A Geometric Study of Single Gimbal Control Moment Gyros — Singularity Problems and Steering Law — Haruhisa Kurokawa Mechanical Engineering Laboratory Report of Mechanical Engineering Laboratory, No. 175, p.108, 1998. A Geometric Study of Single Gimbal Control Moment Gyros — Singularity Problems and Steering Law — by Haruhisa Kurokawa Abstract In this research, a geometric study of singularity of single gimbal CMGs clarified a more serious problem characteristics and steering motion of single gimbal of continuous steering, that is, no steering law can follow Control Moment Gyros (CMGs) was carried out in order all command sequences inside a certain region of the to clarify singularity problems, to construct an effective angular momentum space if the command is given in steering law, and to evaluate this law’s performance. real time. Based on this result, a candidate steering law Passability, as defined by differential geometry effective for rather small space was proposed and verified clarified whether continuous steering motion is possible not only analytically, but also using ground experiments in the neighborhood of a singular system state. which simulated attitude control in space. Topological study of general single gimbal CMGs Similar evaluation of other steering laws and clarified conditions for continuous steering motion over comparison of various system configurations in terms a wider range of angular momentum space. It was shown of the allowed angular momentum region and the that there are angular momentum vector trajectories such system’s weight indicated that the pyramid type single that corresponding gimbal angles cannot be continuous. gimbal CMG system with the proposed steering law is If the command torque, as a function of time, results in one of the most effective candidate torquer for attitude such a trajectory in the angular momentum space, any control, having such advantages as a simple mechanism, steering law neither can follow the command exactly a simpler steering law, and a larger angular momentum nor can be effective. space. A more detailed study of the symmetric pyramid type Keywords Attitude control, Singularity, Momentum exchange device, Inverse kinematics, Steering law ––– i ––– Acknowledgments This research work is a result of projects conducted from cover-to-cover, providing constructive criticism. at the Mechanical Engineering Laboratory, Agency of I would also like to thank my colleague Akio Suzuki Industrial Technology and Science, Ministry of who constructed most of the experimental apparatus, and International Trade and Industry, Japan. Related projects designed and installed controllers for the attitude control. are, “Development of Attitude Control Equipment Prof. Tsuneo Yoshikawa of Kyoto University helped (FY1982–1987)“, “Attitude Control System for Large me when we started the project of attitude control by Space Structures (FY1988–1993)”, and “High Precision CMGs. Discussions held with Dr. Nazareth Bedrossian Position and Attitude Control in Space (FY1993–1997)”. and Dr. Joseph Paradiso of the Charles Stark Draper Laboratory (CSDL) were invaluable. They gave me I wish to acknowledge my debt to many people. Prof. valuable suggestions with various research papers in this Nobuyuki Yajima of the Institute of Space and field. Astronautical Science (ISAS) are earnestly thanked for Dr. Mark Lee Ford as a visiting researcher of our inspiring me with this theme, as well as for collaborations laboratory spent his precious hours for me to correct during his tenure as a division head of our laboratory. I expressions in English. would extend thanks to the late Prof. Toru Tanabe, I would like to thank all the above people, other formerly of the University of Tokyo for his guidance in colleagues sharing other research projects, and the the culmination of this work into a dissertation. In Mechanical Engineering Laboratory (MEL) and the finishing this work, the following professors guided me, directors especially the Director General Dr. Kenichi Assoc. Prof. Shinichi Nakasuka of the University Tokyo, Matuno and the former Department Head Dr. Kiyofumi Prof. Hiroki Matsuo of ISAS, Prof. Shinji Suzuki, Prof. Matsuda for allowing me to continue this research. Yoshihiko Nakamura,Assoc. Prof. Ken Sasaki of the Finally, I thank my wife and daughters for their patience University of Tokyo. particularly during some hectic months. Many discussions with Dr. Shigeru Kokaji of our laboratory proved invaluable. He patiently listened to my abstract explanation of geometry and provided Haruhisa Kurokawa valuable suggestions. Furthermore, he assisted me by soldering and checking circuits, and reviewed this paper June 7, 1997 ––– ii ––– Contents Abstract ............................................................................................................................................................ i Acknowledgments ...........................................................................................................................................ii Terms ........................................................................................................................................................... viii Nomenclature ................................................................................................................................................. ix List of Figures ................................................................................................................................................. x List of Tables ............................................................................................................................................... xiii Chapter 1 Introduction .............................................................................................. 1 1.1 Research Background ..................................................................................................................................... 1 1.2 Scope of Discussion ........................................................................................................................................ 3 1.3 Outline of this Thesis ...................................................................................................................................... 4 Chapter 2 Characteristics of Control Moment Gyro Systems ............................... 5 2.1 CMG Unit Type ............................................................................................................................................. 5 2.2 System Configuration .................................................................................................................................... 5 2.2.1 Single Gimbal CMGs ............................................................................................................................ 6 2.2.2 Two Dimensional System and Twin Type System ................................................................................ 7 2.2.3 Configuration of Double Gimbal CMGs............................................................................................... 7 2.3 Three Axis Attitude Control ........................................................................................................................... 7 2.3.1 Block Diagram ...................................................................................................................................... 8 2.3.2 CMG Steering Law ............................................................................................................................... 8 2.3.3 Momentum Management ...................................................................................................................... 8 2.3.4 Maneuver Command ............................................................................................................................. 8 2.3.5 Disturbance ........................................................................................................................................... 8 2.3.6 Angular Momentum Trajectory ............................................................................................................. 8 2.4 Comparison and Selection ............................................................................................................................. 9 2.4.1 Performance Index ................................................................................................................................ 9 2.4.2 Component Level Comparison ............................................................................................................. 9 2.4.3 System Level Comparison .................................................................................................................... 9 2.4.4 Work Space Size and Weight ................................................................................................................ 9 Chapter 3 General Formulation .............................................................................. 11 3.1 Angular Momentum and Torque ................................................................................................................... 11 3.2 Steering Law ................................................................................................................................................. 12 3.3 Singular Value Decomposition and I/O Ratio............................................................................................... 12 3.4 Singularity ....................................................................................................................................................