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NOVEL STEERING AND CONTROL ALGORITHMS FOR SINGLEGIMBAL CONTROL MOMENT GYROSCOPES By FREDERICK A. LEVE 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 2010 2010 Frederick A. Leve Dedicated to my mother for always being there to support me ACKNOWLEDGMENTS I would like to thank first my advisor Dr. Norman FitzCoy for providing me with the guidance and knowledge for this great research I undertook. Second, I would like to thank my committee members Dr. Warren Dixon, Dr. Anil Rao, Dr. William Hager from UF, and Dr. Scott Erwin from the Air Force Research Lab Space Vehicles Directorate. My committee comprises the expertise in the areas of research that would provide me the best opportunity for my research. Last, but not least, I would like to thank my colleagues in my research lab who provided input throughout my time as a graduate student that aided in this research: Dr. Andy Tatsch, Shawn Allgeier, Vivek Nagabushnan, Josue Munoz, Takashi Hiramatsu, Andrew Waldrum, Sharan Asundi, Dante Buckley, Jimmy Tzu Yu Lin, Shawn Johnson, Katie Cason, and Dr. William Mackunis. TABLE OF CONTENTS page ACKNOWLEDGMENTS ................... ............... 4 LIST O FTABLES ..................... ................. 8 LIST OF FIGURES .................... ................. 9 ABSTRACT .................... ................... .. 13 CHAPTER 1 INTRODUCTION .................... ............... 15 1.1 History and Background ................... ........ 15 1.1.1 Gyroscopic Rate Determination .. .. 15 1.1.2 Spin Stabilized Spacecraft ..... .. .. 15 1.1.3 Spacecraft Attitude Control through Gyrostats .... 15 1.1.4 3axis Attitude Control of Spacecraft ... 16 1.1.5 SingleGimbal Control Moment Gyroscopes (SGCMGs) 17 1.1.6 DoubleGimbal Control Moment Gyroscopes (DGCMGs) 17 1.1.7 VariableSpeed Control Moment Gyroscopes (VSCMGs) 18 1.2 Problem Statement .................. ........... 18 2 DYNAMIC MODELS ... .............. ............ .. 20 2.1 Dynamic Formulation ... .............. ......... .. 20 2.2 Singular Surface Equations ........................ 25 2.2.1 Elliptic Singularities .. 26 2.2.1.1 External singularities . ... 27 2.2.1.2 Elliptic internal singularities 29 2.2.2 Hyperbolic Singularities ... 29 2.2.2.1 Nondegenerate hyperbolic singularities ... 30 2.2.2.2 Degenerate hyperbolic singularities ... 30 2.2.3 GimbalLock.. ........................... 30 2.3 Singularities for SGCMGs Mathematically Defined .... 31 3 CONTROL MOMENT GYROSCOPE ARRANGEMENTS .... 34 3.1 Common SGCMG Arrangements ... 34 3.1.1 R ooftop . .. 34 3.1.2 Box ... . 35 3.1.3 3 B ox . ... 44 3.1.4 Scissor Pair . 45 3.1.5 Pyram id . .. 46 3.2 Choice of Arrangement ............................ 48 3.3 S im ulation . . 50 4 SURVEY OF STEERING ALGORITHMS ................ ... 54 4.1 MoorePenrose PseudoInverse ................ ....... 55 4.2 Singularity Avoidance Algorithms ... ..... 55 4.2.1 Constrained Steering Algorithms ..... 56 4.2.2 Null Motion Algorithms ......................... 56 4.2.2.1 Local gradient (LG) ................. .. 56 4.2.2.2 Global avoidance/Preferred trajectory tracking 57 4.2.2.3 Generalized Inverse Steering Law (GISL) ... 58 4.3 Singularity Escape Algorithms ... 59 4.3.0.4 Singularity Robust (SR) inverse ... 59 4.3.0.5 Generalized Singularity Robust (GSR) inverse 60 4.3.0.6 Singular Direction Avoidance (SDA) ... 60 4.3.0.7 Feedback Steering Law (FSL) .. 62 4.3.0.8 Singularity Penetration with UnitDelay (SPUD) 63 4.4 Singularity Avoidance and Escape Algorithms .. 64 4.4.0.9 Preferred gimbal angles . ... 64 4.4.0.10 Optimal steering law (OSL) ... 64 4.5 Other Steering Algorithms ................ ......... 66 4.6 Steering Algorithm Computation Comparison .. 66 5 STEERING ALGORITHMHYBRID STEERING LOGIC .. 68 5.1 Hybrid Steering Logic ................. .......... 68 5.1.1 Internal Singularity Metrics ... 68 5.1.2 Hybrid Steering Logic Formulation .. 69 5.2 Lyapunov Stability Analysis .......................... 71 5.3 Numerical Simulation ................. ...... .. .. 76 5.3.1 Case 1: At Zero Momentum Configuration 6 = [0 0 0 0]' deg 79 5.3.1.1 Local gradient simulation results ... 80 5.3.1.2 Singular Direction Avoidance simulation results 82 5.3.1.3 Hybrid Steering Logic simulation results ... 85 5.3.2 Case 2: Near Elliptic External Singularity 6 = [105 105 105 105]T d e g . . 8 5 5.3.2.1 Local gradient simulation results ... 88 5.3.2.2 Singular Direction Avoidance simulation results 91 5.3.2.3 Hybrid Steering Logic simulation results ... 94 5.3.3 Case 3: Near Hyperbolic Internal Singularities J = [15 105 195 75]T deg . .. 96 5.3.3.1 Local gradient simulation results ... 96 5.3.3.2 Singular Direction Avoidance simulation results 99 5.3.3.3 Hybrid Steering Logic simulation results ... 102 5.4 Hybrid Steering Logic Summary ... ... ... 104 6 CONTROL ALGORITHMORTHOGONAL TORQUE COMPENSATION .... 106 6.1 Attitude Controller with OTC ................ .......... 106 6.2 Lyapunov Stability Analysis ................ .......... 107 6.3 Num erical Sim ulation .. .. .. .. .. .. .. 110 6.3.1 Case : 60 = [0 0 0 0]T deg ....................... 111 6.3.2 Case II a: 6o = [90 90 90 90]T deg ... 116 6.3.3 Case II b (HSL/OTC): = [90 90 90 90]T deg 123 6.4 Orthogonal Torque Compensation Summary ... 127 7 SCALABILITY ISSUES FOR SGCMGS ........... ........... 129 7.1 Scalability Problems with SGCMG Hardware ..... 129 7.2 Effect of Igw on Torque Error ........... ... .. .......... 130 7.3 Num erical Sim ulation ............................. 132 7.3.1 Case l: Kg, = 0 . 134 7.3.2 Case Il: Kg = 2 ................. ........ 138 7.4 Effect of g,, on Torque Torque Amplification .... 142 7.5 Sum m ary . . 143 8 CO NCLUSIO N . . 144 APPENDIX A RIGID BODY DYNAMICS FORMULATION FOR CONTROL MOMENT GYROSCOPE ACTUATORS (SGCMG/VSCMG) ..........................147 A .1 Assum options . . 147 A .2 Dynam ics . . 147 B MOMENTUM ENVELOPE CODE ......................... 151 C CONTROL MOMENT GYROSCOPE ACTUATOR SPECIFICATIONS 156 REFERENC ES . . .. 157 BIOGRAPHICAL SKETCH ................................ 164 LIST OF TABLES Table page 31 M odel Param eters . .. 50 41 Algorithm Flops m = row(A) and n = column(A). 66 51 M odel Param eters . .. 78 52 Performance Comparisons for Case I: Zero Momentum ... 85 53 Performance Comparisons for Case II: Elliptic Singularity ... 96 54 Performance Comparisons for Case III: Hyperbolic Singularity 104 61 M odel Param eters . . 110 62 Hybrid Steering Logic Parameters ..... .. ...... 123 63 Performance Comparisons ............................. 128 71 M odel Param eters .. .. .. .. .. .. .. .. 133 72 M odel Param eters .. .. .. .. .. .. .. .. 142 C1 OfftheShelf CMG Specifications ..... .. .... .. 156 LIST OF FIGURES Figure page 21 Rigid body with a constant c.m ............................ 21 22 Gimbal frame Tg) of IMPAC SGCMG (Patent Pending) ... 22 23 Singularity shown when CMG torque vectors lie in a plane (IMPAC SGCMGs Patent Pending) ............. ...... ................ 25 24 Singularities for SGCMGs .............................. 27 25 External singular surfaces for a fourSGCMG pyramid ... 28 26 Internal singular surfaces for a fourSGCMG pyramid ..... 30 31 FourSGCMG rooftop arrangement ......... ....... ........ 34 32 FourSGCMG box arrangement ......... ....... .......... 35 33 Angular momentum envelope for a fourSGCMG box arrangement. ...... ..36 34 Planes of torque for a fourCMG rooftop arrangement ... 37 35 Torque planes traced out for a fourSGCMG rooftop arrangement ... 39 36 Angular momentum envelope with plotted angular momentum combinations for the fourSGCMG box arrangement . .. 42 37 Degenerate hyperbolic singularities for the fourSGCMG box arrangement 43 38 Singular surfaces showing 1 ho singularity free region ..... 45 39 3 Orthogonal scissor pairs of SGCMGs . 46 310 Planes of angular momentum and torque for a fourSGCMG pyramid 47 311 FourSGCMG pyramid arrangement ......... ........ ........ 47 312 Optimization process block diagram ..... ..... 49 313 Singular surfaces for the optimized arrangement at the Euler angles 0* = [170.2 13.6 85.5 168.0]T deg and = [17.7 167.0 304.3 92.5]T deg 51 314 Gimbal rates for the optimized and pyramid arrangements ... 51 315 Torque error for the optimized and pyramid arrangements ... 52 316 Singularity measure for the optimized and pyramid arrangements ... 52 317 Optimization cost for the optimized and pyramid arrangements ... 53 41 Outer and inner loops of GNC system 42 Steering algorithms ........ 51 Zeromomentum configuration of a 52 Simulation results for LG with aco = momentum ............. 53 Simulation results for LG with aco = momentum (contd.) ........ 54 Simulation results for SDA with co zero momentum .......... 55 Simulation results for SDA with co zero momentum (contd.) ...... 56 Simulation results for HSL with aco 1 at zero momentum ........ 57 Simulation results for HSL with aco 1 at zero momentum (contd.) . 58 Simulation results for LG with aco = elliptic singularities ......... 59 Simulation results for LG with aco = elliptic singularities (contd.) . 510 Simulation results for SDA with aco near elliptic singularities ...... 511 Simulation results for SDA with aco near elliptic singularities (contd.) 512 Simulation results for HSL with aco 1 near elliptic singularities ..... 513 Simulation results for HSL with aco 1 near elliptic singularities (contd.) 514 Simulation results for LG with aco = hyperbolic singularities ....... 515 Simulation results for LG with aco = hyperbolic singularities (contd.) . 516 Simulation results for SDA with ao near hyperbolic singularities . fourSGCMG pyramid arrangement .. a = b = 1 a = b = = 0.01,3o = = 0.01,3o = =0.01,3o = 2 =0.01,3o = 2 a = b = a = b = = 0.01,/3o = = 0.01,/3o = = 0.01, o =2 = 0.01,3o = 2 a = b = a = b = = 0.01,/3o = = 0and 2 = = 0and 2 = S=b = 12 = S=b = 12 = ,a=l,b= , a= 1, b = = 0 and 12 = 0 and 12 a =b =2 = a= b= /2 _,a =l,b= , a =, b= = 0 and 12 = 0 and 12 0, a = 0, b 3o 1 at zero 3o = 1 at zero 0, and /1 = 1 at 0, and /1 = 1 at 3, and 11 = 12 = 3, and 11 = 12 = = o = 1 near = o = 1 near 0 and p = 1 S0, and 1 = 1 3, and p1 = 12 = \, and 11 = 12 = = o = 1 near = o = 1 near 0, and p = 1 56 79 80 81 83 84 86 87 89 90 92 93 94 95 97 98 100 _) _) 517 Simulation results for SDA with co = 0.01, /o = 0, a = 0, b = 0, and = = 1 near hyperbolic singularities (contd.) ... 101 518 Simulation results for HSL with ao = 0.01, /o = 2, a = 1, b = 3, and p1 = 12 = 1 near hyperbolic singularities ..... ... 103 519 Simulation results for HSL with co = 0.01, /o = 2, a = 1, b = 3, and /i = 12 = 1 near hyperbolic singularities (contd.) ... 104 61 Satellite attitude control system block diagram ... 106 62 G im bal rates . ...... 112 63 O utput torque . ...... 113 64 Vector elements of the error quaternion ... 114 65 Singularity m measure . . 115 66 Singularity parameter (OTC) ............................ 116 67 Quaternion error difference: (A) eGSR eSDA (B) eSDA/OTC eSDA .. 116 68 G im bal rates . . ..... 118 69 O utput torque . ...... 119 610 Vector elements of the error quaternion ... 120 611 Singularity m measure . . 121 612 Gim ballock m measure ................................ 122 613 Singularity parameter (OTC) ............................ 123 614 G im bal rates . ...... 124 615 O utput torque .. .. .. .. .. .. .. .. ...... 125 616 Vector elements of the error quaternion ... 126 617 Singularity m measure . . 126 618 Gim ballock m measure ................................ 127 619 Singularity parameter (OTC) ............................ 127 71 Offtheshelf CM G s . . 129 72 Gim bal rates for Kg, = 0 ...............................135 73 Gimbal accelerations for Kg = 0 .... .. .... .. 136 11 Torque error for Kg, =0 .. . Singularity measure for Kg, Gimbal rates for Kg, =2 ... Gimbal accelerations for Kg, Torque error for Kg = 2 .. . Singularity measure for Kgw . . 1 3 7 . . 1 3 8 . . 1 3 9 . . 1 4 0 . . 1 4 1 . 14 1 74 75 76 77 78 79 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NOVEL STEERING AND CONTROL ALGORITHMS FOR SINGLEGIMBAL CONTROL MOMENT GYROSCOPES By Frederick A. Leve August 2010 Chair: Norman FitzCoy Major: Aerospace Engineering The research presented in this manuscript attempts to first systematically solve the SGCMG steering and control problem. To accomplish this, a better understanding of singularities associated with SGCMGs is required. Next based on a better understanding of singularities, a Hybrid Steering Logic (HSL) is developed and compared to the legacy methods singular direction avoidance (SDA) and local gradient (LG) methods. The HSL is shown analytically and numerically to outperform these legacy methods for a fourSGCMG pyramid arrangement in terms of attitude tracking precision. However, all of the methods are susceptible to gimballock. A method referred to as Orthogonal Torque Compensation (OTC) is developed for singularities with gimballock in SGCMGs, which are known to present a challenge to most steering algorithms. Orthogonal Torque Compensation conditions the attitude control torque by adding torque error orthogonal to the singular direction when at singularity. This method can be combined with any steering algorithm including HSL and is proven analytically to be stable and escape singularities with gimballock. Finally, the problems with scaling systems of SGCMGs are discussed. It is found that scaling produces an increase in the gimbalwheel assembly inertia of the SGCMGs which in turn increases the effect of the dynamics associated with these inertias. Through analysis and numerical simulations, it is shown that more significant gimbalflywheel inertia reduces the performance by increasing torque error and reducing torque amplification of SGCMGs. Since most of the legacy algorithms used for singularity avoidance and escape use the gimbal rates for control, the performance is degraded when the dynamics from the gimbalwheel assembly inertia are increased. This degraded performance is shown to also be dependent on the ratio of gimbalwheel assembly inertia to nominal SGCMG flywheel angular momentum. The overall result here is that the same legacy steering algorithms that use gimbal rates for control cannot be used for systems of SGCMGs of a reduced scale. CHAPTER 1 INTRODUCTION 1.1 History and Background 1.1.1 Gyroscopic Rate Determination The behavior of gyroscopic systems comes from the principle of conservation of angular momentum. Both rate determination and rate generation are possible through this gyroscopic behavior. For example, determination of attitude rates can be found by mechanical gyroscopes. From these devices, spacecraft angular rates are inferred from their reaction onto the gimbals of the mechanical gyroscope. The first known gyroscopes were passive and the first of these was developed by Johann Bohnenberger in 1817 [1, 2]. They were later developed for the navy as the naval gyrocompass [3]. There are now passive gyroscopes used for determining angular rates developed via microelectrical and mechanical systems (MEMS) that are smaller than the human eye can detect. 1.1.2 Spin Stabilized Spacecraft Early spacecraft did not have active attitude control but where spin stabilized about their major axis of inertia by initiating a spin after launch. The majoraxis rule is required for directional stability (i.e., the spacecraft must spin about its major axis) [4]. This provided gyroscopic stability or directional stability to the spacecraft which was defined as stability provided the spacecraft does not have energy dissipation capability (i.e., a rigid body). However, a spacecraft is truly not a rigid body (e.g., flexible booms, solar arrays, internal movable parts, and outgassing) which may dissipate energy and become asymptotically stable. 1.1.3 Spacecraft Attitude Control through Gyrostats A gyrostat is any rigid body that has attached to it a wheel that through the conservation of angular momentum provides either rotational stability or control. To provide attitude stability, constant speed flywheels known as momentum wheels (MWs) were added internal to the spacecraft to provide gyroscopic stiffness which in turn supplies attitude stability [5]. This system is an example of a gyrostat. The gyrostat was the first fundamental dynamical system that considered a spinning flywheel within or attached to a rigid body. In this type of system the flywheel imparts angular momentum stiffness to the body internally through the principle of the conservation of angular momentum. The use of momentum wheels for gyroscopic stability has found its use for other vehicles than spacecraft (e.g., boats, trains, buses [6]). 1.1.4 3axis Attitude Control of Spacecraft When active attitude control was needed, momentum wheels were exchanged for reaction wheels (RWs) which provide a reaction torque on the spacecraft through flywheel accelerations. These devices were then used in combination with thrusters to provide attitude control to spacecraft [7]. Further innovation came to spacecraft attitude control through the introduction of gimbaled MWs known as control moment gyroscopes (CMGs). They have been used in satellite attitude control for decades due to their high precision and property of torque amplification (i.e., larger torque output onto the spacecraft than the input torque from the gimbal motors). Typically, they have been used for large satellites that require high agility while maintaining pointing precision and have even found their uses onboard the international space station (ISS) [8]. Control moment gyroscopes produce a gyroscopic torque through rotation of angular momentum about one or two gimbal axes. Momentum wheels, RWs and CMGs all are known as momentum exchange devices because their torque is produced through redistribution of angular momentum from the CMGs to the spacecraft. Control moment gyroscopes come in two classes: 1) those with a single controllable degree of freedom and 2) those with multiple controllable degrees of freedom. The choice of a CMG or attitude actuator in general depends on the needs of the mission. All CMGs have specific challenges associated with their use. The challenges of CMGs are as follows: single DOF CMGs known as singlegimbal control moment gyroscopes (SGCMGs) suffer from instantaneous internal singularities (i.e., situations within the performance envelope where a given torque cannot be produced); multiple DOF CMGs known doublegimbal control moment gyroscopes (DGCMGs) are mechanically complex and have singularities known as gimballock when their gimbals align thus eliminating their extra DOF; other multiple DOF CMGs known as variablespeed control moment gyroscopes (VSCMGs) are difficult to mitigate induced vibration (i.e., due to the variation of flywheel speeds) and require more complicated control laws, motor driver circuitry, and larger flywheel motors. 1.1.5 SingleGimbal Control Moment Gyroscopes (SGCMGs) Singlegimbal control moment gyroscopes have a long heritage of flight on larger satellites and the ISS [915]. They are known to have the highest torque amplification of all CMGs, are less mechanically complex than DGCMGs and have less mathematically complex dynamics than VSCMGs. These actuators suffer from internal singularities that must be handled onthefly, where torque cannot be generated in a specific direction. There is no single method that has been proven to avoid all internal singularities while tracking an arbitrary torque perfectly (i.e., without the use of torque error or constraining the torque). Thus, there is merit in finding alternate solutions to control of SGCMGs for attitude control. 1.1.6 DoubleGimbal Control Moment Gyroscopes (DGCMGs) Doublegimbal control moment gyroscopes contain two controllable DOFs through their two gimbal axes. They are the most mechanically complex of CMGs although, the redundancy in the additional gimbal may lead to less than three DGCMGs required for 3axis control. The benefit of this redundancy is lost for DGCMGs when they encounter a gimballock singularity. Gimballock is encountered when the gimbal axes are aligned and are no longer linearly independent. As a consequence, the extra controllable DOF in this case is lost. Effective methods already exist for avoiding gimballock singularities associated with DGCMGs [1620]. 1.1.7 VariableSpeed Control Moment Gyroscopes (VSCMGs) Variablespeed control moment gyroscopes utilize an extra controllable DOF through flywheel accelerations. As a consequence of the extra DOF, the flywheel motors must be larger and it is more troublesome to isolate unwanted induced vibration. In addition, this extra degree of freedom makes a system of two or more noncollinear VSCMGs free from singularities through the extra degree of freedom, (i.e., at CMG internal singularities, the needed torque is provided by flywheel accelerations). Several algorithms have been developed that are effective in reducing the amount of flywheel accelerations used and thus providing better torque amplification [2130]. In addition, methods have been developed to use the VSCMG's extra controllable DOF to spin down the flywheels and store their kinetic energy. These methods are known as the integrated power and attitude control system (IPACS) and the flywheel attitude control and energy transmission system (FACETS) in literature [3134]. 1.2 Problem Statement A new paradigm that requires highly agile small spacecraft is ongoing through efforts by government agencies and labs such as Operationally Responsive Space (ORS), Air Force Research Laboratory (AFRL), the National Reconnaissance Organization (NRO), and National Aeronautics and Space Administration (NASA) to perform such missions as intelligence, surveillance and reconnaissance (ISR), space situational awareness (SSA), and space science missions (e.g., the imaging of gamma ray bursts) [3537]. Many of these missions are in LEO and require higher agility and attitude precision to track targets on earth. Attitude control systems (ACSs) based on reaction control devices (e.g., thrusters) can achieve great agility but cannot meet the pointing requirements and space needed for propellant storage on small satellites [38]. Singlegimbal control moment gyroscopes are being considered the actuator of choice to provide higher agility to smaller satellites based on their perceived torque amplification. However, many problems exist that must be solved prior to using SGCMGs for small satellite attitude control. Traditionally, SGCMGs have been oversized for their mission and the angular momentum envelope constrained to avoid internal singularities on larger satellites. For smaller satellite systems, however, the extra volume and mass needed for oversized SGCMGs may be unacceptable. Therefore, small satellite SGCMGs should utilize more of the entire angular momentum envelope where singularities may be encountered in the momentum space (both internal and external). Thus, legacy steering algorithms from larger satellite applications may not provide the same performance for systems of a smaller scale requiring a new approach for steering and control of SGCMGs. For the succeeding chapters: Chapter 2 discusses and reviews the fundamentals of CMG dynamics and describes the different forms of singularities associated with SGCMGs; Chapter 3 describes the possible arrangements for systems of SGCMGs and their desirable and undesirable qualities; Chapter 4 provides the background on previously published methods of steering algorithms for implementation of SGCMGs; Chapter 5 discusses the development of the Hybrid Steering Logic (HSL) for SGCMGs; Chapter 6 discusses the development of OrthogonalTorqueCompensation (OTC) for gimballock escape of SGCMGs; Chapter 7 discusses the performance degradation encountered when scaling systems of SGCMGs; and Chapter 8 provides conclusions of the research. CHAPTER 2 DYNAMIC MODELS 2.1 Dynamic Formulation The dynamic formulation presented in this section addresses momentum exchange devices for attitude control systems where the angular momentum of the spacecraft system (i.e., the spacecraft and system of CMGs) is assumed constant. This is in contrast to reaction control devices (e.g., thrusters) and/or energy dissipation devices (e.g., magnet torquers) which change the angular momentum or energy of the system. It is assumed that the center of mass (c.m.) of each CMG lies along the gimbal axis. This is equivalent to stating that the rotation of the gimbalflywheel system about the gimbal axis does not move the position of the c.m. of the system. It is also assumed that the spacecraftCMG system is a rigid body; and this system is absent of friction and external torques. For a coordinateless derivation of the dynamics see Appendix A. The first assumption can be visualized by treating the CMGs as cylinders and having their spin axis along their c.m. as shown in Figure 21. The total centroidal angular momentum of the spacecraftCMG system in the spacecraft body frame is H = Jw + h (21) which is composed of the spacecraft centroidal angular momentum Jc, composed of the spacecraft centroidal inertia J, and angular velocity w and the angular momentum h contributed from the CMGs. Considering the spacecraft modeled as a rigid body, its centroidal inertia is composed of both constant and timevarying inertias and is expressed as n J, = J + mi(rril rir,) + CBG BG (22) i=1 f Figure 21. Rigid body with a constant c.m. where J represents the constant spacecraftCMG system inertia in referencing position about the spacecraft's c.m; mi(rTril rrT) are the parallel axis terms, with mass mi of the ith gimbalflywheel assembly and its c.m., position ri with respect to the c.m. of the spacecraft, and CBG IgwCG, are the time varying inertias from rotation of the gimbalflywheel system inertia Igw. The angular momentum contributed from the ith CMG in a gimbal frame .g) is expressed as IwQ hi= 0 (23) lgw6i which consists of angular momentum from the flywheel (I/Q) and that from the gimbalwheel system (IgwB). It should be noted that the CMG angular momentum expression in Eq.(23) is based on an SGCMG or VSCMG only and the following development of angular momentum for CMGs will be for a multiple gimbal CMG. The resultant angular momentum from the CMG system in the body frame is found through the summation of the contributions of angular momentum from all CMGs rotated from their respective gimbal frames into the spacecraft body frame; i.e., n h = CBGhi (24) where n is the number of SGCMGs, CBG, is the direction cosine matrix (DCM) from the gimbal frame Fg) shown in Figure 22 to spacecraft body frame F3. 6i Figure 22. Gimbal frame T) of IMPAC SGCMG (Patent Pending) With the assumption of no external torques and frictionless devices, the total angular momentum is constant, thus the inertial time derivative of Eq.(24) shows the redistribution of the systems's angular momentum (i.e., on the mechanism by which the torque is produced by CMGs). Differentiation of Eq.(21) yields dH d(Jcj + h) dt dt X (Jxc + h) = 0 (25) The time derivative of the spacecraft angular momentum yields C = Jc + (26) dt The spacecraft inertia is assumed to only vary by the gimbal angles of the CMGs thus making the second term on the right hand side of Eq.(26) S a(Jw) d6 k JW 96ASj (27) j=1 j=1 where A cE R3Xn is the Jacobian matrix resulting from the coupling of the spacecraft and CMG kinematics from the jth gimbal of a multiple gimbal CMG and n is the number of CMGs. The angular momentum of the CMG system is a function of the flywheel angular velocities Q e Rx I gimbal angles 6j e Rnxl, and gimbal rates 6j e Rnxl, respectively. The time derivative of the CMGs angular momentum can be expressed as dh Oh d6 9h d6 j h dl = h = 6 +) + (28) dt Y. dt 6 dt 96 dt 2dt where the individual Jacobian matrices are, A eI Rs n (29) d6j Oh BJ = E R3X" (210) h6 C= E R 3xn (211) Assuming a constant flywheel speed f = 0 and a single gimbal (k= 1) configuration, then Eq.(25) can be rewritten as dH S= Jc + wxJc + wh +DX = 0 (212) dt where, DX= [(A + A2) B] =h +J = T (213) The general equations for the SGCMG output torque in terms of a given internal control torque 7 is expressed as, DX =  wX>h = T (214) where T is the total torque output from the system of SGCMGs. It should be noted that 6 and 6 are kinematically coupled, thus it is not possible to find both states simultaneously when mapping D e R3s"s3 onto T (i.e., only one gimbal state can be chosen as a control variable). For SGCMG systems that contain significant flywheel angular momentum and gyroscopic torque, the dynamics of the gimbalwheel assembly inertias can be considered insignificant (i.e., A1 z 0 and B z 0). For such systems, it is customary to neglect the inertia variations due to the gimbal motion (i.e., Jc = 0) resulting in the composite D reducing to A2 e Ra3n. Therefore, the Jacobian D is simply A2 and the solution of the output torque from the SGCMGs is contributed solely from the gimbal rates as h = hoA26 = ho[il, 2, ...rn]6 (215) where A2 = hoA and , is the torque vector direction of the ith CMG as shown in Figure 22. The coefficient matrix in Eq.(215) is 3 x n and when n > 3 the system is overactuated. When this matrix becomes rank deficient, the system is said to be singular. Physically, when these singularities occur, the torque vector directions of each SGCMG in the body frame lie in a plane as shown in Figure 23. For convenience, from this point until Chapter 7, it is assumed A2 = A. Figure 23. Singularity shown when CMG torque vectors lie in a plane (IMPAC SGCMGs Patent Pending) 2.2 Singular Surface Equations It is customary to define an orthonormal basis {hi, ,, } as shown in Figure 22 where ii is the spin axis of the flywheel, f, is the SGCMG torque direction, and 6, is the gimbal axis direction. Therefore, the singular direction s c R3X1 is defined from {s e R3 : sTi = 0} (216) This constraint constitutes a maximum (or minimum) projection of fi onto s. There is a fundamental assumption that ho, is equal to ho (i.e., the magnitude of nominal angular momentum is the same for all SGCMGs in the system). For a given singular direction s # 6, (i.e., which only occurs for DGCMGs and for rooftop arrangements), the conditions for singularity are sTi = 0 and sThi 0 (217) If we define A sTfi, then the torque and spin axis directions can be expressed as I x s T = s f 6i, i = 1,...,n (218) 113, x s h X (8, X s )8i, =1, ..., n (219) II6, x sl Combining Eqs.(218) and (219), the total normalized angular momentum from the SGCMGs is expressed as n n x S) X S S s x ,, = (220) 1i=6 i=x s It is important to note that when s = 6i Eqs.(218)(220) are indeterminate. The locus of total normalized angular momentum h from Eq.(220) for all s e RR3 and all e, 0 (i.e., s not collinear to 6,) produces the external singular surface known as the angular momentum envelope shown in Figure 25 for a fourSGCMG pyramid arrangement. Similarly, each of the four internal singular surfaces shown in Figure 26 for a fourSGCMG pyramid arrangement are found by setting one of the ce < 0. Matlab code for both of these surfaces can be seen in Appendix C. Singularities for SGCMGs can be classified into the groups/subgroups as shown in Figure 24. 2.2.1 Elliptic Singularities Elliptic singularities are those in which null solutions to the gimbal angles do not exist for a specific point of CMG angular momentum space. Null motion is a continuous set of null solutions for gimbal angles (i.e., there is a continuous transfer from one null solution to the next) that does not change the CMG's angular momentum and thus, does not produce any motion to the spacecraft. Since elliptic singularities do not have null solutions, the angular momentum must be perturbed thus inducing torque error to the system to escape from these singularities. Elliptic singularities are not limited to internal singularities; (e.g., all external singularities are elliptic and hence cannot be avoided or escaped through null motion). I External Internal Internal GimbalLock GimbalLock I )>n>Iil. Nondegenerate GimbalLock GimbalLock Figure 24. Singularities for SGCMGs 2.2.1.1 External singularities External singularities also known as saturation singularities are associated with the maximum projection of CMG angular momentum in any direction. These singularities cannot be avoided by null motion and therefore by definition are elliptic. These singularities occur on the surface of the angular momentum envelope and an example of this surface for a fourSGCMG pyramid is showed in Figure 25. When these singularities are encountered, the CMGs are unable to produce any more angular momentum in the saturated direction. External singularities are addressed a priori in the design process through sizing of the CMG actuators. 3 2 1.,  ,' ' ,,. .. ""  5 20 4 Y(ho) X(h) Figure 25. External singular surfaces for a fourSGCMG pyramid Consider a fourSGCMG pyramid arrangement, Eq.(221) can be used to express the angular momentum as c(0)s(1) c(2) + c(O)s(6) + c(4) h = ho c(1) c(0)s(62) c(63)+ c()s(64) (221) s(O)(s() + s(2) +s(63) +s(6)) where 0 is the skew angle and 6, is the ith gimbal angle. Further consider the set of gimbal angles 5es = [90 90 90 90] deg, then the momentum vector becomes 0 0 h(6es)= 0 0 (222) hos(O)(s(61) + s(2) + s() + s(64)) 4hos() It is clear that there is only one set of gimbal angles 6es = [90 90 90 90] deg that will give the angular momentum in Eq.(222). Therefore, null solutions do not exist, and this angular momentum vector corresponds to the elliptic saturation singularity along the zaxis. 2.2.1.2 Elliptic internal singularities Elliptic singularities which lie on the internal singular surfaces such as that shown for the fourSGCMG pyramid arrangement in Figure 26 are referred to as elliptic internal singularities. Unlike external singularities, these singularities cannot be accounted for in the design process; furthermore, since they occur instantaneously, they cannot be generally avoided. 2.2.2 Hyperbolic Singularities Hyberbolic singularities are those in which null motion is possible. Thus, all hyperbolic singularities are therefore internal (i.e., these singularities occur on the internal singular surfaces). The points on the internal singular surface corresponding to a hyperbolic singularity have null solutions of gimbal angles, corresponding to the null space of the Jacobian matrix. The null solutions are typically chosen to avoid the singular configurations of the system. Shown in Figure 26, is an example of this surface for a fourSGCMG pyramid. Singularities occur only when the point on this surface corresponds to a singular Jacobian matrix (i.e., there may be nonsingular sets of gimbal angles at this point on the surface). When these singularities are encountered, the SGCMG torque vector directions lie in a plane and as a consequence there is no torque available out of the plane. These singularities, like elliptic singularities, are instantaneous and must be handled on the fly. For a fourSGCMG pyramid arrangement with angular momentum in Eq.(221), a set of gimbal angles 6hs = [180 90 0 90]T deg is an hyperbolic singularity that has the following momentum vector, 0 h(hs) = 2ho (223) 2hos(0) It is clear that there multiple solutions (i.e., null solutions) to Eq.(223). A null solution of the gimbal angles satisfying 61 = 63 = 90 o and s(64) = (62) = substituted into Eq.(221) will also satisfy Eq.(223). 4. 3 2 0 . , 0  N 1  " 2  3  4 5 0 Y(ho) 1 5 2 X(ho) Figure 26. Internal singular surfaces for a fourSGCMG pyramid 2.2.2.1 Nondegenerate hyperbolic singularities NonDegenerate hyperbolic internal singularities are those in which null motion is possible and some of the null solutions are nonsingular providing the possibility of singularity avoidance. 2.2.2.2 Degenerate hyperbolic singularities Degenerate hyperbolic internal singularities occur when the null solutions to gimbal angles correspond to singular sets of gimbal angles leaving no room for avoidance or escape. These singularities are also considered impassable and therefore are handled in a similar manner to elliptic singularities when approached. 2.2.3 GimbalLock Gimballock for SGCMGs occurs at singularity when the mapped output torque vector is in the singular direction. When this occurs, the system becomes trapped in this ; ~ar singular configuration with only a few methods that are capable of escape from it. One such method is known as the Generalized Singularity Robust (GSR) Inverse [39, 40]. This method has been shown numerically to escape gimballock of SGCMGs but not analytically and there is no formal proof to suggest that it is always successful. 2.3 Singularities for SGCMGs Mathematically Defined To quantifiy the effectiveness of avoiding internal singularities through null motion, we must define their forms (i.e., hyperbolic and elliptic) mathematically. Typically, topology and differential geometry are used to represent hyperbolic and elliptic internal singularities as surfaces or manifolds [41, 42]. The behavior of these internal singularities can also be explained through the use of linear algebra [43]. To accomplish this, a Taylor series expansion of the SGCMG angular momentum about a singular configuration gives h(6) h(65) = [ A 6 ,2 A+ H. O.T.1 (224) i= 1 where h(65) is the angular momentum at a singular set of gimbal angles 6s, A6j 6, 6s, and n is the number of SGCMGs in the system. The first term on the righthand side (RHS) of Eq.