ARMATURE BRUSHLESS DC MOTOR Figure 51. Two types of reaction wheels produced by Ithaco. Wheels that have a nominal spin rate of zero1 are called reaction wheels. Wheels that have a constant momentum are called momentum wheels. Moment wheels are used to create a bias in the system which resists small parasitic environmental torques. Sometimes these wheels are also mounted in a gimbaled system that has one or two degrees of freedom. This allows the momentum wheels (with their constant momentum) to change the direction of the their momentum vector. Magnetic torque rods Magnetic torque rods are another method of attitude control and are shown below in Figure 52. They are based on the idea of generating torque by magnetic fields. Here the rods produce a magnetic field which interacts with the magnetic field of the Earth. Obviously this method of attitude control produces small control forces compared to 1. This statement is not always true. Sometimes wheels are biased by a small amount to fight fric tion in the wheels and bearings. These zerocrossing problems, dead zone around zero rpm, can effect the accuracy of a satellite when it is gathering scientific data. 41In) (151in) Figure 52. Torque rods produced by Ithaco. reaction wheels. Also the maximum amount of control effort produced by this method is dependant on the altitude of the spacecraft since the Earth's magnetic field is stronger the closer the spacecraft is to the center of the Earth. In addition to small torque capabilities, electromagnetic shielding to protect satellite components must be considered. This has the additional disadvantage of adding weight and increasing the complexity of the design process. Gas jets Gas jets or control jets (shown in Figure 53) are the most powerful hardware available for attitude control, but these are not the most desirable to use for two reasons. First they consume fuel which is a limited, nonrenewable resource, and second they do not have variable control but rather "onoff'. Thus bangbang control schemes are created to provide pulsewidth modulation control efforts to mimic a variable type of device. Proportional Derivative Control First let's start off with one of the simplest to design and computationally efficient to implement, the proportional derivative (PD) controller. Wie and Barba [41] developed several computationally efficient control schemes for large angle maneuvers. Many of Figure 53. Spacecraft engine produced by Boeing. these stabilizing schemes utilize quaternion and angular velocity feedback. The use of quaternion representation allows for more realistic, large angle maneuver control schemes. These schemes are formulated based on Lyapunov analysis, which produces a range of positive stable gains for that control law. Thus in order to meet desired performance, engineers must iterate through a significant number of gain combinations to obtain the desired response in a clean simulation. However, even when a satisfactory response is finally obtained, there are no guarantees how the satellite will behave in the presence of disturbances, noise, or uncertainties. Bong Wie [22] defines three different PD controllers utilizing quaternions in his book where the origin of the quaternion system is (0,0,0,1). u = KqC C [51] K = kl C = diag(c1, c2, c3) [52] k K = C = diag(cl, c2, c3) [53] q4 K = ksgn(q4)1 C = diag(c1, c2, C3) [54] where u is the control effort and all k's and c's are positive. Sliding Mode Control Theory and Background This section will present an overview of sliding mode to the reader. Further in depth explanation can be obtained from Slotine [42] or DeCarlo et al. [43]. What is Sliding Mode? Nonlinear model based control systems offer a level of dynamic capabilities which linear techniques are incapable of providing when dealing with parameter uncertainties and unmodeled dynamics. Sliding mode, which has been studied in the Soviet Union for many years, is categorized as a variable structure control. This control structure has excellent stability, robustness, and disturbance rejection characteristics. Sliding mode is not a new control technique either, many researchers have utilized its robust properties to control a variety of systems such as missiles [44], mechanical systems [45], robotic manipulators [46][47][48],and submarines [49][50][51][52]. Theory First let's look at a simple example of controlling a system. Given the system below, y = u(t) [55] u(t) = k y(t) [56] a solution to this differential equation can be found and is plotted in Figure 54. Notice that neither of the two solutions drive the system to zero at the origin, but rather remains oscillating with a stable behavior. The trick is to switch between the two solutions depend ing on which quadrant the system is in. u(t) = k y(t) if yy < 0[57] k2 y(t) otherwise y (a) (b) Figure 54. Two possible solutions for the double integrator, where (a) is k = k1 and (b) is k = k2. These plots are typically referred to as phase portraits. They plot position and velocity in this example, but later (when we are doing controls with full state feed back) the portraits will represent