Nonlinear H∞ Filtering on the Special Orthogonal Group SO(3) using Vector Directions
Abstract
The problem of H∞ 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∞ filter is derived which respects a given upper bound γ 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 π/2 radians in terms of the axis-angle representation. The approach builds on earlier results on attitude estimation, in particular nonlinear H∞ 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, γ, 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∞ 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.