NOVEL STEERING AND CONTROL ALGORITHMS FOR SINGLE-GIMBAL 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 2 Dedicated to my mother for always being there to support me 3 ACKNOWLEDGMENTS I would like to thank first my advisor Dr. Norman Fitz-Coy 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. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................. 4 LIST OF TABLES ...................................... 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 3-axis Attitude Control of Spacecraft ................. 16 1.1.5 Single-Gimbal Control Moment Gyroscopes (SGCMGs) ...... 17 1.1.6 Double-Gimbal Control Moment Gyroscopes (DGCMGs) ...... 17 1.1.7 Variable-Speed 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 Non-degenerate hyperbolic singularities .......... 30 2.2.2.2 Degenerate hyperbolic singularities ............ 30 2.2.3 Gimbal-Lock ............................... 30 2.3 Singularities for SGCMGs Mathematically Defined ............. 31 3 CONTROL MOMENT GYROSCOPE ARRANGEMENTS ............ 34 3.1 Common SGCMG Arrangements ....................... 34 3.1.1 Rooftop ................................. 34 3.1.2 Box .................................... 35 3 3.1.3 4 Box .................................. 44 3.1.4 Scissor Pair ............................... 45 3.1.5 Pyramid ................................. 46 3.2 Choice of Arrangement ............................ 48 3.3 Simulation .................................... 50 5 4 SURVEY OF STEERING ALGORITHMS ..................... 54 4.1 Moore-Penrose Pseudo-Inverse ....................... 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 Unit-Delay (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 ALGORITHM-HYBRID 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 δ = [0 0 0 0]T 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 δ = [105 105 105 105]T deg .................................... 85 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 δ = [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 6 CONTROL ALGORITHM-ORTHOGONAL TORQUE COMPENSATION .... 106 6.1 Attitude Controller with OTC .......................... 106 6.2 Lyapunov Stability Analysis .......................... 107 6.3 Numerical Simulation ............................. 110 T 6.3.1 Case I: δ0 = [0 0 0 0] deg ....................... 111 T 6.3.2 Case II a: δ0 = [90 90 90 90] deg .................. 116 T 6.3.3 Case II b (HSL/OTC): δ0 = [90 90 90 90] 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 Numerical Simulation ............................. 132 7.3.1 Case I: Kgw = 0 ............................. 134 7.3.2 Case II: Kgw = 2 ............................ 138 7.4 Effect of Igw on Torque Torque Amplification ................. 142 7.5 Summary .................................... 143 8 CONCLUSION .................................... 144 APPENDIX A RIGID BODY DYNAMICS FORMULATION FOR CONTROL MOMENT GYROSCOPE ACTUATORS (SGCMG/VSCMG) .......................... 147 A.1 Assumptions .................................. 147 A.2 Dynamics .................................... 147 B MOMENTUM ENVELOPE CODE ......................... 151 C CONTROL MOMENT GYROSCOPE ACTUATOR SPECIFICATIONS ...... 156 REFERENCES ....................................... 157 BIOGRAPHICAL SKETCH ................................ 164 7 LIST OF TABLES Table page 3-1 Model Parameters .................................. 50 4-1 Algorithm Flops m = row(A) and n = column(A) ................. 66 5-1 Model Parameters .................................. 78 5-2 Performance Comparisons for Case I: Zero Momentum ............. 85 5-3 Performance Comparisons for Case II: Elliptic Singularity ............ 96 5-4 Performance Comparisons for Case III: Hyperbolic Singularity .......... 104 6-1 Model Parameters .................................. 110 6-2 Hybrid Steering Logic Parameters ......................... 123 6-3 Performance Comparisons ............................. 128 7-1 Model Parameters .................................. 133 7-2 Model Parameters .................................. 142 C-1 Off-the-Shelf CMG Specifications .......................... 156 8 LIST OF FIGURES Figure page 2-1 Rigid body with a constant c.m. ........................... 21 2-2 Gimbal frame FGi of IMPAC SGCMG (Patent Pending) .............. 22 2-3 Singularity shown when CMG torque vectors lie in a plane (IMPAC SGCMGs Patent Pending) ................................... 25 2-4 Singularities for SGCMGs .............................. 27 2-5 External singular surfaces for a four-SGCMG pyramid .............. 28 2-6 Internal singular surfaces for a four-SGCMG pyramid ............... 30 3-1 Four-SGCMG rooftop arrangement ......................... 34 3-2 Four-SGCMG box arrangement ........................... 35 3-3 Angular momentum envelope for a four-SGCMG box arrangement. ....... 36 3-4 Planes of torque for a four-CMG rooftop arrangement .............. 37 3-5 Torque planes traced out for a four-SGCMG rooftop arrangement ........ 39 3-6 Angular momentum envelope with plotted angular momentum combinations for the four-SGCMG box arrangement ....................... 42 3-7 Degenerate hyperbolic singularities for the four-SGCMG box arrangement ... 43 3-8 Singular surfaces showing 1h0 singularity free region ............... 45 3-9 3 Orthogonal scissor pairs of SGCMGs ...................... 46 3-10 Planes of angular momentum and torque for a four-SGCMG pyramid ...... 47 3-11 Four-SGCMG pyramid arrangement ........................ 47 3-12 Optimization process block diagram ........................ 49 3-13 Singular surfaces for the optimized arrangement at the Euler angles θ∗ = [170.2 13.6 85.5 168.0]T deg and ϕ∗ = [17.7 167.0 304.3 92.5]T deg ...... 51 3-14 Gimbal rates