Scaled control moment gyroscope dynamics effects on performance

The majority of the literature that discusses the dynamics of control moment gyroscopes (CMG) contains formulations that are not derived from first principles and make simplifying assumptions early in the derivation, possibly neglecting important contributions. For small satellites, additional dynamics that are no longer negligible are shown to cause an increase in torque error and loss of torque amplification. The goal of the analysis presented here is to provide the reader with a complete and general analytical derivation of the equations for dynamics of a spacecraft with n-CMG and to discuss the performance degradation imposed to CMG actuators when scaling them for small satellites. The paper first derives the equations of motion from first principles for a very general case of a spacecraft with n-CMG. Each contribution of the dynamics is described with its effect on the performance of CMG and its significance on scaled CMG performance is addressed. It is shown analytically and verified numerically, that CMG do not scale properly with performance and care must be taken in their design to trade performance, size, mass, and power when reducing their scale.