Nonlinear $H_{\infty}$ Filtering on the Special Orthogonal Group $SO(3)$ using Vector Directions

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.