  The problem of $H_{\infty}$ filtering for attitude estimation using rotation
matrices and vector measurements is studied. Starting from a storage function
on the Special Orthogonal Group $SO(3)$, a dissipation inequality is
considered, and a deterministic nonlinear $H_{\infty}$ filter is derived which
respects a given upper bound $\gamma$ on the energy gain from exogenous
disturbances and initial estimation errors to a generalized estimation error.
The results are valid for all estimation errors which correspond to an angular
error of less than $\pi/2$ radians in terms of the axis-angle representation.
The approach builds on earlier results on attitude estimation, in particular
nonlinear $H_{\infty}$ filtering using quaternions, and proposes a novel filter
developed directly on $SO(3)$. The proposed filter employs the same innovation
term as the Multiplicative Extended Kalman Filter (MEKF), as well as a matrix
gain updated in accordance with a Riccati-type gain update equation. However,
in contrast to the MEKF, the proposed filter has an additional tuning gain,
$\gamma$, which enables it to be more aggressive during transients. The filter
is simulated for different conditions, and the results are compared with those
obtained using the continuous-time quaternion MEKF and the Geometric
Approximate Minimum Energy (GAME) filter. Simulations indicate competitive
performance. In particular, the GAME filter has the best transient performance,
followed by the proposed $H_{\infty}$ filter and the quaternion MEKF. All three
filters have similar steady-state performance. Therefore, the proposed filter
can be seen as a MEKF variant which achieves better transient performance
without significant degradation in steady-state noise rejection.