(224) contains the ith column of the Jacobian matrix f, = associated with the ith SGCMG's torque direction. The second term on the RHS contains the partial derivative of the Jacobian matrix's ith column with respect to the ith gimbal angle ,i s. Furthermore, from Eq.(215), the RHS of Eq.(224) can be transformed through the realization of the following operations: a2hi ?91 h;=i = hi (225) where () denotes a unit vector. Next, Eq.(225) is substituted into Eq.(224) and the inner product of the result with the singular direction s obtained from null(AT) yields sT[h() h(6s)] 1= hs Ab (226) i=1 The first term on the RHS of Eq.(224) has zero contribution because of the definition of the singular direction (i.e, ATs = 0). Equation (226) can be written more compactly as sT[h(6) h(6s)] = ATPA6 (227) 2 where P is the singularity projection matrix defined as P = diag(h's). By definition, null motion does not affect the total system angular momentum and which equates to h(6) = h(6s). Consequently, the lefthand side (LHS) of Eq.(227) is zero (i.e., A6TPA6 = 0). Null motion is expressed in terms of the basis N = null(A) concatenated in matrix form as follows (nr(A)) A6= Aivi = NA (228) i=1 where A is a column matrix of the scaling components of the null space basis vectors vi and N e Rnx(nr(A)) is the dimension of the null space basis for any system of SGCMGs with r(A) = rank(A). Substituting Eq.(228) into Eq.(227) observing the null motion constraint yields ATQA = 0 (229) As a result of this analysis, a matrix Q is defined as Q = NTPN (230) Therefore, when away from singularity Q e RIi; when at a rank 2 singularity, Q e R2x2; and when at a rank 1 singularity, Q e RS3x. The eigenvalues of the Q matrix determine whether a singularity is hyperbolic or elliptic. If Q is definite (i.e., has all positive or negative eigenvalues), it does not contain a null space since a nonzero null vector A does not exist that satisfies Eq.(229) [44]. Therefore, situations where the matrix Q is definite constitute elliptic singularities. When Q is semidefinite (i.e., it has at least one zero eigenvalue), then a null space exists since there exists a A 7 0 that satisfies Eq.(229). Therefore, null motion is possible near singularity and the possibility of singularity avoidance may hold (i.e., does not for degeneratehyperbolic singularities) [43]. If the matrix Q is indefinite (i.e., the eigenvalues are positive and negative), the result of Eq.(229) has the possibility of being equal to zero. Therefore, both of these situations constitute an hyperbolic singularity. The tools developed in this chapter for describing the existence of elliptic singularities and hyperbolic internal singularities in a system of SGCMGs are used in the next chapter to more specifically discuss which of these internal singularities exist in common arrangements of SGCMGs. In addition, Chapter 5 introduces a novel steering algorithm known as the Hybrid Steering Logic (HSL) which uses these tools in its derivation. CHAPTER 3 CONTROL MOMENT GYROSCOPE ARRANGEMENTS 3.1 Common SGCMG Arrangements Several common SGCMG arrangements have been studied. Typically, the factors that determine the choice of a specific SGCMG arrangement are: (i) available volume (ii) desirable angular momentum envelope, and (iii) associated singularities. In this chapter, we examine the common arrangements and use the tools developed in Chapter 2 to characterize their singularities. 3.1.1 Rooftop The rooftop arrangement shown in Figure 31 has two sets of parallel SGCMGs, each with parallel gimbal axes where 0 is the skew angle relating the planes of torque. A fourSGCMG rooftop arrangement is shown in Figure 31. X B y Figure 31. FourSGCMG rooftop arrangement Since these arrangements are free from elliptic internal singularities, they have a significant flight heritage on satellites and thus, their control is well understood [45]. However, degenerate hyperbolic singularities which are also impassable still exist and like elliptic singularities cannot be addressed through the use of null motion. In addition, there are degenerate cases of hyperbolic singularities for the rooftop arrangement when the Jacobian matrix is rank 1 which may provide difficulty to singularity escape laws that regulate the smallest singular value. 3.1.2 Box The box arrangement is a subset of the rooftop arrangement and has two parallel sets of two SGCMGs with an angle of 90 deg between the two planes of torque as shown in Figure 32. This arrangement is given its name because the planes of angular momentum can form a box [46]. Like the rooftop array, there are situations where this arrangement may have a rank 1 Jacobian. 'N X Figure 32. FourSGCMG box arrangement The angular momentum envelope for the fourSGCMG box arrangement is shown in Figure 33. The angular momentum envelope for all rooftop arrangements is an ellipsoidial surface and thus, there is not equal momentum saturation in all directions. Rooftop arrangements are chosen for their compactness and the fact that they are free from elliptic internal singularities. Analysis provided in the literature proves that though these arrangements are free from elliptic internal singularities. However, they are not free from the impassable degenerate hyperbolic singularities [47]. 1  2,, : .. '.== L, : ' 22 2 3 % 1 0* Y(ho) 4 2 1 X(ho) Figure 33. Angular momentum envelope for a fourSGCMG box arrangement. In Chapter 2, it was shown that the definiteness of the matrix Q determines if a system of SGCMGs is at an elliptic singularity. For a system of four SGCMGs, the largest Q can be is RR2x2 excluding the case when the Jacobian goes rank 1 which will be discussed later. Therefore, excluding a rank 1 Jacobian, and if the determinant of Q is strictly positive, then the system is at an elliptic singularity (e.g., AIA2 > 0 where A1 2 are an eigenvalues of Q). Consequently, when det(Q) < 0, the system is at an hyperbolic singularity. There are a few general cases where singularities may occur for a fourSGCMG rooftop arrangement. The first case occurs when the torque vectors lying in the same plane of torque are either parallel or antiparallel as shown in Figures 34 A and B where 6 62 + 1800, 63 64 1800 and r is the axis intersecting the two planes of torque. /l W / W A Torque planes with parallel torque B Torque planes with one parallel and vectors antiparallel torque vectors Figure 34. Planes of torque for a fourCMG rooftop arrangement A four CMG rooftop system in the configuration of Figure 34 A has a Jacobian c(O)c(61) c(O)c(6G) c(O)c(3) c(O)c(63) A = s(61) s(61) s(63) s(63) (31) s(O)c((6) s(O)c(61) s(0)c(63) s(0)c(63) with angular momentum represented in the spacecraftbody frame (i.e., where the gimbals are enumerated counterclockwise beginning at the spacecraft body xaxis) c(0)s(61) c(0)s(63) h = h + h2+ h3 h4= 2ho c(61) 2ho c(63) (32) s(0)s(61) s(0)s(63) The singular direction for this case is found by cross product of ^1 and ^3 s()s(01 + 63) = 2s(0)c(0)c(61)c(63) (33) c(0)s(1 63) with the resultant projection matrix c(61) 0 0 c(6i) P = 2hos(O)c(O) 0 ) 0 0 0 0 and the nullspace of the Jacobian concatenated in 1 0 1 0 0 0 0 0 c(63) 0 0 c(3) matrix form is (34) (35) The configuration in Figure 34 has the det(Q) = 16s2(O)c2(O)c(61)c(3). It should be noted that when P and Q are definite (i.e., e = [+ + ++] or c = [ ] where ci = sgn(hfj s)), the system is at a external singularity. It was discussed in Chapter 2 that external (or saturation) singularities are elliptic. Therefore, if the matrix P is definite, then the matrix Q is also definite. The det(Q) = 16s2(0)c2(0)c( )c((3) will only be positive when sgn(c(l6)) z sgn(c(O3)) (i.e., at saturation singularity). The saturation singularity is not an internal singularity and therefore neither the rooftop and box arrangements contain elliptic internal singularities for this case. It can be verified that the variations of these cases such as (f1 = t2), (f2 = 4), (f1 = 2) and (f2 = 4) all have det(Q) = 0 and therefore are hyperbolic internal singularities. The other case when the Jacobian of a fourCMG rooftop is singular occurs when the torque vectors of the two parallel SGCMGs lie in the direction of the intersecting torque planes r shown in Figures 35. r r A Torque planes with parallel torque B Torque planes with antiparallel torque vectors along r vectors along r Figure 35. Torque planes traced out for a fourSGCMG rooftop arrangement For Figure 35 A, the Jacobian is c(0)c(63) A = S(63) s(0)cc(s3) with angular momentum vector (0)C(64) 0 s(64) 1 s(0)C(64) 0 c(0) c(0)s(63) h = hi h2 h3 h4= 2ho 0 ho c(63) s()_ s(O)s(63) assuming that the intersection of the planes of torque r = [0 shown in Figure 35. For this situation, the singular direction intersection of the planes and is found to be (36) c(O)s(64) Sho c(64) (37) s(0)s(64) 1 O]T for this arrangement is orthogonal to the s(0) s= 0 c(0) with the resulting projection matrix (38) 1 0 0 1 P = 2hos(0)c(0)c(63) 0 0 0 0 and the null space basis for the Jacobian concatenated N = null(A) 0 0 0 0 0 0 0 0 in matrix form is (39) 1 s(64 3)/C(3) S0 ) (C(64)/C63)) (310) For this case, det(Q) = 2hosin2(20) > 0 which is an elliptic singularity but is not known yet to be one that is internal or external. For the antiparallel case when 63 = 180 + 64 = 2700 deg, the det(Q) = 2hosin2(20) < 0 and the singularity is hyperbolic. Notice that the diagonal entries of P that are zero correspond to the gimbal axis of that roofside being along the singular direction s of the system. Recall, from Chapter 2 that s = Si is a special case that happens only for DGCMGs and rooftop arrangements. For example, consider the case when 63 = 180 + 64 = 2700 and an angular momentum of 0 h= h2 (311) 0 is desired along with, 61 = 62. The result of Eq.(311) is c(61) = c(2) = 2. There are two solutions for any possible value of h2 inside the momentum envelope due to symmetry and thus there is one null solution. Both of these solutions are singular and abide by 63 = 1800 + 64 = 2700 and therefore the null solutions that exist do not help in escape from the singularity. Thus, this is a case of a degenerate hyperbolic singularity at that specific point on the momentum space. In addition, the value of d = 0 (i.e., where m = det(AAT)) for both 63 = 1800 + 4 = 2700 and for 61 and 62 free, and thus there is no set of gimbal angles that will provide a change in m (i.e., no null solutions to escape singularity). To determine if the other cases of singularity when 63 = 64 = 900 deg are elliptic, we check if null motion exists orthogonal to the singularity (i.e., d 6 0). The result of this case when 63 = 64 = 900 and 61 and 62 are free is consistent with Eq.(36). It is found that for any choice of 61 and 62 gives d = 0 and therefore this is a family of elliptic singularities because not only does det(Q) > 0 but also d = 0. To visualize where the external singularities occur for this case, the SGCMG angular momentum is plotted for all variations of 61 and 62 of a fourSGCMG box arrangement shown in Figure 36. 3 ..1. 2 . '  , 3 ' 0 2 3 Figure 36. Angular momentum envelope with plotted angular momentum combinations for the fourSGCMG box arrangement In Figure 36, all possible combinations of angular momentum are plotted onto the angular momentum envelope in black for 61 and 62 and when 63 = 64 = 90 In this figure, every combination of this situation is an external singularity. It can be shown from symmetry that all permutations of this case have the same result in that they are external singularities. Therefore, rooftop arrangements do not contain elliptic internal singularities. In addition, the case when the Jacobian for a fourSGCMG rooftop arrangement approaches rank 1 is a subset of this family. This family of configurations is defined as the cases where at least two torque vectors are parallel along the intersection of the two torque planes r and the outcome of the det(Q) is not dependent on the gimbal angles. The rank 1 Jacobian case is a degenerate hyperbolic singularity for when the torque vectors are all antiparallel (i.e., 0 or 2ho) and a external singularity when all the angular momentum vectors are parallel (i.e., 4 ho). These results confirm those obtained via topology and differential geometry by Kurokawa [41]. In reference [41], it was stated that any rooftop arrangements with no less than six units are free from internal impassable surfaces (i.e., elliptic internal singularities not including external singularities). Kurokawa concluded that there are impassible internal surfaces in the fourSGCMG rooftop arrangements that correspond to the singular direction s not contained in the plane spanned by the two gimbal axes 81 and 62. This is exactly the degenerate case shown in Eqs.(36)(39) where s = 31. The degenerate hyperbolic singularities for these arrangements lie on two circles with radius 2ho when 61 = 62 and 63 = 1800 + 64 = 2700 and at zero momentum Oho when 61 = 62 and 63 = 1800 + 4 = 2700 shown as an example for the box arrangement in Figure 37 [41]. 2, 2 2 0 2 0 X(ho) 2 2Y(ho) Figure 37. Degenerate hyperbolic singularities for the fourSGCMG box arrangement The curves and point in Figure 37 are compartmentalized and not spread throughout the entire angular momentum envelope unlike elliptic singularities and thus, constrained steering algorithms exist to avoid these regions while providing singularity avoidance using null motion. 3.1.3 3 Box The 3 box arrangement is a subset of the box arrangement in which one of the SGCMGs is not used and left as a spare. This arrangement has the longest heritage of flight due to its conservative nature. For this arrangement, a pseudoinverse is not required to obtain a solution to the gimbal rates since the gimbal rates are found directly from the inverse of a 3 x 3 Jacobian matrix [48]. The combination of this arrangement and the constrained angular momentum steering law limiting controllable SGCMG angular momentum to a 1 ho radius of the angular momentum envelope, ensure the safest control of a system of SGCMGs [49]. This design although safe, may not be practical for small satellite applications because the SGCMGs must be oversized to provide the desired performance (i.e., they do not take advantage of the entire momentum envelope). 15 2 2 3 1 5 Y(ho) 3 3 X(h0) 2 1 Y(ho) X(ho) A External singular surface for 3 box B Internal singular surface 3 box Figure 38. Singular surfaces showing 1 ho singularity free region In this arrangement, there is a 1 ho radius of the angular momentum envelope that is guaranteed to be singularity free [48] which is shown by the red circle drawn on the external and internal singular surfaces of Figure 38. It should be mentioned that because there is no longer a null space (i.e., using only 3 of the 4 SGCMGs), any angular momentum point on the singular surfaces correspond to the location of a elliptic singularity. This makes the constraining to the 1 ho sphere of angular momentum imperative. 3.1.4 Scissor Pair The scissorpair arrangement has three sets of collinear pairs of two SGCMGs orthogonal to each other. This arrangement is constrained to have 61 = 62 at all times for both CMGs to avoid internal singularities. With this arrangement shown in Figure 39, full three axis control is possible with a full rank Jacobian matrix as long as it does not extend past the maximum angular momentum of the system. As a consequence of the constraint for these pairs, only onethird of the entire angular momentum envelope is utilized which will be troublesome for use on small satellites (i.e., six SGCMGs needed for 2ho of angular momentum). Use of these arrangements was found to conserve power when a single gimbal motor is used for each scissor pair [50]. Also, analysis has shown that scissor pairs may be beneficial for space robotics application since their torque is unidirectional [51]. T T Figure 39. 3 Orthogonal scissor pairs of SGCMGs Due to the gimbal angle constraint associated with this arrangement, internal singularities do not exist here. Also because of the gimbal angle constraint 61 = 62, three orthogonal scissorpairs contain only external singularities that occur when one or more of the pairs has an overall zero torque vector (i.e., undefined torque for scissor pair). When this occurs, the Jacobian matrix contains a column of zeros for the associated pair. 3.1.5 Pyramid Pyramid arrangements of SGCMGs have independent planes of angular momentum and torque which form a pyramid. As a consequence of these independent planes of torque, these arrangements will never have the Jacobian matrix with rank less than 2. This is shown in Figure 310 for a fourSGCMG pyramid cluster. In terms of small satellite constraints and when utilizing the entire momentum envelope, the four SGCMG arrangement seems practical among all previously discussed arrangements for platforms requiring high torque and slew rates with near equal momentum saturation in three directions . Figure 310. Planes of angular momentum and torque for a fourSGCMG pyramid Figure 311. FourSGCMG pyramid arrangement Control of these arrangements is more complicated than rooftop and box arrangements due to the presence of elliptic internal singularities because null motion solutions do not exist. In addition, elliptic singularities do not have continuous gimbal trajectories associated the corresponding continuous angular momentum trajectories [47]. These arrangements are studied for their desirable momentum envelope (i.e., it is possible to get a near spherical angular momentum envelope with a skew angle of 0 = 54.74 deg) [52]. If high agility is what is needed and there are more relaxed pointing requirements, the pyramid may provide benefits over the other arrangements. Even if this is not the case, if this arrangement is hosted on a small satellite and the attitude error induced from the torque error provided by the singularity escape of elliptic singularities is on the same order of the attitude determination sensors and/or methods, then the torque error used for singularity escape will be inconsequential. 3.2 Choice of Arrangement Beyond the common arrangements of SGCMGs previously discussed, it is difficult to choose the arrangement of SGCMGs through shaping of the angular momentum envelope of the system. This is due to the locations of where internal singularities lie within the angular momentum envelope denoted by the internal singular surfaces (e.g., see Figure 26). These singularities are dispersed and may cover the entire angular momentum envelope leaving only very small singularityfree areas. Formulating the problem as a parameter optimization as in [53] can only provide the optimal arrangement for a given set of slews and initial gimbal angles which makes the problem more constrained than useful. For example, we can express the gimbal axes relation to the spacecraft body frame in terms of the Euler angles, two of which are the optimized constants inclination angle 0i, spacing angle 0i, and the third is the gimbal angle 6,. The DCM that is used to transform from the body to the gimbal frame is CGB = C3 i)C2(0i)C3(i) (312) The angular momentum of the SGCMGs is transformed from the gimbal TFg to the spacecraft body frame Fb through this DCM as n h= CBGhi (313) i=1 which is consistent with Eq.(24). Therefore, holding the spacing and inclination angles constant, the resultant angular momentum of the CMG system is an instantaneous function of only the gimbal angles for SGCMGs. Considering this and the truncated dynamic model of SGCMGs from Chapter 2, the cost function M = f(m2 + aTeTTe b6 T) dt Jt, (314) where a and b are scalars making the cost function unitless and Tre = hoA6, can optimize the system with respect to minimal torque error through the choice of the Euler angles for a given slew, slew time, and initial conditions of the gimbal angles. This procedure for the parameter optimization is shown in Figure 312. Dynamics Solved First Cost Figure 312. Optimization process block diagram 3.3 Simulation An example simulation of a resttorest attitude maneuver has the parameters in Table 31. This simulation will shown the benefit of different arrangements on performing this maneuver (i.e., tracking the torque from the controller). It should be noted that the initial conditions of the gimbal angles although are the same for every arrangement, they produce a different initial CMG angular momentum. Table 31. Model Parameters Variable Value /100 2.0 1.5 S2.0 900 60! 1.5 60 1000 60 [0 0 0 0]T eo [0.04355 0.08710 0.04355 0.99430]T eo [0 0 0]T ho 128 k 0.05 c 0.15 a 1 b 1 At 0.02 e., 0.0001 Units kgm2 deg deg/s Nms 1/s2 1/s 1/N2 m2s2 1/s2 S deg The results were simulated using the following eigenaxis control logic [54] 7 = 2kJe cJw + wxJw (315) combined with a fourthorder RungaKutta integrator at a timestep At until the steadystate error tolerance of the error quaternion eigenangle ess was reached. The simulation compares the optimized solution to the fourSGCMG pyramid arrangement at a skew angle 0 = 54.74 deg. The results for this example at initial conditions 60, eo, and wo have the solution for the system's singular surfaces with calculated arrangement Euler angles shown in Figure 313. 2 1 0 0 N i N  A External singular surface B Internal singular surface Figure 313. Singular surfaces for the optimized arrangement at the Euler angles 0* = [170.2 13.6 85.5 168.0]T deg and 0* = [17.7 167.0 304.3 92.5]T deg The gimbal rates for the optimized arrangement in Figure 314 A are approximately the same magnitude than that for the pyramid arrangement, although they have a smoother transient response. 400 _d61/dt 200 300 _d82/dt 100 d3 /dt 200 d 3dt 0 .d1d g Id64/dt d68/dt S100 100 _d82,/dt 200 d83/dt /10 300 4 100 0 20 40 60 0 20 40 60 Times(s) Times(s) A Gimbal rates for optimized arrangement B Gimbal rates for pyramid arrangement Figure 314. Gimbal rates for the optimized and pyramid arrangements The torque error shown in Figure 315 A for the optimized case is smaller magnitude than that for the pyramid arrangement due to the area under the curves thus, more torque error is added during the maneuver for the pyramid arrangement. E z 0.04 0.02 0 0.02 0 20 40 60 Times(s) A Torque error for optimized arrangement 0 20 40 60 Times(s) B Torque error for pyramid arrangement Figure 315. Torque error for the optimized and pyramid arrangements For the optimized method, the singularity measure is far from singularity initially and does not encounter it as shown in Figure 316 A. This is in contrast to that for the pyramid arrangement shown in Figure 316 B, which starts out initially far from singularity and then encounters singularity several times during the maneuver. The negative quadratic term present in the cost function of Eq.(314) for this singularity does not weight distance from singularity as high as torque error which can be seen when comparing Figures 316 A and B to 315 A and B. A ment 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 20 40 60 0 20 40 60 Times(s) Times(s) Singularity measure for optimized arrange B Singularity measure for pyramid arrangement Figure 316. Singularity measure for the optimized and pyramid arrangements Finally, the cost function of the optimized arrangement in Figure 317 A is less than that for the pyramid in B due to the are under the curves. 60 50 50 40 40 2 30 30 20 20 10o 10 o 0 0 20 40 60 0 20 40 60 Times(s) Times(s) A Cost for optimized arrangement B Cost for pyramid arrangement Figure 317. Optimization cost for the optimized and pyramid arrangements These simulations support the idea that if it where mechanically possible to reconfigure the gimbalaxis arrangements in a timely manner, and the initial gimbal angles and maneuver of interest were known, a solution to the optimal CMG arrangement can be found. In addition, these simulations prove that you cannot simply choose an optimal arrangement because the problem is not only dependent on the attitude maneuver, but also dependent on the initial conditions of the gimbal angles. If a desired arrangement is known while onorbit and there was a mechanical way to reconfigure the SGCMG gimbal axes, such as in the Honeywell patent [55], then there would be merit in finding an algorithm that was successful in reorienting the gimbal axes of the CMG arrangement. Although, no algorithm exists to reorient the SGCMG gimbal axes while keeping spacecraft unperturbed. Also, such an algorithm would still require angular momentum offloading due to the nature of SGCMGs. CHAPTER 4 SURVEY OF STEERING ALGORITHMS A guidance, navigation, and control (GNC) system is composed of the loops shown in Figure 41 Figure 41. Outer and inner loops of GNC system The outer most loop of a spacecraft GNC system concerns the navigation (i.e., provides the state knowledge) and is usually the minimum loop needed for any mission. The second most outer loop is concerns the guidance of the system (i.e., provides the desired trajectories) (e.g., trajectories avoiding pointing a star camera towards the sun). A loop inner to the guidance loop concerns the control of the system (i.e., generates an error of the state knowledge from the navigation loop with the desired trajectories from the guidance loop to be minimized). The inner most loop concerns the distribution of the desired control to the systems actuators (e.g., what thrusters need to fire, what reaction wheels or CMGs need to move). Steering algorithms are concerned with the inner most loop of Figure 41 when the differential equation relating the control to the actuators is singular. When this equation is singular, the steering law realizes a solution. 4.1 MoorePenrose PseudoInverse An early method used to map the gimbal rates from the required output torque uses the minimum twonorm least squares solution also known as the MoorePenrose pseudoinverse. The solution of the gimbal rates using this pseudoinverse mapping has the form 6 A+h= AT(AAT)lh (41) ho ho where A+ is the MoorePenrose pseudoinverse, ho is the magnitude of SGCMG angular momentum, h is the SGCMG output torque, and 6 are the gimbal rates. The MoorePenrose pseudoinverse, however, is singular when the Jacobian matrix A has rank i 3 [56]. It might seem intuitive that the addition of more SGCMG actuators increases the possibility of having full rank, but the performance is not equally increased for all of CMG arrangements. This is because there are 2" singular configurations for any given singular direction of a system containing n SGCMGs [41]. Also, the MoorePenrose pseudoinverse and variations of it cause the system to move toward singular states when performing discrete time control [47]. To handle cases when singularities may be encountered, steering algorithms are used. Steering algorithms can be broken down into the following groups as shown in Figure 42 4.2 Singularity Avoidance Algorithms Singularity avoidance algorithms, are methods which steer the gimbals of the SGCMGs away from internal singularities. These methods either constrain the angular momentum envelope and/or gimbal angles, or apply null motion to avoid singularity encounters. As discussed in Chapter 2, a method that uses only null motion cannot avoid or escape elliptic internal singularities [44, 47]. Figure 42. Steering algorithms 4.2.1 Constrained Steering Algorithms Constrained steering algorithms either constrain the gimbal angles and/or useable angular momentum to avoid singularities. These steering laws are a form of singularity avoidance that takes into account the locations of singularities a priori. As a consequence of not using the entire angular momentum space, these steering logics are typically more effective for systems where the SGCMGs are oversized. Honeywell has patented methods that do not explicitly use null space but that implicitly do so by creating constraints that keep the gimbals away from singularity without needing to recognize their presence explicitly [5759]. A simple example is the steering logic for scissored pairs in Chapter 3, where mere constraints are used to keep the array out of trouble. This method is able to guarantee singularity avoidance and an a available torque but reduces the available workspace of the system by requiring it to be singularity free [52, 60]. 4.2.2 Null Motion Algorithms 4.2.2.1 Local gradient (LG) Singularity avoidance algorithms known as local gradient (LG) methods use null motion to keep the Jacobian matrix from becoming singular. This is accomplished through choice of the null vector d to maximize an objective function that relates the distance from singularity such as the Jacobian matrix condition number, smallest singular value [61, 62], or the singularity measure m which is expressed as m = /det(AAT) (42) An example of the null vector calculation is 9f (8m) 1 (Om) T d = V6f ( m (43) am ( 6 m2 (86) where the objective function f = 1/m [42, 63, 64]. Minimization of this objective function maximizes the distance from singularity by maximizing the singularity measure. The LG methods, however, cannot avoid or escape elliptic internal singularities because they apply only null motion [44, 64]. The null vector d can be arbitrary, although the projection matrix which maps it onto the null space is constrained. 4.2.2.2 Global avoidance/Preferred trajectory tracking A way of choosing the null motion vector to steer gimbals to alternate nonsingular configurations before maneuvering is known as preferred trajectory tracking [63, 6567]. Preferred trajectory tracking is a global method that calculates nonsingular gimbal trajectories offline. The gimbals converge to these trajectories using null motion to minimize an error (6 6*). The gimbal rates using this method are 1 6 = A+h + [1 A+A](6 6*) (44) ho where 6* are the preferred trajectories and A is the singularity parameter defined by A = Xoexppm2 (45) with constants A0 and p. Since this method calculates the preferred trajectories offline, it is not realtime implementable. Also, preferred tracking relies entirely on null motion and thus will be unable to escape elliptic internal singularities. 4.2.2.3 Generalized Inverse Steering Law (GISL) The Generalized Inverse Steering Law (GISL) provides a pseudoinverse which is a variation of the MoorePenrose pseudoinverse. This method defines another Jacobian matrix B which has each of its columns orthogonal to the associated column of the original Jacobian matrix A (i.e., aiLbi, not necessarily aZbj where A = [al a2 a3 a4] and B = [b, b2 b3 b4]) [68]. Therefore, as an example for a fourCMG pyramid arrangement, the matrix A and B have the following form cOcos(61) sin(62) cOcos(63) sin(64) A = sin(61) cOcos(62) sin(63) cOcos(64) (46) s0cos(61) sOcos(62) s0cos(63) sOcos(64) and cOsin(6) cos(62) csin(63) cos(64) B = cos(61) cOsin(62) cos(63) cOsin(64) (47) sOsin(61) sOsin(62) sOsin(63) sOsin(64) where cO, sO are the cosine and sine of the pyramid skew angle 0 and 6, are the gimbal angles, respectively. The pseudoinverse of this steering law with the discussed matrices is AGISL = (A + B)T(A(A +B) )1 (48) It is important to note that this pseudoinverse does not eliminate the problem of internal singularities. The GISL adds null motion from the addition of B and therefore couples the null and forced solution into a single inverse and thus, it is not able to avoid elliptic internal singularities. Proof: Claim: The GISL provides only null motion through B 6 = AGISL = (A + B)T(A(A + B)T)l The torque error is h AJ = A(A + B)T(A(A + B)T)h i h = h h = 0 If the matrix B = aA then AGISL = ((1 + )A)T((1 + )AAT)1 = A Therefore, the matrix B whose components are orthogonal to A must only provide null motion and those that are along A vanish. Because, the GISL or any generalized inverse used for singularity avoidance only adds null motion, it is unable to avoid elliptic internal singularities. 4.3 Singularity Escape Algorithms Singularity escape methods, known as pseudoinverse solutions, add torque error to pass through or escape internal singularity [39, 40, 69, 70]. These methods do not take into consideration the type of internal singularity that is being approached when adding torque error. 4.3.0.4 Singularity Robust (SR) inverse The Singularity Robust (SR) inverse is a variation of the MoorePenrose pseudoinverse [69] where, a positive definite matrix Al composed of an identity matrix scaled by the singularity parameter in Eq.(45) is added to the positive semidefinite matrix AAT. The pseudoinverse of this method has the form ASR = AT(AAT +Al)1 (49) The SR inverse is able to escape both hyperbolic and elliptic singularities [44], although, is ineffective in gimballock escape. To overcome this situation, a modified SR inverse known as the Generalized Singularity Robust (GSR) pseudoinverse was developed [39, 40]. 4.3.0.5 Generalized Singularity Robust (GSR) inverse The GSR inverse approach replaces the constant diagonal positive definite matrix Al with a timevarying positive definite symmetric matrix AE 1 C1 2 E= c1 1 3 ci = cosin(wit + i,) (410) C2 C3 1 where the offdiagonal terms of E are time dependent trigonometric functions with frequency uw and phase shift 0i. The GSR inverse provides a means of escape of the gimballock configuration associated with a system of SGCMGs. The GSR pseudoinverse has the form AGSR = AT(AAT + E)1 (411) and like the SR inverse, is guaranteed to avoid both hyperbolic and elliptic internal singularities. 4.3.0.6 Singular Direction Avoidance (SDA) Another modification of the SR inverse known as the Singular Direction Avoidance (SDA) only applies torque error in the singular direction and therefore reduces the amount of torque error needed for singularity escape. The SDA method decomposes the Jacobian matrix using a singular value decomposition (SVD) to determine its singular values. The matrix of singular values is regulated with the addition of error to the smallest singular value o3 so that the pseudoinverse is defined. The pseudoinverse using SDA has the form 0 0 0 0 ASDA = V 2= VYSDAUT (412) 3 +A 0 0 where o, are the singular values. Regulating only the smallest singular value, reduces the amount of torque error added and creates smoother gimbal rate trajectories when compared to the SR and GSR inverses [70]. This is obvious when the SR inverse decomposed using SVD as 0 "2 0 ASR = V 22A UT = ASR = VYSRUT (413) where all the singular values are regulated and hence there is torque error in all directions. It is clear from Eqs.(412) and (413) that SR inverse and SDA are susceptible to gimballock because when the output torque is along the singular direction h oc s = u3 then it is in the null(ISDAUT) and null(lSRUT) thus encountering gimballock as no consequence to the size of the torque error added from A. Without the perturbations to the Jacobian matrix that are not gimbal state dependent at gimballock the system remains locked in a singular configuration. Recall from Chapter 3, that for a four CMG pyramid arrangement, the rank is never less than two and therefore it is acceptable to regulate only the smallest singular value. However, if the skew angle is made close to 0, 90, 180, or 270 deg (i.e., box or planar arrangement), the Jacobian matrix for these arrangements will have at least two small singular values when near singularity and regulation of the smallest singular value may be ineffective. 4.3.0.7 Feedback Steering Law (FSL) The Feedback Steering Law (FSL) provides a solution to the gimbal rates without using an inverse. This method is derived from a minimization of the torque error which is similar to how the SR inverse is derived. The optimization for defining FSL has the following structure T 1 Te KI 0 Te min (414) ,6R4 2 0 K where K1 and K2 are positive definite gain matrices, and Te = h A6. This minimization reduces to the SR inverse when K1 = 1 and K2 = Al and where A = Aoexppm2) from Eq.(45). The FSL method has K2 = 1 and K1 = K(s) as a compensator. The compensator is derived from an H, minimization w (s)[1 +AK(s)]1 mmin (415) K(s)GR3x3 W2K(s)[l AK(s)]1/ where wi(s) and w2 are weighting matrices. The wi(s) matrix is defined below wi(s) = : ] (416) CK 0 where AK, BK, and CK are matrices associated with statespace model of the system. The w2 matrix is constant and is W2 = 14x4 (417) W where the constant w bounds the gimbal rates. The statespace model of the system has the form ^ = AKX + BiKTe (41 8) 6 = CKX The output matrix CK is an explicit function of the CMG gimbal angles expressed as CK = ATb2P (419) with b as a positive scalar associated with the bandwidth and P is the steadystate solution to the Riccati equation of the statespace system in Eq.(418). Using the feedback of the system, Eq(418) will provide a solution to the gimbal rates that does not require a pseudoinverse. It should be noted that the system may start out stable, however, the observability of the system may be lost resulting in instability, due to CK'S explicit dependence on the gimbal angle (i.e., H(s) = CK(sl AK)1BK where CK constant). For further information on the development of this method, please see [71]. This steering algorithm was shown to go unstable for certain values of CK corresponding to specific gimbal angles sets (see [72]). 