How Nonlinear Is It? A Tutorial on Nonlinearity of Orbit and Attitude Dynamics

Some observations are made on coordinate selection for the two most fundamental problems of astrodynamics, namely orbit dynamics and attitude dynamics, and some interesting connections and analogies between the two problems are explored. While an infinity of coordinate choices are feasible for each of these problems, we review four coordinate choices for each problem, including several that lead to governing differential equations that are regular, and in some cases, rigorously linear. Some methodology is introduced that has a universal flavor with implications for dynamical systems broadly. We show how dramatic qualitative and quantitative alteration of the mathematical description of the motion can be accomplished by introduction of redundant coordinates to describe the evolution of the dynamics in a higher dimensioned space. Some observations are made on an analogy between the two central problems of astrodynamics: Spacecraft attitude dynamics and orbit dynamics. One of the regularizing transformations studied for orbital dynamics is motivated directly by this analogy. Finally broadly applicable analytical and computational developments are presented that provide a measure of nonlinearity over a worst-case region of state space in the vicinity of a reference trajectory; this measure is used to evaluate the several coordinate choices.