  We revisit the gradient based nonlinear attitude complementary filters
(observers) on the Special Orthogonal group SO(3) and provide explicit
solutions of the norm of the attitude estimation error dynamics. One smooth and
two non-smooth attitude observers are considered. The stability and performance
properties of theses attitude observers can be easily deduced from the obtained
explicit solutions. We show that the smooth complementary attitude filter,
previously proposed in the literature, is not Input-to-State-Stable (ISS) with
respect to bounded measurement disturbances while the non-smooth versions are.
We also show that the non-smooth versions of the attitude complementary filter
exhibit better convergence rates. Some simulations are provided to illustrate
our results.

Optimization and Control