4.3.0.8 Singularity Penetration with UnitDelay (SPUD) The Singularity Penetration with Unit Delay (SPUD) algorithm escapes singularity through reuse of the previous gimbal rate command when at a certain threshold of singularity [73]. The previous command is saved through a zerothorder hold to the system. Escape of a singularity is always possible unless the system is initially at the threshold of singularity, then there is no previous command to use for singularity avoidance. Also, SPUD is not intended for attitude tracking maneuvers. The SPUD algorithm accumulates attitude tracking error while escaping singularity and there are no guarantees on how long it will take to escape singularity and how large the torque disturbance will be on the spacecraft as its performance is directly associated with the system and the choice of singularity threshold. 4.4 Singularity Avoidance and Escape Algorithms Singularity avoidance and escape algorithms avoid singularities through null motion whenever possible and use torque error for escape when they are not. 4.4.0.9 Preferred gimbal angles Preferred gimbal angles are a set of initial gimbal angles for SGCMGs that can be reached by null motion. These angles are preferred since maneuvers originating from them avoid a singular configuration [74]. This set of angles is found by backwards integration of the Eq.(49) and the attitude equations of motion. It has been shown that this method cannot avoid singularities if the initial set of gimbal angles is 60 = [45 45 45 45]T deg [74]. Since the null space projection matrix is undefined at singularity, the SR inverse is used in place of the MoorePenrose pseudoinverse of A in ,n as 6n = [1 ASRA]d (420) As a result, this causes the system to add torque error when at singularity. In practice, this method acts as an offline optimization which determines the initial set(s) of the gimbal angles that will give singularity free maneuver(s). However, it is not possible to go from one to any point in gimbal space through null motion itself because there will never be n dimensions of null space. 4.4.0.10 Optimal steering law (OSL) The InnerProduct Index (IPI) combined with the optimal steering law (OSL) is used to determine a steering algorithm that produces minimum torque error while both avoiding and escaping internal singularities [75]. The singularity index is added to the minimization min [cV(6+6At) 16TW +eTRle] (421) 6ER4,e6R3 2 where At is the onestep time delay, V(6 + 6At) is the IPI, re = h A6 (i.e., torque error), c is a positive scalar, and W and R1 are positive definite weighting matrices. The IPI is approximated by a Taylor series expansion up to the 2nd order as OVT 1 6 1 2V V(6 +65 t) V + TAt +6TV &6t2 (422) a +6 0 2 962 6 where the IPI V is expressed as a sum of square of inner products of the column vectors of the Jacobian. 4 V= (a aj)2 (423) i=J1,ij The result of the minimization in Eq.(421) using this approximation of V is 6 = HAT(AHAT + R)l + [HAT(AH1AT + R)AH1 H1]g (424) where the Hessian matrix H is defined as H = cAt2ggT + W (425) with gradient g = a The weighting matrix R shown previously in the minimization of Eq.(421), is expressed as 00 0 R =U 0 0 0 U (426) 0 0Aoexp " where o3 is the smallest of the singular values of the Jacobian matrix, A0 and p are positive scalars and U is the unitary matrix made up of the left singular vectors from the singular value decomposition of the Jacobian matrix A. This addition of torque error into the gimbal rate state equation is analogous to the SDA method except that it is also added to the free response solution [70]. It should be noted that this steering algorithm does not consider the form of internal singularities and therefore, does not truly minimize the amount of torque error for singularity escape. This is because nondegenerate hyperbolic singularities are avoidable through null motion without the use of torque error. At a nondegenerate hyperbolic singularity R is nonzero and thus torque error is still added (see [72]). 4.5 Other Steering Algorithms Other published steering algorithms that have not been discussed can be found in the references [44, 61, 7681]. These methods include mathematical techniques such as neural networks, optimization, and gametheory. 4.6 Steering Algorithm Computation Comparison An analysis comparing the computation for the implementation of the mentioned steering algorithms is difficult due to lack of information on how some were coded in literature. For example, some of these algorithms are offline and may require a large number of memory calls and stored memory but not as many flops. It is however, useful to quantifying some of the previously discussed steering algorithms in terms of floating point operations that are not calculated offiline. These are shown in Table 41 for algorithms where flops make a good comparison. In this table, the metric of comparison is an approximate number of flops per time step. Table 41. Algorithm Flops m = row(A) and n = column(A) Variable Value MP O(m4) LG O(m4) GISL O(m4) SR O(m4) GSR O(m4) SDA 6(nm3) FSL O(mn2) SPUD O(m4) Optimal Steering O(m4) 3+ (nm3) It should be mentioned that many of the steering algorithms discussed have the same order of magnitude of flops (e.g., MP, LG, GISL, SR, GSR, and SPUD) due to the approximate number of flops for a GaussJordan matrix inverse. It is assumed that the calculation of the gradient vector for LG, and OSL is in memory and that the flops associated with then are on a lower order of magnitude that an SVD or GaussJordan matrix inverse. The OSL has to do both SVD for calculation of the R matrix and the GaussJordan matrix inverse and therefore has O(m4) + O(nm3) flops. The addition of O(m4) + 6(nm3) is inserted for the approximate flops of OSL because depending on the amount of SGCMGs this algorithm is working for O(m4) < O(nm3). The FSL has the lowest number of flops because it does not require and SVD or inverse, although it is also not an exact mapping as previously discussed. CHAPTER 5 STEERING ALGORITHMHYBRID STEERING LOGIC 5.1 Hybrid Steering Logic Existing steering logics (see Chapter 4) do not explicitly consider the type of singularity that is being encountered and thus, do not completely address attitude tracking performance of SGCMG attitude control systems. A proposed method known as the Hybrid Steering Logic (HSL) which utilizes the knowledge of the type of singularity encountered (i.e., elliptic or hyperbolic singularities) to improve the attitude tracking performance of the SGCMG attitude control system, is developed for a fourSGCMG pyramid arrangement at a skew angle 0 = 54.74 deg. By using a hybrid approach, HSL acts as an LG method (i.e., null motion for singularity avoidance) at hyperbolic singularity and an SDA method (i.e., pseudoinverse solutions for singularity escape) at elliptic singularity. Also, because HSL is developed for a fourSGCMG pyramid arrangement, there is no existence of degeneratehyperbolic singularities [41]. The challenge is to develop the appropriate singularity metrics such that the LG and SDA components of the hybrid strategy do not counteract each other during operation. 5.1.1 Internal Singularity Metrics The singularity metrics developed are of similar form as the singularity parameter in Eq.(45) with the addition of terms relating to the form of the actual singularity. a = oaexpaexppm (51) / = Po expbexp2m (52) where a, b, p/, p2, ao and /o are positive scalar constants and m is the singularity measure as defined in Eq.(42). Away from singularity, a fourSGCMG pyramid arrangement at a skew angle 0 = 54.75 deg, has the matrix Q e R. At singularity this SGCMG arrangement has Q e R2x2 (see Chapter ??) and therefore, the det(Q) will be zero or negative (i.e., Q is negative semidefinite or indefinite) for hyperbolic singularities and positive (i.e., Q is definite) for elliptic singularities. Taking this into account, parameters a and p are defined as a = Qo det(Q) (53) 1 1 = (54) SQo det(Q) a where Qo is a scalar value chosen on the same order of magnitude of det(Q) but greater to scale the response of a and /. It is difficult to analytically define Qo since it depends on the maximum value of det(Q) (i.e., det(Q) varies with gimbal angle and therefore the maximum must span all combinations of the gimbal angles) which is of high dimensionality and highly nonlinear. However, through simulation of a fourSGCMG pyramid arrangement at a skew angle 0 = 54.75 deg, it was found that Idet(Q) < 1 and therefore we define Qo 1. In addition, it is important to note that the constant parameters a, b, p1, PJ, ao, and /o are used to morph the HSL steering logic into the respective LG and SDA methods when appropriate: (e.g., if the parameters a = b = ao = 0 and /o a 0 then the HSL method is the LG method). Therefore, the choice of metrics a and / in this way ensures that null motion will be added when approaching a hyperbolic singularity and torque error with less null motion will be added when approaching an elliptic singularity. It should be noted that when using HSL det(Q) is normalized by the nominal angular momentum ho. 5.1.2 Hybrid Steering Logic Formulation The proposed steering logic is defined as 6 = ASDAa + /3[1 A+A]d (55) ho where ASDA a is 0 0 0 0 ASDAa = V (2 UT (56) 00 0 (T32 2+ 0 0 If it is assumed that the analytic function for the gradient vector d is derived offline and the calculation of it at each timestep is less than that for SVD, this algorithm has the same number of flops on order as SDA from Table 41 of 6(nm3) from the SVD. The difference between the conventional ASDA and ASDA,a is the parameter that regulates o3. In ASDA, the regulation parameter is 7 (i.e., different from A in Chapter 4 by using m instead of m2) which is 7 = 70exppm (57) with positive constants 70 and p, but with ASDAa the singularity parameter is a defined in Eq.(51) which regulates the amount of induced torque error in the vicinity of elliptic singularities. Through a SVD decomposition of A, Eq.(55) can be written as 1 [1 6 = ASDAah + /[1 V V]d (58) h0 OT 0 Here, the null motion projection matrix is expressed as a function of nonsingular matrices V. Also, very robust numerical algorithms exist for computing the SVD, so its computational risk in a realtime implementation is not particularly high. The scalar that regulates the magnitude of the null motion is 3. The null vector d is in the direction of the gradient of f = det(AAT) = m2 and maximizes the distance from singularity. This choice of this objective function reduces the computation needed for the gradient (i.e., the derivative of (det(AAT)) is less computationally intensive than the derivative /(det(AAT)) and ensures that the addition of null motion will not approach infinity at the region of singularity for cases such as f = 1 and then t = It m W65 m2 5 " should be mentioned that the null vector is a nonlinear function of the gimbal angles and is simplified due to the symmetry of the four CMG pyramid arrangement. To prove the feasibility of HSL, a stability analysis is conducted. 5.2 Lyapunov Stability Analysis The candidate Lyapunov function V = w eTKJ + eTe +(1 e4)2 (59) 2 is chosen for this analysis and can be rewritten as V = zTMz (510) where z = [WT eT (1 e4)]T and M = diag(~K'J, 1,1). Consequently the Lyapunov function is bounded as Ami, z12 < V < Amax lz12 (511) where Ami, and Amax denote the minimum and maximum eigenvalues of M. This bound will become useful later in the analysis. A resttorest quaternion regulator controller is given by Eq.(512) for the internal control torque 7, is chosen for its flown heritage and the fact that it yields an global asymptotic stable control solution proven through LaSalle's Invariant Theorem [63]. 7 = Ke Cw + XJ (512) Gain matrices K = 2kJ and C = cJ of Eq.(512) are positive definite and symmetric. Assuming rigid body dynamics, the spacecraft's angular momentum is given by H =Jw +h The rotational equations of motion come from taking the inertia time derivative of Eq.(513) as Lj = l[Tact oXJ] (514) with SGCMG output torque h= xh = hoA (515) where w is the spacecraft angular velocity, J is the spacecraft centroidal inertia, H is the total system angular momentum, and h is the angular momentum from the CMGs. The spacecraft's angular velocity and the CMG angular moment are governed by Eqs.(514) and (515) respectively, where Tact is the actual control torque (i.e., may differ due to induced torque error for singularity escape). It is assumed here that the contribution to the dynamics from the gimbalflywheel assembly inertias is negligible and therefore J is constant. The actual control torque Tact based on the mapping of the gimbal rates is Tact= hoA6 xh= A(ASDAah +3[1 V 1 0 VT]d)wh (516) 0T 0 and needs to be considered in the Lyapunov analysis for stability of the attitude controller/steering algorithm combination. When simplified, Eq.(516) becomes 00 0 Tact =U 0 0 0 UT[  wXh] + (517) 0 0  3r+a where the stability of the system is affected by the torque perturbation matrix CHSL from ASDAa defined as (513) 00 0 EHSL = U 0 0 0 UT (518) 0 0 a+ o +a The spacecraft attitude error kinematics is governed by 1 1 e= wxe + Ie4 (519) 2 2 e4 Te (520) where e is the quaternion error vector elements and e4 is its scalar element. The time derivative of the Lyapunov function is 1 1 V = wK [rat WJ] + 2eT[2 Xe+ Iwe4] + 2(1 e4) w e (521) 2 2 2 Equation(521) can be reduced by substituting in the expression for ,act from Eq.(517) and the desired control torque vector 7 from Eq.(512). The time derivative of the Lyapunov function now yields V = WcTK [C EHSL(C+ HX )]w K 6HSLKe (522) or more compactly V = 2kTW + TJ HSLtJ +C1 (523) 2k 2k where e1 = w TJ HSLHX + WTJIEHSLJe (524) 2k Since T2a < 1, Eq.(518) can be used to rewrite Eq.(523) as 10 0 V < C TjU 0 1 0 UTJW + e1 = _(TR + e (525) 2k 0 0 A1 and can be further bounded as V < Allz112 + (526) where A1 is the minimum eigenvalue of the positive semidefinite matrix R, (i.e., A0 = 0 at singularity), ( = 2, and z is defined in Eq.(510). Substituting Eq.(511) into Eq.(526) yields V < AV + (527) A max The solution to the differential equation in Eq.(527) in a Volterra integral form is to V < V(0)exp( t) + ,(t c1 ,T d(528) The error can be bounded from Eq.(528) as 12 < V(0) X t) 1 t(t7))(7)d z 12 < exp +, 1 t exp maxt (529) Amin Amin Jto At singularity when A = 0, the error is z2 < V m c1()dT (530) where Vs is the error at time the occurrence of singularity at time ts. Stability cannot be proven from Eq.(530). It is assumed that the system will not remain locked in singularity except for the special case of gimballock. If components of the torque needed for stability are actually in the singular direction, periods of instability may occur at singularity. The duration of this instability is dependent on the selection and sizing of the singularity parameters a and / which provide torque error for singularity escape and/or null motion for singularity avoidance. From a practical perspective, stability cannot be proven for this Lyapunov function at singularity, since at singularity, there is no torque available in the singular direction. For the special case of a singularity with gimballock, the angular velocity of the spacecraft is constant assuming the absence of friction and external torques in the system. In this case, the contribution from c1 to the error is bounded and even sometimes zero. This can be shown by evaluating the expression for the angular acceleration at gimballock which is S= 0 = J1[0 + wH] (531) It is clear that Eq.(531) is satisfied only when the product w" H = 0 which is only true when w is parallel to H, w = 0, or H = 0. When w is parallel to H or H = 0, 1i = wTJEHSLJe which is a bounded sinusoid whose integral is also a bounded sinusoid. Therefore, for these two cases, the error is bounded at gimballock. When w = 0, c0 = 0 and the error is simply locked at Vs. It should be mentioned that at times away from singularity, the error monotonically decreases because the contribution from cl to the error becomes negligible. Care needs to be taken in the design of the singularity parameter so that the minimum steadystate error is achieved while meeting the constraints of the actuators. The steadystate error assuming that the systems has a singularity free period towards the end of the maneuver (i.e., does not end at singularity) is iiz(0)l12 < 1 r, z(oo) exp ax) exp)mx ())d? (532) Amin ts This expression is indeterminate so application of LHopital's rule to Eq.(533) yields d 1 t ( A 1 ) lim dt Am, J exp x (00o) (533) t*oo dxp t) Amin i dt x which suggests that a sufficiently large value of c (i.e., larger () will lower the amount of steadystate error giving you a uniformlyultimately bounded (UUB) result away from singularity. When the maneuver is finished, the effect of e1 on the error will become a constant assuming the maneuver ends at rest. It should be noted that away from singularity the size of e1 exponentially decreases due to the behavior of EHSL. The difference in impact of HSL rather than SR inverse on stability can be observed from the magnitude of the positive semidefinite matrix EHSL in Eq.(518) compared to the matrix shown in Eq.(534). The ESR matrix has a larger norm and therefore has a worse UUB even for sufficiently large values of (. From comparing Eqs.(56) and (518), the SDA method has a similar amount of torque error added when compared to HSL, although it will add this torque error whenever the singularity approached not taking into account the form. 0 0 SR = U 0 0 (534) 0 0 2 U3+ . The above results are only for the attitude controller/steering algorithm combination. For example, an attitude controller whose torque trajectory was chosen to avoid the occurrence of singularities may not have the periods of possible instabilities at singularity and thus may provide better stability performance. However, no realtime controller of this form exists (i.e., one that ensures singularity avoidance) and thus was not considered in the following simulations. 5.3 Numerical Simulation To evaluate the performance of the proposed HSL against heritage steering logics (i.e., LG and SDA), simulations were performed using a fourSGCMG pyramidal arrangement with a skew angle of 0 = 54.74 deg. To ensure a fair comparison, the control logic and satellite model were identical for all simulations. For each steering algorithm, three different scenarios were simulated: (1) starting in a zeromomentum configuration 6 = [0 0 0 0]T deg (i.e., far from singularity); (2) starting near an elliptic external singularity 6 = [105 105 105 105]T deg; and (3) starting near an hyperbolic singularity 6 = [15 105 195 75]T deg. The singularity conditions were verified for each case by observing the singularity measure defined in Eq.(42). For these simulations, the following performance measures were compared: (i) the transient response of the error quaternion, (ii) the amount and duration of singularity encounter, (iii) the magnitude of gimbal rate, (iv) the amount of torque error (i.e., h hoA6) for singularity escape, and (v) null motion contribution. Additionally, a, 3, and det(Q) are also considered. The Jacobian associated with this pyramidal configuration is c(0)c(61) S(2) c(0)C(63) S(6 A = s(61) c(O)c(2) s(3) c(0)c(4) (535) s(0)c((1) s(O)c(62) s(O)c(63) s(O)c(64) and the associated angular momentum vector is c(0)s(1) C(62) + C(O)S(3) c(4) h = ho c(6) c(0)s(62) c(63) c()s(64) (536) s(O)(s(6) +S(62) + (3) +S(4)) All simulations are performed using a fourthorder fixed time step Runga Kutta with the parameters shown in Table 51. The actuator parameters chosen for this simulation are based on the Honeywell M95 SGCMGs, which are sized for the satellite system chosen for simulation [82]. Table 51. Model Parameters Variable Value Units /100 2 1.5 \ J 2 900 60 kgm \1.5 60 1000/ 0 54.74 deg eo [0.04355 0.08710 0.04355 0.99430]T w0 [0 0 0]T deg/s ho 128 Nms k 0.05 1/s2 c 0.15 1/s mo 0.5 ess 0.001 deg At 0.02 sec It should be noted that care must be taken when numerically defining the singular direction since s = 0 when the system has a full rank Jacobian. Because the rank is numerically determined, a tolerance should be set on the singularity measure to determine what is considered full rank. For the results presented in this paper, rank deficiency for the HSL was defined as m < mo where for this simulation mo = 0.5. The simulations terminate when the steady state error ess defined in Eq.(537) is achieved. es = min[2sinl(llel ), 27 2sinl(le)] (537) The magnitude of ess given in Table 51 is based off reference [38]. 5.3.1 Case 1: At Zero Momentum Configuration 6 = [0 0 0 0]T deg The first set of simulations has initial gimbal angles at 6 = [0 0 0 0]T deg which represents a scenario starting far away from singularities. Figure 51 shows this configuration which is also a typical startup configuration for a fourSGCMG pyramid arrangement. 11,3 h i:h x z li I Y Figure 51. Zeromomentum configuration of a fourSGCMG pyramid arrangement 5.3.1.1 Local gradient simulation results The parameters for the LG simulation are: ao = a = b = p/ = 0 and P2 = o = 1. Figures 52 A and B show that the LG method was able to perform the maneuver to the given error tolerance ess without inducing torque error. The absence of torque error in Figure 52 B is due to the zero value of singularity metric a in Figure 52 C (i.e., LG is an exact mapping). The null motion shown in Figure 53 B is small but significant when compared to the total output gimbal rates in Figure 52 A. This is a consequence of the singularity metric 3 in Figure 53 D. Figures 53 C and D show that the maneuver was completed without singularity encounter. 0.04 0.02 _e2 0 e3 S0.02 0.04 0.06 0.08 0 20 40 60 8( Times(s) A Quaternion error vector elements 0.1 0.05 s 0 0.05 0.1 x 1014 4 3 2 1 0 20 40 60 80 Times(s) B Torque Error 0.3367 0.3366 0.3366 0.3365 0.3365 0 20 40 60 80 0 20 40 60 Times(s) Times(s) C Alpha D Beta Figure 52. Simulation results for LG with ao = a = b = / = 0 and P2 = Ao momentum 1 at zero dS1Idt _d82/dt d83/dt _d84/dt *^ ^  d ,/d ............... At 0 20 40 60 Times(s) A CMG gimbal rates 1.0891 1.089 1.0889 E 1.0888 1.0887 1.0886 0 20 40 Times(s) B Null motion 0.05 0.1 0 0 20 40 60 Times(s) C Singularity measure 20 40 Times(s) D det(Q) 60 80 60 80 Figure 53. Simulation results for LG with ao = a = b momentum (contd.) p = 0 and 2 = 3 = 1 at zero 1 n n2 n3 n4 ........ H, 0.05 \ 0.05 5.3.1.2 Singular Direction Avoidance simulation results The parameters for the SDA simulation are: ao = 0.01, / = a = b = 2 = 0, and p/ = 1. This method shows similar results in the transient response of the error states in Figure 54 A to that for LG in Figure 52 A with the exception of nonzero torque error seen in Figure 54 B. Also, this method had a slower rate of convergence to the steadystate error ess than LG as evident from the time in simulation in Figure 52 A. This is due to the small nonzero value of the singularity metric a, shown in Figure 54 C. Figure 55 B shows a zero null motion contribution to the gimbal rates in Figure 55 A for SDA. Figures 55 C and D and Figures 53 C and D are almost equivalent because the system started far away from singularity. 0.02\ _e 0 e, < 0.02 0.04 0.06 0.08 0 20 40 60 80 Times(s) A Quaternion error vector elements 3.367 3.3665 3.366 3.3655 C x 103 20 40 60 20 40 60 Times(s) C Alpha 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 20 40 60 80 Times(s) B Torque Error S1 0.05 H, 0.05 0.1 0 20 40 60 Times(s) D Beta Figure 54. Simulation results for SDA with ao = 0.01, /o at zero momentum = a = b = 2 = 0, and /p = 1 L dS1Idt _ d82/dt d83/dt S d 4/dt 0.05 0.1 0 20 40 60 Times(s) A CMG gimbal rates 20 40 60 Times(s) B Null motion 1.089 1.0889 1.0888 1.0887 1.0886 1.0885 V 0 20 40 60 Times(s) C Singularity measure 0.05 0.05 0.1 0 Figure 55. Simulation results for SDA with co = 0.01, /o at zero momentum (contd.) 20 40 60 Times(s) D det(Q) =a = b = 2 = 0, and p = 1 0.1 0.05 "1 n"2 3 "4 H, 5.3.1.3 Hybrid Steering Logic simulation results The parameters for the HSL simulation are: ao = 0.01, /o = 2, a = 1, b = 3, and pi = p2 = 1. In Figure 56 A, the HSL method shows similar results to that of the SDA shown in Figure 54 A, with exception to the faster rate of convergence of the transient error response. However, the torque error in Figure 56 B added into the system is smaller than that of SDA in Figures 54 B and null motion in Figure 57 B is smaller than that of the LG method in 52 B. This is due to the nonzero value for both singularity metrics a and 3 in Figures 56 C and D. Singularity was not encountered in this simulations as is shown by a value m > 0.5 in Figure 57 C and a zero value of det(Q) in 57 D. For Case I at zeromomentum, Table 52, compares the rootmean squared (RMS) gimbal rates (deg/s) and tracking performance in terms of RMS torque error (Nm) for LG, SDA, and HSL. In this table it is shown that all three methods have approximately the same performance which is expected for a fourSGCMG pyramid arrangement at zeromomentum, far from singularity. The steadystate error for LG or any of the other methods is nonzero as a consequence of the controller's performance is captured here. Table 52. Performance Comparisons for Case I: Zero Momentum Steering Algorithm sRMS TeRMS LG 5.7366 2.2437e06 SDA 5.7279 3.5902 HSL (mo = 0.5) 5.7317 2.2573 5.3.2 Case 2: Near Elliptic External Singularity 6 = [105 105 105 105]T deg The second set of simulations starts at initial gimbal angles 6 = [105 105 105 105]T deg, which represents a scenario near an elliptic external singularity at (i.e., 15 deg for each SGCMG away from the external singularity 6 = [90 90 90 90]T deg). 0.02 0 a 0.02 0.04 0.06 0.08 E 8 z S6 4 2 0 20 40 60 Times(s) A Quaternion error vector element 0 20 40 60 80 Times(s) C Alpha 80 0 20 40 60 80 Times(s) s B Torque Error 0.0335 0.0335 ~0.0335 0.0335 0.0335 0 Figure 56. Simulation results for HSL with ro = 0.01, /o P1 = p2 = 1 at zero momentum 20 40 60 80 Times(s) D Beta 2, a= l, b= 3, and _d8 /dt _d82/dt d83/dt * d84/dt 0 20 40 60 Times(s) A CMG gimbal rates 0 20 40 Times(s) B Null motion 0.05 0.1 0 0 20 40 60 Times(s) C Singularity measure Figure 57. Simulation results for HSL with co = 0.01, /o l = P 2 = 1 at zero momentum (contd.) 20 40 Times(s) D det(Q) =2, a= 1, b 60 80 60 80 3, and H, 0.05 5.3.2.1 Local gradient simulation results The parameters for the LG simulation are: o = a = b = p = 0 and 2 = /3 = 1. The plots in Figure 58 A show that the LG method appears to have successfully performed the maneuver as shown in Figures 58 A and B. However, this is misleading since nonimplementable gimbal rates and accelerations are required to do so as shown in Figure 59 A. The singularity metrics a 0 as expected for this method and 3 = 1 at the singularity encounter. Even though = 1 at singularity, null motion at the exact time of singularity encounter is zero as shown in Figure 59 because the gradient vector d for LG is zero at elliptic singularities (i.e., no gradient vector exists that is in the direction away from singularity). Also, the singularity, verified to be elliptic from the positive value of det(Q) in Figure 59 D, was escaped immediately with the help of the nonimplementable gimbal rates and accelerations, shown by Figure 59 C. x 1012 0.02\ e O_ e 0, a 0.02 0.04 0.06 0.08 0 20 40 60 8 Times(s) A Quaternion error vector elements ) 20 40 60 80 Times(s) B Torque Error 0.1 0.05 0 0.05 0.1 0 20 40 Times(s) C Alpha 0.9 0.8 0.7 0.6 0.5 0.4 0.3 60 80 0 Figure 58. Simulation results for LG with ao = a = b elliptic singularities 20 40 Times(s) D Beta /i1 = 0 and p2 60 80 :/3 =1 near 5000 5000 _d81/dt _d82/dt d83/dt d. 4/dt 0 20 40 60 81 Times(s) A CMG gimbal rates f n "1 n2 3 0 _ Y 2 0 20 40 60 80 Times(s) B Null motion 1.2 1 0.8 E0.6 0.4 0.2 0 20 40 60 Times(s) C Singularity measure Figure 59. Simulation results for LG with elliptic singularities (contd.) 0.3 0.25  0.2 _ 0.15 0.1 0.05 0 0 ao = a = b 20 40 Times(s) D det(Q) /1 = 0 and P2 ~ 60 80 S=/3 1 near 5.3.2.2 Singular Direction Avoidance simulation results The parameters for the SDA simulation are: ao = 0.01, /o = a = b = /2 = 0, and 1 = 1. The transient response of the error for the SDA method shown in Figure 510 A is comparable to that of the LG method in Figure 58 A, but with implementable gimbal rates and accelerations as shown in Figure 511 A. The SDA method escapes the elliptic external singularity as shown in Figure 511 C at the expense of significant torque error shown in Figure 510 B. The torque error scaled by the singularity metric a shown in Figure 510 C decreases away from singularity as shown in Figure 511 C. As expected for SDA, the singularity metric / in Figure 510 C is zero resulting in zero null motion as shown in Figure 511 B. In contrast to the LG method, for SDA, the system lingers in singularity for around 15 seconds before escaping as shown in Figure 511 C. Elliptic singularity for this simulation is verified by the positive value of det(Q) in Figure 511 D. 0.02\ e. 0 e, a 0.02 0.04 0.06 0.08 0 50 100 Times(s) A Quaternion error vector elements 50 100 Times(s) C Alpha 0 50 100 Times(s) B Torque Error H, 0.05 0.1 0 50 Times(s) D Beta Figure 510. Simulation results for SDA with ao = 0.01, /o = a = b = 2 = 0, and / = 1 near elliptic singularities 0.05  da1/dt _d82/dt _d83/dt d6 4/dt d64/dt 50 11 Times(s) A CMG gimbal rates 50 10 Times(s) C Singularity measure 0.05 0.05 0.1 "1 n"2 3 "4 50 Times(s) B Null motion 50 Times(s) D det(Q) Figure 511. Simulation results for SDA with co = 0.01, /o = a = b = 2 = 0, and / = 1 near elliptic singularities (contd.) 5.3.2.3 Hybrid Steering Logic simulation results The parameters for the HSL simulation are: ao = 0.01, /o = 2, a = 1, b = 3, and /i = /p2 = 1. The results for HSL shown in Figures 512 and 513 are almost identical to the corresponding results of SDA for this simulation. The only difference between HSL and SDA simulated results, lies in the nonzero singularity metrics a and / in Figures 512 C and D. Due to the choice of the HSL parameters a, b, / ,, pL2 I, ao, /3, the threshold for singularity m < 0.5, and Qo, the HSL acts as the SDA at an elliptic singularity 0.04 0.02\A _e, O .0 e, a 0.02 0.04 0.06 0.08 0 50 100 Times(s) A Quaternion error vector elements x 103 0 20 40 60 80 100 Times(s) B Torque Error 0 20 40 60 80 100 0 20 40 60 80 100 Times(s) Times(s) C Alpha D Beta 12. Simulation results for HSL with ao = 0.01, /o = 2, a = 1, b = 3, and 1 = P2 = 1 near elliptic singularities IJ Figure 5 0.03 0.02 _d82/dt _d83/dt _d84/dt 20 40 60 80 100 Times(s) A CMG gimbal rates 0 20 40 60 80 100 Times(s) C Singularity measure S103 15 10 a 5 "1 n2 n "3 n4 5 ..' 0 20 40 60 80 100 Times(s) B Null motion 0.3 0.25 S0.2 S0.15 0.1 0.05 0 0 20 40 60 80 100 Times(s) D det(Q) Figure 513. Simulation results for HSL with co = 0.01,/3o 1 = P2 = 1 near elliptic singularities (contd.) 2, a = 1, b = 3, and For Case II near an elliptic singularity, Table 53 compares the RMS gimbal rates (deg/s) and tracking performance in terms of RMS torque error (Nm) for LG, SDA, and HSL. In this table the LG method is said to have an infinite RMS gimbal rate to point out that it failed for elliptic singularity. Also, it is shown that SDA and HSL were successful in completing the maneuver while escaping singularity. Both SDA and HSL had approximately the same performance for elliptic singularity with the exception of slightly better tracking performance for HSL. NIL 'C' Table 53. Performance Comparisons for Case II: Elliptic Singularity Steering Algorithm sRMS TeRMS LG c0 7.7159e06 SDA 8.2564 29.8989 HSL (mo = 0.5) 8.1366 26.6946 5.3.3 Case 3: Near Hyperbolic Internal Singularities J = [15 105 195 75]T deg The final set of simulations starts at initial gimbal angles 6 = [15 105 195 75]T deg which represents a scenario near an hyperbolic singularity at (i.e., a distance 15 deg from each CMG away from the singularity at 6 = [0 90 180 90]T deg). 5.3.3.1 Local gradient simulation results The parameters for the LG simulation are: ao = a = b = p = 0 and /2 = / = 1. The transient response of the error for the LG method in Figure 514 A is identical to that for the other two cases. This is because the LG method is an exact mapping evident from a = 0 in Figure 514 C and has no torque error associated with it in theory as shown in Figure 514 B. The null motion in Figure 515 B makes up almost the entire contribution of the gimbal rates in Figure 515 A due to the nonzero value of 3 in Figure 514 D. The LG method by itself is able to avoid the hyperbolic singularity swiftly and remain away as shown in Figure 514 C and D. 0.02\/ e 0 e, < 0.02 0.04 0.06 0.08 0 20 40 60 8 Times(s) A Quaternion error vector elements 0.1 0.05 0 0.05 0.1 0 20 40 Times(s) C Alpha 60 80 x 1013 3 2.5 2 E 1.5 1 0.5 0 20 40 60 80 Times(s) B Torque Error 0 20 40 Times(s) D Beta 60 80 Figure 514. Simulation results for LG with ao = a = hyperbolic singularities b = 1 = 0 and 2 = Po = 1 near _d81/dt _d83/dt d6 4/dt d684/dt 20 15 _10 05 0 5 n2 3 "4 20 40 60 80 0 20 40 60 80 Times(s) Times(s) A CMG gimbal rates B Null motion 0 20 40 60 80 Times(s) C Singularity measure Figure 515. 0 0.01 0.02 0.03 0.04 0.05 0.06 0 Simulation results for LG with ao = a = b hyperbolic singularities (contd.) 20 40 Times(s) D det(Q) /i = 0 and P2 60 80 = /o = 1 near 5.3.3.2 Singular Direction Avoidance simulation results The parameters for the SDA simulation are: /o = a = b = p2 = 0,/1 = 1 and co = 0.01. The transient response of the error for the SDA method in Figure 516 A is different in the rate of convergence to ess, but on the same order of magnitude to that for the LG method. However, the gimbal rates for SDA shown in Figure 517 B are an order of magnitude smaller than that for the LG method. This method escapes the hyperbolic singularity successfully with torque error as shown in Figures 516 B as a consequence of the nonzero value of a in Figure 516 C. The singularity metric P in 516 D is zero because SDA does not use null motion. Added torque error for singularity escape versus null motion for singularity avoidance is the trade off between SDA and LG. The singularity in this simulation is verified to be hyperbolic from the negative result shown in Figure 517 D. Also, the SDA method did not escape by what is considered singularity in Figure 517 C by the threshold m < 0.5. However, this did not affect the decaying of the errors transient response. This is due to the fact that the torque error is scaled by the needed output torque being mapped and therefore, is not seen to have a significant effect towards the end of the maneuver. 0.02\X e 0__ e, a 0.02 0.04 0.06 0.08 0 50 100 Times(s) A Quaternion error vector elements 0 50 100 Times(s) C Alpha 0 50 100 Times(s) B Torque Error H, 0.05 0.1 0 50 Times(s) D Beta Figure 516. Simulation results for SDA with ao = 0.01, /o = 0, a = 0, b = 0, and / = 1 near hyperbolic singularities 100 0.05 _d82/dt _d83/dt d6 4/dt d684/dt 0.05 0.05 0.1 50 Times(s) A CMG gimbal rates 0.01 0.02 o 0.03 50 1 Times(s) C Singularity measure "1 n"2 3 "4 50 Times(s) B Null motion 50 100 Times(s) D det(Q) Figure 517. Simulation results for SDA with co = 0.01, /o near hyperbolic singularities (contd.) = 0, a = 0, b = 0, and p = 1 0.04 0.05 0 5.3.3.3 Hybrid Steering Logic simulation results The parameters for the HSL simulation are: ao = 0.01, /o = 2, a = 1, b = 3, and p 1 = 12 = 1. The transient response of the error in Figure 518 A is almost identical to the LG method for this case and has a faster rate of convergence to ess than SDA. This is attributed to the nonzero values of the singularity metrics a and / in Figures 518 C and D which provide an order of magnitude less null motion for singularity avoidance than LG and orders of magnitude less torque error than SDA for this case shown in Figure 519 B and 518 A when avoiding the hyperbolic singularity verified in Figures 519 C and D. Unlike SDA, HSL escaped and then avoided the singularity which is due to the addition of null motion for this method (see Figures 516 C and 517 C). Therefore, HSL relies more on null motion for singularity avoidance rather than soley trying to pass through the hyperbolic singularities as SDA, SR, and GSR do. Precision in attitude tracking with the threat of hyperbolic singularities while still being able to escape elliptic singularities is the strength of HSL. 102 0.04 0.02 e 0 e e3 < 0.02 0.04 0.06 0.08 0 20 40 60 80 Times(s) A Quaternion error vector elements E 0.04 z P 0.03 0.02 0.01 0 20 40 60 80 Times(s) B Torque Error 0.07 0.06 0.05 0 20 40 60 80 0 20 40 60 80 Times(s) Times(s) C Alpha D Beta Figure 518. Simulation results for HSL with co = 0.01,/o = 11 = 1/2 = 1 near hyperbolic singularities For Case III near a hyperbolic singularity, Table 53 compares the RMS gimbal rates (deg/s) and tracking performance in terms of RMS torque error (Nm) for LG, SDA, and HSL. In this table the LG method has the largest RMS gimbal rate among the three methods, which is needed for singularity avoidance. Also, LG performed the method with the best tracking performance among the three methods which is an expected result for an exact method. The HSL had better tracking performance in terms of RMS torque error than SDA as a consequence of the larger gimbal rates needed for null motion singularity avoidance. This is an expected strength of HSL at hyperbolic singularity. 