error and error velocity. By switching, we can drive the system to the origin in a stable manor, as shown in Figure 55. Slotine took this idea and developed a systematic method for designing sliding mode controllers. Looking at Figure 55, we can develop the following table: Table 52: Summary of the switching law. Quadrant y y u(t) 1 + + k2 2 + k 3 + kl 4 k2 This table can be summarized by the switching law. sgn(s) =1 s(y,y)>0 l1 s(y, y) < 0 where s(y,y) = y+ [y [58] [59] yy Figure 55. Switching between the two controllers results in driving the system to zero in a stable manor. is called your sliding surface and the control effort is given by: u(t) = K sgn(s) [510] where K is typically a scalar or diagonal matrix of positive definite gains. In order to design a controller, we will now replace our states (y, y) with our errors (x, x). This will now (as previously stated) drive our system errors to zero in a stable manor. Second, we will modify the control effort equation to include a feed forward aspect as follows. u(t) = uK sgn(s) [511] The term u is called the equivalent control effort. It is model based, thus if we have a perfect model of the system, we can calculate the required control effort to produce the desired results. However, that is never the case. There are always disturbances and unmodelled dynamics that influence our system. Thus the job of the second term in [511] is to reduce the error and error velocity produced by the equivalent control in a stable manor. Also note that the X is a weighting factor between the error and error velocity. This term can be adjusted depending on which is more important, and its effects are depicted s ,,x>i S Y s=O, k>1 y s = O, 0 = 45' \ x = 1 s=0, x<1 Figure 56. Graphical representation of how x effects the sliding surface's orientation in the phase plane. The dotted lines represent the sliding surface (s = 0). below in Figure 56. Typically this term is constant, but there is no reason it could not be dynamic. Simple Example For a simple example, we will develop a sliding mode controller for a massspring dampener system. The dynamic equations for this system is mx + cx + kx = u. [512] Our sliding surface and equivalent control are s = x + x, [513] s = x +x = 0 [514] m(u cxkx)xd + Xx = 0 [515] u = m(x )+cx + kx [516] Now looking at [516] we notice that if there is no error in the modelling of the system or disturbances, then this control effort is sufficient to control the system. Also if we had a model of any disturbances in the system, we could include them in our equivalent control term. However in this example, there are no disturbances included in the system or we do not have a model for any disturbances seen by the system during runtime. Unfortunately there will always be error in modelling, noise, or disturbances. Thus we must include another term to account for these problems. u = uK sgn(s) [517] Other Sliding Mode Designs for Spacecraft Many authors have suggested many types of control architectures for satellite attitude control [5359]. However, the one of the most interesting and simpler to implement types of control is sliding mode. The use of sliding mode control is not new to satellite attitude control. Lo and Chen [60] designed a sliding mode controller scheme which avoids the inverse of the inertia matrix, and smooths the control effort of the controller. Such a strategy provides for a more efficient use of fuel. Their simulations involved small uncertainties in the spacecraft inertia matrix and a sinusoidal disturbance. Simulations showed favorable results. The control effort was u = ox Jw +Jvd Jq ksgn(s) [518] where the vd a dd for all intensive purposes. Boskovic et al. [61] and [62] developed a sliding mode controller, but specifically designed the controller such that it did not saturate the control effort. They were able to produce accurate results, and even tested the controller on a system with a larger inertia matrix than what the controller was designed with. Unfortunately they drew an incorrect conclusion that the controller is independent of the inertia matrix it is presented with. Sliding mode will be able to control a system with a larger inertia matrix, but may not control a system with a smaller inertia matrix. This is because the control effort is model based, and thus it is possible to make a satellite go unstable by issuing too much control effort. The simulations utilized a square wave disturbance and small uncertainties in the inertia matrix. The controller developed was as follows: ki u = xJ XJq diag sgn(sl) sgn(s2) sgn(3)] [519] Crassidis et al. [63] developed an optimal variablestructure (sliding mode) controller for largeangle maneuvers on satellites. A cost function was developed, which when minimized, resulted in the optimal control effort. The the minimization lead to a two point boundary value problem. The cost function and the resulting optimal control effort were 1 ~T~ n(co) = ft (pq q+ o)dt [520] u = xJ( + J(X sgn(q4)[E(q)E(qd)(d (qd)E(q)W(]) +d ksgn(s) [521] Simulations of the controller on the MAP satellite showed favorable results. Coleman and