  We revisit the nonlinear complimentary filter on $SO(3)$, previously proposed
in the literature, and provide the (time-explicit) solution to the matrix ODE
governing the attitude estimation error in the absence of measurement errors.
The stability and performance properties of this filter can be easily deduced
from the obtained closed-from solution. Thereafter, we consider two nonlinear
complimentary filters (with state-dependent gains) which are shown to exhibit
improved stability and performance proprieties compared to the traditional
filter. We perform robustness analysis for the three discussed attitude filters
on $SO(3)$ with respect to attitude and angular velocity measurement errors.
Specifically, we show that the state-dependent-gain filters may exhibit
improved robustness to gyro measurement disturbances and a better disturbance
attenuation levels. Simulation results are performed to confirm the obtained
theoretical results.