103 2, a=l, b 3, and 2.5 2 _n "1 "2 S1 d6 dt 1. .............d64ldt : 0, 4 Foa 1 44 S1 2 0.5 0 20 40 60 80 0 20 40 60 80 Times(s) Times(s) A CMG gimbal rates B Null motion 0 1.2 1 0.02 E 0.8 0.04 0.6 0.06 0.4 0 20 40 60 80 0 20 40 60 80 Times(s) Times(s) C Singularity measure D det(Q) Figure 519. Simulation results for HSL with co = 0.01, /0 = 2, a = 1, b = 3, and P1 = P2 = 1 near hyperbolic singularities (contd.) Table 54. Performance Comparisons for Case III: Hyperbolic Singularity Steering Algorithm sRMS TeRMS LG 10.3905 1.4742e05 SDA 6.3611 14.4937 HSL (mo = 0.5) 9.9330 4.7925 5.4 Hybrid Steering Logic Summary The HSL was found numerically to preserve attitude tracking precision in the presence of hyperbolic singularities, act comparably to SDA in the presence of elliptic singularities, and perform better than SDA away from singularity. The performance of this algorithm is attributed to the new singularity metrics, which allow smooth transition between singularity avoidance using LG and singularity escape using SDA. By reducing the times where torque error is induced for singularity escape, this method provides improved attitude tracking performance. Analytic and simulated results show that HSL has many benefits over the two other methods for singularity avoidance and escape. These benefits are: it can be implemented realtime; although SVD may be computationally intensive, it removes the need for an inverse and provides all the information needed for HSL; numerically robust algorithms exist for SVD; HSL induces less torque error than SDA by itself; and finally, the HSL provides a nonsingular expression that can start at singularity. The HSL is not successful in avoiding gimballock because null motion is nonexistent at elliptic singularities and SDA fails at gimballock (see Chapter 4). 105 CHAPTER 6 CONTROL ALGORITHMORTHOGONAL TORQUE COMPENSATION 6.1 Attitude Controller with OTC Traditionally the control law and steering algorithm are separated for attitude control systems using SGCMGs as shown in Figure 61. This is done to facilitate understanding of the attitude control system and actuator dynamics separately. However, considering the steering algorithm separate from the control law may reduce the possibility of an increase in the performance in the system. W"s Separated! (d 7 /^ l qe 7 WWWWWWW h,,/ q, q  q,w Figure 61. Satellite attitude control system block diagram Many steering logics by themselves are incapable of avoiding gimbal lock. Gimballock occurs when the required torque for an attitude maneuver is along the singular direction. This produces a local minimum condition where the gimbal rate solution is zero while the required torque is still not met. Openloop methods that provide a gimbal trajectory free of this condition exist; examples of such methods are forward propagation from preferred gimbal angles, global steering, and optimal control [65, 66, 74, 83]. These methods are time consuming and cannot guarantee a solution exists for the constraints provided. Realtime solutions to gimballock avoidance exist such as the GSR inverse which uses offdiagonal dither components in its perturbation matrix to escape gimballock (see Chapter 4). There is no formal proof that these methods will always be successful in avoiding gimballock. Through the use of nonlinear control, an orthogonal torque 106 compensation (OTC) methodology can be augmented with a suitable steering and control algorithm to also avoid or escape gimballock. Through this nonlinear control framework, stability can be proven and the steering algorithm can be chosen separately in contrast to GSR which relies entirely on handling gimballock avoidance/escape through the steering algorithm. Openloop methods such as optimal control for gimballock escape or avoidance may not find a feasible solution or a solution at all depending on how the cost function and constraints are formulated. It is possible that combination of an optimal control maneuver with a pseudoinverse method (e.g., SR inverse) will drive the system toward the vicinity of singularity as the maneuver is completed. This may occur since the required gimbal rates are not only scaled from the distance to singularity, but also by the needed output torque from the SGCMG system. As the next resttorest maneuver is needed the torque may be required about the singular direction. When this occurs, the maneuver could cause the local minimum previously discussed. 6.2 Lyapunov Stability Analysis For the cases considered, OTC will be a modification to the quaternion regulator control logic from reference [54] shown in Eq.(512). It should be noted that this modification could, in theory, work with any control algorithm which in turn can be combined with any steering algorithm for SGCMGs. Therefore, it is not restricted to any steering algorithm or the quaternion regulator control law if the proper stability analysis is carried out. The quaternion regulator control logic assumes perfect information and has the following nominal form h = Ke + Cw + H (61) where K = 2kJ and C = cJ are positivedefinite symmetric gain matrices based on the spacecraft's centroidal inertia J, e is the vector elements of the quaternion error vector, 107 w is the spacecraft angular velocity, and H is the total spacecraft system centroidal angular momentum from Eq.(21). Recall from Eq.(412) that the Jacobian's left singular vectors U is an orthonormal basis for the output torque h. This basis is composed of a unit vector in the direction of the singular direction ull when at singularity, and two unit vectors orthogonal to the singular direction, u_ and u, (i.e., even when A is nonsingular, the basis from U still exists). Utilizing this basis in the formation of the output torque yields h* = Pull + u + Cu, h (62) with coefficients p= hTll h = h Tu + ag(m) (63) S= iun bg(m) The quantity g(m) is a augmentation to the orthogonal to the singular direction components of torque that is an explicit function of the singularity measure. It will be referenced as the OTC singularity parameter and a and b are switching elements defined by 1 if h u > 0 ba 1 if nun < 0 Substituting Eq.(62) into Eq.(523) and bounding yields, 108 10 0 V =< TJU O 1 0 UTJW + 2 = _ TR2W + 2 (64) 2k 0 0 A1 where C2 = C1 + g(m)wTKl(au1 + bun) and R2 = R1 in Eq.(525) with the singularity parameter A from Eq.(45) in place of a for the HSL in Eq.(51). Similar to the Lyapunov analysis in Chapter 5 for HSL, the error z is bounded with a Volterra integral expression as  z 2 < V(0) exp( x ( 1 exp(_ (t) r) d (65) Amin Amin to Since the SGCMG output torque will always have components orthogonal to the singular direction when near singularity, it is assumed that a system using OTC will never encounter gimballock up to a specific size of lel and A from Eq.(45) where g(m) = Allel (66) Therefore, situations of singularity other than those with gimballock are of concern. When at singularity, the expression for the error is Iz12 < + e r)d (67) because the error z is based off the transient term of the Lyapunov equation Vs from Eq.(72) and the dynamic term containing the effect of the torque error added for gimballock escape ft C2(r)dr. Recall from Chapter 5, while using HSL, that when singularity occurs with the exception of gimballock, there may be a period of instability and it is assumed that the maneuver does not end at singularity. With this in mind, the steadystate error of the of a system away from singularity using SDA combined with OTC is bounded as 109 iz(oo)l12 < m 2(00) (68) through the use of L'Hospital's rule as in Eq.(533). The result of OTC is UUB for sufficiently large the choices of c rather than (. This is true because short periods of instability may arise, but the negative semidefinite term of Eq.(64) becomes negative definite away from singularity and will become dominate for sufficiently large values of c. With the choice of g(m) in Eq.(66), whenever there is an attitude error and the system is in proximity to a singularity, there will be torque error added orthogonal to the singular direction, and thus gimballock will be escaped. 6.3 Numerical Simulation For the steering algorithms of SDA, GSR, and SDA with OTC (SDA/OTC) augmented to the attitude controller, two cases were simulated for a fourSGCMG pyramidal cluster at 8 = 54.74 deg and the model parameters in Table 61: (1) a zaxis maneuver starting at initially at the zero momentum configuration from Chapter 5 (i.e., 6 = [0 0 0 O]T deg) and (2) a zaxis maneuver starting at gimballock configuration (i.e., 6 = [90 90 90 90]T deg). Both cases use the same control gains applied to a pyramidal arrangement of fourSGCMGs. Also, the simulation was propagated with a discrete fourthorder RungaKutta at a timestep of At = 0.02 sec. Table 61. Model Parameters Variable Value Units J 13x3 kgm2 0 54.74 deg eo [0 0 0.3] wo [0 0 0]T deg/s ho 1 Nms k 2 1/s2 c 10 1/s ei 0.1 rand(1) Ao 0.1 At 0.02 sec 110 6.3.1 Case 1: 60 = [0 0 0 0]T deg For a zaxis maneuver originating from an initial zeromomentum configuration, Figures 62 show that SDA, GSR, and the SDA/OTC appear identical. This is what is expected for a maneuver far from singularity. The results for the torque Figure 63 confirm this because the transient response for this case (i.e., away from singularity) is short. The transient response of the output torque shown in Figure 63 D of SDA/OTC has significant jitter but with a small magnitude. This jitter has negligible effect on the gimbal rates shown in Figure 62 D which is due to the mapping of the output torque onto the gimbal rates. The difference in quaternion error and singularity measure in Figures 64 and 65 are small. This should not be surprising since for SDA, GSR, and SDA/OTC, the contributions of torque error are designed to be significant only when the system is close to a singularity. The OTC singularity parameter shown in Fig. 66, while initially nonzero for this case, converges to zero rapidly. The fact that this parameter is nonzero initially and there is no significant differences in the quaternion error responses as shown in Figure 67, might suggest that the torque error from the SDA method itself was dominant. In addition, it should be noted that the difference in quaternion error responses while small (108), is not on the order of machine precision (1016) or (1032). V 0) ro o 0.3 0.25 0.2 ~0.15 S0.1 0.05 0 0 0.3r 0.25 0.2 S0.15 0.1 0.05 0 0 80 100 0.5 0 1 2 3 4 times) B SDA (transient response) 0.5 0 20 40 60 times) C GSR 80 100 0.5 0 1 2 3 4 times) D GSR (transient response) 0.5 co 0 20 40 60 times) E SDA/OTC 80 100 0.5 0 1 2 3 4 times) F SDA/OTC (transient response) Figure 62. Gimbal rates 112 20 40 60 times) A SDA 0.3r 0.25 S0.2 S0.15 S0.1 0.05 1.2 1 0.8 E Z z0.6 0 S0.4 0.2 0 0 20 40 60 80 100 times) A SDA 1.2 1 0.8 E z0.6 0 S0.4 0.2 0 0 20 40 60 80 100 times) C GSR 1.2 1 . 0.8 E z 0.6 0 o c 0.4 0.2 0 20 40 60 80 100 times) E SDA/OTC times) B SDA (transient response) 0.05 E  0 0 o 0.05' 0 1 2 3 4 5 times) D GSR (transient response) times) F SDA/OTC (transient response) Figure 63. Output torque 113 0.3 0.25 0.2 a 0.15 0.1 0.05 0 0 20 40 60 80 100 times) A SDA 0.3 0.25 0.2 a< 0.15 0.1 0.05 0 0 20 40 60 80 100 times) B GSR 0.3 0.25 0.2 a 0.15 0.1 0.05 0 20 40 60 80 100 times) C SDA/OTC Figure 64. Vector elements of the error quaternion 1.095 E 1.09 1.085 1.08 0 50 100 times) A SDA 1.1 1.095 E 1.09 1.085 1.08 0 50 100 times) B GSR 1.1 1.095 E 1.09 1.085 1.08 0 50 100 times) C SDA/OTC Figure 65. Singularity measure 115 0 10 20 times) Figure 66. Singularity parameter (OTC) x 109 0 1 2 3 0 10 2 0 10 20 times) A eGSR eSDA 30 40 0 10 20 30 40 times) B eSDA/OTC eSDA Figure 67. Quaternion error difference: (A) eGSR eSDA (B) eSDA/OTC eSDA 6.3.2 Case II a: 60 = [90 90 90 90]T deg For this case, the gimbals are initially oriented such that the system is in a gimballock configuration. Figure 68 A shows that the gimbal rates of the SDA method are unchanged throughout the simulation since the system starts in a gimballock configuration and SDA cannot generate the necessary commands to escape. The gimbal rates for GSR and SDA/OTC (Fig. 68 B and D), however, are nonzero because the addition of the torque error has provided the system with the ability to escape gimballock. In addition, the controller approaches the original quaternion regulator controller as the system moves away from singularity. 116 30 40 The transient response of the gimbal rates for the GSR and SDA/OTC in Figure 68 C and E are both oscillatory with GSR having the higher amplitude and duration. This is attributed to the fact that unlike OTC, the functions adding torque error in GSR for gimballock escape are not clearly visualized (i.e., depend on the combination of sinusoids with possible different frequencies and phases for dither) when mapped to the gimbal rates. Figures 69 and 610 show, respectively, the required torque and attitude error. In both cases, the results show a similar trend as the gimbal rates for GSR, and SDA/OTC approach zero;the GSR and SDA/OTC were able to generate the torque required to drive the attitude error to zero. An examination of the singularity measures shown in Figure 611 reaffirms the responses shown in Figures 68 through 610 where the SDA remains at singularity unlike GSR and the SDA/OTC which escape singularity but transition back to it as the maneuver is completed. This transition back to singularity is common for all pseudoinverse steering algorithms, which work by approaching a singular configuration and then making a rapid transition for escape [47]. Recall previously from 6.1, that it was stated that it is possible to end in the vicinity of a singularity when the maneuver was completed; this is an example of such a case shown in Figure 610 and 611. The measure of how far the system of SGCMGs is from gimbal lock can be found as the norm IIATh 0. Both the GSR and the SDA/OTC were successful in escaping gimballock as shown in Figure 612. It should be noted that because this measure is a function of h, it goes to zero as the maneuver is completed. The OTC singularity parameter is shown in Figure 613. It has a nonzero initial value and converges rapidly to zero which makes it effective for helping in singularity escape. 117 0.1 0.05 0.1 50 100 times) A SDA 0 0.5 0.5 2I 2.5 3 0.0.5 0 50 100 0 1 2 3 4 5 times) times) B GSR C GSR (transient response) 0.5 o 1 1 2 I 2.5 0.5 20 2 40 60 80 100 0 1 2 3 4 5 times) times) D SDA/OTC E SDA/OTC (transient response) Figure 68. Gimbal rates 118 118 1 0.8 E z S0.6 0 0.4 0 20 40 60 80 100 times) A SDA 0 20 40 60 80 100 times) B GSR 1.2 1 0.8 E z 0.6 0 S0.4 0.2 0 20 40 60 80 100 times) D SDA/OTC times) C GSR (transient response) 0 1 2 3 4 times) E SDA/OTC (transient response) Figure 69. Output torque 119 3 0.3 0.25 0.2 W 0.15 0.1 0.05 0 0 20 40 60 80 100 times) A SDA 0.3 0.25 0.2 a 0.15 0.1 0.05 0 20 40 60 80 100 times) B GSR 0 20 40 60 80 100 times) C SDA/OTC Figure 610. Vector elements of the error quaternion 120 0.01 0.008 0.006 E 0.004 0.002 00 50 100 times) A SDA 0.8 0.6 E 0.4 0.2 0 20 40 60 80 100 times) B GSR 0.6 E 0.4 0.2 0 20 40 60 80 100 times) C SDA/OTC Figure 611. Singularity measure 0.1 0.05 I I 0.05 0.1 0.2 0.15 0.1 0.05 n 1020 10 20 times) A SDA 0 10 20 times) B GSR .2 5 .1 15 0 1 20, 0 10 20 times) C SDA/OTC 30 40 30 40 30 40 Figure 612. Gimballock measure 122 0.025 0.02 E 0.015 0.01 0.005 0 10 20 30 40 times) Figure 613. Singularity parameter (OTC) 6.3.3 Case II b (HSL/OTC): 60 = [90 90 90 90]T deg Recall, from 6.1 that OTC can be used in combination with any steering algorithm for gimballock avoidance/escape. This case verifies through simulation that this is indeed true by comparing HSL/OTC to GSR starting at gimballock (60 = [90 90 90 90]T deg). The HSL parameters are shown in Table 62. Table 62. Hybrid Steering Logic Parameters Variable Value ao 0.01 0o 2 Pil 1 P2 1 a 1 b 3 mo 0.5 With the exception of the initial transient, the gimbal rates for GSR and those of HSL/OTC in Figure 614, are approximately the same magnitude. 123 C 0 U.0 0.5 1.5 S0o V 2 V \ o "o I 2.5 30. 0 50 100 0 1 2 3 4 5 times) times) A GSR B GSR (transient response) 3 0.5 2 1 0 0 2 0.5 0 20 40 60 80 100 0 1 2 3 4 5 times) times) C HSL/OTC D HSL/OTC (transient response) Figure 614. Gimbal rates The transient response of the gimbal rates for the GSR in Figure 614 B is highly oscillatory and not as smooth as that for HSL/OTC (compare with Figure 614 D). This is due to oscillatory behavior of the ditherused for gimballock escape that may be of any duration depending on the frequencies and phases of the offdiagonal components of the GSR perturbation matrix E. The HSL method acts as a SDA method, but when combined with OTC will avoid/escape a singularity at the speed of the parameters chosen for A in Eq.(66) in the which the duration will be understood for all singularities and their combinations to the norm of quaternion error. Figures 615 and 616 show, respectively, the required torque and attitude error. In both cases, the results show that both methods (GSR and HSL/OTC) were successful in meeting the required torque and completing the attitude maneuver. 20 40 60 times) A GSR 1.2 1 0.8 E z0.6 0 c 0.4 0.2 0 20 40 60 0 20 40 60 times) C HSL/OTC 80 100 B GSR (transient response) 80 100 0 1 2 3 4 times) D HSL/OTC (transient response) Figure 615. Output torque The singularity parameters for both methods in Figure 617, escape singularity although they transition back to it as the maneuver is completed. Recall, it was mentioned previously that a maneuver can be completed (i.e., e 0) while the gimbal angles settle into a singular configuration; Figure 611 and 617 show this trend (compare with Figure 611). Figure 618 shows instantaneous escape from gimballock for both methods. 125 1.2 1 .0.8 E S0.6  0.4 0.2 n 40 60 80 100 40 60 80 100 times) A GSR 0.3 0.25 0.2 0.15 0.1 0.05 0 C 20 40 60 times) B HSL/OTC 80 100 Figure 616. Vector elements of the error quaternion 40 60 80 100 0 20 40 60 times) times) A GSR B HSL/OTC Figure 617. Singularity measure The OTC singularity parameter shown in Figure 619 has an initial nonzero value and converges rapidly to zero similarly to Figure 613, which makes it effective for helping in singularity escape. The Table 63 compares the rootmean squared (RMS) gimbal rates (rad/s), tracking performance in terms of RMS torque error (Nm), and pointing performance in terms of the norm of the steadystate error quaternion for GSR, SDA/OTC, and HSL/OTC. From Table 63 it can be shown that the choice of the singularity threshold mo has an effect on the tracking and pointing performance of the HSL method combined 126 0.3 0.25 0.2 a 0.15 0.1 0.05 0 20  ' 0.15 0.15 SI 0.1 0.1 0.05 0.05 0 0 10 20 30 40 0 10 20 30 40 times) times) A GSR B HSL/OTC Figure 618. Gimballock measure 0.025 0.02 E 0.015 0.01 0.005 0 10 20 30 40 times) Figure 619. Singularity parameter (OTC) with OTC. In fact, when this value is mo = 0.5 for this model and with the set of control gains that differ from the model in Chapter 5, the tracking and pointing performance of HSL/OTC is actually worse. This is expected as shown by the Lyapunov analysis in 6.2 where the steadystate error of SDA is dependent on the torque error added into the system; and a larger threshold value of mo will increase the steadystate error. 6.4 Orthogonal Torque Compensation Summary Orthogonal torque compensation (OTC) methodology was developed to ensure escape from all singularities, particularly scenarios involving gimballock configurations. The compensation methodology can be incorporated with any attitude controller/steering 127 C Table 63. Performance Comparisons Steering Algorithm 6RMS TeRMS I essII GSR 7.7901 10.1864 0.0024 SDA/OTC 7.3146 10.4080 0.0020 HSL/OTC (mo = 0.5) 7.1693 6.1819 0.0038 HSL/OTC (mo = 0.05) 7.0041 3.0416 2.1474e09 HSL/OTC (mo = 0.005) 6.9166 2.9852 1.8843e09 logic combination and was shown through analysis to ensure stability with sufficiently large choice of the controller gain c. Since the compensator was designed to work with any attitude controller, then it is compatible with any steering algorithms. This could prove very beneficial for steering algorithms like HSL which reduce the amount of torque error at hyperbolic singularities (see Chapter 5). The OTC was also demonstrated through numerical simulation where it was shown to be effective in escaping gimballock with near zero steadystate attitude error. These simulations were based on a fourSGCMG pyramidal arrangement using an quaternion regulator controller combined with the steering algorithms SDA and HSL and compared with GSR. 128 CHAPTER 7 SCALABILITY ISSUES FOR SGCMGS 7.1 Scalability Problems with SGCMG Hardware Currently available CMG actuators are shown in Figure 71 with specifications from Table C1 in Appendix C do not meet the power, mass, and volume requirements for satellites smaller than the microsat class. Currently, development of CMG hardware underway will meet some of the constraints for these smaller classes of satellites. New steering algorithms to complement these newly developed CMG hardware is not being emphasized and will have a major effect on how systems of miniature CMGs perform. This chapter highlights the effect of scaling on the performance of miniature CMGs. L./ Power and Mass Figure 71. Offtheshelf CMGs 129 Torque "0 to 50 Nm CM s Be^^ ing^^^ Developedl^^ Power ~ 0 to 20 W Mass 0 to 15 kg 7.2 Effect of g,, on Torque Error The gimbal accelerations are kinematically dependent on the choice of the gimbal rates and as a consequence, only one of them can be used as a control variable. Therefore, the gimbal rates are considered as measurable quantities and the gimbal accelerations are the control. The solution to the gimbal accelerations as a control is defined as = BT(BBT)1[T A26] (71) where A = A1 + A2 from Eqs.(27) and (A26) and T = h + A16 equivalent to Eq.(213). This solution is considered an exact solution but for some cases may be highly oscillatory and/or unstable for the gimbal rates and accelerations. A Lyapunov analysis is presented below. It was stated previously that the direct solution in Eq.(71) may be unstable. To prove this we start with the given candidate Lyapunov function V1 = w'K1Jc + eTe + (1 e4)2 (72) 2 Taking the time derivative, yields V = TK1[T wx H] + Te (73) For the system to be globally asymptotically stable (i.e., h, e, and w > 0 as t oo) T = A + B6 = Ke + Cw w H (74) Next, a second candidate Lyapunov function is required to analyze the behavior of the gimbal rates and accelerations as time approaches infinity. 2 (7 V2 = _T6 (75) 2 130 Taking the time derivative utilizing Eq.(71), we obtain V2= 6 6 = TBT(BB T) [T A6] (76) From the previous Lyapunov analysis where h, e, and w 0 as t oo it can be assumed that V2= TBT(BBT) 1A (h )6TS (77) Igw where B = IgwB, A = hoA, and S = BT(BBT)A. Matrix S is semiindefinite and therefore the gimbal rates can be unstable. Furthermore, a Lyapunov analysis of V,+ V2 shows that the ratio ( ') plays a key role in the stability of the whole system. Next, we consider the use of the SR inverse where the gimbal rates are found from 6sR = ASR(T Igw^) (78) ho with ASR AT(AAT + A)1 (79) where A is the singularity parameter defined in Eq.(45). Assuming the SR inverse is used to apply the gimbal rates as a control variable we find that the torque error is expressed as Te = Tct T = hoA6sR + IgwB T (710) Furthermore, AASR = [ + A(AAT)]1 (711) Away from singularity, a series expansion of Eq.(711) with only the linear terms yields AASR I A(AAT)~ This series expansion is convergent if away from singularity because the term I(AAT)11 < 1. Substituting Eq.(712) into the torque error, Eq.(710) we have Te I/gw,(AAT)1 A(AAT)1T (713) It can be seen that the torque error may be amplified by the magnitude of the gimbalflywheel inertia Igw. Furthermore, if 6 or Ig, is considered negligible then the torque error is only affected by the singularity parameter A, the distance from singularity which is related to the determinant of (AAT)1, and T. It should be noted that an increase in g,, is followed by a decrease in AAT, but its effectiveness in lowering the torque error requires a large ratio of 1 (i.e., effective when >> 1 which could be thought of as being a system of RWs). The eigenaxis control logic from Eq.(512) is used to define the torque needed for a given maneuver to be mapped onto the gimbal states. The SGCMG system proposed in this analysis assumes that it is selfcontained and therefore the metric of the hosted algorithm performance is independent of the control logic chosen as long as it meets the constraints of the SGCMG actuators. Therefore, no generality is lost for the choice of the control logic in the analysis. 7.3 Numerical Simulation The cases compared here are the SR Inverse and a filtered gimbalacceleration control law based on Oh and Vadali [84]. The filtered gimbal acceleration control law has the following form 6= K6(6sR 6) 6R (714) 132 (712) where K6 is the gain matrix that sizes the amount of gimbal acceleration utilized for control and 6SR and 6SR are the gimbal rates and accelerations from the SR inverse and the time derivative of that rate. The effect of the gimbalflywheel inertia is scaled in the simulation by the gain Kg, (i.e., gw, = Kg1wgw) where Kg, = 0 signifies that their is no torque or angular momentum contributed from the gimbal dynamics. Simulations of these two steering algorithms were compared by scaling g,, through two different values of Kg,: i) Kg, = 0 and ii) Kg, = 2. The model parameters for the nominal satellite inertia J and gimbalflywheel inertia g,, are based on a fourSGCMG pyramidal arrangement sized for a 1 U CubeSat. Both simulations were for a maneuver of 1800 about the zaxis. The initial gimbal angles for all simulations are 60 = [90 90 90 90]T deg corresponding to a elliptic saturation singularity about the zaxis. This set of initial gimbal angles along with the required maneuver will force the system to enter gimballock (i.e., ATh = 0) and accumulate a steady state attitude error. This situation was chosen to test the system to its performance limit. The parameters that were used for all of the results are shown in Table 71. Table 71. Model Parameters Variable Value Units 533.8 0 0 Js 0 533.8 0 x 106 kgm2 0 0 895.6 0 52 deg eo [0 0 1 0]T Wo [0 0 0]T deg/s ho 4.486 x 104 Nms 7gw 5.154 x 106 kgm2 k 10 1/s2 c 50 1/s K6 10 14x4 Ao 0.5 /P 10 60 [0 0 0 0]T deg/s 133 7.3.1 Case I: Kg,= 0 For this case, Figures 72 and 73 show that the gimbal rates and accelerations are quite similar for both the SR inverse and filtered acceleration control law, except at the very beginning. Since the filtered acceleration control law also uses an SR inverse in its formulation, it is reasonable to assume that the differences can be attributed to the filter (see Eq.(714) ). The torque errors in Figure 74 are initially larger for the filtered acceleration algorithm as compared to the SR inverse by itself. Again, this is most likely due to the lower initial gimbal rates and accelerations attributed to the filter. Also, the torque error for both cases has a steady state offset where the system encounters gimballock. 0 5 Time(s) A SR Inverse 5 1 Co o 1 1 Time(s) C Filtered gimbal acceleration 5 5 0 2 4 6 8 Time(s) B SR Inverse (transient response) 0 2 4 6 8 10 Time(s) D Filtered gimbal acceleration (transient re sponse) Figure 72. Gimbal rates for Kg, = 0 135 300 200 100 0 100 200 300 Time(s) A SR Inverse ) 0 20 40 6( Time(s) C Filtered gimbal acceleration 300 200 100 0 100 200 300 0 2 4 6 8 Time(s) B SR Inverse (transient response) 50 50 0 2 4 6 8 D sponse) Time(s) Filtered gimbal acceleration (transient re Figure 73. Gimbal accelerations for Kgw 136 1 0 E 1 z 2 3 Time(s) A SR Inverse x 103 E 4 ' 6 8 10 0 20 40 Time(s) C Filtered gimbal acceleration Figure 74. Torque error for Kg, = 0 Time(s) B SR Inverse (transient response) x 103 0 5 10 Time(s) D Filtered gimbal acceleration (transient re sponse) 137 The singularity measure m shown in Figure 75 is identical for both methods in these plots. The only discrepancy between the gimbal rates and accelerations of the two methods for this cases was at the very beginning of the maneuver. Therefore, the differences in m would not be obvious in these plots. The value of m here is shown to transit away from but return to singularity in Figure 75 for both methods. This is common to steering algorithms of the SR inverse type. Time(s) Time(s) A SR Inverse B Filtered gimbal acceleration Figure 75. Singularity measure for Kg, = 0 7.3.2 Case II: Kg = 2 The initial gimbal rates and accelerations are less for the filtered acceleration algorithm than for SR inverse itself. This is the shown in Figures 76 and 77. 138 0.01 0 0 V 0.01 5 0.02 0 20 40 60 0 20 40 Time(s) Time(s) A SR Inverse B SR Inverse (transient response) Sx 103 1 1 0 u 1 1 2 0 20 40 60 0 10 20 30 40 50 Time(s) Time(s) C Filtered gimbal acceleration D Filtered gimbal acceleration (transient re sponse) Figure 76. Gimbal rates for Kg, = 2 A closer look at the transient response for the gimbal rates and accelerations of the two methods is shown in Figures 76 B and D and 77 B and D. It is gathered from Figure 76 B and 77 B that the steadystate response for the gimbal rates and accelerations of the SR and filtered acceleration law inverse is nonzero. The steadystate response for the gimbal rates and accelerations of the filtered acceleration law although nonzero, is considerably smaller than that for the SR inverse which in turn prevents the torque error from diverging. However, Figure 78 shows the torque error for the SR inverse appears to diverge where as the torque error for the filtered acceleration algorithm remains bounded. 139 LUU 100 r 0 _ 100 200 200 Time(s) A SR Inverse 10 20 30 40 50 Time(s) B SR Inverse (transient response) 0.06 0.04 0.02 0 0.02 0.04 0.06 0 20 40 60 Time(s) C Filtered gimbal acceleration Figure 77. Gimbal accelerations for Kg, 0 20 40 Time(s) D Filtered gimbal acceleration (transient re sponse) Furthermore, it appears as though the precision is improved with larger Ig, for the filtered acceleration algorithm. This may be due to the fact that it relies less on the SR inverse which would be the source of the torque error in this example. 140 l~r~7~ 0 20 40 Time(s) A SR Inverse Figure 78. Torque error for Kg, 0 20 40 Time(s) B Filtered gimbal acceleration The singularity measures shown in Figure 79 are identical except that the singularity is approached quicker for SR inverse. It should be noted that as the value of Ig, is increased, the singularity measure approaches singularity later in the simulation for the filtered steering algorithm (i.e., larger gimbal rates lead to larger gimbal angle excursions which in turn, make the system approach singularity quicker). x 1010 x 1010 Time(s) Time(s) A SR Inverse B Filtered gimbal acceleration Figure 79. Singularity measure for Kgw = 2 The gimbalflywheel assembly inertia will also have a degrading effect on the actual torque amplification of SGCMG actuators. This will be explained in the next section. 7.4 Effect of Igw on Torque Torque Amplification The torque amplification of a single SGCMG can be described by its output torque divided by the input torque as ,Tout ll lh x6 + gw1,, 6 h (715) 1ini, I Ilw X h + lgwJ ll From Eq.(715) it is seen that as the gimbalflywheel inertia Igw oo, the other terms in the equation become less dominant and the torque amplification converges to one. This is undesirable for SGCMGs because at the point that the torque amplification converges to one, the system essentially becomes a reactionwheel system and the benefits of using SGCMGs are lost. Fortunately, a system of SGCMGs of this scale does not exist. The scaling of SGCMGs does reduce the torque amplification. To show actually how much the torque amplification is degraded by scaling, the value of torque amplification is calculated for the IMPAC SGCMGs in Figure 22 with the parameters in Table 72. Table 72. Model Parameters Variable Value Units 533.8 0 0 Js 0 533.8 0 x 106 kgm2 0 0 895.6 0 52 deg ho 4.486 x 104 Nms Igw 5.154 x 106 kgm2 wmax 3 deg/s 6max 1 rad/s 6max 1 rad/s2 For this example, IIotl Ia +holn + /w max 'out hJomax 'gwmax 43.4 (71 6) 117n 1 Wmaxho + Igw6max Therefore, there is a significant value of torque amplification even when scaling as seen by the result in Eq.(716). 142 7.5 Summary Singularities from systems utilizing SGCMGs cannot be easily scaled when describing the algorithms for their control. Just as the performance on a hardware level for SGCMGs will eventually flatline with scaling, so will the use of current steering algorithms for singularity avoidance. This chapter showed that some current steering algorithms may have difficulty when the gimbalwheel assembly inertia becomes significant. In addition, this chapter also showed that as a consequence of significant values for gimbalwheel assembly inertia, the performance of the SGCMG system is affected by the torque amplification approaching 1 as g,, oo. 143 CHAPTER 8 CONCLUSION Control of spacecraft attitude with singlegimbal control moment gyroscopes (SGCMGs) is difficult and becomes more so with the scaling of these actuators to small satellites. The research presented in this manuscript began with a discussion of the dynamic model for systems containing CMGs and their singularities. For SGCMG systems, singularities were classified and tools were developed to quantify the form of the singularity. These tools provided insights into these singularities (i.e., singular surfaces) and were used to quantify them mathematically. The singularities associated with SGCMGs were discussed in detail and classified by the tools developed. Through this discussion, it was shown a complete explanation of SGCMG singularities is absent from the literature. For example, it was found that the special case of where the singular direction s is along a gimbal axis 6 can occur for rooftop arrangements when the rank of the Jacobian is 2. This was further shown to be a degenerate case which could lead to degenerate hyperbolic singularities that were previously neglected in the literature for systems of SGCMGs. Using linear algebra, it was proven that rooftop arrangements are free of elliptic internal singularities but still contained elliptic external singularities (i.e., all elliptic singularities do not have null motion are are thus impassable by null motion) associated with angular momentum saturation. Furthermore, degenerate hyperbolic singularities were shown to exist for rooftop arrangements (i.e., degenerate hyperbolic singularities contain only singular null solutions exist and are also impassable by null motion). It was shown that selecting an arrangement of SGCMGS through choice of a desirable angular momentum envelope is difficult. Thus, a method of offline optimization was suggested in a very constrained case that will provide the best SGCMG arrangement in terms of Euler angles. However, this method is not that applicable for real spacecraft design, although, it suggested that the current common configurations do not necessarily have the best performance. Next, it was shown that legacy steering algorithms, which can be categorized into the three families of singularity avoidance, singularity escape, and singularity avoidance and escape, did not consider the form of internal singularity (i.e., hyperbolic or elliptic). This was shown to be problematic when precise attitude tracking is required because the same amount of torque error was used for both hyperbolic and elliptic singularities. A Hybrid Steering Logic (HSL) was developed that takes into account the form of singularity. This algorithm uses this knowledge to apply null motion from a local gradient (LG) method for singularity avoidance when near a hyperbolic singularity and torque error from Singular Direction Avoidance (SDA) when near a elliptic singularities. Through analytic derivations and numerical simulations, HSL was shown to perform better (i.e., lower torque error at hyperbolic singularities than pseudoinverse methods and the ability to escape both elliptic and hyperbolic singularities unlike LG methods) than legacy methods for precise attitude tracking when using a fourSGCMG pyramid arrangement of SGCMGs. Also, HSL was shown to have computation of flops on the same order as many legacy methods. Gimballock was shown to be a special case of singularity when the output torque lies in the singular direction of the Jacobian. With the exception of the GeneralizedSingularity Robust (GSR) inverse, legacy steering algorithms are known to be ineffective in escaping gimballock. To provide other steering algorithms with the same benefit as GSR, a attitude controller augmentation defined as Orthogonal Torque Compensation (OTC) was developed. This method was shown to be successful in escaping gimballock by adding orthogonal components of torque error when at singularity. This method was combined with two separate steering algorithms, simulated, and compared to GSR where it was shown numerically to have a much smoother transient response for the gimbal rates. 