Godbole [64] conducted a performance trade study between fuzzy logic, PID, and sliding mode control. The controllers were tuned for and tested on three similar linear plants. In all three cases, the sliding mode controller outperformed the fuzzy logic controller. This is a typical result, where sliding mode tends to provide a superior model based performance compared to fuzzy logic, assuming the model is accurate enough. However all of these authors utilized simple satellite dynamics with a simple external disturbance in their simulations. Fuel slosh, which is coupled with the dynamics of the satellite, is a more complex problem. It is difficult to accurately model the real disturbance and requires a control system that can account for this uncertainty. Satellite Controller Design For This Work This section will cover the sliding mode controller developed for this research. Controller Derivation This formulation, which is a fullstate feedback technique, utilizes the existing dynamic model and compensates for uncertainty while formulating a control effort that tracks the desired trajectories. The equations of motion for a satellite's attitude (in quaternion space) are given by [22] Jio = QJm +f+ u [522] 1 1 qq + q4m [523] where 0 03 (02 q 2 = 03 0 ( q = q [524] _(2 (0 0 q Also f in [522] represents the modelling error and potential disturbances in the system. This term will provide a feed forward aspect to the controller and should yield better results. The q is a vector composed of the three imaginary elements of a quaternion. However, in this formulation, we can take the state variable to be A or small euler angle orientation, and equate this to q. 6y = 2 qy [525] Thus the sliding surface (s) is given by: s = + x = x + x = o+ kq [526] where the error terms are defined as: X = X Xdesired [527] In order to calculate the equivalent control effort (u), we need to take the derivative of the sliding surface and set it equal to zero. s = co+ q [528] Now multiplying both sides by the inertia matrix (J) will allow us to substitute in the equa tions of motion ([522] and [523]). Js = JC+ Jq = Jo+ Jq Jdesired [529] Js = QJa+f+ + +Jq J desired [530] The hat over particular variables denotes that they are estimates, since accurate values for the inertia matrix and disturbances may not be known. Since the sliding surface does not move, its time derivative is zero, and thus the equation becomes Js = J+f+u + JqJOdesired = 0. [531] Now solving for the equivalent control effort (u) from [531] yields: u = QJ f Jq + Jmdesired. [532] The quaternion error rate in the above equation is defined as: 1~ ~ 1~ ~ q 2q + q4co [533] 2 2 0 (03 (02 03= 03 0 C [534] (2 (1 0 The estimated disturbance in the above equation is equal to the estimated solar snap and fuel slosh disturbance. f = fss +ffs [535] where f,, is the disturbance due to solar snap and ff, is the disturbance due to fuel slosh. Finally, the sliding mode control effort is given by: u = uKsgn(s) [536] The K term in [536] will be discussed when stability of the controller is covered. This switching term provides the appropriate control action to quickly drive the trajectories back onto the sliding surface. Since real actuators do not have an infinite bandwidth, the sign function tends to cause excessive chatter in the control effort. The sign function is typically replaced by a saturation function. The saturation function not only smooths the control response, it also reduces the amount of fuel used by the controller. Fuel Slosh Disturbance The effects of the fuel slosh are incorporated into the controller by the ffs term in the controller. A discrete state space system is used to track the effects of the fuel slosh: xfs = Axfs +Bu [537] ffs = Cxfs [538] where the state vector is: Xf= x [539] x where x is the position of the fuel slosh and x is its velocity. The input to the system u is the previous control effort used by the satellite. Also the state space matrices are discrete time representations to reduce the computational load of the controller. Stability of Controller The following Lyapunov candidate function is proposed. This function is clearly positive definite. V sTJs [540] 2 Taking its time derivative and substituting in previous equations of motion and control effort gives: V = sTJs [541] where the true dynamics of the system are inserted into the equations are Js = c x Jo + u +f+ hJq [542] The modelled dynamics, which define the controller output are given by: u = uKsgn(s) and u = cxJcwkJq [543] After some algebra, the dynamics cancel each other out, and the only terms left are the dif ference between the true disturbance forces and the estimated/modelled ones and the last terms on the control effort equation. Thus [541] becomes: V = sT(FKsgn(s)) [544] where F = f [545] Note that any differences between the true inertia matrix and the real inertia matrix, plus any other modelling error or disturbance, is contained in f Where f is the true distur bances, modelling errors, unmodelled higher order terms, etc. For a stable system, the lyapunov derivative needs to be negative definite. All variables are known except for the scalar gain value K. Two of the three remaining terms are positive definite, K and F. Thus we need to do further simplifications and algebra to solve for the gain. sTF sTKsgn(s) < 0 [546] sTFK(s) <0 [547] where (s) = sTsgn(s) [548] This operation is a cross between an absolute value operation (if it were a scalar) and a norm of a vector (if the sign function was not present). Thus this operation will be denoted by the angled brackets and not confused with either absolute value or norm of a vector. A quick example is shown below, and note that the result will always be scalar and positive. 