An intrinsic Cram\'er-Rao bound on SO(3) for (dynamic) attitude filtering

TL;DR

This paper establishes an intrinsic Cramér-Rao bound on SO(3) for attitude estimation, valid regardless of parameterization, and demonstrates its application to nonlinear filtering, showing the bound matches the covariance from the Invariant EKF.

Contribution

It introduces an intrinsic, coordinate-free Cramér-Rao bound on SO(3) applicable to attitude filtering, and links it to the Invariant EKF's covariance in continuous-time systems.

Findings

The bound is valid for any estimator on SO(3).

The intrinsic CRB matches the covariance of the Invariant EKF.

The bound can be computed online for attitude estimation.

Abstract

In this note an intrinsic version of the Cram\'er-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 $\hat R$ of $R\in SO(3)$ we give indeed a lower bound on the quantity $E(\log(R\hat R^T))$, that is, the estimation error expressed in terms of group multiplication, whereas the usual estimation error $E(\hat R-R)$ 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 Cram\'er-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.