145 Finally, the problems with scaling SGCMGs were discussed. It was shown that the performance of SGCMGs is degraded (i.e., a lower torque amplification) and same legacy algorithms previously used on larger SGCMGs could be ineffective for scaled SGCMGs. A mathematical proof was used to show that with the increase in the gimbalflywheel assembly inertia Ig, compared to the flywheel angular momentum ho causes this degradation in performance and the ineffectiveness of SGCMG control with use of the Singularity Robust (SR) inverse. The utility of scaled SGCMGs is still viable because the approximate SGCMG torque amplification for a single acutator was shown to be on the order of 50 which is far more than the less than onetoone ratio for systems of reaction wheels. 146 APPENDIX A RIGID BODY DYNAMICS FORMULATION FOR CONTROL MOMENT GYROSCOPE ACTUATORS (SGCMG/VSCMG) A.1 Assumptions The dynamic formulation for single gimbal and variable speed control moment gyroscope (CMG) actuators assumes the absence of friction and external torque in the system (spacecraft including CMGs). In addition, it is also assumed that the center of mass (cm) of each CMG is along its gimbal axis and therefore does not affect the position of the overall cm of the system. These assumptions are valid for current stateoftheart CMGs. A.2 Dynamics The centroidal angular momentum of the system consisting of that from the spacecraft and a single CMG is H = hw + hG + hs/c (A1) with contributions from the flywheel hw, gimbal hG, and the spacecraft hs/c. The flywheel and gimbal angular momentum are expressed as hw = Iwih6 (A2) and hG = g66 (A3) where the gimbal frame basis [6, , 6] is related to the spacecraft bodyfixed basis through a 321 rotation through the angles [6, 0, b] by h = (s6s c c6CcO)ebl (S6Cb C6SCO)eb2 (CSO)eb3 = h bl + h2eb2 + h3b3 (A4) 147 i = c, s',e, + s' s,e, + c06b3 = tlebl + t26b2 + t36b3 6 = (c6s s6c ceO)bl (c6c s6s~ceO)b2+ (s6sO)eb3 = d1bl +d2b2 +d3b3 (A6) where c(.) = cos(.) and s(.) = sin(.) and [ebl, 6b2, 6b3] is the basis for the spacecraft body frame. Therefore, equivalent vector components for these angular moment shown in the spacecraft bodyfixed basis are hw = Iwj(hlbl + h2b2 + h3b3) (A7) and hG = Ig3(dleb d2b2 d3eb3) (A8) where Iw and Ig3 are the first and third components of the flywheel and gimbal inertias. The angular momentum from the spacecraft is expressed as the tensor product of the spacecraft centroidal inertia dyadic Jc with the inertial spacecraft angular velocity w. hs/c = Jc. (A9) The spacecraft centroidal inertia dyadic is Jc = IG + JO mGW (rC ricl r 0 rc) (A10) where re is the position of the cm of a CMG's cm from the cm of the system expressed as rc = rlebl + rc2eb2 + rc3eb3 (A1 1) and the static spacecraft inertia dyadic Jo is made up of constant inertias (i.e., assuming that the cm of the CMGs lies along the gimbal axis) and the inertias due 148 (A5) to the gimbalwheel assembly IG are time varying due to the rotation about the gimbal axis. The expression of the static spacecraft inertia dyadic is 3 3 Jo = #bi eb (A12) i=1 j 1 where (ebi ebj) ebi = 0 and (ebi ebj) ebj ey= bi. It is assumed that the gimbalwheel assembly inertia is aligned with the principle axes and can be expressed as IGW =Igh h + lg2T 0 + g36 0 6 (A13) where 3 3 6 6 = hihJb, 0 ib (A1 4) i=1 j 1 3 3 S0+ = titibi 0 bj (A15) i=1 j=1 3 3 6 = Y d d, dj bi bJ (A16) i=1 j 1 The equations of motion (EOM) assuming torque free motion (i.e., no external torques) are found through taking the inertial time derivative of the total system centroidal angular momentum in Eq.(A1) as dH, o dH ,i M H +l w x Hc =0 (A17) where He = [(al + a21)6 b i + c Q] Jc W (A18) The final EOM for a single CMG that has a single gimbal is [(all + a21)6 + b1j + c] +Jc +W + w x H= 0 (A1 9) The Jacobian matrices all, a21, bl, and cl are 149 ha OIG a21  96 9h bi = 96 (A20) (A21) (A22) 8h c = A (A23) where the CMG angular momentum h = hw + hG. For a system of CMGs with a single gimbal, the EOM concatenated into matrix which is a consequence of Eq.(A19), is expressed as [(A1 + A2)6 + B6 + CQ] + JcW + 0x He = 0 (A24) where for n CMGs the Jacobian matrices are represented as A = [all, a12, a13, ...ain] A2 = [a21, a22, a23, ...a2n] B = [b b ,, b3,...b] C = [c, C2, c3 ...cn] (A25) (A26) (A27) (A28) This concludes the development of the EOM for a rigid body spacecraft system of n CMGs which contain a single gimbal. 150 APPENDIX B MOMENTUM ENVELOPE CODE function [hx,hy,hz] Momentum_Envelope_PM(th,si,h0,int_ext) % % This code is generate the singularity surfaces for a % any general SGCMG cluster with skew angle theta or % inclination angle phi(i) and spacing angle si(i) % where i = num_CMG. % % The angles th(i) and si(i) are the Euler angles relating the % spin axis of each CMG to the body frames Xaxis % %Frederick Leve %Last updated: 07/08/08 %_______________________________________    % % This function simulated the CMG algorithms % _   % INPUTS: % hO = nominal SGCMG angular momentum (could be vector if each % CMG does not have the same angular momentum % % th = vector of inclination angles % si = vector of spacing angles % OUTPUTS: % hx = angular momentum of envelope in the xdirection % hy = angular momentum of envelope in the xdirection 151 % hz = angular momentum of envelope in the xdirection %_____________________________________________________________ % %epsilon parameter vector for surface generation %to show internal singular surface make one epsilon 1 instead of 1 num_CMG = length(si); if length(hO) if int_ext % external singular surface eps = ones(num_CMG,1); elseif intext % internal singular surface % eps = [ones(num_CMGl,1);1]; % eps = [1 1 1 1]; eps = [1 1 1 1]; else display('intext must be either 0 or 1... for internal or external singular surface') end else min_h0 = min(h0); if intext == 0 % external singular surface for i eps(i) end elseif int % internal Sl:num_CMG = hO(i)/min(hO); .ext ==1 singular surface 152 for i = l:num_CMG1 eps(i) = h0(i)/min(h0); end eps(num_CMG) = hO(num_CMG)/min(hO); else display('intext must be either 0 or 1... for internal or external singular surface') end end for 1 = l:num_CMG % The transformation C1 is about the inclination angle phi(i) C1(:,:,l) = [cos(th(l)+3*pi/2) 0 sin(th(1)+3*pi/2); 0 1 0; sin(th(l)+3*pi/2) 0 cos(th(1)+3*pi/2)]; C2 (:, :,1) [cos(si(l)) sin(si(l)) 0; sin(si(l)) cos(si(l)) 0; g(:,l) = transpose(C1(:,:,l)*C2(:,:,1))*[1;0;0]; end 87 %total angular momentum at the singular states corresponding to singular %direction u H = zeros(3,1); n = 100; [x,y,z] sphere(n); %number of grid point for unit sphere %generate the unit sphere (domain of u) redlight = 5; trafficlight = zeros( n+1 n+1 ); 153 for i = l:n+l for j = l:n+l u = [ x(i,j) ; y(i,j) ; z(i,j) ]; for k = l:num_CMG %this is the cosine of angle %between %both of vectors since unit norm u_dotgk = abs(u'*g(:,k)); if ( udot_gk > 0.95 ) traffic_light(i,j) red_light; end for i = l:n+l for j = l:n+l u = [ x(i,j) ; y(i,j) ; z(i,j) ]; %compose the %singularity vector u for k = l:num_CMG end H = H + eps(k)/norm(cross( g(:,k), u ) )... *cross( cross( g(:,k), u ) g(:,k) ); 154 end hx(i,j) = H(1); hy(i,j) = H(2); hz(i,j) = H(3); H = zeros(3,1); %parse out the components of the %momentum vector for later %surface or mesh plotting. end end surfl(hz,hy,hx); alpha(0.05); 155 APPENDIX C CONTROL MOMENT GYROSCOPE ACTUATOR SPECIFICATIONS Table C1. 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Advances in the Astronautical Sciences, 123, 2006. [84] H.S. Oh and SR Vadali. Feedback control and steering laws for spacecraft using single gimbal control moment gyros. Master's thesis, Texas A& M University, 1988. 163 BIOGRAPHICAL SKETCH Frederick Aaron Leve was born in Hollywood, Florida, in 1981. In August 2000 he was accepted into the University of Floridas Department of Aerospace Engineering in the College of Engineering where he pursued his bachelors degrees in Mechanical and Aerospace Engineering. After completing his bachelors degrees in May 2005, he was accepted into the masters program in aerospace engineering at the University of Florida. While in the masters program, he received two awards in academia. In January 2007, he received the American Institute of Aeronautics and Astronautic's Abe Zarem Award for Distinguished Achievement in Astronautics. For this award he was invited to Valencia, Spain, where he competed in the International Astronautical Federations International Astronautical Congress Student Competition. Here he received the silver Herman Oberth medal in the graduate category. He completed the masters program in May 2008 and continued on to his PhD. In May 2006, he was accepted to the Air Force Research Lab (AFRL) Space Scholars Program, where spent his summer conducting space research. After space scholars, he was employed as a student temporary employee at AFRL where he received the Civilian Quarterly Award for all of AFRL in his category. Currently he works in the Guidance, Navigation, and Control group at AFRL Space Vehicles Directorate. His interests include, applied math, satellite attitude control, satellite pursuit evasion, astrodynamics, and orbit relative motion. PAGE 1 NOVELSTEERINGANDCONTROLALGORITHMSFORSINGLEGIMBALCONTROLMOMENTGYROSCOPESByFREDERICKA.LEVEADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2010 PAGE 2 2010FrederickA.Leve 2 PAGE 3 Dedicatedtomymotherforalwaysbeingtheretosupportme 3 PAGE 4 ACKNOWLEDGMENTS IwouldliketothankrstmyadvisorDr.NormanFitzCoyforprovidingmewiththeguidanceandknowledgeforthisgreatresearchIundertook.Second,IwouldliketothankmycommitteemembersDr.WarrenDixon,Dr.AnilRao,Dr.WilliamHagerfromUF,andDr.ScottErwinfromtheAirForceResearchLabSpaceVehiclesDirectorate.Mycommitteecomprisestheexpertiseintheareasofresearchthatwouldprovidemethebestopportunityformyresearch.Last,butnotleast,Iwouldliketothankmycolleaguesinmyresearchlabwhoprovidedinputthroughoutmytimeasagraduatestudentthataidedinthisresearch:Dr.AndyTatsch,ShawnAllgeier,VivekNagabushnan,JosueMunoz,TakashiHiramatsu,AndrewWaldrum,SharanAsundi,DanteBuckley,JimmyTzuYuLin,ShawnJohnson,KatieCason,andDr.WilliamMackunis. 4 PAGE 5 TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 13 CHAPTER 1INTRODUCTION ................................... 15 1.1HistoryandBackground ............................ 15 1.1.1GyroscopicRateDetermination .................... 15 1.1.2SpinStabilizedSpacecraft ....................... 15 1.1.3SpacecraftAttitudeControlthroughGyrostats ............ 15 1.1.43axisAttitudeControlofSpacecraft ................. 16 1.1.5SingleGimbalControlMomentGyroscopes(SGCMGs) ...... 17 1.1.6DoubleGimbalControlMomentGyroscopes(DGCMGs) ...... 17 1.1.7VariableSpeedControlMomentGyroscopes(VSCMGs) ...... 18 1.2ProblemStatement ............................... 18 2DYNAMICMODELS ................................. 20 2.1DynamicFormulation ............................. 20 2.2SingularSurfaceEquations .......................... 25 2.2.1EllipticSingularities ........................... 26 2.2.1.1Externalsingularities .................... 27 2.2.1.2Ellipticinternalsingularities ................. 29 2.2.2HyperbolicSingularities ........................ 29 2.2.2.1Nondegeneratehyperbolicsingularities .......... 30 2.2.2.2Degeneratehyperbolicsingularities ............ 30 2.2.3GimbalLock ............................... 30 2.3SingularitiesforSGCMGsMathematicallyDened ............. 31 3CONTROLMOMENTGYROSCOPEARRANGEMENTS ............ 34 3.1CommonSGCMGArrangements ....................... 34 3.1.1Rooftop ................................. 34 3.1.2Box .................................... 35 3.1.33 4Box .................................. 44 3.1.4ScissorPair ............................... 45 3.1.5Pyramid ................................. 46 3.2ChoiceofArrangement ............................ 48 3.3Simulation .................................... 50 5 PAGE 6 4SURVEYOFSTEERINGALGORITHMS ..................... 54 4.1MoorePenrosePseudoInverse ....................... 55 4.2SingularityAvoidanceAlgorithms ....................... 55 4.2.1ConstrainedSteeringAlgorithms ................... 56 4.2.2NullMotionAlgorithms ......................... 56 4.2.2.1Localgradient(LG) ..................... 56 4.2.2.2Globalavoidance/Preferredtrajectorytracking ...... 57 4.2.2.3GeneralizedInverseSteeringLaw(GISL) ......... 58 4.3SingularityEscapeAlgorithms ........................ 59 4.3.0.4SingularityRobust(SR)inverse .............. 59 4.3.0.5GeneralizedSingularityRobust(GSR)inverse ...... 60 4.3.0.6SingularDirectionAvoidance(SDA) ............ 60 4.3.0.7FeedbackSteeringLaw(FSL) ............... 62 4.3.0.8SingularityPenetrationwithUnitDelay(SPUD) ...... 63 4.4SingularityAvoidanceandEscapeAlgorithms ................ 64 4.4.0.9Preferredgimbalangles ................... 64 4.4.0.10Optimalsteeringlaw(OSL) ................. 64 4.5OtherSteeringAlgorithms ........................... 66 4.6SteeringAlgorithmComputationComparison ................ 66 5STEERINGALGORITHMHYBRIDSTEERINGLOGIC ............. 68 5.1HybridSteeringLogic ............................. 68 5.1.1InternalSingularityMetrics ....................... 68 5.1.2HybridSteeringLogicFormulation .................. 69 5.2LyapunovStabilityAnalysis .......................... 71 5.3NumericalSimulation ............................. 76 5.3.1Case1:AtZeroMomentumConguration=[0000]Tdeg ... 79 5.3.1.1Localgradientsimulationresults .............. 80 5.3.1.2SingularDirectionAvoidancesimulationresults ...... 82 5.3.1.3HybridSteeringLogicsimulationresults .......... 85 5.3.2Case2:NearEllipticExternalSingularity=[105105105105]Tdeg .................................... 85 5.3.2.1Localgradientsimulationresults .............. 88 5.3.2.2SingularDirectionAvoidancesimulationresults ...... 91 5.3.2.3HybridSteeringLogicsimulationresults .......... 94 5.3.3Case3:NearHyperbolicInternalSingularities=[15105195)]TJ /F4 11.955 Tf 9.3 0 Td[(75]Tdeg ................................ 96 5.3.3.1Localgradientsimulationresults .............. 96 5.3.3.2SingularDirectionAvoidancesimulationresults ...... 99 5.3.3.3HybridSteeringLogicsimulationresults .......... 102 5.4HybridSteeringLogicSummary ....................... 104 6 PAGE 7 6CONTROLALGORITHMORTHOGONALTORQUECOMPENSATION .... 106 6.1AttitudeControllerwithOTC .......................... 106 6.2LyapunovStabilityAnalysis .......................... 107 6.3NumericalSimulation ............................. 110 6.3.1CaseI:0=[0000]Tdeg ....................... 111 6.3.2CaseIIa:0=[90909090]Tdeg .................. 116 6.3.3CaseIIb(HSL/OTC):0=[90909090]Tdeg ............ 123 6.4OrthogonalTorqueCompensationSummary ................ 127 7SCALABILITYISSUESFORSGCMGS ...................... 129 7.1ScalabilityProblemswithSGCMGHardware ................ 129 7.2EffectofIgwonTorqueError .......................... 130 7.3NumericalSimulation ............................. 132 7.3.1CaseI:Kgw=0 ............................. 134 7.3.2CaseII:Kgw=2 ............................ 138 7.4EffectofIgwonTorqueTorqueAmplication ................. 142 7.5Summary .................................... 143 8CONCLUSION .................................... 144 APPENDIX ARIGIDBODYDYNAMICSFORMULATIONFORCONTROLMOMENTGYROSCOPEACTUATORS(SGCMG/VSCMG) .......................... 147 A.1Assumptions .................................. 147 A.2Dynamics .................................... 147 BMOMENTUMENVELOPECODE ......................... 151 CCONTROLMOMENTGYROSCOPEACTUATORSPECIFICATIONS ...... 156 REFERENCES ....................................... 157 BIOGRAPHICALSKETCH ................................ 164 7 PAGE 8 LISTOFTABLES Table page 31ModelParameters .................................. 50 41AlgorithmFlopsm=row(A)andn=column(A) ................. 66 51ModelParameters .................................. 78 52PerformanceComparisonsforCaseI:ZeroMomentum ............. 85 53PerformanceComparisonsforCaseII:EllipticSingularity ............ 96 54PerformanceComparisonsforCaseIII:HyperbolicSingularity .......... 104 61ModelParameters .................................. 110 62HybridSteeringLogicParameters ......................... 123 63PerformanceComparisons ............................. 128 71ModelParameters .................................. 133 72ModelParameters .................................. 142 C1OfftheShelfCMGSpecications .......................... 156 8 PAGE 9 LISTOFFIGURES Figure page 21Rigidbodywithaconstantc.m. ........................... 21 22GimbalframeFGiofIMPACSGCMG(PatentPending) .............. 22 23SingularityshownwhenCMGtorquevectorslieinaplane(IMPACSGCMGsPatentPending) ................................... 25 24SingularitiesforSGCMGs .............................. 27 25ExternalsingularsurfacesforafourSGCMGpyramid .............. 28 26InternalsingularsurfacesforafourSGCMGpyramid ............... 30 31FourSGCMGrooftoparrangement ......................... 34 32FourSGCMGboxarrangement ........................... 35 33AngularmomentumenvelopeforafourSGCMGboxarrangement. ....... 36 34PlanesoftorqueforafourCMGrooftoparrangement .............. 37 35TorqueplanestracedoutforafourSGCMGrooftoparrangement ........ 39 36AngularmomentumenvelopewithplottedangularmomentumcombinationsforthefourSGCMGboxarrangement ....................... 42 37DegeneratehyperbolicsingularitiesforthefourSGCMGboxarrangement ... 43 38Singularsurfacesshowing1h0singularityfreeregion ............... 45 393OrthogonalscissorpairsofSGCMGs ...................... 46 310PlanesofangularmomentumandtorqueforafourSGCMGpyramid ...... 47 311FourSGCMGpyramidarrangement ........................ 47 312Optimizationprocessblockdiagram ........................ 49 313SingularsurfacesfortheoptimizedarrangementattheEulerangles=[170.213.685.5168.0]Tdegand=[17.7167.0304.392.5]Tdeg ...... 51 314Gimbalratesfortheoptimizedandpyramidarrangements ............ 51 315Torqueerrorfortheoptimizedandpyramidarrangements ............ 52 316Singularitymeasurefortheoptimizedandpyramidarrangements ........ 52 317Optimizationcostfortheoptimizedandpyramidarrangements ......... 53 9 PAGE 10 41OuterandinnerloopsofGNCsystem ....................... 54 42Steeringalgorithms ................................. 56 51ZeromomentumcongurationofafourSGCMGpyramidarrangement ..... 79 52SimulationresultsforLGwith0=a=b=1=0and2=0=1atzeromomentum ...................................... 80 53SimulationresultsforLGwith0=a=b=1=0and2=0=1atzeromomentum(contd.) ................................. 81 54SimulationresultsforSDAwith0=0.01,0=a=b=2=0,and1=1atzeromomentum ................................... 83 55SimulationresultsforSDAwith0=0.01,0=a=b=2=0,and1=1atzeromomentum(contd.) ............................... 84 56SimulationresultsforHSLwith0=0.01,0=2,a=1,b=3,and1=2=1atzeromomentum ................................. 86 57SimulationresultsforHSLwith0=0.01,0=2,a=1,b=3,and1=2=1atzeromomentum(contd.) ............................ 87 58SimulationresultsforLGwith0=a=b=1=0and2=0=1nearellipticsingularities .................................. 89 59SimulationresultsforLGwith0=a=b=1=0and2=0=1nearellipticsingularities(contd.) ............................. 90 510SimulationresultsforSDAwith0=0.01,0=a=b=2=0,and1=1nearellipticsingularities ............................... 92 511SimulationresultsforSDAwith0=0.01,0=a=b=2=0,and1=1nearellipticsingularities(contd.) .......................... 93 512SimulationresultsforHSLwith0=0.01,0=2,a=1,b=3,and1=2=1nearellipticsingularities .............................. 94 513SimulationresultsforHSLwith0=0.01,0=2,a=1,b=3,and1=2=1nearellipticsingularities(contd.) ......................... 95 514SimulationresultsforLGwith0=a=b=1=0and2=0=1nearhyperbolicsingularities ................................ 97 515SimulationresultsforLGwith0=a=b=1=0and2=0=1nearhyperbolicsingularities(contd.) ........................... 98 516SimulationresultsforSDAwith0=0.01,0=0,a=0,b=0,and=1nearhyperbolicsingularities ............................. 100 10 PAGE 11 517SimulationresultsforSDAwith0=0.01,0=0,a=0,b=0,and=1nearhyperbolicsingularities(contd.) ........................ 101 518SimulationresultsforHSLwith0=0.01,0=2,a=1,b=3,and1=2=1nearhyperbolicsingularities ............................ 103 519SimulationresultsforHSLwith0=0.01,0=2,a=1,b=3,and1=2=1nearhyperbolicsingularities(contd.) ....................... 104 61Satelliteattitudecontrolsystemblockdiagram .................. 106 62Gimbalrates ..................................... 112 63Outputtorque ..................................... 113 64Vectorelementsoftheerrorquaternion ...................... 114 65Singularitymeasure ................................. 115 66Singularityparameter(OTC) ............................ 116 67Quaternionerrordifference:(A)eGSR)]TJ /F7 11.955 Tf 11.96 0 Td[(eSDA(B)eSDA=OTC)]TJ /F7 11.955 Tf 11.96 0 Td[(eSDA ........ 116 68Gimbalrates ..................................... 118 69Outputtorque ..................................... 119 610Vectorelementsoftheerrorquaternion ...................... 120 611Singularitymeasure ................................. 121 612Gimballockmeasure ................................ 122 613Singularityparameter(OTC) ............................ 123 614Gimbalrates ..................................... 124 615Outputtorque ..................................... 125 616Vectorelementsoftheerrorquaternion ...................... 126 617Singularitymeasure ................................. 126 618Gimballockmeasure ................................ 127 619Singularityparameter(OTC) ............................ 127 71OfftheshelfCMGs ................................. 129 72GimbalratesforKgw=0 ............................... 135 73GimbalaccelerationsforKgw=0 .......................... 136 11 PAGE 12 74TorqueerrorforKgw=0 ............................... 137 75SingularitymeasureforKgw=0 .......................... 138 76GimbalratesforKgw=2 ............................... 139 77GimbalaccelerationsforKgw=2 .......................... 140 78TorqueerrorforKgw=2 ............................... 141 79SingularitymeasureforKgw=2 .......................... 141 12 PAGE 13 AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyNOVELSTEERINGANDCONTROLALGORITHMSFORSINGLEGIMBALCONTROLMOMENTGYROSCOPESByFrederickA.LeveAugust2010Chair:NormanFitzCoyMajor:AerospaceEngineering TheresearchpresentedinthismanuscriptattemptstorstsystematicallysolvetheSGCMGsteeringandcontrolproblem.Toaccomplishthis,abetterunderstandingofsingularitiesassociatedwithSGCMGsisrequired. Nextbasedonabetterunderstandingofsingularities,aHybridSteeringLogic(HSL)isdevelopedandcomparedtothelegacymethodssingulardirectionavoidance(SDA)andlocalgradient(LG)methods.TheHSLisshownanalyticallyandnumericallytooutperformtheselegacymethodsforafourSGCMGpyramidarrangementintermsofattitudetrackingprecision.However,allofthemethodsaresusceptibletogimballock. AmethodreferredtoasOrthogonalTorqueCompensation(OTC)isdevelopedforsingularitieswithgimballockinSGCMGs,whichareknowntopresentachallengetomoststeeringalgorithms.OrthogonalTorqueCompensationconditionstheattitudecontroltorquebyaddingtorqueerrororthogonaltothesingulardirectionwhenatsingularity.ThismethodcanbecombinedwithanysteeringalgorithmincludingHSLandisprovenanalyticallytobestableandescapesingularitieswithgimballock. Finally,theproblemswithscalingsystemsofSGCMGsarediscussed.ItisfoundthatscalingproducesanincreaseinthegimbalwheelassemblyinertiaoftheSGCMGswhichinturnincreasestheeffectofthedynamicsassociatedwiththeseinertias.Throughanalysisandnumericalsimulations,itisshownthatmoresignicantgimbalywheelinertiareducestheperformancebyincreasingtorqueerrorand 13 PAGE 14 reducingtorqueamplicationofSGCMGs.Sincemostofthelegacyalgorithmsusedforsingularityavoidanceandescapeusethegimbalratesforcontrol,theperformanceisdegradedwhenthedynamicsfromthegimbalwheelassemblyinertiaareincreased.ThisdegradedperformanceisshowntoalsobedependentontheratioofgimbalwheelassemblyinertiatonominalSGCMGywheelangularmomentum.TheoverallresulthereisthatthesamelegacysteeringalgorithmsthatusegimbalratesforcontrolcannotbeusedforsystemsofSGCMGsofareducedscale. 14 PAGE 15 CHAPTER1INTRODUCTION1.1HistoryandBackground 1.1.1GyroscopicRateDetermination Thebehaviorofgyroscopicsystemscomesfromtheprincipleofconservationofangularmomentum.Bothratedeterminationandrategenerationarepossiblethroughthisgyroscopicbehavior.Forexample,determinationofattituderatescanbefoundbymechanicalgyroscopes.Fromthesedevices,spacecraftangularratesareinferredfromtheirreactionontothegimbalsofthemechanicalgyroscope.TherstknowngyroscopeswerepassiveandtherstofthesewasdevelopedbyJohannBohnenbergerin1817[ 1 2 ].Theywerelaterdevelopedforthenavyasthenavalgyrocompass[ 3 ].Therearenowpassivegyroscopesusedfordeterminingangularratesdevelopedviamicroelectricalandmechanicalsystems(MEMS)thataresmallerthanthehumaneyecandetect.1.1.2SpinStabilizedSpacecraft Earlyspacecraftdidnothaveactiveattitudecontrolbutwherespinstabilizedabouttheirmajoraxisofinertiabyinitiatingaspinafterlaunch.Themajoraxisruleisrequiredfordirectionalstability(i.e.,thespacecraftmustspinaboutitsmajoraxis)[ 4 ].Thisprovidedgyroscopicstabilityordirectionalstabilitytothespacecraftwhichwasdenedasstabilityprovidedthespacecraftdoesnothaveenergydissipationcapability(i.e.,arigidbody).However,aspacecraftistrulynotarigidbody(e.g.,exiblebooms,solararrays,internalmovableparts,andoutgassing)whichmaydissipateenergyandbecomeasymptoticallystable.1.1.3SpacecraftAttitudeControlthroughGyrostats Agyrostatisanyrigidbodythathasattachedtoitawheelthatthroughtheconservationofangularmomentumprovideseitherrotationalstabilityorcontrol.Toprovideattitudestability,constantspeedywheelsknownasmomentumwheels(MWs) 15 PAGE 16 wereaddedinternaltothespacecrafttoprovidegyroscopicstiffnesswhichinturnsuppliesattitudestability[ 5 ].Thissystemisanexampleofagyrostat.Thegyrostatwastherstfundamentaldynamicalsystemthatconsideredaspinningywheelwithinorattachedtoarigidbody.Inthistypeofsystemtheywheelimpartsangularmomentumstiffnesstothebodyinternallythroughtheprincipleoftheconservationofangularmomentum.Theuseofmomentumwheelsforgyroscopicstabilityhasfounditsuseforothervehiclesthanspacecraft(e.g.,boats,trains,buses[ 6 ]).1.1.43axisAttitudeControlofSpacecraft Whenactiveattitudecontrolwasneeded,momentumwheelswereexchangedforreactionwheels(RWs)whichprovideareactiontorqueonthespacecraftthroughywheelaccelerations.Thesedeviceswerethenusedincombinationwiththrusterstoprovideattitudecontroltospacecraft[ 7 ].FurtherinnovationcametospacecraftattitudecontrolthroughtheintroductionofgimbaledMWsknownascontrolmomentgyroscopes(CMGs).Theyhavebeenusedinsatelliteattitudecontrolfordecadesduetotheirhighprecisionandpropertyoftorqueamplication(i.e.,largertorqueoutputontothespacecraftthantheinputtorquefromthegimbalmotors).Typically,theyhavebeenusedforlargesatellitesthatrequirehighagilitywhilemaintainingpointingprecisionandhaveevenfoundtheirusesonboardtheinternationalspacestation(ISS)[ 8 ].Controlmomentgyroscopesproduceagyroscopictorquethroughrotationofangularmomentumaboutoneortwogimbalaxes.Momentumwheels,RWsandCMGsallareknownasmomentumexchangedevicesbecausetheirtorqueisproducedthroughredistributionofangularmomentumfromtheCMGstothespacecraft.Controlmomentgyroscopescomeintwoclasses:1)thosewithasinglecontrollabledegreeoffreedomand2)thosewithmultiplecontrollabledegreesoffreedom. ThechoiceofaCMGorattitudeactuatoringeneraldependsontheneedsofthemission.AllCMGshavespecicchallengesassociatedwiththeiruse.ThechallengesofCMGsareasfollows:singleDOFCMGsknownassinglegimbalcontrol 16 PAGE 17 momentgyroscopes(SGCMGs)sufferfrominstantaneousinternalsingularities(i.e.,situationswithintheperformanceenvelopewhereagiventorquecannotbeproduced);multipleDOFCMGsknowndoublegimbalcontrolmomentgyroscopes(DGCMGs)aremechanicallycomplexandhavesingularitiesknownasgimballockwhentheirgimbalsalignthuseliminatingtheirextraDOF;othermultipleDOFCMGsknownasvariablespeedcontrolmomentgyroscopes(VSCMGs)aredifculttomitigateinducedvibration(i.e.,duetothevariationofywheelspeeds)andrequiremorecomplicatedcontrollaws,motordrivercircuitry,andlargerywheelmotors.1.1.5SingleGimbalControlMomentGyroscopes(SGCMGs) SinglegimbalcontrolmomentgyroscopeshavealongheritageofightonlargersatellitesandtheISS[ 9 15 ].TheyareknowntohavethehighesttorqueamplicationofallCMGs,arelessmechanicallycomplexthanDGCMGsandhavelessmathematicallycomplexdynamicsthanVSCMGs.Theseactuatorssufferfrominternalsingularitiesthatmustbehandledonthey,wheretorquecannotbegeneratedinaspecicdirection.Thereisnosinglemethodthathasbeenproventoavoidallinternalsingularitieswhiletrackinganarbitrarytorqueperfectly(i.e.,withouttheuseoftorqueerrororconstrainingthetorque).Thus,thereismeritinndingalternatesolutionstocontrolofSGCMGsforattitudecontrol.1.1.6DoubleGimbalControlMomentGyroscopes(DGCMGs) DoublegimbalcontrolmomentgyroscopescontaintwocontrollableDOFsthroughtheirtwogimbalaxes.TheyarethemostmechanicallycomplexofCMGsalthough,theredundancyintheadditionalgimbalmayleadtolessthanthreeDGCMGsrequiredfor3axiscontrol.ThebenetofthisredundancyislostforDGCMGswhentheyencounteragimballocksingularity.Gimballockisencounteredwhenthegimbalaxesarealignedandarenolongerlinearlyindependent.Asaconsequence,theextracontrollableDOFinthiscaseislost.EffectivemethodsalreadyexistforavoidinggimballocksingularitiesassociatedwithDGCMGs[ 16 20 ]. 17 PAGE 18 1.1.7VariableSpeedControlMomentGyroscopes(VSCMGs) VariablespeedcontrolmomentgyroscopesutilizeanextracontrollableDOFthroughywheelaccelerations.AsaconsequenceoftheextraDOF,theywheelmotorsmustbelargeranditismoretroublesometoisolateunwantedinducedvibration.Inaddition,thisextradegreeoffreedommakesasystemoftwoormorenoncollinearVSCMGsfreefromsingularitiesthroughtheextradegreeoffreedom,(i.e.,atCMGinternalsingularities,theneededtorqueisprovidedbyywheelaccelerations).Severalalgorithmshavebeendevelopedthatareeffectiveinreducingtheamountofywheelaccelerationsusedandthusprovidingbettertorqueamplication[ 21 30 ].Inaddition,methodshavebeendevelopedtousetheVSCMG'sextracontrollableDOFtospindowntheywheelsandstoretheirkineticenergy.Thesemethodsareknownastheintegratedpowerandattitudecontrolsystem(IPACS)andtheywheelattitudecontrolandenergytransmissionsystem(FACETS)inliterature[ 31 34 ].1.2ProblemStatement AnewparadigmthatrequireshighlyagilesmallspacecraftisongoingthrougheffortsbygovernmentagenciesandlabssuchasOperationallyResponsiveSpace(ORS),AirForceResearchLaboratory(AFRL),theNationalReconnaissanceOrganization(NRO),andNationalAeronauticsandSpaceAdministration(NASA)toperformsuchmissionsasintelligence,surveillanceandreconnaissance(ISR),spacesituationalawareness(SSA),andspacesciencemissions(e.g.,theimagingofgammaraybursts)[ 35 37 ].ManyofthesemissionsareinLEOandrequirehigheragilityandattitudeprecisiontotracktargetsonearth.Attitudecontrolsystems(ACSs)basedonreactioncontroldevices(e.g.,thrusters)canachievegreatagilitybutcannotmeetthepointingrequirementsandspaceneededforpropellantstorageonsmallsatellites[ 38 ]. Singlegimbalcontrolmomentgyroscopesarebeingconsideredtheactuatorofchoicetoprovidehigheragilitytosmallersatellitesbasedontheirperceivedtorqueamplication.However,manyproblemsexistthatmustbesolvedpriortousing 18 PAGE 19 SGCMGsforsmallsatelliteattitudecontrol.Traditionally,SGCMGshavebeenoversizedfortheirmissionandtheangularmomentumenvelopeconstrainedtoavoidinternalsingularitiesonlargersatellites.Forsmallersatellitesystems,however,theextravolumeandmassneededforoversizedSGCMGsmaybeunacceptable.Therefore,smallsatelliteSGCMGsshouldutilizemoreoftheentireangularmomentumenvelopewheresingularitiesmaybeencounteredinthemomentumspace(bothinternalandexternal).Thus,legacysteeringalgorithmsfromlargersatelliteapplicationsmaynotprovidethesameperformanceforsystemsofasmallerscalerequiringanewapproachforsteeringandcontrolofSGCMGs. Forthesucceedingchapters:Chapter 2 discussesandreviewsthefundamentalsofCMGdynamicsanddescribesthedifferentformsofsingularitiesassociatedwithSGCMGs;Chapter 3 describesthepossiblearrangementsforsystemsofSGCMGsandtheirdesirableandundesirablequalities;Chapter 4 providesthebackgroundonpreviouslypublishedmethodsofsteeringalgorithmsforimplementationofSGCMGs;Chapter 5 discussesthedevelopmentoftheHybridSteeringLogic(HSL)forSGCMGs;Chapter 6 discussesthedevelopmentofOrthogonalTorqueCompensation(OTC)forgimballockescapeofSGCMGs;Chapter 7 discussestheperformancedegradationencounteredwhenscalingsystemsofSGCMGs;andChapter 8 providesconclusionsoftheresearch. 19 PAGE 20 CHAPTER2DYNAMICMODELS2.1DynamicFormulation Thedynamicformulationpresentedinthissectionaddressesmomentumexchangedevicesforattitudecontrolsystemswheretheangularmomentumofthespacecraftsystem(i.e.,thespacecraftandsystemofCMGs)isassumedconstant.Thisisincontrasttoreactioncontroldevices(e.g.,thrusters)and/orenergydissipationdevices(e.g.,magnettorquers)whichchangetheangularmomentumorenergyofthesystem. Itisassumedthatthecenterofmass(c.m.)ofeachCMGliesalongthegimbalaxis.Thisisequivalenttostatingthattherotationofthegimbalywheelsystemaboutthegimbalaxisdoesnotmovethepositionofthec.m.ofthesystem.ItisalsoassumedthatthespacecraftCMGsystemisarigidbody;andthissystemisabsentoffrictionandexternaltorques.ForacoordinatelessderivationofthedynamicsseeAppendix A .TherstassumptioncanbevisualizedbytreatingtheCMGsascylindersandhavingtheirspinaxisalongtheirc.m.asshowninFigure 21 ThetotalcentroidalangularmomentumofthespacecraftCMGsysteminthespacecraftbodyframeis H=Jc!+h(2) whichiscomposedofthespacecraftcentroidalangularmomentumJc!composedofthespacecraftcentroidalinertiaJcandangularvelocity!andtheangularmomentumhcontributedfromtheCMGs. Consideringthespacecraftmodeledasarigidbody,itscentroidalinertiaiscomposedofbothconstantandtimevaryinginertiasandisexpressedas Jc=J+mi(rTiri1)]TJ /F8 11.955 Tf 11.95 0 Td[(rirTi)+nXi=1CBGiIgwCTBGi(2) 20 PAGE 21 Figure21. Rigidbodywithaconstantc.m. whereJrepresentstheconstantspacecraftCMGsysteminertiainreferencingpositionaboutthespacecraft'sc.m;mi(rTiri1)]TJ /F8 11.955 Tf 12.56 0 Td[(rirTi)aretheparallelaxisterms,withmassmioftheithgimbalywheelassemblyanditsc.m.,positionriwithrespecttothec.m.ofthespacecraft,andCBGiIgwCTBGiarethetimevaryinginertiasfromrotationofthegimbalywheelsysteminertiaIgw.TheangularmomentumcontributedfromtheithCMGinagimbalframeFGiisexpressedas hi=266664Iw0Igw_i377775(2) whichconsistsofangularmomentumfromtheywheel(Iw)andthatfromthegimbalwheelsystem(Igw_i).ItshouldbenotedthattheCMGangularmomentumexpressioninEq.( 2 )isbasedonanSGCMGorVSCMGonlyandthefollowingdevelopmentofangularmomentumforCMGswillbeforamultiplegimbalCMG.The 21 PAGE 22 resultantangularmomentumfromtheCMGsysteminthebodyframeisfoundthroughthesummationofthecontributionsofangularmomentumfromallCMGsrotatedfromtheirrespectivegimbalframesintothespacecraftbodyframe;i.e., h=nXi=1CBGihi(2) wherenisthenumberofSGCMGs,CBGiisthedirectioncosinematrix(DCM)fromthegimbalframeFGishowninFigure 22 tospacecraftbodyframeFB. Figure22. GimbalframeFGiofIMPACSGCMG(PatentPending) Withtheassumptionofnoexternaltorquesandfrictionlessdevices,thetotalangularmomentumisconstant,thustheinertialtimederivativeofEq.( 2 )showstheredistributionofthesystems'sangularmomentum(i.e.,onthemechanismbywhichthetorqueisproducedbyCMGs).DifferentiationofEq.( 2 )yields dH dt=d(Jc_!+h) dt+!(Jc_!+h)=0(2) 22 PAGE 23 Thetimederivativeofthespacecraftangularmomentumyields d(Jc!) dt=Jc_!+_Jc!(2) ThespacecraftinertiaisassumedtoonlyvarybythegimbalanglesoftheCMGsthusmakingthesecondtermontherighthandsideofEq.( 2 ) _Jc!kXj=1@(Jc!) @jdj dt=kXj=1Aj1_j(2) whereAj12R3nistheJacobianmatrixresultingfromthecouplingofthespacecraftandCMGkinematicsfromthejthgimbalofamultiplegimbalCMGandnisthenumberofCMGs. TheangularmomentumoftheCMGsystemisafunctionoftheywheelangularvelocities2Rn1,gimbalanglesj2Rn1,andgimbalrates_j2Rn1,respectively.ThetimederivativeoftheCMGsangularmomentumcanbeexpressedas dh dt=_h=kXj=1@h @jdj dt+@h @_jdj dt+@h @d dt(2) wheretheindividualJacobianmatricesare, Aj2=@h @j2R3n(2) Bj=@h @_j2R3n(2) C=@h @2R3n(2) Assumingaconstantywheelspeed_=0andasinglegimbal(k=1)conguration,thenEq.( 2 )canberewrittenasdH dt=Jc_!+!Jc!+!h+D_X=0(2) 23 PAGE 24 where, D_X=[(A1+A2)B]264_375=_h+_Jc!=T(2) ThegeneralequationsfortheSGCMGoutputtorqueintermsofagiveninternalcontroltorqueisexpressedas, D_X=)]TJ /F10 11.955 Tf 9.29 0 Td[()]TJ /F10 11.955 Tf 11.95 0 Td[(!h=T(2) whereTisthetotaltorqueoutputfromthesystemofSGCMGs.Itshouldbenotedthat_andarekinematicallycoupled,thusitisnotpossibletondbothstatessimultaneouslywhenmappingD2R33nontoT(i.e.,onlyonegimbalstatecanbechosenasacontrolvariable).ForSGCMGsystemsthatcontainsignicantywheelangularmomentumandgyroscopictorque,thedynamicsofthegimbalwheelassemblyinertiascanbeconsideredinsignicant(i.e.,A10andB0).Forsuchsystems,itiscustomarytoneglecttheinertiavariationsduetothegimbalmotion(i.e.,_Jc=0)resultinginthecompositeDreducingtoA22R3n.Therefore,theJacobianDissimplyA2andthesolutionoftheoutputtorquefromtheSGCMGsiscontributedsolelyfromthegimbalratesas _h=h0^A2_=h0[^1,^2,...^n]_(2) whereA2=h0^A2and^iisthetorquevectordirectionoftheithCMGasshowninFigure 22 .ThecoefcientmatrixinEq.( 2 )is3nandwhenn>3thesystemisoveractuated.Whenthismatrixbecomesrankdecient,thesystemissaidtobesingular.Physically,whenthesesingularitiesoccur,thetorquevectordirectionsofeachSGCMGinthebodyframelieinaplaneasshowninFigure 23 Forconvenience,fromthispointuntilChapter 7 ,itisassumed^A2=A. 24 PAGE 25 Figure23. SingularityshownwhenCMGtorquevectorslieinaplane(IMPACSGCMGsPatentPending) 2.2SingularSurfaceEquations Itiscustomarytodeneanorthonormalbasisf^hi,^i,^igasshowninFigure 22 where^hiisthespinaxisoftheywheel,^iistheSGCMGtorquedirection,and^iisthegimbalaxisdirection. Therefore,thesingulardirections2R31isdenedfrom fs2R3:sT^i=0g(2) Thisconstraintconstitutesamaximum(orminimum)projectionof^hiontos.Thereisafundamentalassumptionthath0iisequaltoh0(i.e.,themagnitudeofnominalangularmomentumisthesameforallSGCMGsinthesystem).Foragivensingulardirections6=^i(i.e.,whichonlyoccursforDGCMGsandforrooftoparrangements),theconditionsforsingularityare sT^i=0andsT^hi6=0(2) Ifwedenei,sT^hi,thenthetorqueandspinaxisdirectionscanbeexpressedas ^i=i^is jj^isjj,s6=^i,i=1,...,n(2) 25 PAGE 26 ^hi=^i^i=i(^is)^i jj^isjj,s6=^i,i=1,...,n(2) CombiningEqs.