On deterministic attitude observers on the Special Orthogonal group SO(3)

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