An intrinsic Cramér-Rao bound on SO(3) for (dynamic) attitude filtering

In this note an intrinsic version of the Cramér-Rao bound on estimation accuracy is established on the Special Orthogonal group SO(3). It is intrinsic in the sense that it does not rely on a specific choice of coordinates on SO(3): the result is derived using rotation matrices, but remains valid when using other parameterizations, such as quaternions. For any estimator R̂ of R ∈ SO(3) we give indeed a lower bound on the covariance matrix of log (RR̂^T), that is, the estimation error expressed in terms of group multiplication, whereas the usual estimation error (R̂-R) is meaningless on SO(3). The result is first applied to Whaba´s problem. Then, we consider the problem of a continuous-time nonlinear deterministic system on SO(3) with discrete measurements subject to additive isotropic Gaussian noise, and we derive a lower bound to the estimation error covariance matrix. We prove the intrinsic Cramér-Rao bound coincides with the covariance matrix returned by the Invariant EKF, and thus can be computed online. This is in sharp contrast with the general case, where the bound can only be computed if the true trajectory of the system is known.