( 2 )and( 2 ),thetotalnormalizedangularmomentumfromtheSGCMGsisexpressedas ^h=nXi=1^hi=nXi=1^i^i=i(^is)^i jj^isjj(2) Itisimportanttonotethatwhens=^iEqs.( 2 )( 2 )areindeterminate. Thelocusoftotalnormalizedangularmomentum^hfromEq.( 2 )foralls2R3andalli6=0(i.e.,snotcollinearto^i)producestheexternalsingularsurfaceknownastheangularmomentumenvelopeshowninFigure 25 forafourSGCMGpyramidarrangement.Similarly,eachofthefourinternalsingularsurfacesshowninFigure 26 forafourSGCMGpyramidarrangementarefoundbysettingoneofthei<0.MatlabcodeforbothofthesesurfacescanbeseeninAppendix C .SingularitiesforSGCMGscanbeclassiedintothegroups/subgroupsasshowninFigure 24 .2.2.1EllipticSingularities EllipticsingularitiesarethoseinwhichnullsolutionstothegimbalanglesdonotexistforaspecicpointofCMGangularmomentumspace.Nullmotionisacontinuoussetofnullsolutionsforgimbalangles(i.e.,thereisacontinuoustransferfromonenullsolutiontothenext)thatdoesnotchangetheCMG'sangularmomentumandthus,doesnotproduceanymotiontothespacecraft.Sinceellipticsingularitiesdonothavenullsolutions,theangularmomentummustbeperturbedthusinducingtorqueerrortothesystemtoescapefromthesesingularities.Ellipticsingularitiesarenotlimitedtointernalsingularities;(e.g.,allexternalsingularitiesareellipticandhencecannotbeavoidedorescapedthroughnullmotion). 26 PAGE 27 Figure24. SingularitiesforSGCMGs 2.2.1.1Externalsingularities ExternalsingularitiesalsoknownassaturationsingularitiesareassociatedwiththemaximumprojectionofCMGangularmomentuminanydirection.Thesesingularitiescannotbeavoidedbynullmotionandthereforebydenitionareelliptic.ThesesingularitiesoccuronthesurfaceoftheangularmomentumenvelopeandanexampleofthissurfaceforafourSGCMGpyramidisshowedinFigure 25 .Whenthesesingularitiesareencountered,theCMGsareunabletoproduceanymoreangularmomentuminthesaturateddirection.ExternalsingularitiesareaddressedaprioriinthedesignprocessthroughsizingoftheCMGactuators. 27 PAGE 28 Figure25. ExternalsingularsurfacesforafourSGCMGpyramid ConsiderafourSGCMGpyramidarrangement,Eq.( 2 )canbeusedtoexpresstheangularmomentumash=h0266664)]TJ /F7 11.955 Tf 9.3 0 Td[(c()s(1))]TJ /F7 11.955 Tf 11.96 0 Td[(c(2)+c()s(3)+c(4)c(1))]TJ /F7 11.955 Tf 11.95 0 Td[(c()s(2))]TJ /F7 11.955 Tf 11.95 0 Td[(c(3)+c()s(4)s()(s(1)+s(2)+s(3)+s(4))377775(2) whereistheskewangleandiistheithgimbalangle.Furtherconsiderthesetofgimbalangleses=[90909090]deg,thenthemomentumvectorbecomes h(es)=26666400h0s()(s(1)+s(2)+s(3)+s(4))377775=266664004h0s()377775(2) Itisclearthatthereisonlyonesetofgimbalangleses=[90909090]TdegthatwillgivetheangularmomentuminEq.( 2 ).Therefore,nullsolutionsdonotexist,andthis 28 PAGE 29 angularmomentumvectorcorrespondstotheellipticsaturationsingularityalongthezaxis.2.2.1.2Ellipticinternalsingularities EllipticsingularitieswhichlieontheinternalsingularsurfacessuchasthatshownforthefourSGCMGpyramidarrangementinFigure 26 arereferredtoasellipticinternalsingularities.Unlikeexternalsingularities,thesesingularitiescannotbeaccountedforinthedesignprocess;furthermore,sincetheyoccurinstantaneously,theycannotbegenerallyavoided.2.2.2HyperbolicSingularities Hyberbolicsingularitiesarethoseinwhichnullmotionispossible.Thus,allhyperbolicsingularitiesarethereforeinternal(i.e.,thesesingularitiesoccurontheinternalsingularsurfaces).Thepointsontheinternalsingularsurfacecorrespondingtoahyperbolicsingularityhavenullsolutionsofgimbalangles,correspondingtothenullspaceoftheJacobianmatrix.Thenullsolutionsaretypicallychosentoavoidthesingularcongurationsofthesystem.ShowninFigure 26 ,isanexampleofthissurfaceforafourSGCMGpyramid.SingularitiesoccuronlywhenthepointonthissurfacecorrespondstoasingularJacobianmatrix(i.e.,theremaybenonsingularsetsofgimbalanglesatthispointonthesurface).Whenthesesingularitiesareencountered,theSGCMGtorquevectordirectionslieinaplaneandasaconsequencethereisnotorqueavailableoutoftheplane.Thesesingularities,likeellipticsingularities,areinstantaneousandmustbehandledonthey.ForafourSGCMGpyramidarrangementwithangularmomentuminEq.( 2 ),asetofgimbalangleshs=[18090090]Tdegisanhyperbolicsingularitythathasthefollowingmomentumvector, h(hs)=2666640)]TJ /F4 11.955 Tf 9.29 0 Td[(2h02h0s()377775(2) 29 PAGE 30 Itisclearthattheremultiplesolutions(i.e.,nullsolutions)toEq.( 2 ).Anullsolutionofthegimbalanglessatisfying1=3=90ands(4)=)]TJ /F7 11.955 Tf 9.3 0 Td[(s(2)=1 c()substitutedintoEq.( 2 )willalsosatisfyEq.( 2 ). Figure26. InternalsingularsurfacesforafourSGCMGpyramid 2.2.2.1Nondegeneratehyperbolicsingularities NonDegeneratehyperbolicinternalsingularitiesarethoseinwhichnullmotionispossibleandsomeofthenullsolutionsarenonsingularprovidingthepossibilityofsingularityavoidance.2.2.2.2Degeneratehyperbolicsingularities Degeneratehyperbolicinternalsingularitiesoccurwhenthenullsolutionstogimbalanglescorrespondtosingularsetsofgimbalanglesleavingnoroomforavoidanceorescape.Thesesingularitiesarealsoconsideredimpassableandthereforearehandledinasimilarmannertoellipticsingularitieswhenapproached.2.2.3GimbalLock GimballockforSGCMGsoccursatsingularitywhenthemappedoutputtorquevectorisinthesingulardirection.Whenthisoccurs,thesystembecomestrappedinthis 30 PAGE 31 singularcongurationwithonlyafewmethodsthatarecapableofescapefromit.OnesuchmethodisknownastheGeneralizedSingularityRobust(GSR)Inverse[ 39 40 ].ThismethodhasbeenshownnumericallytoescapegimballockofSGCMGsbutnotanalyticallyandthereisnoformalprooftosuggestthatitisalwayssuccessful.2.3SingularitiesforSGCMGsMathematicallyDened Toquantiytheeffectivenessofavoidinginternalsingularitiesthroughnullmotion,wemustdenetheirforms(i.e.,hyperbolicandelliptic)mathematically.Typically,topologyanddifferentialgeometryareusedtorepresenthyperbolicandellipticinternalsingularitiesassurfacesormanifolds[ 41 42 ].Thebehavioroftheseinternalsingularitiescanalsobeexplainedthroughtheuseoflinearalgebra[ 43 ].Toaccomplishthis,aTaylorseriesexpansionoftheSGCMGangularmomentumaboutasingularcongurationgives h())]TJ /F8 11.955 Tf 11.95 0 Td[(h(S)=nXi=1@hi @iSii+1 2@2hi @2iSi2i+H.O.T.(2) whereh(S)istheangularmomentumatasingularsetofgimbalanglesS,i=i)]TJ /F3 11.955 Tf 11.95 0 Td[(Si,andnisthenumberofSGCMGsinthesystem. Thersttermontherighthandside(RHS)ofEq.( 2 )containstheithcolumnoftheJacobianmatrix^i=@hi @ijSi,associatedwiththeithSGCMG'storquedirection.ThesecondtermontheRHScontainsthepartialderivativeoftheJacobianmatrix'sithcolumnwithrespecttotheithgimbalangle@2hi @2ijSi.Furthermore,fromEq.( 2 ),theRHSofEq.( 2 )canbetransformedthroughtherealizationofthefollowingoperations: @2hi @2i=@^i @i=)]TJ /F7 11.955 Tf 9.3 0 Td[(hi^hi=)]TJ /F8 11.955 Tf 9.29 0 Td[(hi(2) where^()denotesaunitvector.Next,Eq.( 2 )issubstitutedintoEq.( 2 )andtheinnerproductoftheresultwiththesingulardirectionsobtainedfromnull(AT)yields 31 PAGE 32 sT[h())]TJ /F8 11.955 Tf 11.96 0 Td[(h(S)]=)]TJ /F4 11.955 Tf 10.5 8.08 Td[(1 2nXi=1hTis2i(2) ThersttermontheRHSofEq.( 2 )haszerocontributionbecauseofthedenitionofthesingulardirection(i.e,ATs=0).Equation( 2 )canbewrittenmorecompactlyas sT[h())]TJ /F8 11.955 Tf 11.96 0 Td[(h(S)]=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(1 2TP(2) wherePisthesingularityprojectionmatrixdenedasP=diag(hTis). Bydenition,nullmotiondoesnotaffectthetotalsystemangularmomentumandwhichequatestoh()=h(S).Consequently,thelefthandside(LHS)ofEq.( 2 )iszero(i.e.,TP=0).NullmotionisexpressedintermsofthebasisN=null(A)concatenatedinmatrixformasfollows =(n)]TJ /F5 7.97 Tf 6.59 0 Td[(r(A))Xi=1ii=N(2) whereisacolumnmatrixofthescalingcomponentsofthenullspacebasisvectorsiandN2Rn(n)]TJ /F5 7.97 Tf 6.59 0 Td[(r(A))isthedimensionofthenullspacebasisforanysystemofSGCMGswithr(A)=rank(A).SubstitutingEq.( 2 )intoEq.( 2 )observingthenullmotionconstraintyields TQ=0(2) Asaresultofthisanalysis,amatrixQisdenedasQ=NTPN(2) Therefore,whenawayfromsingularityQ2R11;whenatarank2singularity,Q2R22;andwhenatarank1singularity,Q2R33.TheeigenvaluesoftheQmatrixdeterminewhetherasingularityishyperbolicorelliptic.IfQisdenite(i.e.,hasallpositiveornegativeeigenvalues),itdoesnotcontainanullspacesinceanonzeronullvector 32 PAGE 33 doesnotexistthatsatisesEq.( 2 )[ 44 ].Therefore,situationswherethematrixQisdeniteconstituteellipticsingularities. WhenQissemidenite(i.e.,ithasatleastonezeroeigenvalue),thenanullspaceexistssincethereexistsa6=0thatsatisesEq.( 2 ).Therefore,nullmotionispossiblenearsingularityandthepossibilityofsingularityavoidancemayhold(i.e.,doesnotfordegeneratehyperbolicsingularities)[ 43 ].IfthematrixQisindenite(i.e.,theeigenvaluesarepositiveandnegative),theresultofEq.( 2 )hasthepossibilityofbeingequaltozero.Therefore,bothofthesesituationsconstituteanhyperbolicsingularity. ThetoolsdevelopedinthischapterfordescribingtheexistenceofellipticsingularitiesandhyperbolicinternalsingularitiesinasystemofSGCMGsareusedinthenextchaptertomorespecicallydiscusswhichoftheseinternalsingularitiesexistincommonarrangementsofSGCMGs.Inaddition,Chapter 5 introducesanovelsteeringalgorithmknownastheHybridSteeringLogic(HSL)whichusesthesetoolsinitsderivation. 33 PAGE 34 CHAPTER3CONTROLMOMENTGYROSCOPEARRANGEMENTS3.1CommonSGCMGArrangements SeveralcommonSGCMGarrangementshavebeenstudied.Typically,thefactorsthatdeterminethechoiceofaspecicSGCMGarrangementare:(i)availablevolume(ii)desirableangularmomentumenvelope,and(iii)associatedsingularities.Inthischapter,weexaminethecommonarrangementsandusethetoolsdevelopedinChapter 2 tocharacterizetheirsingularities.3.1.1Rooftop TherooftoparrangementshowninFigure 31 hastwosetsofparallelSGCMGs,eachwithparallelgimbalaxeswhereistheskewanglerelatingtheplanesoftorque.AfourSGCMGrooftoparrangementisshowninFigure 31 Figure31. FourSGCMGrooftoparrangement Sincethesearrangementsarefreefromellipticinternalsingularities,theyhaveasignicantightheritageonsatellitesandthus,theircontroliswellunderstood[ 45 ].However,degeneratehyperbolicsingularitieswhicharealsoimpassablestillexistandlikeellipticsingularitiescannotbeaddressedthroughtheuseofnullmotion.Inaddition,therearedegeneratecasesofhyperbolicsingularitiesfortherooftoparrangementwhen 34 PAGE 35 theJacobianmatrixisrank1whichmayprovidedifcultytosingularityescapelawsthatregulatethesmallestsingularvalue.3.1.2Box TheboxarrangementisasubsetoftherooftoparrangementandhastwoparallelsetsoftwoSGCMGswithanangleof90degbetweenthetwoplanesoftorqueasshowninFigure 32 .Thisarrangementisgivenitsnamebecausetheplanesofangularmomentumcanformabox[ 46 ].Liketherooftoparray,therearesituationswherethisarrangementmayhavearank1Jacobian. Figure32. FourSGCMGboxarrangement TheangularmomentumenvelopeforthefourSGCMGboxarrangementisshowninFigure 33 .Theangularmomentumenvelopeforallrooftoparrangementsisanellipsoidialsurfaceandthus,thereisnotequalmomentumsaturationinalldirections. Rooftoparrangementsarechosenfortheircompactnessandthefactthattheyarefreefromellipticinternalsingularities.Analysisprovidedintheliteratureprovesthatthoughthesearrangementsarefreefromellipticinternalsingularities.However,theyarenotfreefromtheimpassabledegeneratehyperbolicsingularities[ 47 ]. 35 PAGE 36 Figure33. AngularmomentumenvelopeforafourSGCMGboxarrangement. InChapter 2 ,itwasshownthatthedenitenessofthematrixQdeterminesifasystemofSGCMGsisatanellipticsingularity.ForasystemoffourSGCMGs,thelargestQcanbeisR22excludingthecasewhentheJacobiangoesrank1whichwillbediscussedlater.Therefore,excludingarank1Jacobian,andifthedeterminantofQisstrictlypositive,thenthesystemisatanellipticsingularity(e.g.,12>0where1,2areaneigenvaluesofQ).Consequently,whendet(Q)0,thesystemisatanhyperbolicsingularity.ThereareafewgeneralcaseswheresingularitiesmayoccurforafourSGCMGrooftoparrangement.TherstcaseoccurswhenthetorquevectorslyinginthesameplaneoftorqueareeitherparallelorantiparallelasshowninFigures 34 AandBwhere12+180,34+180andristheaxisintersectingthetwoplanesoftorque. 36 PAGE 37 ATorqueplaneswithparalleltorquevectors BTorqueplaneswithoneparallelandantiparalleltorquevectors Figure34. PlanesoftorqueforafourCMGrooftoparrangement 37 PAGE 38 AfourCMGrooftopsysteminthecongurationofFigure 34 AhasaJacobian A=266664c()c(1)c()c(1))]TJ /F7 11.955 Tf 9.3 0 Td[(c()c(3))]TJ /F7 11.955 Tf 9.3 0 Td[(c()c(3)s(1)s(1))]TJ /F7 11.955 Tf 9.3 0 Td[(s(3))]TJ /F7 11.955 Tf 9.3 0 Td[(s(3)s()c(1)s()c(1)s()c(3)s()c(3)377775(3) withangularmomentumrepresentedinthespacecraftbodyframe(i.e.,wherethegimbalsareenumeratedcounterclockwisebeginningatthespacecraftbodyxaxis) h=h1+h2+h3+h4=2h0266664c()s(1))]TJ /F7 11.955 Tf 9.3 0 Td[(c(1)s()s(1)377775+2h0266664)]TJ /F7 11.955 Tf 9.29 0 Td[(c()s(3)c(3)s()s(3)377775(3) Thesingulardirectionforthiscaseisfoundbycrossproductof^1and^3 s=266664s()s(1+3))]TJ /F4 11.955 Tf 9.3 0 Td[(2s()c()c(1)c(3)c()s(1)]TJ /F3 11.955 Tf 11.96 0 Td[(3)377775(3) withtheresultantprojectionmatrix P=2h0s()c()266666664c(1)0000c(1)0000)]TJ /F7 11.955 Tf 9.3 0 Td[(c(3)0000)]TJ /F7 11.955 Tf 9.3 0 Td[(c(3)377777775(3) andthenullspaceoftheJacobianconcatenatedinmatrixformis N=266666664)]TJ /F4 11.955 Tf 9.3 0 Td[(10100)]TJ /F4 11.955 Tf 9.3 0 Td[(101377777775(3) 38 PAGE 39 ThecongurationinFigure 34 hasthedet(Q)=)]TJ /F4 11.955 Tf 9.3 0 Td[(16s2()c2()c(1)c(3).ItshouldbenotedthatwhenPandQaredenite(i.e.,=[++++]or=[)276()276(\000]wherei=sgn(^his)),thesystemisataexternalsingularity.ItwasdiscussedinChapter 2 thatexternal(orsaturation)singularitiesareelliptic.Therefore,ifthematrixPisdenite,thenthematrixQisalsodenite.Thedet(Q)=)]TJ /F4 11.955 Tf 9.3 0 Td[(16s2()c2()c(1)c(3)willonlybepositivewhensgn(c(1))6=sgn(c(3))(i.e.,atsaturationsingularity).Thesaturationsingularityisnotaninternalsingularityandthereforeneithertherooftopandboxarrangementscontainellipticinternalsingularitiesforthiscase.Itcanbeveriedthatthevariationsofthesecasessuchas(^1=)]TJ /F8 11.955 Tf 10.31 0 Td[(^2),(^2=)]TJ /F8 11.955 Tf 10.31 0 Td[(^4),(^1=)]TJ /F8 11.955 Tf 10.31 0 Td[(^2)and(^2=^4)allhavedet(Q)=0andthereforearehyperbolicinternalsingularities. TheothercasewhentheJacobianofafourCMGrooftopissingularoccurswhenthetorquevectorsofthetwoparallelSGCMGslieinthedirectionoftheintersectingtorqueplanesrshowninFigures 35 ATorqueplaneswithparalleltorquevectorsalongr BTorqueplaneswithantiparalleltorquevectorsalongr Figure35. TorqueplanestracedoutforafourSGCMGrooftoparrangement ForFigure 35 A,theJacobianis 39 PAGE 40 A=266664)]TJ /F7 11.955 Tf 9.3 0 Td[(c()c(3))]TJ /F7 11.955 Tf 9.29 0 Td[(c()c(4)00)]TJ /F7 11.955 Tf 9.3 0 Td[(s(3))]TJ /F7 11.955 Tf 9.29 0 Td[(s(4)11s()c(3)s()c(4)00377775(3) withangularmomentumvector h=h1+h2+h3+h4=2h0266664c()0s()377775+h0266664)]TJ /F7 11.955 Tf 9.29 0 Td[(c()s(3)c(3)s()s(3)377775+h0266664)]TJ /F7 11.955 Tf 9.3 0 Td[(c()s(4)c(4)s()s(4)377775(3) assumingthattheintersectionoftheplanesoftorquer=[010]TforthisarrangementshowninFigure 35 .Forthissituation,thesingulardirectionisorthogonaltotheintersectionoftheplanesandisfoundtobe s=266664s()0c()377775(3) withtheresultingprojectionmatrix P=2h0s()c()c(3)2666666641000010000000000377777775(3) andthenullspacebasisfortheJacobianconcatenatedinmatrixformis N=null(A)=266666664)]TJ /F4 11.955 Tf 9.29 0 Td[(1s(4)]TJ /F3 11.955 Tf 11.96 0 Td[(3)=c(3)100)]TJ /F4 11.955 Tf 9.3 0 Td[((c(4)=c(3))01377777775(3) 40 PAGE 41 Forthiscase,det(Q)=2h0sin2(2)>0whichisanellipticsingularitybutisnotknownyettobeonethatisinternalorexternal.Fortheantiparallelcasewhen3=180+4=270deg,thedet(Q)=2h0sin2(2)<0andthesingularityishyperbolic.NoticethatthediagonalentriesofPthatarezerocorrespondtothegimbalaxisofthatroofsidebeingalongthesingulardirectionsofthesystem. Recall,fromChapter 2 thats=^iisaspecialcasethathappensonlyforDGCMGsandrooftoparrangements.Forexample,considerthecasewhen3=180+4=270andanangularmomentumof h=2666640h20377775(3) isdesiredalongwith,1=)]TJ /F3 11.955 Tf 9.3 0 Td[(2.TheresultofEq.( 3 )isc(1)=c(2)=)]TJ /F5 7.97 Tf 6.59 0 Td[(h2 2.Therearetwosolutionsforanypossiblevalueofh2insidethemomentumenvelopeduetosymmetryandthusthereisonenullsolution.Bothofthesesolutionsaresingularandabideby3=180+4=270andthereforethenullsolutionsthatexistdonothelpinescapefromthesingularity.Thus,thisisacaseofadegeneratehyperbolicsingularityatthatspecicpointonthemomentumspace.Inaddition,thevalueofdm d=0(i.e.,wherem=p det(AAT))forboth3=180+4=270andfor1and2free,andthusthereisnosetofgimbalanglesthatwillprovideachangeinm(i.e.,nonullsolutionstoescapesingularity). Todetermineiftheothercasesofsingularitywhen3=4=90degareelliptic,wecheckifnullmotionexistsorthogonaltothesingularity(i.e.,dm d6=0).Theresultofthiscasewhen3=4=90and1and2arefreeisconsistentwithEq.( 3 ).Itisfoundthatforanychoiceof1and2givesdm d=0andthereforethisisafamilyofellipticsingularitiesbecausenotonlydoesdet(Q)>0butalsodm d=0.Tovisualizewherethe 41 PAGE 42 externalsingularitiesoccurforthiscase,theSGCMGangularmomentumisplottedforallvariationsof1and2ofafourSGCMGboxarrangementshowninFigure 36 Figure36. AngularmomentumenvelopewithplottedangularmomentumcombinationsforthefourSGCMGboxarrangement InFigure 36 ,allpossiblecombinationsofangularmomentumareplottedontotheangularmomentumenvelopeinblackfor1and2andwhen3=4=90.Inthisgure,everycombinationofthissituationisanexternalsingularity.Itcanbeshownfromsymmetrythatallpermutationsofthiscasehavethesameresultinthattheyareexternalsingularities.Therefore,rooftoparrangementsdonotcontainellipticinternalsingularities.Inaddition,thecasewhentheJacobianforafourSGCMGrooftoparrangementapproachesrank1isasubsetofthisfamily.Thisfamilyofcongurationsisdenedasthecaseswhereatleasttwotorquevectorsareparallelalongtheintersectionofthetwotorqueplanesrandtheoutcomeofthedet(Q)isnotdependentonthegimbalangles.Therank1Jacobiancaseisadegeneratehyperbolicsingularityforwhenthe 42 PAGE 43 torquevectorsareallantiparallel(i.e.,0or2h0)andaexternalsingularitywhenalltheangularmomentumvectorsareparallel(i.e.,4h0). TheseresultsconrmthoseobtainedviatopologyanddifferentialgeometrybyKurokawa[ 41 ].Inreference[ 41 ],itwasstatedthatanyrooftoparrangementswithnolessthansixunitsarefreefrominternalimpassablesurfaces(i.e.,ellipticinternalsingularitiesnotincludingexternalsingularities).KurokawaconcludedthatthereareimpassibleinternalsurfacesinthefourSGCMGrooftoparrangementsthatcorrespondtothesingulardirectionsnotcontainedintheplanespannedbythetwogimbalaxes^1and^2.ThisisexactlythedegeneratecaseshowninEqs.( 3 )( 3 )wheres=^1.Thedegeneratehyperbolicsingularitiesforthesearrangementslieontwocircleswithradius2h0when1=2and3=180+4=270andatzeromomentum0h0when1=)]TJ /F3 11.955 Tf 9.3 0 Td[(2and3=180+4=270shownasanexamplefortheboxarrangementinFigure 37 [ 41 ]. Figure37. DegeneratehyperbolicsingularitiesforthefourSGCMGboxarrangement ThecurvesandpointinFigure 37 arecompartmentalizedandnotspreadthroughouttheentireangularmomentumenvelopeunlikeellipticsingularitiesandthus, 43 PAGE 44 constrainedsteeringalgorithmsexisttoavoidtheseregionswhileprovidingsingularityavoidanceusingnullmotion.3.1.33 4Box The3 4boxarrangementisasubsetoftheboxarrangementinwhichoneoftheSGCMGsisnotusedandleftasaspare.Thisarrangementhasthelongestheritageofightduetoitsconservativenature.Forthisarrrangement,apseudoinverseisnotrequiredtoobtainasolutiontothegimbalratessincethegimbalratesarefounddirectlyfromtheinverseofa33Jacobianmatrix[ 48 ].ThecombinationofthisarrangementandtheconstrainedangularmomentumsteeringlawlimitingcontrollableSGCMGangularmomentumtoa1h0radiusoftheangularmomentumenvelope,ensurethesafestcontrolofasystemofSGCMGs[ 49 ].Thisdesignalthoughsafe,maynotbepracticalforsmallsatelliteapplicationsbecausetheSGCMGsmustbeoversizedtoprovidethedesiredperformance(i.e.,theydonottakeadvantageoftheentiremomentumenvelope). 44 PAGE 45 AExternalsingularsurfacefor3 4box BInternalsingularsurface3 4box Figure38. Singularsurfacesshowing1h0singularityfreeregion Inthisarrangement,thereisa1h0radiusoftheangularmomentumenvelopethatisguaranteedtobesingularityfree[ 48 ]whichisshownbytheredcircledrawnontheexternalandinternalsingularsurfacesofFigure 38 .Itshouldbementionedthatbecausethereisnolongeranullspace(i.e.,usingonly3ofthe4SGCMGs),anyangularmomentumpointonthesingularsurfacescorrespondtothelocationofaellipticsingularity.Thismakestheconstrainingtothe1h0sphereofangularmomentumimperative.3.1.4ScissorPair ThescissorpairarrangementhasthreesetsofcollinearpairsoftwoSGCMGsorthogonaltoeachother.Thisarrangementisconstrainedtohave1=)]TJ /F3 11.955 Tf 9.3 0 Td[(2atalltimesforbothCMGstoavoidinternalsingularities.WiththisarrangementshowninFigure 39 ,fullthreeaxiscontrolispossiblewithafullrankJacobianmatrixaslongasitdoesnotextendpastthemaximumangularmomentumofthesystem.Asaconsequenceoftheconstraintforthesepairs,onlyonethirdoftheentireangularmomentumenvelopeisutilizedwhichwillbetroublesomeforuseonsmallsatellites(i.e.,sixSGCMGsneededfor2h0ofangularmomentum).Useofthesearrangementswasfoundtoconservepowerwhenasinglegimbalmotorisusedforeachscissorpair[ 50 ].Also,analysishasshown 45 PAGE 46 thatscissorpairsmaybebenecialforspaceroboticsapplicationsincetheirtorqueisunidirectional[ 51 ]. Figure39. 3OrthogonalscissorpairsofSGCMGs Duetothegimbalangleconstraintassociatedwiththisarrangement,internalsingularitiesdonotexisthere.Alsobecauseofthegimbalangleconstraint1=)]TJ /F3 11.955 Tf 9.3 0 Td[(2,threeorthogonalscissorpairscontainonlyexternalsingularitiesthatoccurwhenoneormoreofthepairshasanoverallzerotorquevector(i.e.,undenedtorqueforscissorpair).Whenthisoccurs,theJacobianmatrixcontainsacolumnofzerosfortheassociatedpair.3.1.5Pyramid PyramidarrangementsofSGCMGshaveindependentplanesofangularmomentumandtorquewhichformapyramid.Asaconsequenceoftheseindependentplanesoftorque,thesearrangementswillneverhavetheJacobianmatrixwithranklessthan2.ThisisshowninFigure 310 forafourSGCMGpyramidcluster. Intermsofsmallsatelliteconstraintsandwhenutilizingtheentiremomentumenvelope,thefourSGCMGarrangementseemspracticalamongallpreviouslydiscussedarrangementsforplatformsrequiringhightorqueandslewrateswithnearequalmomentumsaturationinthreedirections. 46 PAGE 47 Figure310. PlanesofangularmomentumandtorqueforafourSGCMGpyramid Figure311. FourSGCMGpyramidarrangement Controlofthesearrangementsismorecomplicatedthanrooftopandboxarrangementsduetothepresenceofellipticinternalsingularitiesbecausenullmotionsolutionsdonotexist.Inaddition,ellipticsingularitiesdonothavecontinuousgimbaltrajectoriesassociatedthecorrespondingcontinuousangularmomentumtrajectories[ 47 ].Thesearrangementsarestudiedfortheirdesirablemomentumenvelope(i.e.,itispossibletogetanearsphericalangularmomentumenvelopewithaskewangleof=54.74deg)[ 52 ].Ifhighagilityiswhatisneededandtherearemorerelaxedpointingrequirements,thepyramidmayprovidebenetsovertheotherarrangements.Evenifthisisnotthe 47 PAGE 48 case,ifthisarrangementishostedonasmallsatelliteandtheattitudeerrorinducedfromthetorqueerrorprovidedbythesingularityescapeofellipticsingularitiesisonthesameorderoftheattitudedeterminationsensorsand/ormethods,thenthetorqueerrorusedforsingularityescapewillbeinconsequential.3.2ChoiceofArrangement BeyondthecommonarrangementsofSGCMGspreviouslydiscussed,itisdifculttochoosethearrangementofSGCMGsthroughshapingoftheangularmomentumenvelopeofthesystem.Thisisduetothelocationsofwhereinternalsingularitiesliewithintheangularmomentumenvelopedenotedbytheinternalsingularsurfaces(e.g.,seeFigure 26 ).Thesesingularitiesaredispersedandmaycovertheentireangularmomentumenvelopeleavingonlyverysmallsingularityfreeareas.Formulatingtheproblemasaparameteroptimizationasin[ 53 ]canonlyprovidetheoptimalarrangementforagivensetofslewsandinitialgimbalangleswhichmakestheproblemmoreconstrainedthanuseful.Forexample,wecanexpressthegimbalaxesrelationtothespacecraftbodyframeintermsoftheEulerangles,twoofwhicharetheoptimizedconstantsinclinationanglei,spacinganglei,andthethirdisthegimbalanglei.TheDCMthatisusedtotransformfromthebodytothegimbalframeis CGiB=C3(i)C2(i)C3(i)(3) TheangularmomentumoftheSGCMGsistransformedfromthegimbalFGitothespacecraftbodyframeFBthroughthisDCMas h=nXi=1CBGihi(3) whichisconsistentwithEq.( 2 ).Therefore,holdingthespacingandinclinationanglesconstant,theresultantangularmomentumoftheCMGsystemisaninstantaneous 48 PAGE 49 functionofonlythegimbalanglesforSGCMGs.ConsideringthisandthetruncateddynamicmodelofSGCMGsfromChapter 2 ,thecostfunction M=Zti+1ti()]TJ /F7 11.955 Tf 9.3 0 Td[(m2+aeTe+b_T_)dt(3) whereaandbarescalarsmakingthecostfunctionunitlessande=_h)]TJ /F7 11.955 Tf 12.61 0 Td[(h0A_,canoptimizethesystemwithrespecttominimaltorqueerrorthroughthechoiceoftheEuleranglesforagivenslew,slewtime,andinitialconditionsofthegimbalangles.ThisprocedurefortheparameteroptimizationisshowninFigure 312 Figure312. Optimizationprocessblockdiagram 49 PAGE 50 3.3Simulation AnexamplesimulationofaresttorestattitudemaneuverhastheparametersinTable 31 .Thissimulationwillshownthebenetofdifferentarrangementsonperformingthismaneuver(i.e.,trackingthetorquefromthecontroller).Itshouldbenotedthattheinitialconditionsofthegimbalanglesalthougharethesameforeveryarrangement,theyproduceadifferentinitialCMGangularmomentum. Table31. ModelParametersVariableValueUnits J0@100)]TJ /F4 11.955 Tf 9.3 0 Td[(2.01.5)]TJ /F4 11.955 Tf 9.3 0 Td[(2.0900)]TJ /F4 11.955 Tf 9.29 0 Td[(601.5)]TJ /F4 11.955 Tf 9.3 0 Td[(6010001Akgm20[0000]Tdege0[0.04355)]TJ /F4 11.955 Tf 11.95 0 Td[(0.087100.043550.99430]T\000!0[000]Tdeg=sh0128Nmsk0.051=s2c0.151=sa11=N2m2s2b11=s2t0.02sess0.0001deg Theresultsweresimulatedusingthefollowingeigenaxiscontrollogic[ 54 ] =)]TJ /F4 11.955 Tf 9.3 0 Td[(2kJe)]TJ /F7 11.955 Tf 11.95 0 Td[(cJ!+!J!(3) combinedwithafourthorderRungaKuttaintegratoratatimesteptuntilthesteadystateerrortoleranceoftheerrorquaternioneigenangleesswasreached.ThesimulationcomparestheoptimizedsolutiontothefourSGCMGpyramidarrangementataskewangle=54.74deg. Theresultsforthisexampleatinitialconditions0,e0,and!0havethesolutionforthesystem'ssingularsurfaceswithcalculatedarrangementEuleranglesshowninFigure 313 50 PAGE 51 AExternalsingularsurface BInternalsingularsurface Figure313. SingularsurfacesfortheoptimizedarrangementattheEulerangles=[170.213.685.5168.0]Tdegand=[17.7167.0304.392.5]Tdeg ThegimbalratesfortheoptimizedarrangementinFigure 314 Aareapproximatelythesamemagnitudethanthatforthepyramidarrangement,althoughtheyhaveasmoothertransientresponse. AGimbalratesforoptimizedarrangement BGimbalratesforpyramidarrangement Figure314. Gimbalratesfortheoptimizedandpyramidarrangements ThetorqueerrorshowninFigure 315 Afortheoptimizedcaseissmallermagnitudethanthatforthepyramidarrangementduetotheareaunderthecurvesthus,moretorqueerrorisaddedduringthemaneuverforthepyramidarrangement. 51 PAGE 52 ATorqueerrorforoptimizedarrangement BTorqueerrorforpyramidarrangement Figure315. Torqueerrorfortheoptimizedandpyramidarrangements Fortheoptimizedmethod,thesingularitymeasureisfarfromsingularityinitiallyanddoesnotencounteritasshowninFigure 316 A.ThisisincontrasttothatforthepyramidarrangementshowninFigure 316 B,whichstartsoutinitiallyfarfromsingularityandthenencounterssingularityseveraltimesduringthemaneuver.ThenegativequadratictermpresentinthecostfunctionofEq.( 3 )forthissingularitydoesnotweightdistancefromsingularityashighastorqueerrorwhichcanbeseenwhencomparingFigures 316 AandBto 315 AandB. ASingularitymeasureforoptimizedarrangement BSingularitymeasureforpyramidarrangement Figure316. Singularitymeasurefortheoptimizedandpyramidarrangements 52 PAGE 53 Finally,thecostfunctionoftheoptimizedarrangementinFigure 317 AislessthanthatforthepyramidinBduetotheareunderthecurves. ACostforoptimizedarrangement BCostforpyramidarrangement Figure317. Optimizationcostfortheoptimizedandpyramidarrangements Thesesimulationssupporttheideathatifitwheremechanicallypossibletorecongurethegimbalaxisarrangementsinatimelymanner,andtheinitialgimbalanglesandmaneuverofinterestwereknown,asolutiontotheoptimalCMGarrangementcanbefound.Inaddition,thesesimulationsprovethatyoucannotsimplychooseanoptimalarrangementbecausetheproblemisnotonlydependentontheattitudemaneuver,butalsodependentontheinitialconditionsofthegimbalangles. IfadesiredarrangementisknownwhileonorbitandtherewasamechanicalwaytoreconguretheSGCMGgimbalaxes,suchasintheHoneywellpatent[ 55 ],thentherewouldbemeritinndinganalgorithmthatwassuccessfulinreorientingthegimbalaxesoftheCMGarrangement.Although,noalgorithmexiststoreorienttheSGCMGgimbalaxeswhilekeepingspacecraftunperturbed.Also,suchanalgorithmwouldstillrequireangularmomentumofoadingduetothenatureofSGCMGs. 53 PAGE 54 CHAPTER4SURVEYOFSTEERINGALGORITHMS Aguidance,navigation,andcontrol(GNC)systemiscomposedoftheloopsshowninFigure 41 Figure41. OuterandinnerloopsofGNCsystem TheoutermostloopofaspacecraftGNCsystemconcernsthenavigation(i.e.,providesthestateknowledge)andisusuallytheminimumloopneededforanymission.Thesecondmostouterloopisconcernstheguidanceofthesystem(i.e.,providesthedesiredtrajectories)(e.g.,trajectoriesavoidingpointingastarcameratowardsthesun).Aloopinnertotheguidanceloopconcernsthecontrolofthesystem(i.e.,generatesanerrorofthestateknowledgefromthenavigationloopwiththedesiredtrajectoriesfromtheguidancelooptobeminimized).Theinnermostloopconcernsthedistributionofthedesiredcontroltothesystemsactuators(e.g.,whatthrustersneedtore,whatreactionwheelsorCMGsneedtomove).Steeringalgorithmsareconcernedwiththeinnermost 54 PAGE 55 loopofFigure 41 whenthedifferentialequationrelatingthecontroltotheactuatorsissingular.Whenthisequationissingular,thesteeringlawrealizesasolution.4.1MoorePenrosePseudoInverse AnearlymethodusedtomapthegimbalratesfromtherequiredoutputtorqueusestheminimumtwonormleastsquaressolutionalsoknownastheMoorePenrosepseudoinverse.Thesolutionofthegimbalratesusingthispseudoinversemappinghastheform _=1 h0A+_h=1 h0AT(AAT))]TJ /F2 7.97 Tf 6.59 0 Td[(1_h(4) whereA+istheMoorePenrosepseudoinverse,h0isthemagnitudeofSGCMGangularmomentum,_histheSGCMGoutputtorque,and_arethegimbalrates.TheMoorePenrosepseudoinverse,however,issingularwhentheJacobianmatrixAhasrank3[ 56 ].ItmightseemintuitivethattheadditionofmoreSGCMGactuatorsincreasesthepossibilityofhavingfullrank,buttheperformanceisnotequallyincreasedforallofCMGarrangements.Thisisbecausethereare2nsingularcongurationsforanygivensingulardirectionofasystemcontainingnSGCMGs[ 41 ].Also,theMoorePenrosepseudoinverseandvariationsofitcausethesystemtomovetowardsingularstateswhenperformingdiscretetimecontrol[ 47 ]. Tohandlecaseswhensingularitiesmaybeencountered,steeringalgorithmsareused.SteeringalgorithmscanbebrokendownintothefollowinggroupsasshowninFigure 42 4.2SingularityAvoidanceAlgorithms Singularityavoidancealgorithms,aremethodswhichsteerthegimbalsoftheSGCMGsawayfrominternalsingularities.Thesemethodseitherconstraintheangularmomentumenvelopeand/orgimbalangles,orapplynullmotiontoavoidsingularityencounters.AsdiscussedinChapter 2 ,amethodthatusesonlynullmotioncannotavoidorescapeellipticinternalsingularities[ 44 47 ]. 55 PAGE 56 Figure42. Steeringalgorithms 4.2.1ConstrainedSteeringAlgorithms Constrainedsteeringalgorithmseitherconstrainthegimbalanglesand/oruseableangularmomentumtoavoidsingularities.Thesesteeringlawsareaformofsingularityavoidancethattakesintoaccountthelocationsofsingularitiesapriori.Asaconsequenceofnotusingtheentireangularmomentumspace,thesesteeringlogicsaretypicallymoreeffectiveforsystemswheretheSGCMGsareoversized.Honeywellhaspatentedmethodsthatdonotexplicitlyusenullspacebutthatimplicitlydosobycreatingconstraintsthatkeepthegimbalsawayfromsingularitywithoutneedingtorecognizetheirpresenceexplicitly[ 57 59 ].AsimpleexampleisthesteeringlogicforscissoredpairsinChapter 3 ,wheremereconstraintsareusedtokeepthearrayoutoftrouble.Thismethodisabletoguaranteesingularityavoidanceandanaavailabletorquebutreducestheavailableworkspaceofthesystembyrequiringittobesingularityfree[ 52 60 ].4.2.2NullMotionAlgorithms4.2.2.1Localgradient(LG) Singularityavoidancealgorithmsknownaslocalgradient(LG)methodsusenullmotiontokeeptheJacobianmatrixfrombecomingsingular.Thisisaccomplishedthroughchoiceofthenullvectordtomaximizeanobjectivefunctionthatrelatesthe 56 PAGE 57 distancefromsingularitysuchastheJacobianmatrixconditionnumber,smallestsingularvalue[ 61 62 ],orthesingularitymeasuremwhichisexpressedas m=p det(AAT)(4) Anexampleofthenullvectorcalculationisd=rf=@f @m@m @T=)]TJ /F4 11.955 Tf 9.3 0 Td[(1 m2@m @T(4) wheretheobjectivefunctionf=1=m[ 42 63 64 ].Minimizationofthisobjectivefunctionmaximizesthedistancefromsingularitybymaximizingthesingularitymeasure.TheLGmethods,however,cannotavoidorescapeellipticinternalsingularitiesbecausetheyapplyonlynullmotion[ 44 64 ].Thenullvectordcanbearbitrary,althoughtheprojectionmatrixwhichmapsitontothenullspaceisconstrained.4.2.2.2Globalavoidance/Preferredtrajectorytracking Awayofchoosingthenullmotionvectortosteergimbalstoalternatenonsingularcongurationsbeforemaneuveringisknownaspreferredtrajectorytracking[ 63 65 67 ].Preferredtrajectorytrackingisaglobalmethodthatcalculatesnonsingulargimbaltrajectoriesofine.Thegimbalsconvergetothesetrajectoriesusingnullmotiontominimizeanerror()]TJ /F10 11.955 Tf 11.96 0 Td[().Thegimbalratesusingthismethodare _=1 h0A+_h+[1)]TJ /F8 11.955 Tf 11.95 0 Td[(A+A]()]TJ /F10 11.955 Tf 11.96 0 Td[()(4) wherearethepreferredtrajectoriesandisthesingularityparameterdenedby =0exp)]TJ /F11 7.97 Tf 6.59 0 Td[(m2(4) withconstants0and.Sincethismethodcalculatesthepreferredtrajectoriesofine,itisnotrealtimeimplementable.Also,preferredtrackingreliesentirelyonnullmotionandthuswillbeunabletoescapeellipticinternalsingularities. 57 PAGE 58 4.2.2.3GeneralizedInverseSteeringLaw(GISL) TheGeneralizedInverseSteeringLaw(GISL)providesapseudoinversewhichisavariationoftheMoorePenrosepseudoinverse.ThismethoddenesanotherJacobianmatrixBwhichhaseachofitscolumnsorthogonaltotheassociatedcolumnoftheoriginalJacobianmatrixA(i.e.,ai?bi,notnecessarilyai)]TJ 1.33 1.32 Td[()]TJ /F6 11.955 Tf .33 .17 Td[(?bjwhereA=[a1a2a3a4]andB=[b1b2b3b4])[ 68 ].Therefore,asanexampleforafourCMGpyramidarrangement,thematrixAandBhavethefollowingform A=266664)]TJ /F7 11.955 Tf 9.29 0 Td[(ccos(1)sin(2)ccos(3))]TJ /F7 11.955 Tf 9.3 0 Td[(sin(4))]TJ /F7 11.955 Tf 9.3 0 Td[(sin(1))]TJ /F7 11.955 Tf 9.3 0 Td[(ccos(2)sin(3)ccos(4)scos(1)scos(2)scos(3)scos(4)377775(4) and B=266664)]TJ /F7 11.955 Tf 9.3 0 Td[(csin(1))]TJ /F7 11.955 Tf 9.29 0 Td[(cos(2)csin(3)cos(4)cos(1))]TJ /F7 11.955 Tf 9.3 0 Td[(csin(2))]TJ /F7 11.955 Tf 9.29 0 Td[(cos(3)csin(4)ssin(1)ssin(2)ssin(3)ssin(4)377775(4) wherec,sarethecosineandsineofthepyramidskewangleandiarethegimbalangles,respectively.Thepseudoinverseofthissteeringlawwiththediscussedmatricesis AGISL=(A+B)T(A(A+B)T))]TJ /F2 7.97 Tf 6.59 0 Td[(1(4) Itisimportanttonotethatthispseudoinversedoesnoteliminatetheproblemofinternalsingularities.TheGISLaddsnullmotionfromtheadditionofBandthereforecouplesthenullandforcedsolutionintoasingleinverseandthus,itisnotabletoavoidellipticinternalsingularities. Proof: Claim:TheGISLprovidesonlynullmotionthroughB 58 PAGE 59 _=AGISL=(A+B)T(A(A+B)T))]TJ /F2 7.97 Tf 6.58 0 Td[(1_h Thetorqueerroris _h)]TJ /F8 11.955 Tf 11.95 0 Td[(A_=A(A+B)T(A(A+B)T))]TJ /F2 7.97 Tf 6.59 0 Td[(1_h)]TJ /F8 11.955 Tf 13.48 2.66 Td[(_h=_h)]TJ /F8 11.955 Tf 13.48 2.66 Td[(_h=0 IfthematrixB=A then AGISL=((1+)A)T((1+)AAT))]TJ /F2 7.97 Tf 6.59 0 Td[(1=A+ Therefore,thematrixBwhosecomponentsareorthogonaltoAmustonlyprovidenullmotionandthosethatarealongAvanish.Because,theGISLoranygeneralizedinverseusedforsingularityavoidanceonlyaddsnullmotion,itisunabletoavoidellipticinternalsingularities.4.3SingularityEscapeAlgorithms Singularityescapemethods,knownaspseudoinversesolutions,addtorqueerrortopassthroughorescapeinternalsingularity[ 39 40 69 70 ].Thesemethodsdonottakeintoconsiderationthetypeofinternalsingularitythatisbeingapproachedwhenaddingtorqueerror.4.3.0.4SingularityRobust(SR)inverse TheSingularityRobust(SR)inverseisavariationoftheMoorePenrosepseudoinverse[ 69 ]where,apositivedenitematrix1composedofanidentitymatrixscaledbythesingularityparameterinEq.( 4 )isaddedtothepositivesemidenitematrixAAT.Thepseudoinverseofthismethodhastheform ASR=AT(AAT+1))]TJ /F2 7.97 Tf 6.58 0 Td[(1(4) TheSRinverseisabletoescapebothhyperbolicandellipticsingularities[ 44 ],although,isineffectiveingimballockescape.Toovercomethissituation,amodiedSRinverseknownastheGeneralizedSingularityRobust(GSR)pseudoinversewasdeveloped[ 39 40 ]. 59 PAGE 60 4.3.0.5GeneralizedSingularityRobust(GSR)inverse TheGSRinverseapproachreplacestheconstantdiagonalpositivedenitematrix1withatimevaryingpositivedenitesymmetricmatrixE E=266664112113231377775,i=0sin(!it+i)(4) wheretheoffdiagonaltermsofEaretimedependenttrigonometricfunctionswithfrequency!iandphaseshifti.TheGSRinverseprovidesameansofescapeofthegimballockcongurationassociatedwithasystemofSGCMGs.TheGSRpseudoinversehastheform AGSR=AT(AAT+E))]TJ /F2 7.97 Tf 6.59 0 Td[(1(4) andliketheSRinverse,isguaranteedtoavoidbothhyperbolicandellipticinternalsingularities.4.3.0.6SingularDirectionAvoidance(SDA) AnothermodicationoftheSRinverseknownastheSingularDirectionAvoidance(SDA)onlyappliestorqueerrorinthesingulardirectionandthereforereducestheamountoftorqueerrorneededforsingularityescape.TheSDAmethoddecomposestheJacobianmatrixusingasingularvaluedecomposition(SVD)todetermineitssingularvalues.Thematrixofsingularvaluesisregulatedwiththeadditionoferrortothesmallestsingularvalue3sothatthepseudoinverseisdened.ThepseudoinverseusingSDAhastheform 60 PAGE 61 ASDA=V2666666641 10001 20003 32+000377777775=VSDAUT(4) whereiarethesingularvalues.Regulatingonlythesmallestsingularvalue,reducestheamountoftorqueerroraddedandcreatessmoothergimbalratetrajectorieswhencomparedtotheSRandGSRinverses[ 70 ].ThisisobviouswhentheSRinversedecomposedusingSVDas ASR=V2666666641 12+0002 22+0003 32+000377777775UT=ASR=VSRUT(4) whereallthesingularvaluesareregulatedandhencethereistorqueerrorinalldirections.ItisclearfromEqs.