1 3 4] sgn 3 3 4] = 8 [549] Thus the scalar gain K is K sT F K > [550] (S) In order to guarantee the above inequality, we will add a little offset (r1) that will always be positive. K +s [551] (s) Now substituting this gain back into [547] results in: sTF ( F + (S) This equation has two results depending on whether s is positive or negative. If s is nega tive then the two sTF's sum together and the entire equation is negative definite. If s is positive, then the two cancel each other out and the equation is still negative definite. 2sTFq(s)<0 or q(s)<0 [553] Saturation The sign function used in the sliding mode controller leads to an excessive amount of chattering in the control effort. A common modification to the sliding mode controller is to change the sign function to a saturation function with the following properties. sat(a) = < a < a [554] else sgn(a) Thus our sliding mode control equation changes to u = uKsat( [555] where the ( is a scalar value that can be used to adjust when the saturation function will saturate. All other sliding mode equations remain the same. Comparing The Two Types of Controller Slotine's sliding mode formulation and a PD controller have some very similar attributes when the sliding mode is in the sliding region. Simplifying the system some, assume that the equivalent control effort (u) is zero, since this is really a feed forward term. Thus the control effort is now: u = Ksat( [556] The sliding surface s is very similar to a PD control effort. s x = x + kx [557] where x is a rate error and x is a position error. When the errors are on the sliding surface (meaning when the errors are small, they are in the saturation bounds), the saturation func tion returns a scaled version of the values passed to it (as opposed to a 1 or 1 when out side of the saturation bounds). Thus the control effort becomes scaled errors and scaled rate error multiplied by a gain K. u=K [558] u = (x + x [559] Then the PD parameters can be written as follows: hK 1 k T, [560] Conclusion This chapter introduced PD and sliding mode theory for satellite attitude control. An in depth discussion of sliding mode design, stability, and a comparison between sliding mode 82 and PD was provided. This laid the foundation for the similarities and differences between the two controller which will soon manifest themselves when the results are presented. CHAPTER 6 SIMULATION AND RESULTS This chapter will discuss the simulation, the gains chosen for the PD controller, and finally discuss the results. The first section will describe the specifics of the satellite and the simulation. All of the variable from proceeding chapters will be filled in and explained. Next the simulation, equations of motion, and the controllers that were developed are covered. Finally the results of the simulation are discussed. Simulation Dynamics and Parameters The simulation used to evaluate the performance of the controllers was written in C++. The first simulations were written in Matlab because of the ease of working with vectors and matrices. The problem with Matlab, however, was the amount of time each simulation took. Once the simulation were rewritten for C++, they were magnitudes faster. The reader is referred to Appendix C, where some of the mathematical code developed is presented. Satellite The satellite shown in Figure 61 and Figure 62 is Earth Observing 1 (EO1), and will be the satellite used in all simulations. EO1 was primarily constructed by Swales Figure 61. EO1 satellite with solar panel deployed. Figure 62. Diagram of satellite's internal components. Aerospace and is part of the New Millennium Program which is dedicated to validating new technologies. The spacecraft was launched aboard a Boeing Delta II rocket from Vandenberg Air Force Base on November 21, 2000. Some of the new technology being tested is the Advanced Land Imager, hyperspectral imaging spectrometer called Hyperion, X band phased array, pulsed plasma thrusters, and formation flying with Landsat 7 spacecraft which is already in orbit. Table 61: Technical Specifications for the EO1 spacecraft Dry Mass 568 kg Volume of Bus 1.5m x 1.5m x 2m Inertia Tensor [443 179 429] kg m^2 ACS Zero Momentum, 3 axis stabilized Command and Data Han Mongoose V, Rad Hard at 12 Mhz, dling Bus Architecture RISC Architecture Solar Arrays 3 panel / Si w/GaAs / Articulating Bus Structure Hexagonal with aluminum honeycomb Propulsion 1 fuel tank with 4 thrusters Propellent Capacity 23 kg Mission Design Life 1.5 years 