A COMPARISON STUDY ON CONTROL MOMENT GYROSCOPE ARRAYS AND STEERING LAWS CHITIIRAN KRISHNA MOORTHY A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE GRADUATE PROGRAM IN EARTH AND SPACE SCIENCE YORK UNIVERSITY TORONTO, ONTARIO DECEMBER 2019 ©CHITIIRAN KRISHNA MOORTHY, 2019 i Abstract Current reaction wheels and magnetorquers for microsatellite are limited by low slew rate and heavily depends on orbital parameters for coverage area. Control moment gyroscope (CMG) clusters offer an alternative solution for high slew rates and rapid retargeting. Though CMGs are often used in large space missions, their use in microsatellites is limited due to the stringent mass budget. Most literature reports only on pyramid configuration, and there are no definite cross comparison studies between various CMG clusters and steering laws. In this research, a generic tool in Matlab and Simulink is developed to further understand CMG configurations and steering laws for a microsat mission. Various steering laws necessary for mitigating singularities in CMG clusters are compared in two distinct missions. The simulation results were evaluated based on the pointing accuracy, platform jitter, and pointing stability achieved by the spacecraft for each combination of CMG clusters, steering laws and trajectories. The simulation results demonstrate that the pyramid cluster is marginally better than the rooftop cluster in pointing accuracy. The comparison of steering laws shows that, counterintuitively, Singularity Robust steering law, which passes through singularities, outperforms both Moore-Penrose and Local Gradient methods for almost all evaluation criteria for the two missions it was tested on. The simulation results would aid systems engineers in designing low-cost actuation systems and corresponding control software, which can increase the data acquisition rate of remote sensing missions. ii Acknowledgments Firstly, I would like to express my immense gratitude to Dr. Regina Lee for guidance from my first step in masters and her patience throughout my master’s degree. Her kind approach and farsightedness have prepared me for obstacles that I could not have anticipated both in academic endeavours and graduate student duties. I appreciate Dr. Alexander Frias for his help in his guidance in simulation. We have spent valuable hours in-person or through emails, which helped me climb the technical learning curve faster. I would also like to thanks Dr. Guy Benari for his guidance in solidifying the master’s core concept. I would like to thank my fellow colleagues in NanoSat Research Lab, who was always there to lighten up the room and provide valuable thoughts while discussing specific of my thesis. Some subsections would have been harder without them sharing their expertise with the lab. I’m honoured Microsat Systems Canada Inc (MSCI) and Natural Science and Engineering Reseach Council, NSERC, has funded this knowledge hunt. Lastly, I would like to thank my parents and loved ones for their support in my journey far from home. iii Table of Contents Abstract ...................................................................................................................................... i Acknowledgments.................................................................................................................... iii Table of Contents ..................................................................................................................... iv List of Tables .......................................................................................................................... vii List of Figures .......................................................................................................................... ix List of Acronyms ................................................................................................................... xiii 1 Introduction ....................................................................................................................... 1 1.1 Problem Statement ..................................................................................................... 1 1.2 Satellite actuators ....................................................................................................... 2 Reaction wheel .................................................................................................... 3 Torque rods ......................................................................................................... 4 Momentum wheel ............................................................................................... 6 Control Moment Gyroscope ............................................................................... 7 1.3 Research objective...................................................................................................... 9 1.4 Thesis Contributions ................................................................................................ 11 1.5 Thesis outline ........................................................................................................... 12 2 Principles of Control Moment Gyroscopes (CMG) ........................................................ 14 2.1 Background .............................................................................................................. 14 iv 2.2 Advantages and disadvantages of CMG .................................................................. 17 2.3 Angular momentum envelope .................................................................................. 20 2.4 Singularities in CMG momentum space .................................................................. 23 2.5 Market survey of commercial-grade CMG .............................................................. 27 3 Simulation model of CMG configurations ..................................................................... 30 3.1 Reference frames ...................................................................................................... 30 3.2 DC motor model ....................................................................................................... 32 3.3 Single gimbal control moment gyroscope model, SGCMG .................................... 35 3.4 Control moment gyroscope cluster model ............................................................... 37 Rooftop cluster .................................................................................................. 38 Pyramid cluster ................................................................................................. 39 3.5 Control law ............................................................................................................... 41 3.6 Steering law .............................................................................................................. 43 Moore-Penrose Pseudoinverse .......................................................................... 45 Singularity robust inverse ................................................................................. 45 Local gradient ................................................................................................... 46 3.7 Mission profiles ........................................................................................................ 47 Ground communication maneuver .................................................................... 47 Sun vector avoidance trajectory ........................................................................ 52 4 Simulation results and analysis ....................................................................................... 58 v 4.1 Evaluation criteria .................................................................................................... 58 4.2 CMG performance in Ground Communication Maneuver ...................................... 62 Pointing Accuracy ............................................................................................. 63 Jitter................................................................................................................... 66 Pointing Stability .............................................................................................. 69 4.3 CMG performance in sun vector avoidance maneuver ............................................ 75 Pointing Accuracy ............................................................................................. 76 Jitter................................................................................................................... 79 Pointing Stability .............................................................................................. 82 5 Final Remarks ................................................................................................................. 89 5.1 Future works ............................................................................................................. 93 6 References ....................................................................................................................... 96 vi List of Tables Table 1 CMG types based on the configuration of gimbal axis ............................................. 15 Table 2 Commercially-off-the-shelf CMGs for large satellites .............................................. 28 Table 3 Commercially-off-the-shelf CMGs compatible with small satellites. ....................... 29 Table 4 Keplerian Elements of NEOSSat ..............................................................................