( 4 )and( 4 )thatSRinverseandSDAaresusceptibletogimballockbecausewhentheoutputtorqueisalongthesingulardirection_h/s=u3thenitisinthenull(SDAUT)andnull(SRUT)thusencounteringgimballockasnoconsequencetothesizeofthetorqueerroraddedfrom.WithouttheperturbationstotheJacobianmatrixthatarenotgimbalstatedependentatgimballockthesystemremainslockedinasingularconguration. RecallfromChapter 3 ,thatforafourCMGpyramidarrangement,therankisneverlessthantwoandthereforeitisacceptabletoregulateonlythesmallestsingularvalue.However,iftheskewangleismadecloseto0,90,180,or270deg(i.e.,boxorplanararrangement),theJacobianmatrixforthesearrangementswillhaveatleasttwosmallsingularvalueswhennearsingularityandregulationofthesmallestsingularvaluemaybeineffective. 61 PAGE 62 4.3.0.7FeedbackSteeringLaw(FSL) TheFeedbackSteeringLaw(FSL)providesasolutiontothegimbalrateswithoutusinganinverse.ThismethodisderivedfromaminimizationofthetorqueerrorwhichissimilartohowtheSRinverseisderived.TheoptimizationfordeningFSLhasthefollowingstructure min_2R41 2264e_375T264K100K2375264e_375(4) whereK1andK2arepositivedenitegainmatrices,ande=_h)]TJ /F8 11.955 Tf 12.04 0 Td[(A_.ThisminimizationreducestotheSRinversewhenK1=1andK2=1andwhere=0exp)]TJ /F11 7.97 Tf 6.58 0 Td[(m2)fromEq.( 4 ). TheFSLmethodhasK2=1andK1=K(s)asacompensator.ThecompensatorisderivedfromanH1minimization minK(s)2R330B@w1(s)[1+AK(s)])]TJ /F2 7.97 Tf 6.58 0 Td[(1w2K(s)[1+AK(s)])]TJ /F2 7.97 Tf 6.59 0 Td[(11CA1(4) wherew1(s)andw2areweightingmatrices.Thew1(s)matrixisdenedbelow w1(s)=264AKBKCK0375(4) whereAK,BK,andCKarematricesassociatedwithstatespacemodelofthesystem.Thew2matrixisconstantandis w2=1 w144(4) wheretheconstantwboundsthegimbalrates.Thestatespacemodelofthesystemhastheform 62 PAGE 63 _^x=AK^x+BKe_=CK^x(4) TheoutputmatrixCKisanexplicitfunctionoftheCMGgimbalanglesexpressedas CK=ATb!2 P(4) withbasapositivescalarassociatedwiththebandwidthand PisthesteadystatesolutiontotheRiccatiequationofthestatespacesysteminEq.( 4 ).Usingthefeedbackofthesystem,Eq( 4 )willprovideasolutiontothegimbalratesthatdoesnotrequireapseudoinverse.Itshouldbenotedthatthesystemmaystartoutstable,however,theobservabilityofthesystemmaybelostresultingininstability,duetoCK'sexplicitdependenceonthegimbalangle(i.e.,H(s)=CK(s1)]TJ /F8 11.955 Tf 12.41 0 Td[(AK))]TJ /F2 7.97 Tf 6.59 0 Td[(1BKwhereCK6=constant).Forfurtherinformationonthedevelopmentofthismethod,pleasesee[ 71 ].ThissteeringalgorithmwasshowntogounstableforcertainvaluesofCKcorrespondingtospecicgimbalanglessets(see[ 72 ]).4.3.0.8SingularityPenetrationwithUnitDelay(SPUD) TheSingularityPenetrationwithUnitDelay(SPUD)algorithmescapessingularitythroughreuseofthepreviousgimbalratecommandwhenatacertainthresholdofsingularity[ 73 ].Thepreviouscommandissavedthroughazerothorderholdtothesystem.Escapeofasingularityisalwayspossibleunlessthesystemisinitiallyatthethresholdofsingularity,thenthereisnopreviouscommandtouseforsingularityavoidance.Also,SPUDisnotintendedforattitudetrackingmaneuvers.TheSPUDalgorithmaccumulatesattitudetrackingerrorwhileescapingsingularityandtherearenoguaranteesonhowlongitwilltaketoescapesingularityandhowlargethetorquedisturbancewillbeonthespacecraftasitsperformanceisdirectlyassociatedwiththesystemandthechoiceofsingularitythreshold. 63 PAGE 64 4.4SingularityAvoidanceandEscapeAlgorithms Singularityavoidanceandescapealgorithmsavoidsingularitiesthroughnullmotionwheneverpossibleandusetorqueerrorforescapewhentheyarenot.4.4.0.9Preferredgimbalangles PreferredgimbalanglesareasetofinitialgimbalanglesforSGCMGsthatcanbereachedbynullmotion.Theseanglesarepreferredsincemaneuversoriginatingfromthemavoidasingularconguration[ 74 ].ThissetofanglesisfoundbybackwardsintegrationoftheEq.( 4 )andtheattitudeequationsofmotion.Ithasbeenshownthatthismethodcannotavoidsingularitiesiftheinitialsetofgimbalanglesis0=[45)]TJ /F4 11.955 Tf 9.29 0 Td[(4545)]TJ /F4 11.955 Tf 9.3 0 Td[(45]Tdeg[ 74 ].Sincethenullspaceprojectionmatrixisundenedatsingularity,theSRinverseisusedinplaceoftheMoorePenrosepseudoinverseofAin_nas _n=[1)]TJ /F8 11.955 Tf 11.96 0 Td[(ASRA]d(4) Asaresult,thiscausesthesystemtoaddtorqueerrorwhenatsingularity.Inpractice,thismethodactsasanofineoptimizationwhichdeterminestheinitialset(s)ofthegimbalanglesthatwillgivesingularityfreemaneuver(s).However,itisnotpossibletogofromonetoanypointingimbalspacethroughnullmotionitselfbecausetherewillneverbendimensionsofnullspace.4.4.0.10Optimalsteeringlaw(OSL) TheInnerProductIndex(IPI)combinedwiththeoptimalsteeringlaw(OSL)isusedtodetermineasteeringalgorithmthatproducesminimumtorqueerrorwhilebothavoidingandescapinginternalsingularities[ 75 ].Thesingularityindexisaddedtotheminimization min_2R4,e2R3[)]TJ /F7 11.955 Tf 9.3 0 Td[(cV(+_t)+1 2_TW_+eTR)]TJ /F2 7.97 Tf 6.59 0 Td[(1e](4) 64 PAGE 65 wheretistheonesteptimedelay,V(+_t)istheIPI,e=_h)]TJ /F8 11.955 Tf 12.57 0 Td[(A_(i.e.,torqueerror),cisapositivescalar,andWandR)]TJ /F2 7.97 Tf 6.58 0 Td[(1arepositivedeniteweightingmatrices.TheIPIisapproximatedbyaTaylorseriesexpansionuptothe2ndorderas V(+_t)V+@V @T_Tt+1 2_T@2V @2_t2(4) wheretheIPIVisexpressedasasumofsquareofinnerproductsofthecolumnvectorsoftheJacobian. V=1 24Xi=j=1,i6=j(aTiaj)2(4) TheresultoftheminimizationinEq.( 4 )usingthisapproximationofVis _=H)]TJ /F2 7.97 Tf 6.58 0 Td[(1AT(AH)]TJ /F2 7.97 Tf 6.58 0 Td[(1AT+R))]TJ /F2 7.97 Tf 6.59 0 Td[(1_h+[H)]TJ /F2 7.97 Tf 6.59 0 Td[(1AT(AH)]TJ /F2 7.97 Tf 6.59 0 Td[(1AT+R))]TJ /F2 7.97 Tf 6.58 0 Td[(1AH)]TJ /F2 7.97 Tf 6.58 0 Td[(1)]TJ /F8 11.955 Tf 11.96 0 Td[(H)]TJ /F2 7.97 Tf 6.59 0 Td[(1]g(4) wheretheHessianmatrixHisdenedas H=ct2ggT+W(4) withgradientg=@V @T.TheweightingmatrixRshownpreviouslyintheminimizationofEq.( 4 ),isexpressedas R=U266664000000000exp)]TJ /F11 7.97 Tf 6.59 0 Td[(23377775UT(4) where3isthesmallestofthesingularvaluesoftheJacobianmatrix,0andarepositivescalarsandUistheunitarymatrixmadeupoftheleftsingularvectorsfromthesingularvaluedecompositionoftheJacobianmatrixA.ThisadditionoftorqueerrorintothegimbalratestateequationisanalogoustotheSDAmethodexceptthatitisalso 65 PAGE 66 addedtothefreeresponsesolution[ 70 ].Itshouldbenotedthatthissteeringalgorithmdoesnotconsidertheformofinternalsingularitiesandtherefore,doesnottrulyminimizetheamountoftorqueerrorforsingularityescape.Thisisbecausenondegeneratehyperbolicsingularitiesareavoidablethroughnullmotionwithouttheuseoftorqueerror.AtanondegeneratehyperbolicsingularityRisnonzeroandthustorqueerrorisstilladded(see[ 72 ]).4.5OtherSteeringAlgorithms Otherpublishedsteeringalgorithmsthathavenotbeendiscussedcanbefoundinthereferences[ 44 61 76 81 ].Thesemethodsincludemathematicaltechniquessuchasneuralnetworks,optimization,andgametheory.4.6SteeringAlgorithmComputationComparison Ananalysiscomparingthecomputationfortheimplementationofthementionedsteeringalgorithmsisdifcultduetolackofinformationonhowsomewerecodedinliterature.Forexample,someofthesealgorithmsareofineandmayrequirealargenumberofmemorycallsandstoredmemorybutnotasmanyops.Itishowever,usefultoquantifyingsomeofthepreviouslydiscussedsteeringalgorithmsintermsofoatingpointoperationsthatarenotcalculatedofline.TheseareshowninTable 41 foralgorithmswhereopsmakeagoodcomparison.Inthistable,themetricofcomparisonisanapproximatenumberofopspertimestep. Table41. AlgorithmFlopsm=row(A)andn=column(A) VariableValue MPO(m4)LGO(m4)GISLO(m4)SRO(m4)GSRO(m4)SDAO(nm3)FSLO(mn2)SPUDO(m4)OptimalSteeringO(m4)+O(nm3) 66 PAGE 67 Itshouldbementionedthatmanyofthesteeringalgorithmsdiscussedhavethesameorderofmagnitudeofops(e.g.,MP,LG,GISL,SR,GSR,andSPUD)duetotheapproximatenumberofopsforaGaussJordanmatrixinverse.ItisassumedthatthecalculationofthegradientvectorforLG,andOSLisinmemoryandthattheopsassociatedwiththenareonalowerorderofmagnitudethatanSVDorGaussJordanmatrixinverse.TheOSLhastodobothSVDforcalculationoftheRmatrixandtheGaussJordanmatrixinverseandthereforehasO(m4)+O(nm3)ops.TheadditionofO(m4)+O(nm3)isinsertedfortheapproximateopsofOSLbecausedependingontheamountofSGCMGsthisalgorithmisworkingforO(m4) PAGE 68 CHAPTER5STEERINGALGORITHMHYBRIDSTEERINGLOGIC5.1HybridSteeringLogic Existingsteeringlogics(seeChapter 4 )donotexplicitlyconsiderthetypeofsingularitythatisbeingencounteredandthus,donotcompletelyaddressattitudetrackingperformanceofSGCMGattitudecontrolsystems.AproposedmethodknownastheHybridSteeringLogic(HSL)whichutilizestheknowledgeofthetypeofsingularityencountered(i.e.,ellipticorhyperbolicsingularities)toimprovetheattitudetrackingperformanceoftheSGCMGattitudecontrolsystem,isdevelopedforafourSGCMGpyramidarrangementataskewangle=54.74deg.Byusingahybridapproach,HSLactsasanLGmethod(i.e.,nullmotionforsingularityavoidance)athyperbolicsingularityandanSDAmethod(i.e.,pseudoinversesolutionsforsingularityescape)atellipticsingularity.Also,becauseHSLisdevelopedforafourSGCMGpyramidarrangement,thereisnoexistenceofdegeneratehyperbolicsingularities[ 41 ].ThechallengeistodeveloptheappropriatesingularitymetricssuchthattheLGandSDAcomponentsofthehybridstrategydonotcounteracteachotherduringoperation.5.1.1InternalSingularityMetrics ThesingularitymetricsdevelopedareofsimilarformasthesingularityparameterinEq.( 4 )withtheadditionoftermsrelatingtotheformoftheactualsingularity. =0exp)]TJ /F5 7.97 Tf 6.58 0 Td[(a exp)]TJ /F11 7.97 Tf 6.59 0 Td[(1m(5) =0exp)]TJ /F5 7.97 Tf 6.59 0 Td[(b exp)]TJ /F11 7.97 Tf 6.59 0 Td[(2m(5) wherea,b,1,2,0and0arepositivescalarconstantsandmisthesingularitymeasureasdenedinEq.( 4 ).Awayfromsingularity,afourSGCMGpyramidarrangementataskewangle=54.75deg,hasthematrixQ2R.AtsingularitythisSGCMGarrangementhasQ2R22(seeChapter??)andtherefore,thedet(Q) 68 PAGE 69 willbezeroornegative(i.e.,Qisnegativesemideniteorindenite)forhyperbolicsingularitiesandpositive(i.e.,Qisdenite)forellipticsingularities.Takingthisintoaccount,parameters and aredenedas =jQ0)]TJ /F7 11.955 Tf 11.96 0 Td[(det(Q)j(5) =1 jQ0)]TJ /F7 11.955 Tf 11.96 0 Td[(det(Q)j=1 (5) whereQ0isascalarvaluechosenonthesameorderofmagnitudeofdet(Q)butgreatertoscaletheresponseof and .ItisdifculttoanalyticallydeneQ0sinceitdependsonthemaximumvalueofdet(Q)(i.e.,det(Q)varieswithgimbalangleandthereforethemaximummustspanallcombinationsofthegimbalangles)whichisofhighdimensionalityandhighlynonlinear.However,throughsimulationofafourSGCMGpyramidarrangementataskewangle=54.75deg,itwasfoundthatjdet(Q)j<1andthereforewedeneQ01.Inaddition,itisimportanttonotethattheconstantparametersa,b,1,2,0,and0areusedtomorphtheHSLsteeringlogicintotherespectiveLGandSDAmethodswhenappropriate:(e.g.,iftheparametersa=b=0=0and06=0thentheHSLmethodistheLGmethod).Therefore,thechoiceofmetricsandinthiswayensuresthatnullmotionwillbeaddedwhenapproachingahyperbolicsingularityandtorqueerrorwithlessnullmotionwillbeaddedwhenapproachinganellipticsingularity.ItshouldbenotedthatwhenusingHSLdet(Q)isnormalizedbythenominalangularmomentumh0.5.1.2HybridSteeringLogicFormulation Theproposedsteeringlogicisdenedas _=1 h0ASDA,_h+[1)]TJ /F8 11.955 Tf 11.96 0 Td[(A+A]d(5) whereASDA,is 69 PAGE 70 ASDA,=V2666666641 10001 20003 32+000377777775UT(5) IfitisassumedthattheanalyticfunctionforthegradientvectordisderivedofineandthecalculationofitateachtimestepislessthanthatforSVD,thisalgorithmhasthesamenumberofopsonorderasSDAfromTable 41 ofO(nm3)fromtheSVD.ThedifferencebetweentheconventionalASDAandASDA,istheparameterthatregulates3.InASDA,theregulationparameteris(i.e.,differentfrominChapter 4 byusingminsteadofm2)whichis =0exp)]TJ /F11 7.97 Tf 6.59 0 Td[(m(5) withpositiveconstants0and,butwithASDA,thesingularityparameterisdenedinEq.( 5 )whichregulatestheamountofinducedtorqueerrorinthevicinityofellipticsingularities.ThroughaSVDdecompositionofA,Eq.( 5 )canbewrittenas _=1 h0ASDA,_h+[1)]TJ /F8 11.955 Tf 11.96 0 Td[(V264100T0375VT]d(5) Here,thenullmotionprojectionmatrixisexpressedasafunctionofnonsingularmatricesV.Also,veryrobustnumericalalgorithmsexistforcomputingtheSVD,soitscomputationalriskinarealtimeimplementationisnotparticularlyhigh. Thescalarthatregulatesthemagnitudeofthenullmotionis.Thenullvectordisinthedirectionofthegradientoff=)]TJ /F7 11.955 Tf 9.3 0 Td[(det(AAT)=)]TJ /F7 11.955 Tf 9.29 0 Td[(m2andmaximizesthedistancefromsingularity. Thischoiceofthisobjectivefunctionreducesthecomputationneededforthegradient(i.e.,thederivativeof()]TJ /F7 11.955 Tf 9.3 0 Td[(det(AAT))islesscomputationallyintensivethanthe 70 PAGE 71 derivativep (det(AAT))andensuresthattheadditionofnullmotionwillnotapproachinnityattheregionofsingularityforcasessuchasf=1 mandthen@f @=)]TJ /F2 7.97 Tf 13.7 4.7 Td[(1 m2@m @.ItshouldbementionedthatthenullvectorisanonlinearfunctionofthegimbalanglesandissimpliedduetothesymmetryofthefourCMGpyramidarrangement.ToprovethefeasibilityofHSL,astabilityanalysisisconducted.5.2LyapunovStabilityAnalysis ThecandidateLyapunovfunction V=1 2!TK)]TJ /F2 7.97 Tf 6.59 0 Td[(1J!+eTe+(1)]TJ /F7 11.955 Tf 11.96 0 Td[(e4)2(5) ischosenforthisanalysisandcanberewrittenas V=zTMz(5) wherez=[!TeT(1)]TJ /F7 11.955 Tf 12.37 0 Td[(e4)]TandM=diag(1 2K)]TJ /F2 7.97 Tf 6.58 0 Td[(1J,1,1).ConsequentlytheLyapunovfunctionisboundedas minjjzjj2Vmaxjjzjj2(5) whereminandmaxdenotetheminimumandmaximumeigenvaluesofM.Thisboundwillbecomeusefullaterintheanalysis. AresttorestquaternionregulatorcontrollerisgivenbyEq.( 5 )fortheinternalcontroltorque,ischosenforitsownheritageandthefactthatityieldsanglobalasymptoticstablecontrolsolutionproventhroughLaSalle'sInvariantTheorem[ 63 ]. =)]TJ /F8 11.955 Tf 9.3 0 Td[(Ke)]TJ /F8 11.955 Tf 11.95 0 Td[(C!+!J!(5) GainmatricesK=2kJandC=cJofEq.( 5 )arepositivedeniteandsymmetric.Assumingrigidbodydynamics,thespacecraft'sangularmomentumisgivenby 71 PAGE 72 H=J!+h(5) TherotationalequationsofmotioncomefromtakingtheinertiatimederivativeofEq.( 5 )as_!=J)]TJ /F2 7.97 Tf 6.59 0 Td[(1[act)]TJ /F10 11.955 Tf 11.95 0 Td[(!J!](5) withSGCMGoutputtorque_h=)]TJ /F10 11.955 Tf 9.3 0 Td[()]TJ /F10 11.955 Tf 11.96 0 Td[(!h=h0A_(5) where!isthespacecraftangularvelocity,Jisthespacecraftcentroidalinertia,Histhetotalsystemangularmomentum,andhistheangularmomentumfromtheCMGs.Thespacecraft'sangularvelocityandtheCMGangularmomentaaregovernedbyEqs.( 5 )and( 5 )respectively,whereactistheactualcontroltorque(i.e.,maydifferduetoinducedtorqueerrorforsingularityescape).ItisassumedherethatthecontributiontothedynamicsfromthegimbalywheelassemblyinertiasisnegligibleandthereforeJisconstant. Theactualcontroltorqueactbasedonthemappingofthegimbalratesis act=)]TJ /F7 11.955 Tf 9.29 0 Td[(h0A_)]TJ /F10 11.955 Tf 11.95 0 Td[(!h=)]TJ /F8 11.955 Tf 9.3 0 Td[(A(ASDA,_h+[1)]TJ /F8 11.955 Tf 11.96 0 Td[(V264100T0375VT]d))]TJ /F10 11.955 Tf 11.96 0 Td[(!h(5) andneedstobeconsideredintheLyapunovanalysisforstabilityoftheattitudecontroller/steeringalgorithmcombination.Whensimplied,Eq.( 5 )becomes act=U26666400000000)]TJ /F11 7.97 Tf 6.59 0 Td[( 23+377775UT[)]TJ /F3 11.955 Tf 9.3 0 Td[()]TJ /F10 11.955 Tf 11.95 0 Td[(!h]+(5) wherethestabilityofthesystemisaffectedbythetorqueperturbationmatrixHSLfromASDA,denedas 72 PAGE 73 HSL=U26666400000000 23+377775UT(5) Thespacecraftattitudeerrorkinematicsisgovernedby _e=)]TJ /F4 11.955 Tf 10.5 8.08 Td[(1 2!e+1 2!e4(5) _e4=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2!Te(5) whereeisthequaternionerrorvectorelementsande4isitsscalarelement.ThetimederivativeoftheLyapunovfunctionis _V=!TK)]TJ /F2 7.97 Tf 6.59 0 Td[(1[act)]TJ /F10 11.955 Tf 11.96 0 Td[(!J!]+2eT[)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2!e+1 2!e4]+2(1)]TJ /F7 11.955 Tf 11.95 0 Td[(e4)1 2!Te(5) Equation( 5 )canbereducedbysubstitutingintheexpressionforactfromEq.( 5 )andthedesiredcontroltorquevectorfromEq.( 5 ).ThetimederivativeoftheLyapunovfunctionnowyields _V=)]TJ /F10 11.955 Tf 9.29 0 Td[(!TK)]TJ /F2 7.97 Tf 6.59 0 Td[(1[C)]TJ /F10 11.955 Tf 11.95 0 Td[(HSL(C+H)]!+!TK)]TJ /F2 7.97 Tf 6.59 0 Td[(1HSLKe(5) ormorecompactly _V=)]TJ /F7 11.955 Tf 13.75 8.09 Td[(c 2k!T!+c 2k!TJ)]TJ /F2 7.97 Tf 6.59 0 Td[(1HSLJ!+1(5) where1=1 2k!TJ)]TJ /F2 7.97 Tf 6.59 0 Td[(1HSLH!+!TJ)]TJ /F2 7.97 Tf 6.59 0 Td[(1HSLJe(5) 73 PAGE 74 Since 23+1,Eq.( 5 )canbeusedtorewriteEq.( 5 )as_V)]TJ /F7 11.955 Tf 26.37 8.09 Td[(c 2k!TJ)]TJ /F2 7.97 Tf 6.58 0 Td[(1U266664100010001377775UTJ!+1=)]TJ /F3 11.955 Tf 9.29 0 Td[(!TR1!+1(5) andcanbefurtherboundedas_V)]TJ /F3 11.955 Tf 21.92 0 Td[(1jjzjj2+1(5) where1istheminimumeigenvalueofthepositivesemidenitematrixR1(i.e.,1=0atsingularity),=c 2k,andjjzjjisdenedinEq.( 5 ).SubstitutingEq.( 5 )intoEq.( 5 )yields_V)]TJ /F3 11.955 Tf 25.72 8.09 Td[(1 maxV+1(5) ThesolutiontothedifferentialequationinEq.( 5 )inaVolterraintegralformisVV(0)exp()]TJ /F28 5.978 Tf 10.1 4.4 Td[(1 maxt)+Ztt0exp()]TJ /F28 5.978 Tf 10.1 4.4 Td[(1 max(t)]TJ /F11 7.97 Tf 6.59 0 Td[())1()d(5) TheerrorcanbeboundedfromEq.( 5 )as jjzjj2V(0) minexp()]TJ /F28 5.978 Tf 10.1 4.4 Td[(1 maxt)+1 minZtt0exp()]TJ /F28 5.978 Tf 10.1 4.4 Td[(1 max(t)]TJ /F11 7.97 Tf 6.58 0 Td[())1()d(5) Atsingularitywhen1=0,theerroris jjzjj2Vs+1 minZtts1()d(5) whereVsistheerrorattimetheoccurrenceofsingularityattimets. StabilitycannotbeprovenfromEq.( 5 ).Itisassumedthatthesystemwillnotremainlockedinsingularityexceptforthespecialcaseofgimballock.Ifcomponentsofthetorqueneededforstabilityareactuallyinthesingulardirection,periodsofinstabilitymayoccuratsingularity.Thedurationofthisinstabilityisdependentontheselectionandsizingofthesingularityparametersandwhichprovidetorque 74 PAGE 75 errorforsingularityescapeand/ornullmotionforsingularityavoidance.Fromapracticalperspective,stabilitycannotbeprovenforthisLyapunovfunctionatsingularity,sinceatsingularity,thereisnotorqueavailableinthesingulardirection. Forthespecialcaseofasingularitywithgimballock,theangularvelocityofthespacecraftisconstantassumingtheabsenceoffrictionandexternaltorquesinthesystem.Inthiscase,thecontributionfrom1totheerrorisboundedandevensometimeszero.Thiscanbeshownbyevaluatingtheexpressionfortheangularaccelerationatgimballockwhichis _!=0=J)]TJ /F2 7.97 Tf 6.59 0 Td[(1[0+!H](5) ItisclearthatEq.( 5 )issatisedonlywhentheproduct!H=0whichisonlytruewhen!isparalleltoH,!=0,orH=0.When!isparalleltoHorH=0,1=!TJ)]TJ /F2 7.97 Tf 6.59 0 Td[(1HSLJewhichisaboundedsinusoidwhoseintegralisalsoaboundedsinusoid.Therefore,forthesetwocases,theerrorisboundedatgimballock.When!=0,1=0andtheerrorissimplylockedatVs.Itshouldbementionedthatattimesawayfromsingularity,theerrormonotonicallydecreasesbecausethecontributionfrom1totheerrorbecomesnegligible.Careneedstobetakeninthedesignofthesingularityparametersothattheminimumsteadystateerrorisachievedwhilemeetingtheconstraintsoftheactuators. Thesteadystateerrorassumingthatthesystemshasasingularityfreeperiodtowardstheendofthemaneuver(i.e.,doesnotendatsingularity)is jjz(1)jj21 minexp()]TJ /F28 5.978 Tf 10.1 4.39 Td[(1 max1)Z1tsexp(1 max)1()d(5) ThisexpressionisindeterminatesoapplicationofL'Hopital'sruletoEq.( 5 )yields limt)177(!1d dt1 minRttsexp(1 max)1d d dtexp(1 maxt)=)]TJ /F3 11.955 Tf 18.21 8.09 Td[(max min11(1)(5) 75 PAGE 76 whichsuggeststhatasufcientlylargevalueofc(i.e.,larger)willlowertheamountofsteadystateerrorgivingyouauniformlyultimatelybounded(UUB)resultawayfromsingularity.Whenthemaneuverisnished,theeffectof1ontheerrorwillbecomeaconstantassumingthemaneuverendsatrest.Itshouldbenotedthatawayfromsingularitythesizeof1exponentiallydecreasesduetothebehaviorofHSL.ThedifferenceinimpactofHSLratherthanSRinverseonstabilitycanbeobservedfromthemagnitudeofthepositivesemidenitematrixHSLinEq.( 5 )comparedtothematrixshowninEq.( 5 ).TheSRmatrixhasalargernormandthereforehasaworseUUBevenforsufcientlylargevaluesof.FromcomparingEqs.( 5 )and( 5 ),theSDAmethodhasasimilaramountoftorqueerroraddedwhencomparedtoHSL,althoughitwilladdthistorqueerrorwheneverthesingularityapproachednottakingintoaccounttheform. SR=U266664 21+000 22+000 23+377775UT(5) Theaboveresultsareonlyfortheattitudecontroller/steeringalgorithmcombination.Forexample,anattitudecontrollerwhosetorquetrajectorywaschosentoavoidtheoccurrenceofsingularitiesmaynothavetheperiodsofpossibleinstabilitiesatsingularityandthusmayprovidebetterstabilityperformance.However,norealtimecontrollerofthisformexists(i.e.,onethatensuressingularityavoidance)andthuswasnotconsideredinthefollowingsimulations.5.3NumericalSimulation ToevaluatetheperformanceoftheproposedHSLagainstheritagesteeringlogics(i.e.,LGandSDA),simulationswereperformedusingafourSGCMGpyramidalarrangementwithaskewangleof=54.74deg.Toensureafaircomparison,thecontrollogicandsatellitemodelwereidenticalforallsimulations.Foreachsteering 76 PAGE 77 algorithm,threedifferentscenariosweresimulated:(1)startinginazeromomentumconguration=[0000]Tdeg(i.e.,farfromsingularity);(2)startingnearanellipticexternalsingularity=[105105105105]Tdeg;and(3)startingnearanhyperbolicsingularity=[15105195)]TJ /F4 11.955 Tf 9.3 0 Td[(75]Tdeg.ThesingularityconditionswereveriedforeachcasebyobservingthesingularitymeasuredenedinEq.( 4 ).Forthesesimulations,thefollowingperformancemeasureswerecompared:(i)thetransientresponseoftheerrorquaternion,(ii)theamountanddurationofsingularityencounter,(iii)themagnitudeofgimbalrate,(iv)theamountoftorqueerror(i.e.,_h)]TJ /F7 11.955 Tf 12.13 0 Td[(h0A_)forsingularityescape,and(v)nullmotioncontribution.Additionally,,,anddet(Q)arealsoconsidered. TheJacobianassociatedwiththispyramidalcongurationis A=266664)]TJ /F7 11.955 Tf 9.3 0 Td[(c()c(1)s(2)c()c(3))]TJ /F7 11.955 Tf 9.3 0 Td[(s(4))]TJ /F7 11.955 Tf 9.3 0 Td[(s(1))]TJ /F7 11.955 Tf 9.3 0 Td[(c()c(2)s(3)c()c(4)s()c(1)s()c(2)s()c(3)s()c(4)377775,(5) andtheassociatedangularmomentumvectoris h=h0266664)]TJ /F7 11.955 Tf 9.3 0 Td[(c()s(1))]TJ /F7 11.955 Tf 11.96 0 Td[(c(2)+c()s(3)+c(4)c(1))]TJ /F7 11.955 Tf 11.95 0 Td[(c()s(2))]TJ /F7 11.955 Tf 11.95 0 Td[(c(3)+c()s(4)s()(s(1)+s(2)+s(3)+s(4))377775(5) AllsimulationsareperformedusingafourthorderxedtimestepRungaKuttawiththeparametersshowninTable 51 .TheactuatorparameterschosenforthissimulationarebasedontheHoneywellM95SGCMGs,whicharesizedforthesatellitesystemchosenforsimulation[ 82 ]. 77 PAGE 78 Table51. ModelParametersVariableValueUnits J0@100)]TJ /F4 11.955 Tf 9.3 0 Td[(21.5)]TJ /F4 11.955 Tf 9.3 0 Td[(2900)]TJ /F4 11.955 Tf 9.3 0 Td[(601.5)]TJ /F4 11.955 Tf 9.3 0 Td[(6010001Akgm254.74dege0[0.043550.087100.043550.99430]T\000!0[000]Tdeg=sh0128Nmsk0.051=s2c0.151=sm00.5\000ess0.001degt0.02sec 78 PAGE 79 Itshouldbenotedthatcaremustbetakenwhennumericallydeningthesingulardirectionsinces=0whenthesystemhasafullrankJacobian.Becausetherankisnumericallydetermined,atoleranceshouldbesetonthesingularitymeasuretodeterminewhatisconsideredfullrank.Fortheresultspresentedinthispaper,rankdeciencyfortheHSLwasdenedasmm0whereforthissimulationm0=0.5.ThesimulationsterminatewhenthesteadystateerroressdenedinEq.( 5 )isachieved. ess=min[2sin)]TJ /F2 7.97 Tf 6.58 0 Td[(1(jjejj),2)]TJ /F4 11.955 Tf 11.96 0 Td[(2sin)]TJ /F2 7.97 Tf 6.58 0 Td[(1(jjejj)](5) ThemagnitudeofessgiveninTable 51 isbasedoffreference[ 38 ].5.3.1Case1:AtZeroMomentumConguration=[0000]Tdeg Therstsetofsimulationshasinitialgimbalanglesat=[0000]Tdegwhichrepresentsascenariostartingfarawayfromsingularities.Figure 51 showsthiscongurationwhichisalsoatypicalstartupcongurationforafourSGCMGpyramidarrangement. Figure51. ZeromomentumcongurationofafourSGCMGpyramidarrangement 79 PAGE 80 5.3.1.1Localgradientsimulationresults TheparametersfortheLGsimulationare:0=a=b=1=0and2=0=1.Figures 52 AandBshowthattheLGmethodwasabletoperformthemaneuvertothegivenerrortoleranceesswithoutinducingtorqueerror.TheabsenceoftorqueerrorinFigure 52 BisduetothezerovalueofsingularitymetricinFigure 52 C(i.e.,LGisanexactmapping).ThenullmotionshowninFigure 53 BissmallbutsignicantwhencomparedtothetotaloutputgimbalratesinFigure 52 A.ThisisaconsequenceofthesingularitymetricinFigure 53 D.Figures 53 CandDshowthatthemaneuverwascompletedwithoutsingularityencounter. AQuaternionerrorvectorelements BTorqueError CAlpha DBeta Figure52. SimulationresultsforLGwith0=a=b=1=0and2=0=1atzeromomentum 80 PAGE 81 ACMGgimbalrates BNullmotion CSingularitymeasure Ddet(Q) Figure53. SimulationresultsforLGwith0=a=b=1=0and2=0=1atzeromomentum(contd.) 81 PAGE 82 5.3.1.2SingularDirectionAvoidancesimulationresults TheparametersfortheSDAsimulationare:0=0.01,0=a=b=2=0,and1=1.ThismethodshowssimilarresultsinthetransientresponseoftheerrorstatesinFigure 54 AtothatforLGinFigure 52 AwiththeexceptionofnonzerotorqueerrorseeninFigure 54 B.Also,thismethodhadaslowerrateofconvergencetothesteadystateerroressthanLGasevidentfromthetimeinsimulationinFigure 52 A.Thisisduetothesmallnonzerovalueofthesingularitymetric,showninFigure 54 C.Figure 55 BshowsazeronullmotioncontributiontothegimbalratesinFigure 55 AforSDA.Figures 55 CandDandFigures 53 CandDarealmostequivalentbecausethesystemstartedfarawayfromsingularity. 82 PAGE 83 AQuaternionerrorvectorelements BTorqueError CAlpha DBeta Figure54. SimulationresultsforSDAwith0=0.01,0=a=b=2=0,and1=1atzeromomentum 83 PAGE 84 ACMGgimbalrates BNullmotion CSingularitymeasure Ddet(Q) Figure55. SimulationresultsforSDAwith0=0.01,0=a=b=2=0,and1=1atzeromomentum(contd.) 84 PAGE 85 5.3.1.3HybridSteeringLogicsimulationresults TheparametersfortheHSLsimulationare:0=0.01,0=2,a=1,b=3,and1=2=1.InFigure 56 A,theHSLmethodshowssimilarresultstothatoftheSDAshowninFigure 54 A,withexceptiontothefasterrateofconvergenceofthetransienterrorresponse.However,thetorqueerrorinFigure 56 BaddedintothesystemissmallerthanthatofSDAinFigures 54 BandnullmotioninFigure 57 BissmallerthanthatoftheLGmethodin 52 B.ThisisduetothenonzerovalueforbothsingularitymetricsandinFigures 56 CandD.Singularitywasnotencounteredinthissimulationsasisshownbyavaluem>0.5inFigure 57 Candazerovalueofdet(Q)in 57 D. ForCaseIatzeromomentum,Table 52 ,comparestherootmeansquared(RMS)gimbalrates(deg/s)andtrackingperformanceintermsofRMStorqueerror(Nm)forLG,SDA,andHSL.InthistableitisshownthatallthreemethodshaveapproximatelythesameperformancewhichisexpectedforafourSGCMGpyramidarrangementatzeromomentum,farfromsingularity.ThesteadystateerrorforLGoranyoftheothermethodsisnonzeroasaconsequenceofthecontroller'sperformanceiscapturedhere. Table52. PerformanceComparisonsforCaseI:ZeroMomentum SteeringAlgorithm_RMSeRMS LG5.73662.2437e06SDA5.72793.5902HSL(m0=0.5)5.73172.2573 5.3.2Case2:NearEllipticExternalSingularity=[105105105105]Tdeg Thesecondsetofsimulationsstartsatinitialgimbalangles=[105105105105]Tdeg,whichrepresentsascenarionearanellipticexternalsingularityat(i.e.,15degforeachSGCMGawayfromtheexternalsingularity=[90909090]Tdeg). 85 PAGE 86 AQuaternionerrorvectorelements BTorqueError CAlpha DBeta Figure56. SimulationresultsforHSLwith0=0.01,0=2,a=1,b=3,and1=2=1atzeromomentum 86 PAGE 87 ACMGgimbalrates BNullmotion CSingularitymeasure Ddet(Q) Figure57. SimulationresultsforHSLwith0=0.01,0=2,a=1,b=3,and1=2=1atzeromomentum(contd.) 87 PAGE 88 5.3.2.1Localgradientsimulationresults TheparametersfortheLGsimulationare:0=a=b=1=0and2=0=1.TheplotsinFigure 58 AshowthattheLGmethodappearstohavesuccessfullyperformedthemaneuverasshowninFigures 58 AandB.However,thisismisleadingsincenonimplementablegimbalratesandaccelerationsarerequiredtodosoasshowninFigure 59 A.Thesingularitymetrics=0asexpectedforthismethodand=1atthesingularityencounter.Eventhough=1atsingularity,nullmotionattheexacttimeofsingularityencounteriszeroasshowninFigure 59 becausethegradientvectordforLGiszeroatellipticsingularities(i.e.,nogradientvectorexiststhatisinthedirectionawayfromsingularity).Also,thesingularity,veriedtobeellipticfromthepositivevalueofdet(Q)inFigure 59 D,wasescapedimmediatelywiththehelpofthenonimplementablegimbalratesandaccelerations,shownbyFigure 59 C. 88 PAGE 89 AQuaternionerrorvectorelements BTorqueError CAlpha DBeta Figure58. SimulationresultsforLGwith0=a=b=1=0and2=0=1nearellipticsingularities 89 PAGE 90 ACMGgimbalrates BNullmotion CSingularitymeasure Ddet(Q) Figure59. SimulationresultsforLGwith0=a=b=1=0and2=0=1nearellipticsingularities(contd.) 90 PAGE 91 5.3.2.2SingularDirectionAvoidancesimulationresults TheparametersfortheSDAsimulationare:0=0.01,0=a=b=2=0,and1=1.ThetransientresponseoftheerrorfortheSDAmethodshowninFigure 510 AiscomparabletothatoftheLGmethodinFigure 58 A,butwithimplementablegimbalratesandaccelerationsasshowninFigure 511 A.TheSDAmethodescapestheellipticexternalsingularityasshowninFigure 511 CattheexpenseofsignicanttorqueerrorshowninFigure 510 B.ThetorqueerrorscaledbythesingularitymetricshowninFigure 510 CdecreasesawayfromsingularityasshowninFigure 511 C.AsexpectedforSDA,thesingularitymetricinFigure 510 CiszeroresultinginzeronullmotionasshowninFigure 511 B.IncontrasttotheLGmethod,forSDA,thesystemlingersinsingularityforaround15secondsbeforeescapingasshowninFigure 511 C.Ellipticsingularityforthissimulationisveriedbythepositivevalueofdet(Q)inFigure 511 D. 91 PAGE 92 AQuaternionerrorvectorelements BTorqueError CAlpha DBeta Figure510. SimulationresultsforSDAwith0=0.01,0=a=b=2=0,and1=1nearellipticsingularities 92 PAGE 93 ACMGgimbalrates BNullmotion CSingularitymeasure Ddet(Q) Figure511. SimulationresultsforSDAwith0=0.01,0=a=b=2=0,and1=1nearellipticsingularities(contd.) 93 PAGE 94 5.3.2.3HybridSteeringLogicsimulationresults TheparametersfortheHSLsimulationare:0=0.01,0=2,a=1,b=3,and1=2=1.TheresultsforHSLshowninFigures 512 and 513 arealmostidenticaltothecorrespondingresultsofSDAforthissimulation.TheonlydifferencebetweenHSLandSDAsimulatedresults,liesinthenonzerosingularitymetricsandinFigures 512 CandD.DuetothechoiceoftheHSLparametersa,b,1,2,,0,0,thethresholdforsingularitym0.5,andQ0,theHSLactsastheSDAatanellipticsingularity AQuaternionerrorvectorelements BTorqueError CAlpha DBeta Figure512. SimulationresultsforHSLwith0=0.01,0=2,a=1,b=3,and1=2=1nearellipticsingularities 94 PAGE 95 ACMGgimbalrates BNullmotion CSingularitymeasure Ddet(Q) Figure513. SimulationresultsforHSLwith0=0.01,0=2,a=1,b=3,and1=2=1nearellipticsingularities(contd.) ForCaseIInearanellipticsingularity,Table 53 ,comparestheRMSgimbalrates(deg/s)andtrackingperformanceintermsofRMStorqueerror(Nm)forLG,SDA,andHSL.InthistabletheLGmethodissaidtohaveaninniteRMSgimbalratetopointoutthatitfailedforellipticsingularity.Also,itisshownthatSDAandHSLweresuccessfulincompletingthemaneuverwhileescapingsingularity.BothSDAandHSLhadapproximatelythesameperformanceforellipticsingularitywiththeexceptionofslightlybettertrackingperformanceforHSL. 95 PAGE 96 Table53. PerformanceComparisonsforCaseII:EllipticSingularity SteeringAlgorithm_RMSeRMS LG17.7159e06SDA8.256429.8989HSL(m0=0.5)8.136626.6946 5.3.3Case3:NearHyperbolicInternalSingularities=[15105195)]TJ /F4 11.955 Tf 9.3 0 Td[(75]Tdeg Thenalsetofsimulationsstartsatinitialgimbalangles=[15105195)]TJ /F4 11.955 Tf 9.3 0 Td[(75]Tdegwhichrepresentsascenarionearanhyperbolicsingularityat(i.e.,adistance15degfromeachCMGawayfromthesingularityat=[090180)]TJ /F4 11.955 Tf 9.3 0 Td[(90]Tdeg).5.3.3.1Localgradientsimulationresults TheparametersfortheLGsimulationare:0=a=b=1=0and2=0=1.ThetransientresponseoftheerrorfortheLGmethodinFigure 514 Aisidenticaltothatfortheothertwocases.ThisisbecausetheLGmethodisanexactmappingevidentfrom=0inFigure 514 CandhasnotorqueerrorassociatedwithitintheoryasshowninFigure 514 B.ThenullmotioninFigure 515 BmakesupalmosttheentirecontributionofthegimbalratesinFigure 515 AduetothenonzerovalueofinFigure 514 D.TheLGmethodbyitselfisabletoavoidthehyperbolicsingularityswiftlyandremainawayasshowninFigure 514 CandD. 96 PAGE 97 AQuaternionerrorvectorelements BTorqueError CAlpha DBeta Figure514. SimulationresultsforLGwith0=a=b=1=0and2=0=1nearhyperbolicsingularities 97 PAGE 98 ACMGgimbalrates BNullmotion CSingularitymeasure Ddet(Q) Figure515. SimulationresultsforLGwith0=a=b=1=0and2=0=1nearhyperbolicsingularities(contd.) 98 PAGE 99 5.3.3.2SingularDirectionAvoidancesimulationresults TheparametersfortheSDAsimulationare:0=a=b=2=0,1=1and0=0.01.ThetransientresponseoftheerrorfortheSDAmethodinFigure 516 Aisdifferentintherateofconvergencetoess,butonthesameorderofmagnitudetothatfortheLGmethod.However,thegimbalratesforSDAshowninFigure 517 BareanorderofmagnitudesmallerthanthatfortheLGmethod.ThismethodescapesthehyperbolicsingularitysuccessfullywithtorqueerrorasshowninFigures 516 BasaconsequenceofthenonzerovalueofinFigure 516 C.Thesingularitymetricin 516 DiszerobecauseSDAdoesnotusenullmotion.AddedtorqueerrorforsingularityescapeversusnullmotionforsingularityavoidanceisthetradeoffbetweenSDAandLG.ThesingularityinthissimulationisveriedtobehyperbolicfromthenegativeresultshowninFigure 517 D.Also,theSDAmethoddidnotescapebywhatisconsideredsingularityinFigure 517 Cbythethresholdm0.5.However,thisdidnotaffectthedecayingoftheerrorstransientresponse.Thisisduetothefactthatthetorqueerrorisscaledbytheneededoutputtorquebeingmappedandtherefore,isnotseentohaveasignicanteffecttowardstheendofthemaneuver. 99 PAGE 100 AQuaternionerrorvectorelements BTorqueError CAlpha DBeta Figure516. SimulationresultsforSDAwith0=0.01,0=0,a=0,b=0,and=1nearhyperbolicsingularities 100 PAGE 101 ACMGgimbalrates BNullmotion CSingularitymeasure Ddet(Q) Figure517. SimulationresultsforSDAwith0=0.01,0=0,a=0,b=0,and=1nearhyperbolicsingularities(contd.) 101 PAGE 102 5.3.3.3HybridSteeringLogicsimulationresults TheparametersfortheHSLsimulationare:0=0.01,0=2,a=1,b=3,and1=2=1.ThetransientresponseoftheerrorinFigure 518 AisalmostidenticaltotheLGmethodforthiscaseandhasafasterrateofconvergencetoessthanSDA.ThisisattributedtothenonzerovaluesofthesingularitymetricsandinFigures 518 CandDwhichprovideanorderofmagnitudelessnullmotionforsingularityavoidancethanLGandordersofmagnitudelesstorqueerrorthanSDAforthiscaseshowninFigure 519 Band 518 AwhenavoidingthehyperbolicsingularityveriedinFigures 519 CandD.UnlikeSDA,HSLescapedandthenavoidedthesingularitywhichisduetotheadditionofnullmotionforthismethod(seeFigures 516 Cand 517 C).Therefore,HSLreliesmoreonnullmotionforsingularityavoidanceratherthansoleytryingtopassthroughthehyperbolicsingularitiesasSDA,SR,andGSRdo.PrecisioninattitudetrackingwiththethreatofhyperbolicsingularitieswhilestillbeingabletoescapeellipticsingularitiesisthestrengthofHSL. 102 PAGE 103 AQuaternionerrorvectorelements BTorqueError CAlpha DBeta Figure518. SimulationresultsforHSLwith0=0.01,0=2,a=1,b=3,and1=2=1nearhyperbolicsingularities ForCaseIIInearahyperbolicsingularity,Table 53 ,comparestheRMSgimbalrates(deg/s)andtrackingperformanceintermsofRMStorqueerror(Nm)forLG,SDA,andHSL.InthistabletheLGmethodhasthelargestRMSgimbalrateamongthethreemethods,whichisneededforsingularityavoidance.Also,LGperformedthemethodwiththebesttrackingperformanceamongthethreemethodswhichisanexpectedresultforanexactmethod.TheHSLhadbettertrackingperformanceintermsofRMStorqueerrorthanSDAasaconsequenceofthelargergimbalratesneededfornullmotionsingularityavoidance.ThisisanexpectedstrengthofHSLathyperbolicsingularity. 103 PAGE 104 ACMGgimbalrates BNullmotion CSingularitymeasure Ddet(Q) Figure519. SimulationresultsforHSLwith0=0.01,0=2,a=1,b=3,and1=2=1nearhyperbolicsingularities(contd.) Table54. PerformanceComparisonsforCaseIII:HyperbolicSingularity SteeringAlgorithm_RMSeRMS LG10.39051.4742e05SDA6.361114.4937HSL(m0=0.5)9.93304.7925 5.4HybridSteeringLogicSummary TheHSLwasfoundnumericallytopreserveattitudetrackingprecisioninthepresenceofhyperbolicsingularities,actcomparablytoSDAinthepresenceofellipticsingularities,andperformbetterthanSDAawayfromsingularity.Theperformanceofthisalgorithmisattributedtothenewsingularitymetrics,whichallowsmooth 104 PAGE 105 transitionbetweensingularityavoidanceusingLGandsingularityescapeusingSDA.Byreducingthetimeswheretorqueerrorisinducedforsingularityescape,thismethodprovidesimprovedattitudetrackingperformance.AnalyticandsimulatedresultsshowthatHSLhasmanybenetsoverthetwoothermethodsforsingularityavoidanceandescape.Thesebenetsare:itcanbeimplementedrealtime;althoughSVDmaybecomputationallyintensive,itremovestheneedforaninverseandprovidesalltheinformationneededforHSL;numericallyrobustalgorithmsexistforSVD;HSLinduceslesstorqueerrorthanSDAbyitself;andnally,theHSLprovidesanonsingularexpressionthatcanstartatsingularity.TheHSLisnotsuccessfulinavoidinggimballockbecausenullmotionisnonexistentatellipticsingularitiesandSDAfailsatgimballock(seeChapter 4 ). 105 PAGE 106 CHAPTER6CONTROLALGORITHMORTHOGONALTORQUECOMPENSATION6.1AttitudeControllerwithOTC TraditionallythecontrollawandsteeringalgorithmareseparatedforattitudecontrolsystemsusingSGCMGsasshowninFigure 61 .Thisisdonetofacilitateunderstandingoftheattitudecontrolsystemandactuatordynamicsseparately.However,consideringthesteeringalgorithmseparatefromthecontrollawmayreducethepossibilityofanincreaseintheperformanceinthesystem. Figure61. Satelliteattitudecontrolsystemblockdiagram Manysteeringlogicsbythemselvesareincapableofavoidinggimballock.Gimballockoccurswhentherequiredtorqueforanattitudemaneuverisalongthesingulardirection.Thisproducesalocalminimumconditionwherethegimbalratesolutioniszerowhiletherequiredtorqueisstillnotmet.Openloopmethodsthatprovideagimbaltrajectoryfreeofthisconditionexist;examplesofsuchmethodsareforwardpropagationfrompreferredgimbalangles,globalsteering,andoptimalcontrol[ 65 66 74 83 ].Thesemethodsaretimeconsumingandcannotguaranteeasolutionexistsfortheconstraintsprovided. RealtimesolutionstogimballockavoidanceexistsuchastheGSRinversewhichusesoffdiagonaldithercomponentsinitsperturbationmatrixtoescapegimballock(seeChapter 4 ).Thereisnoformalproofthatthesemethodswillalwaysbesuccessfulinavoidinggimballock.Throughtheuseofnonlinearcontrol,anorthogonaltorque 106 PAGE 107 compensation(OTC)methodologycanbeaugmentedwithasuitablesteeringandcontrolalgorithmtoalsoavoidorescapegimballock.Throughthisnonlinearcontrolframework,stabilitycanbeprovenandthesteeringalgorithmcanbechosenseparatelyincontrasttoGSRwhichreliesentirelyonhandlinggimballockavoidance/escapethroughthesteeringalgorithm. Openloopmethodssuchasoptimalcontrolforgimballockescapeoravoidancemaynotndafeasiblesolutionorasolutionatalldependingonhowthecostfunctionandconstraintsareformulated. Itispossiblethatcombinationofanoptimalcontrolmaneuverwithapseudoinversemethod(e.g.,SRinverse)willdrivethesystemtowardthevicinityofsingularityasthemaneuveriscompleted.Thismayoccursincetherequiredgimbalratesarenotonlyscaledfromthedistancetosingularity,butalsobytheneededoutputtorquefromtheSGCMGsystem.Asthenextresttorestmaneuverisneededthetorquemayberequiredaboutthesingulardirection.Whenthisoccurs,themaneuvercouldcausethelocalminimumpreviouslydiscussed.6.2LyapunovStabilityAnalysis Forthecasesconsidered,OTCwillbeamodicationtothequaternionregulatorcontrollogicfromreference[ 54 ]showninEq.