  Attitude control systems naturally evolve on nonlinear configuration spaces,
such as S^2 and SO(3). The nontrivial topological properties of these
configuration spaces result in interesting and complicated nonlinear dynamics
when studying the corresponding closed loop attitude control systems. In this
paper, we review some global analysis and simulation techniques that allow us
to describe the global nonlinear stable manifolds of the hyperbolic equilibria
of these closed loop systems. A deeper understanding of these invariant
manifold structures are critical to understanding the global stabilization
properties of closed loop control systems on nonlinear spaces, and these global
analysis techniques are applicable to a broad range of problems on nonlinear
configuration manifolds.