( 5 ).Itshouldbenotedthatthismodicationcould,intheory,workwithanycontrolalgorithmwhichinturncanbecombinedwithanysteeringalgorithmforSGCMGs.Therefore,itisnotrestrictedtoanysteeringalgorithmorthequaternionregulatorcontrollawiftheproperstabilityanalysisiscarriedout.Thequaternionregulatorcontrollogicassumesperfectinformationandhasthefollowingnominalform _h=Ke+C!+!H(6) whereK=2kJandC=cJarepositivedenitesymmetricgainmatricesbasedonthespacecraft'scentroidalinertiaJ,eisthevectorelementsofthequaternionerrorvector, 107 PAGE 108 !isthespacecraftangularvelocity,andHisthetotalspacecraftsystemcentroidalangularmomentumfromEq.( 2 ). RecallfromEq.( 4 )thattheJacobian'sleftsingularvectorsUisanorthonormalbasisfortheoutputtorque_h.Thisbasisiscomposedofaunitvectorinthedirectionofthesingulardirectionukwhenatsingularity,andtwounitvectorsorthogonaltothesingulardirection,u?andun(i.e.,evenwhenAisnonsingular,thebasisfromUstillexists).Utilizingthisbasisintheformationoftheoutputtorqueyields _h=uk+ u?+un6=_h(6) withcoefcients =_h Tuk =_h Tu?+ag(m)=_h Tun+bg(m)(6) Thequantityg(m)isaaugmentationtotheorthogonaltothesingulardirectioncomponentsoftorquethatisanexplicitfunctionofthesingularitymeasure.ItwillbereferencedastheOTCsingularityparameterandaandbareswitchingelementsdenedby a=8><>:1if_h Tu?0)]TJ /F4 11.955 Tf 9.3 0 Td[(1if_h Tu?<0 b=8><>:1if_h Tun0)]TJ /F4 11.955 Tf 9.3 0 Td[(1if_h Tun<0 SubstitutingEq.( 6 )intoEq.( 5 )andboundingyields, 108 PAGE 109 _V=)]TJ /F7 11.955 Tf 26.37 8.09 Td[(c 2k!TJ)]TJ /F2 7.97 Tf 6.59 0 Td[(1U266664100010001377775UTJ!+2=)]TJ /F3 11.955 Tf 9.29 0 Td[(!TR2!+2(6) where2=1+g(m)!TK)]TJ /F2 7.97 Tf 6.58 0 Td[(1(au?+bun)andR2=R1inEq.( 5 )withthesingularityparameterfromEq.( 4 )inplaceoffortheHSLinEq.( 5 ).SimilartotheLyapunovanalysisinChapter 5 forHSL,theerrorzisboundedwithaVolterraintegralexpressionas jjzjj2V(0) minexp()]TJ /F28 5.978 Tf 10.1 4.4 Td[(1 maxt)+1 minZtt0exp()]TJ /F28 5.978 Tf 10.1 4.4 Td[(1 max(t)]TJ /F11 7.97 Tf 6.59 0 Td[())2()d(6) SincetheSGCMGoutputtorquewillalwayshavecomponentsorthogonaltothesingulardirectionwhennearsingularity,itisassumedthatasystemusingOTCwillneverencountergimballockuptoaspecicsizeofjjejjandfromEq.( 4 )whereg(m)=jjejj(6) Therefore,situationsofsingularityotherthanthosewithgimballockareofconcern.Whenatsingularity,theexpressionfortheerroris jjzjj2Vs+Ztts2()d(6) becausetheerrorzisbasedoffthetransienttermoftheLyapunovequationVsfromEq.( 7 )andthedynamictermcontainingtheeffectofthetorqueerroraddedforgimballockescapeRtts2()d.RecallfromChapter 5 ,whileusingHSL,thatwhensingularityoccurswiththeexceptionofgimballock,theremaybeaperiodofinstabilityanditisassumedthatthemaneuverdoesnotendatsingularity.Withthisinmind,thesteadystateerroroftheofasystemawayfromsingularityusingSDAcombinedwithOTCisboundedas 109 PAGE 110 jjz(1)jj2)]TJ /F3 11.955 Tf 30.83 8.08 Td[(max min12(1)(6) throughtheuseofL'Hospital'sruleasinEq.( 5 ). TheresultofOTCisUUBforsufcientlylargethechoicesofcratherthan.Thisistruebecauseshortperiodsofinstabilitymayarise,butthenegativesemidenitetermofEq.( 6 )becomesnegativedeniteawayfromsingularityandwillbecomedominateforsufcientlylargevaluesofc.Withthechoiceofg(m)inEq.( 6 ),wheneverthereisanattitudeerrorandthesystemisinproximitytoasingularity,therewillbetorqueerroraddedorthogonaltothesingulardirection,andthusgimballockwillbeescaped.6.3NumericalSimulation ForthesteeringalgorithmsofSDA,GSR,andSDAwithOTC(SDA/OTC)augmentedtotheattitudecontroller,twocasesweresimulatedforafourSGCMGpyramidalclusterat=54.74degandthemodelparametersinTable 61 :(1)azaxismaneuverstartingatinitiallyatthezeromomentumcongurationfromChapter 5 (i.e.,=[0000]Tdeg)and(2)azaxismaneuverstartingatgimballockconguration(i.e.,=[90909090]Tdeg).BothcasesusethesamecontrolgainsappliedtoapyramidalarrangementoffourSGCMGs.Also,thesimulationwaspropagatedwithadiscretefourthorderRungaKuttaatatimestepoft=0.02sec. Table61. ModelParametersVariableValueUnits J133kgm254.74dege0[000.3]T\000!0[000]Tdeg=sh01Nmsk21=s2c101=si0.1rand(1)\00000.1\000t0.02sec 110 PAGE 111 6.3.1CaseI:0=[0000]Tdeg Forazaxismaneuveroriginatingfromaninitialzeromomentumconguration,Figures 62 showthatSDA,GSR,andtheSDA/OTCappearidentical.Thisiswhatisexpectedforamaneuverfarfromsingularity.TheresultsforthetorqueFigure 63 conrmthisbecausethetransientresponseforthiscase(i.e.,awayfromsingularity)isshort.ThetransientresponseoftheoutputtorqueshowninFigure 63 DofSDA/OTChassignicantjitterbutwithasmallmagnitude.ThisjitterhasnegligibleeffectonthegimbalratesshowninFigure 62 Dwhichisduetothemappingoftheoutputtorqueontothegimbalrates.ThedifferenceinquaternionerrorandsingularitymeasureinFigures 64 and 65 aresmall.ThisshouldnotbesurprisingsinceforSDA,GSR,andSDA/OTC,thecontributionsoftorqueerroraredesignedtobesignicantonlywhenthesystemisclosetoasingularity. TheOTCsingularityparametershowninFig. 66 ,whileinitiallynonzeroforthiscase,convergestozerorapidly.ThefactthatthisparameterisnonzeroinitiallyandthereisnosignicantdifferencesinthequaternionerrorresponsesasshowninFigure 67 ,mightsuggestthatthetorqueerrorfromtheSDAmethoditselfwasdominant.Inaddition,itshouldbenotedthatthedifferenceinquaternionerrorresponseswhilesmall(10)]TJ /F2 7.97 Tf 6.59 0 Td[(8),isnotontheorderofmachineprecision(10)]TJ /F2 7.97 Tf 6.58 0 Td[(16)or(10)]TJ /F2 7.97 Tf 6.58 0 Td[(32). 111 PAGE 112 ASDA BSDA(transientresponse) CGSR DGSR(transientresponse) ESDA/OTC FSDA/OTC(transientresponse) Figure62. Gimbalrates 112 PAGE 113 ASDA BSDA(transientresponse) CGSR DGSR(transientresponse) ESDA/OTC FSDA/OTC(transientresponse) Figure63. Outputtorque 113 PAGE 114 ASDA BGSR CSDA/OTC Figure64. Vectorelementsoftheerrorquaternion 114 PAGE 115 ASDA BGSR CSDA/OTC Figure65. Singularitymeasure 115 PAGE 116 Figure66. Singularityparameter(OTC) AeGSR)]TJ /F29 9.963 Tf 9.96 0 Td[(eSDA BeSDA=OTC)]TJ /F29 9.963 Tf 9.96 0 Td[(eSDA Figure67. Quaternionerrordifference:(A)eGSR)]TJ /F7 11.955 Tf 11.96 0 Td[(eSDA(B)eSDA=OTC)]TJ /F7 11.955 Tf 11.96 0 Td[(eSDA 6.3.2CaseIIa:0=[90909090]Tdeg Forthiscase,thegimbalsareinitiallyorientedsuchthatthesystemisinagimballockconguration.Figure 68 AshowsthatthegimbalratesoftheSDAmethodareunchangedthroughoutthesimulationsincethesystemstartsinagimballockcongurationandSDAcannotgeneratethenecessarycommandstoescape.ThegimbalratesforGSRandSDA/OTC(Fig. 68 BandD),however,arenonzerobecausetheadditionofthetorqueerrorhasprovidedthesystemwiththeabilitytoescapegimballock.Inaddition,thecontrollerapproachestheoriginalquaternionregulatorcontrollerasthesystemmovesawayfromsingularity. 116 PAGE 117 ThetransientresponseofthegimbalratesfortheGSRandSDA/OTCinFigure 68 CandEarebothoscillatorywithGSRhavingthehigheramplitudeandduration.ThisisattributedtothefactthatunlikeOTC,thefunctionsaddingtorqueerrorinGSRforgimballockescapearenotclearlyvisualized(i.e.,dependonthecombinationofsinusoidswithpossibledifferentfrequenciesandphasesfordither)whenmappedtothegimbalrates. Figures 69 and 610 show,respectively,therequiredtorqueandattitudeerror.Inbothcases,theresultsshowasimilartrendasthegimbalratesforGSR,andSDA/OTCapproachzero;theGSRandSDA/OTCwereabletogeneratethetorquerequiredtodrivetheattitudeerrortozero. AnexaminationofthesingularitymeasuresshowninFigure 611 reafrmstheresponsesshowninFigures 68 through 610 wheretheSDAremainsatsingularityunlikeGSRandtheSDA/OTCwhichescapesingularitybuttransitionbacktoitasthemaneuveriscompleted.Thistransitionbacktosingularityiscommonforallpseudoinversesteeringalgorithms,whichworkbyapproachingasingularcongurationandthenmakingarapidtransitionforescape[ 47 ].Recallpreviouslyfrom 6.1 ,thatitwasstatedthatitispossibletoendinthevicinityofasingularitywhenthemaneuverwascompleted;thisisanexampleofsuchacaseshowninFigure 610 and 611 ThemeasureofhowfarthesystemofSGCMGsisfromgimballockcanbefoundasthenormjjAT_hjj)166(!0.BoththeGSRandtheSDA/OTCweresuccessfulinescapinggimballockasshowninFigure 612 .Itshouldbenotedthatbecausethismeasureisafunctionof_h,itgoestozeroasthemaneuveriscompleted. TheOTCsingularityparameterisshowninFigure 613 .Ithasanonzeroinitialvalueandconvergesrapidlytozerowhichmakesiteffectiveforhelpinginsingularityescape. 117 PAGE 118 ASDA BGSR CGSR(transientresponse) DSDA/OTC ESDA/OTC(transientresponse) Figure68. Gimbalrates 118 PAGE 119 ASDA BGSR CGSR(transientresponse) DSDA/OTC ESDA/OTC(transientresponse) Figure69. Outputtorque 119 PAGE 120 ASDA BGSR CSDA/OTC Figure610. Vectorelementsoftheerrorquaternion 120 PAGE 121 ASDA BGSR CSDA/OTC Figure611. Singularitymeasure 121 PAGE 122 ASDA BGSR CSDA/OTC Figure612. Gimballockmeasure 122 PAGE 123 Figure613. Singularityparameter(OTC) 6.3.3CaseIIb(HSL/OTC):0=[90909090]Tdeg Recall,from 6.1 thatOTCcanbeusedincombinationwithanysteeringalgorithmforgimballockavoidance/escape.ThiscaseveriesthroughsimulationthatthisisindeedtruebycomparingHSL/OTCtoGSRstartingatgimballock(0=[90909090]Tdeg).TheHSLparametersareshowninTable 62 Table62. HybridSteeringLogicParametersVariableValue 00.01021121a1b3m00.5 Withtheexceptionoftheinitialtransient,thegimbalratesforGSRandthoseofHSL/OTCinFigure 614 ,areapproximatelythesamemagnitude. 123 PAGE 124 AGSR BGSR(transientresponse) CHSL/OTC DHSL/OTC(transientresponse) Figure614. Gimbalrates ThetransientresponseofthegimbalratesfortheGSRinFigure 614 BishighlyoscillatoryandnotassmoothasthatforHSL/OTC(comparewithFigure 614 D).ThisisduetooscillatorybehavioroftheditherusedforgimballockescapethatmaybeofanydurationdependingonthefrequenciesandphasesoftheoffdiagonalcomponentsoftheGSRperturbationmatrixE.TheHSLmethodactsasaSDAmethod,butwhencombinedwithOTCwillavoid/escapeasingularityatthespeedoftheparameterschosenforinEq.( 6 )inthewhichthedurationwillbeunderstoodforallsingularitiesandtheircombinationstothenormofquaternionerror.Figures 615 and 616 show,respectively,therequiredtorqueandattitudeerror.Inbothcases,theresultsshowthat 124 PAGE 125 bothmethods(GSRandHSL/OTC)weresuccessfulinmeetingtherequiredtorqueandcompletingtheattitudemaneuver. AGSR BGSR(transientresponse) CHSL/OTC DHSL/OTC(transientresponse) Figure615. Outputtorque ThesingularityparametersforbothmethodsinFigure 617 ,escapesingularityalthoughtheytransitionbacktoitasthemaneuveriscompleted.Recall,itwasmentionedpreviouslythatamaneuvercanbecompleted(i.e.,e)166(!0)whilethegimbalanglessettleintoasingularconguration;Figure 611 and 617 showthistrend(comparewithFigure 611 ). Figure 618 showsinstantaneousescapefromgimballockforbothmethods. 125 PAGE 126 AGSR BHSL/OTC Figure616. Vectorelementsoftheerrorquaternion AGSR BHSL/OTC Figure617. Singularitymeasure TheOTCsingularityparametershowninFigure 619 hasaninitialnonzerovalueandconvergesrapidlytozerosimilarlytoFigure 613 ,whichmakesiteffectiveforhelpinginsingularityescape. TheTable 63 comparestherootmeansquared(RMS)gimbalrates(rad/s),trackingperformanceintermsofRMStorqueerror(Nm),andpointingperformanceintermsofthenormofthesteadystateerrorquaternionforGSR,SDA/OTC,andHSL/OTC.FromTable 63 itcanbeshownthatthechoiceofthesingularitythresholdm0hasaneffectonthetrackingandpointingperformanceoftheHSLmethodcombined 126 PAGE 127 AGSR BHSL/OTC Figure618. Gimballockmeasure Figure619. Singularityparameter(OTC) withOTC.Infact,whenthisvalueism0=0.5forthismodelandwiththesetofcontrolgainsthatdifferfromthemodelinChapter 5 ,thetrackingandpointingperformanceofHSL/OTCisactuallyworse.ThisisexpectedasshownbytheLyapunovanalysisin 6.2 wherethesteadystateerrorofSDAisdependentonthetorqueerroraddedintothesystem;andalargerthresholdvalueofm0willincreasethesteadystateerror.6.4OrthogonalTorqueCompensationSummary Orthogonaltorquecompensation(OTC)methodologywasdevelopedtoensureescapefromallsingularities,particularlyscenariosinvolvinggimballockcongurations.Thecompensationmethodologycanbeincorporatedwithanyattitudecontroller/steering 127 PAGE 128 Table63. PerformanceComparisons SteeringAlgorithm_RMSeRMSjjessjj GSR7.790110.18640.0024SDA/OTC7.314610.40800.0020HSL/OTC(m0=0.5)7.16936.18190.0038HSL/OTC(m0=0.05)7.00413.04162.1474e09HSL/OTC(m0=0.005)6.91662.98521.8843e09 logiccombinationandwasshownthroughanalysistoensurestabilitywithsufcientlylargechoiceofthecontrollergainc.Sincethecompensatorwasdesignedtoworkwithanyattitudecontroller,thenitiscompatiblewithanysteeringalgorithms.ThiscouldproveverybenecialforsteeringalgorithmslikeHSLwhichreducetheamountoftorqueerrorathyperbolicsingularities(seeChapter 5 ).TheOTCwasalsodemonstratedthroughnumericalsimulationwhereitwasshowntobeeffectiveinescapinggimballockwithnearzerosteadystateattitudeerror.ThesesimulationswerebasedonafourSGCMGpyramidalarrangementusinganquaternionregulatorcontrollercombinedwiththesteeringalgorithmsSDAandHSLandcomparedwithGSR. 128 PAGE 129 CHAPTER7SCALABILITYISSUESFORSGCMGS7.1ScalabilityProblemswithSGCMGHardware CurrentlyavailableCMGactuatorsareshowninFigure 71 withspecicationsfromTable C1 inAppendix C donotmeetthepower,mass,andvolumerequirementsforsatellitessmallerthanthemicrosatclass.Currently,developmentofCMGhardwareunderwaywillmeetsomeoftheconstraintsforthesesmallerclassesofsatellites.NewsteeringalgorithmstocomplementthesenewlydevelopedCMGhardwareisnotbeingemphasizedandwillhaveamajoreffectonhowsystemsofminiatureCMGsperform.ThischapterhighlightstheeffectofscalingontheperformanceofminiatureCMGs. Figure71. OfftheshelfCMGs 129 PAGE 130 7.2EffectofIgwonTorqueError Thegimbalaccelerationsarekinematicallydependentonthechoiceofthegimbalratesandasaconsequence,onlyoneofthemcanbeusedasacontrolvariable.Therefore,thegimbalratesareconsideredasmeasurablequantitiesandthegimbalaccelerationsarethecontrol.Thesolutiontothegimbalaccelerationsasacontrolisdenedas =BT(BBT))]TJ /F2 7.97 Tf 6.59 0 Td[(1[T)]TJ /F8 11.955 Tf 11.96 0 Td[(A2_](7) whereA=A1+A2fromEqs.( 2 )and( A )andT=_h+A1_equivalenttoEq.( 2 ).Thissolutionisconsideredanexactsolutionbutforsomecasesmaybehighlyoscillatoryand/orunstableforthegimbalratesandaccelerations.ALyapunovanalysisispresentedbelow. ItwasstatedpreviouslythatthedirectsolutioninEq.( 7 )maybeunstable.ToprovethiswestartwiththegivencandidateLyapunovfunction V1=1 2!TK)]TJ /F2 7.97 Tf 6.58 0 Td[(1Jc!+eTe+(1)]TJ /F7 11.955 Tf 11.95 0 Td[(e4)2(7) Takingthetimederivative,yields _V1=!TK)]TJ /F2 7.97 Tf 6.59 0 Td[(1[)]TJ /F8 11.955 Tf 9.3 0 Td[(T)]TJ /F10 11.955 Tf 11.95 0 Td[(!H]+!Te(7) Forthesystemtobegloballyasymptoticallystable(i.e.,_h,e,and!)166(!0ast)166(!1) T=A_+B=Ke+C!)]TJ /F10 11.955 Tf 11.96 0 Td[(!H(7) Next,asecondcandidateLyapunovfunctionisrequiredtoanalyzethebehaviorofthegimbalratesandaccelerationsastimeapproachesinnity. V2=1 2_T_(7) 130 PAGE 131 TakingthetimederivativeutilizingEq.( 7 ),weobtain _V2=_T=_TBT(BBT))]TJ /F2 7.97 Tf 6.59 0 Td[(1[T)]TJ /F8 11.955 Tf 11.96 0 Td[(A_](7) FromthepreviousLyapunovanalysiswhere_h,e,and!)166(!0ast)166(!1itcanbeassumedthat _V2=)]TJ /F8 11.955 Tf 11.59 2.66 Td[(_TBT(BBT))]TJ /F2 7.97 Tf 6.59 0 Td[(1A_=(h0 Igw)_TS_(7) whereB=Igw^B,A=h0^A,andS=)]TJ /F8 11.955 Tf 10.39 2.66 Td[(^BT(^B^BT))]TJ /F2 7.97 Tf 6.59 0 Td[(1^A.MatrixSissemiindeniteandthereforethegimbalratescanbeunstable.Furthermore,aLyapunovanalysisofV1+V2showsthattheratio(h0 Igw)playsakeyroleinthestabilityofthewholesystem. Next,weconsidertheuseoftheSRinversewherethegimbalratesarefoundfrom _SR=1 h0ASR(T)]TJ /F7 11.955 Tf 11.96 0 Td[(Igw^B)(7) with ASR=^AT(^A^AT+1))]TJ /F2 7.97 Tf 6.59 0 Td[(1(7) whereisthesingularityparameterdenedinEq.( 4 ). AssumingtheSRinverseisusedtoapplythegimbalratesasacontrolvariablewendthatthetorqueerrorisexpressedas e=Tact)]TJ /F8 11.955 Tf 11.95 0 Td[(T=h0^A_SR+Igw^B)]TJ /F8 11.955 Tf 11.96 0 Td[(T(7) Furthermore, ^AASR=[1+(^A^AT))]TJ /F2 7.97 Tf 6.58 0 Td[(1])]TJ /F2 7.97 Tf 6.58 0 Td[(1(7) Awayfromsingularity,aseriesexpansionofEq.( 7 )withonlythelineartermsyields 131 PAGE 132 ^AASR1)]TJ /F3 11.955 Tf 11.95 0 Td[((^A^AT))]TJ /F2 7.97 Tf 6.58 0 Td[(1(7) Thisseriesexpansionisconvergentifawayfromsingularitybecausethetermj(^A^AT))]TJ /F2 7.97 Tf 6.58 0 Td[(1j<1.SubstitutingEq.( 7 )intothetorqueerror,Eq.( 7 )wehave eIgw(^A^AT))]TJ /F2 7.97 Tf 6.59 0 Td[(1^B)]TJ /F3 11.955 Tf 11.95 0 Td[((^A^AT))]TJ /F2 7.97 Tf 6.58 0 Td[(1T(7) ItcanbeseenthatthetorqueerrormaybeampliedbythemagnitudeofthegimbalywheelinertiaIgw.Furthermore,iforIgwisconsiderednegligiblethenthetorqueerrorisonlyaffectedbythesingularityparameter,thedistancefromsingularitywhichisrelatedtothedeterminantof(^A^AT))]TJ /F2 7.97 Tf 6.58 0 Td[(1,andT.ItshouldbenotedthatanincreaseinIgwisfollowedbyadecreasein^A^AT,butitseffectivenessinloweringthetorqueerrorrequiresalargeratioofIgw h0(i.e.,effectivewhenIgw h0>>1whichcouldbethoughtofasbeingasystemofRWs). TheeigenaxiscontrollogicfromEq.( 5 )isusedtodenethetorqueneededforagivenmaneuvertobemappedontothegimbalstates.TheSGCMGsystemproposedinthisanalysisassumesthatitisselfcontainedandthereforethemetricofthehostedalgorithmperformanceisindependentofthecontrollogicchosenaslongasitmeetstheconstraintsoftheSGCMGactuators.Therefore,nogeneralityislostforthechoiceofthecontrollogicintheanalysis.7.3NumericalSimulation ThecasescomparedherearetheSRInverseandalteredgimbalaccelerationcontrollawbasedonOhandVadali[ 84 ].Thelteredgimbalaccelerationcontrollawhasthefollowingform =K(_SR)]TJ /F8 11.955 Tf 14.25 2.66 Td[(_)+SR(7) 132 PAGE 133 whereKisthegainmatrixthatsizestheamountofgimbalaccelerationutilizedforcontroland_SRandSRarethegimbalratesandaccelerationsfromtheSRinverseandthetimederivativeofthatrate.TheeffectofthegimbalywheelinertiaisscaledinthesimulationbythegainKgw(i.e.,Igw=Kgw Igw)whereKgw=0signiesthattheirisnotorqueorangularmomentumcontributedfromthegimbaldynamics. SimulationsofthesetwosteeringalgorithmswerecomparedbyscalingIgwthroughtwodifferentvaluesofKgw:i)Kgw=0andii)Kgw=2.ThemodelparametersforthenominalsatelliteinertiaJandgimbalywheelinertiaIgwarebasedonafourSGCMGpyramidalarrangementsizedfora1UCubeSat.Bothsimulationswereforamaneuverof180oaboutthezaxis.Theinitialgimbalanglesforallsimulationsare0=[)]TJ /F4 11.955 Tf 9.3 0 Td[(90)]TJ /F4 11.955 Tf 9.3 0 Td[(90)]TJ /F4 11.955 Tf 9.3 0 Td[(90)]TJ /F4 11.955 Tf 9.3 0 Td[(90]Tdegcorrespondingtoaellipticsaturationsingularityaboutthezaxis.Thissetofinitialgimbalanglesalongwiththerequiredmaneuverwillforcethesystemtoentergimballock(i.e.,AT_h=0)andaccumulateasteadystateattitudeerror.Thissituationwaschosentotestthesystemtoitsperformancelimit.TheparametersthatwereusedforalloftheresultsareshowninTable 71 Table71. ModelParametersVariableValueUnits Js24533.8000533.8000895.63510)]TJ /F2 7.97 Tf 6.58 0 Td[(6kgm252dege0[0010]T\000!0[000]Tdeg=sh04.48610)]TJ /F2 7.97 Tf 6.59 0 Td[(4Nms Igw5.15410)]TJ /F2 7.97 Tf 6.59 0 Td[(6kgm2k101=s2c501=sK10I44\00000.5\00010\000_0[0000]Tdeg=s 133 PAGE 134 7.3.1CaseI:Kgw=0 Forthiscase,Figures 72 and 73 showthatthegimbalratesandaccelerationsarequitesimilarforboththeSRinverseandlteredaccelerationcontrollaw,exceptattheverybeginning.SincethelteredaccelerationcontrollawalsousesanSRinverseinitsformulation,itisreasonabletoassumethatthedifferencescanbeattributedtothelter(seeEq.( 7 )).ThetorqueerrorsinFigure 74 areinitiallylargerforthelteredaccelerationalgorithmascomparedtotheSRinversebyitself.Again,thisismostlikelyduetothelowerinitialgimbalratesandaccelerationsattributedtothelter.Also,thetorqueerrorforbothcaseshasasteadystateoffsetwherethesystemencountersgimballock. 134 PAGE 135 ASRInverse BSRInverse(transientresponse) CFilteredgimbalacceleration DFilteredgimbalacceleration(transientresponse) Figure72. GimbalratesforKgw=0 135 PAGE 136 ASRInverse BSRInverse(transientresponse) CFilteredgimbalacceleration DFilteredgimbalacceleration(transientresponse) Figure73. GimbalaccelerationsforKgw=0 136 PAGE 137 ASRInverse BSRInverse(transientresponse) CFilteredgimbalacceleration DFilteredgimbalacceleration(transientresponse) Figure74. TorqueerrorforKgw=0 137 PAGE 138 ThesingularitymeasuremshowninFigure 75 isidenticalforbothmethodsintheseplots.Theonlydiscrepancybetweenthegimbalratesandaccelerationsofthetwomethodsforthiscaseswasattheverybeginningofthemaneuver.Therefore,thedifferencesinmwouldnotbeobviousintheseplots.ThevalueofmhereisshowntotransitawayfrombutreturntosingularityinFigure 75 forbothmethods.ThisiscommontosteeringalgorithmsoftheSRinversetype. ASRInverse BFilteredgimbalacceleration Figure75. SingularitymeasureforKgw=0 7.3.2CaseII:Kgw=2 TheinitialgimbalratesandaccelerationsarelessforthelteredaccelerationalgorithmthanforSRinverseitself.ThisistheshowninFigures 76 and 77 138 PAGE 139 ASRInverse BSRInverse(transientresponse) CFilteredgimbalacceleration DFilteredgimbalacceleration(transientresponse) Figure76. GimbalratesforKgw=2 AcloserlookatthetransientresponseforthegimbalratesandaccelerationsofthetwomethodsisshowninFigures 76 BandDand 77 BandD.ItisgatheredfromFigure 76 Band 77 BthatthesteadystateresponseforthegimbalratesandaccelerationsoftheSRandlteredaccelerationlawinverseisnonzero.Thesteadystateresponseforthegimbalratesandaccelerationsofthelteredaccelerationlawalthoughnonzero,isconsiderablysmallerthanthatfortheSRinversewhichinturnpreventsthetorqueerrorfromdiverging. However,Figure 78 showsthetorqueerrorfortheSRinverseappearstodivergewhereasthetorqueerrorforthelteredaccelerationalgorithmremainsbounded. 139 PAGE 140 ASRInverse BSRInverse(transientresponse) CFilteredgimbalacceleration DFilteredgimbalacceleration(transientresponse) Figure77. GimbalaccelerationsforKgw=2 Furthermore,itappearsasthoughtheprecisionisimprovedwithlargerIgwforthelteredaccelerationalgorithm.ThismaybeduetothefactthatitrelieslessontheSRinversewhichwouldbethesourceofthetorqueerrorinthisexample. 140 PAGE 141 ASRInverse BFilteredgimbalacceleration Figure78. TorqueerrorforKgw=2 ThesingularitymeasuresshowninFigure 79 areidenticalexceptthatthesingularityisapproachedquickerforSRinverse.ItshouldbenotedthatasthevalueofIgwisincreased,thesingularitymeasureapproachessingularitylaterinthesimulationforthelteredsteeringalgorithm(i.e.,largergimbalratesleadtolargergimbalangleexcursionswhichinturn,makethesystemapproachsingularityquicker). ASRInverse BFilteredgimbalacceleration Figure79. SingularitymeasureforKgw=2 ThegimbalywheelassemblyinertiawillalsohaveadegradingeffectontheactualtorqueamplicationofSGCMGactuators.Thiswillbeexplainedinthenextsection. 141 PAGE 142 7.4EffectofIgwonTorqueTorqueAmplication ThetorqueamplicationofasingleSGCMGcanbedescribedbyitsoutputtorquedividedbytheinputtorqueas jjoutjj jjinjj=jjh_+Igwjj jj!h+Igwjj(7) FromEq.( 7 )itisseenthatasthegimbalywheelinertiaIgw)166(!1,theothertermsintheequationbecomelessdominantandthetorqueamplicationconvergestoone.ThisisundesirableforSGCMGsbecauseatthepointthatthetorqueamplicationconvergestoone,thesystemessentiallybecomesareactionwheelsystemandthebenetsofusingSGCMGsarelost.Fortunately,asystemofSGCMGsofthisscaledoesnotexist. ThescalingofSGCMGsdoesreducethetorqueamplication.Toshowactuallyhowmuchthetorqueamplicationisdegradedbyscaling,thevalueoftorqueamplicationiscalculatedfortheIMPACSGCMGsinFigure 22 withtheparametersinTable 72 Table72. ModelParametersVariableValueUnits Js24533.8000533.8000895.63510)]TJ /F2 7.97 Tf 6.58 0 Td[(6kgm252degh04.48610)]TJ /F2 7.97 Tf 6.59 0 Td[(4NmsIgw5.15410)]TJ /F2 7.97 Tf 6.59 0 Td[(6kgm2!max3deg=s_max1rad=smax1rad=s2 Forthisexample, jjoutjj jjinjj=h0_max+Igwmax !maxh0+Igwmax43.4(7) Therefore,thereisasignicantvalueoftorqueamplicationevenwhenscalingasseenbytheresultinEq.( 7 ). 142 PAGE 143 7.5Summary SingularitiesfromsystemsutilizingSGCMGscannotbeeasilyscaledwhendescribingthealgorithmsfortheircontrol.JustastheperformanceonahardwarelevelforSGCMGswilleventuallyatlinewithscaling,sowilltheuseofcurrentsteeringalgorithmsforsingularityavoidance.Thischaptershowedthatsomecurrentsteeringalgorithmsmayhavedifcultywhenthegimbalwheelassemblyinertiabecomessignicant.Inaddition,thischapteralsoshowedthatasaconsequenceofsignicantvaluesforgimbalwheelassemblyinertia,theperformanceoftheSGCMGsystemisaffectedbythetorqueamplicationapproaching1asIgw)166(!1. 143 PAGE 144 CHAPTER8CONCLUSION Controlofspacecraftattitudewithsinglegimbalcontrolmomentgyroscopes(SGCMGs)isdifcultandbecomesmoresowiththescalingoftheseactuatorstosmallsatellites.TheresearchpresentedinthismanuscriptbeganwithadiscussionofthedynamicmodelforsystemscontainingCMGsandtheirsingularities.ForSGCMGsystems,singularitieswereclassiedandtoolsweredevelopedtoquantifytheformofthesingularity.Thesetoolsprovidedinsightsintothesesingularities(i.e.,singularsurfaces)andwereusedtoquantifythemmathematically. ThesingularitiesassociatedwithSGCMGswerediscussedindetailandclassiedbythetoolsdeveloped.Throughthisdiscussion,itwasshownacompleteexplanationofSGCMGsingularitiesisabsentfromtheliterature.Forexample,itwasfoundthatthespecialcaseofwherethesingulardirectionsisalongagimbalaxis^canoccurforrooftoparrangementswhentherankoftheJacobianis2.ThiswasfurthershowntobeadegeneratecasewhichcouldleadtodegeneratehyperbolicsingularitiesthatwerepreviouslyneglectedintheliteratureforsystemsofSGCMGs.Usinglinearalgebra,itwasproventhatrooftoparrangementsarefreeofellipticinternalsingularitiesbutstillcontainedellipticexternalsingularities(i.e.,allellipticsingularitiesdonothavenullmotionarearethusimpassablebynullmotion)associatedwithangularmomentumsaturation.Furthermore,degeneratehyperbolicsingularitieswereshowntoexistforrooftoparrangements(i.e.,degeneratehyperbolicsingularitiescontainonlysingularnullsolutionsexistandarealsoimpassablebynullmotion). ItwasshownthatselectinganarrangementofSGCMGSthroughchoiceofadesirableangularmomentumenvelopeisdifcult.Thus,amethodofofineoptimizationwassuggestedinaveryconstrainedcasethatwillprovidethebestSGCMGarrangementintermsofEulerangles.However,thismethodisnotthatapplicableforrealspacecraft 144 PAGE 145 design,although,itsuggestedthatthecurrentcommoncongurationsdonotnecessarilyhavethebestperformance. Next,itwasshownthatlegacysteeringalgorithms,whichcanbecategorizedintothethreefamiliesofsingularityavoidance,singularityescape,andsingularityavoidanceandescape,didnotconsidertheformofinternalsingularity(i.e.,hyperbolicorelliptic).Thiswasshowntobeproblematicwhenpreciseattitudetrackingisrequiredbecausethesameamountoftorqueerrorwasusedforbothhyperbolicandellipticsingularities.AHybridSteeringLogic(HSL)wasdevelopedthattakesintoaccounttheformofsingularity.Thisalgorithmusesthisknowledgetoapplynullmotionfromalocalgradient(LG)methodforsingularityavoidancewhennearahyperbolicsingularityandtorqueerrorfromSingularDirectionAvoidance(SDA)whennearaellipticsingularities.Throughanalyticderivationsandnumericalsimulations,HSLwasshowntoperformbetter(i.e.,lowertorqueerrorathyperbolicsingularitiesthanpseudoinversemethodsandtheabilitytoescapebothellipticandhyperbolicsingularitiesunlikeLGmethods)thanlegacymethodsforpreciseattitudetrackingwhenusingafourSGCMGpyramidarrangementofSGCMGs.Also,HSLwasshowntohavecomputationofopsonthesameorderasmanylegacymethods. GimballockwasshowntobeaspecialcaseofsingularitywhentheoutputtorqueliesinthesingulardirectionoftheJacobian.WiththeexceptionoftheGeneralizedSingularityRobust(GSR)inverse,legacysteeringalgorithmsareknowntobeineffectiveinescapinggimballock.ToprovideothersteeringalgorithmswiththesamebenetasGSR,aattitudecontrolleraugmentationdenedasOrthogonalTorqueCompensation(OTC)wasdeveloped.Thismethodwasshowntobesuccessfulinescapinggimballockbyaddingorthogonalcomponentsoftorqueerrorwhenatsingularity.Thismethodwascombinedwithtwoseparatesteeringalgorithms,simulated,andcomparedtoGSRwhereitwasshownnumericallytohaveamuchsmoothertransientresponseforthegimbalrates. 145 PAGE 146 Finally,theproblemswithscalingSGCMGswerediscussed.ItwasshownthattheperformanceofSGCMGsisdegraded(i.e.,alowertorqueamplication)andsamelegacyalgorithmspreviouslyusedonlargerSGCMGscouldbeineffectiveforscaledSGCMGs.AmathematicalproofwasusedtoshowthatwiththeincreaseinthegimbalywheelassemblyinertiaIgwcomparedtotheywheelangularmomentumh0causesthisdegradationinperformanceandtheineffectivenessofSGCMGcontrolwithuseoftheSingularityRobust(SR)inverse.TheutilityofscaledSGCMGsisstillviablebecausetheapproximateSGCMGtorqueamplicationforasingleacutatorwasshowntobeontheorderof50whichisfarmorethanthelessthanonetooneratioforsystemsofreactionwheels. 146 PAGE 147 APPENDIXARIGIDBODYDYNAMICSFORMULATIONFORCONTROLMOMENTGYROSCOPEACTUATORS(SGCMG/VSCMG)A.1Assumptions Thedynamicformulationforsinglegimbalandvariablespeedcontrolmomentgyroscope(CMG)actuatorsassumestheabsenceoffrictionandexternaltorqueinthesystem(spacecraftincludingCMGs).Inaddition,itisalsoassumedthatthecenterofmass(cm)ofeachCMGisalongitsgimbalaxisandthereforedoesnotaffectthepositionoftheoverallcmofthesystem.TheseassumptionsarevalidforcurrentstateoftheartCMGs.A.2Dynamics ThecentroidalangularmomentumofthesystemconsistingofthatfromthespacecraftandasingleCMGis Hc=hW+hG+hS=C(A) withcontributionsfromtheywheelhW,gimbalhG,andthespacecrafthS=C.Theywheelandgimbalangularmomentumareexpressedas hW=Iw1^h(A) andhG=Ig1_^(A) wherethegimbalframebasis[^h,^,^]isrelatedtothespacecraftbodyxedbasisthrougha321rotationthroughtheangles[,, ]by ^h=(ss )]TJ /F7 11.955 Tf 11.22 0 Td[(cc c)^eb1)]TJ /F4 11.955 Tf 11.21 0 Td[((sc )]TJ /F7 11.955 Tf 11.22 0 Td[(cs c)^eb2)]TJ /F4 11.955 Tf 11.22 0 Td[((cs)^eb3=h1^eb1+h2^eb2+h3^eb3(A) 147 PAGE 148 ^=c s^eb1+s s^eb2+c^eb3=t1^eb1+t2^eb2+t3^eb3(A) ^=)]TJ /F4 11.955 Tf 9.3 0 Td[((cs )]TJ /F7 11.955 Tf 10.51 0 Td[(sc c)^eb1+(cc )]TJ /F7 11.955 Tf 10.51 0 Td[(ss c)^eb2+(ss)^eb3=d1^eb1+d2^eb2+d3^eb3(A) wherec()=cos()ands()=sin()and[^eb1,^eb2,^eb3]isthebasisforthespacecraftbodyframe.Therefore,equivalentvectorcomponentsfortheseangularmomentashowninthespacecraftbodyxedbasisare hW=Iw1(h1^eb1+h2^eb2+h3^eb3)(A) andhG=Ig3_(d1^eb1+d2^eb2+d3^eb3)(A) whereIw1andIg3aretherstandthirdcomponentsoftheywheelandgimbalinertias.TheangularmomentumfromthespacecraftisexpressedasthetensorproductofthespacecraftcentroidalinertiadyadicJcwiththeinertialspacecraftangularvelocity!. hS=C=Jc!(A) Thespacecraftcentroidalinertiadyadicis Jc=IG+J0+mGW(rcrc1)]TJ /F8 11.955 Tf 11.96 0 Td[(rcrc)(A) wherercisthepositionofthecmofaCMG'scmfromthecmofthesystemexpressedasrc=rc1^eb1+rc2^eb2+rc3^eb3(A) andthestaticspacecraftinertiadyadicJ0ismadeupofconstantinertias(i.e.,assumingthatthecmoftheCMGsliesalongthegimbalaxis)andtheinertiasdue 148 PAGE 149 tothegimbalwheelassemblyIGaretimevaryingduetotherotationaboutthegimbalaxis.Theexpressionofthestaticspacecraftinertiadyadicis J0=3Xi=13Xj=1Jij^ebi^ebj(A) where(^ebi^ebj)^ebi=0and(^ebi^ebj)^ebj=^ebi.Itisassumedthatthegimbalwheelassemblyinertiaisalignedwiththeprincipleaxesandcanbeexpressedas IGW=Ig1^h^h+Ig2^^+Ig3^^(A) where^h^h=3Xi=13Xj=1hihj^ebi^ebj(A) ^^=3Xi=13Xj=1titj^ebi^ebj(A) ^^=3Xi=13Xj=1didj^ebi^ebj(A) Theequationsofmotion(EOM)assumingtorquefreemotion(i.e.,noexternaltorques)arefoundthroughtakingtheinertialtimederivativeofthetotalsystemcentroidalangularmomentuminEq.( A )as dHc dt=XMc=Hc+!Hc=0(A) whereHc=[(a11+a21)_+b1+c1_]+Jc_!(A) ThenalEOMforasingleCMGthathasasinglegimbalis[(a11+a21)_+b1+c1_]+Jc_!+!Hc=0(A) TheJacobianmatricesa11,a21,b1,andc1are 149 PAGE 150 a11=@h @(A) a21=@IG @!(A) b1=@h @_(A) c1=@h @(A) wheretheCMGangularmomentumh=hW+hG. ForasystemofCMGswithasinglegimbal,theEOMconcatenatedintomatrixwhichisaconsequenceofEq.( A ),isexpressedas [(A1+A2)_+B+C_]+Jc_!+!Hc=0(A) wherefornCMGstheJacobianmatricesarerepresentedasA1=[a11,a12,a13,...a1n](A) A2=[a21,a22,a23,...a2n](A) B=[b1,b2,b3,...bn](A) C=[c1,c2,c3,...cn](A) ThisconcludesthedevelopmentoftheEOMforarigidbodyspacecraftsystemofnCMGswhichcontainasinglegimbal. 150 PAGE 151 APPENDIXBMOMENTUMENVELOPECODE 1 function [hx,hy,hz]=Momentum Envelope PM(th,si,h0,int ext) 2 3 % % 4 % 5 % This code is generate the singularity surfaces for a 6 % any general SGCMG cluster with skew angle theta or 7 % inclination angle phi ( i ) and spacing angle si ( i ) 8 % where i = num CMG. 9 % 10 % The angles th ( i ) and si ( i ) are the Euler angles relating the 11 % spin axis of each CMG to the body frames X )]TJ ET 0 G 0 g 0.133 0.545 0.133 RG 0.133 0.545 0.133 rg BT /F35 9.963 Tf 286 278.35 Td[(axis 12 % % 13 % 14 % Frederick Leve 15 % Last updated : 07/08/08 16 % % 17 % 18 % This function simulated the CMG algorithms 19 % % 20 % INPUTS : 21 % h0 = nominal SGCMG angular momentum ( could be vector if each 22 % CMG does not have the same angular momentum 23 % 24 % th = vector of inclination angles 25 % si = vector of spacing angles 26 % % 27 % OUTPUTS : 28 % hx = angular momentum of envelope in the x )]TJ ET 0 G 0 g 0.133 0.545 0.133 RG 0.133 0.545 0.133 rg BT /F35 9.963 Tf 286 614.71 Td[(direction 29 % hy = angular momentum of envelope in the x )]TJ ET 0 G 0 g 0.133 0.545 0.133 RG 0.133 0.545 0.133 rg BT /F35 9.963 Tf 286 634.5 Td[(direction 151 PAGE 152 30 % hz = angular momentum of envelope in the x )]TJ ET 0 G 0 g 0.133 0.545 0.133 RG 0.133 0.545 0.133 rg BT /F35 9.963 Tf 286 13.85 Td[(direction 31 % % 32 33 % epsilon parameter vector for surface generation 34 % to show internal singular surface make one epsilon )]TJ /F35 9.963 Tf 7.31 0 Td[(1 instead of 1 35 36 37num CMG=length(si); 38 39 if length(h0)==1 40 if int ext==0 41 % external singular surface 42eps=ones(num CMG,1); 43 44 elseif int ext==1 45 % internal singular surface 46 % eps = [ ones ( num CMG )]TJ /F35 9.963 Tf 7.47 0 Td[(1,1);)]TJ /F35 9.963 Tf 7.48 0 Td[(1]; 47 % eps = [1 1 )]TJ /F35 9.963 Tf 7.31 0 Td[(1 1]; 48eps=[11)]TJ /F35 9.963 Tf 7.3 0 Td[(11]; 49 50 else 51display( int ext must be either 0 or 1 ... 52 for internal or external singular surface ) 53 end 54 else 55min h0=min(h0); 56 if int ext==0 57 % external singular surface 58 for i=1:num CMG 59eps(i)=h0(i)/min(h0); 60 end 61 elseif int ext==1 62 % internal singular surface 152 PAGE 153 63 for i=1:num CMG)]TJ /F35 9.963 Tf 7.16 0 Td[(1 64eps(i)=h0(i)/min(h0); 65 end 66eps(num CMG)=)]TJ /F35 9.963 Tf 7.16 0 Td[(h0(num CMG)/min(h0); 67 else 68display( int ext must be either 0 or 1 ... 69 for internal or external singular surface ) 70 end 71 end 72 73 for l=1:num CMG 74 % The transformation C1 is about the inclination angle phi ( i ) 75C1(:,:,l)=[cos(th(l)+3*pi/2)0)]TJ /F35 9.963 Tf 7.16 0 Td[(sin(th(l)+3*pi/2); 76010; 77sin(th(l)+3*pi/2)0cos(th(l)+3*pi/2)]; 78 79C2(:,:,l)=[cos(si(l))sin(si(l))0; 80)]TJ /F35 9.963 Tf 7.16 0 Td[(sin(si(l))cos(si(l))0; 81001]; 82 83g(:,l)=transpose(C1(:,:,l)*C2(:,:,l))*[1;0;0]; 84 end 85 86 87 % total angular momentum at the singular states corresponding to singular 88 % direction u 89H=zeros(3,1); 90n=100; % number of grid point for unit sphere 91[x,y,z]=sphere(n); % generate the unit sphere ( domain of u ) 92 93red light=5; 94traffic light=zeros(n+1,n+1); 95 153 PAGE 154 96 for i=1:n+1 97 for j=1:n+1 98 99u=[x(i,j);y(i,j);z(i,j)]; 100 101 for k=1:num CMG 102 103 % this is the cosine of angle 104 % between vectors since 105 % both of unit norm 106u dot gk=abs(u*g(:,k)); 107 108 if (u dot gk0.95) 109traffic light(i,j)=red light; 110 end 111 112 end 113 end 114 end 115 116 117 for i=1:n+1 118 for j=1:n+1 119 120u=[x(i,j);y(i,j);z(i,j)]; % compose the 121 % singularity vector u 122 123 for k=1:num CMG 124 125H=H+eps(k)/norm(cross(g(:,k),u)) ... 126*cross(cross(g(:,k),u),g(:,k)); 127 128 end 154 PAGE 155 129 130hx(i,j)=H(1); % parse out the components of the 131hy(i,j)=H(2); % momentum vector for later 132hz(i,j)=H(3); % surface or mesh plotting. 133H=zeros(3,1); 134 end 135 end 136 137surfl(hz,hy,hx); 138alpha(0.05); 155 PAGE 156 APPENDIXCCONTROLMOMENTGYROSCOPEACTUATORSPECIFICATIONS TableC1. 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[84] H.S.OhandSRVadali.Feedbackcontrolandsteeringlawsforspacecraftusingsinglegimbalcontrolmomentgyros.Master'sthesis,TexasA&MUniversity,1988. 163 PAGE 164 BIOGRAPHICALSKETCH FrederickAaronLevewasborninHollywood,Florida,in1981.InAugust2000hewasacceptedintotheUniversityofFloridasDepartmentofAerospaceEngineeringintheCollegeofEngineeringwherehepursuedhisbachelorsdegreesinMechanicalandAerospaceEngineering.AftercompletinghisbachelorsdegreesinMay2005,hewasacceptedintothemastersprograminaerospaceengineeringattheUniversityofFlorida.Whileinthemastersprogram,hereceivedtwoawardsinacademia.InJanuary2007,hereceivedtheAmericanInstituteofAeronauticsandAstronautic'sAbeZaremAwardforDistinguishedAchievementinAstronautics.ForthisawardhewasinvitedtoValencia,Spain,wherehecompetedintheInternationalAstronauticalFederationsInternationalAstronauticalCongressStudentCompetition.HerehereceivedthesilverHermanOberthmedalinthegraduatecategory.HecompletedthemastersprograminMay2008andcontinuedontohisPhD.InMay2006,hewasacceptedtotheAirForceResearchLab(AFRL)SpaceScholarsProgram,wherespenthissummerconductingspaceresearch.Afterspacescholars,hewasemployedasastudenttemporaryemployeeatAFRLwherehereceivedtheCivilianQuarterlyAwardforallofAFRLinhiscategory.CurrentlyheworksintheGuidance,Navigation,andControlgroupatAFRLSpaceVehiclesDirectorate.Hisinterestsinclude,appliedmath,satelliteattitudecontrol,satellitepursuitevasion,astrodynamics,andorbitrelativemotion. 164 