Hybrid Performance Bounds for Lie Group and Integer Parameters: Application to GNSS Attitude Estimation

Kurzfassung

The estimation of a mixture of real- and integer-valued parameter vectors appears in different control and signal processing applications. In some particular cases, for instance when dealing with covariance or rotation matrices, the real parameter must be constrained by geometrical properties. This article considers the general estimation problem containing both Lie group and integer parameters, for which the corresponding performance bound is a important missing tool. Leveraging recent results on the Cramér-Rao bound (CRB) for a mixture of real/integer parameters and the CRB for matrix Lie groups, the aim is twofold: i) to derive new performance lower bounds (McAulay-Seidman bound and CRB) on the Lie group-integer space, and ii) to obtain a closed-form CRB expression for multi-antenna Global Navigation Satellite Systems attitude estimation, where observations depend on both an integer carrier phase ambiguity vector and a rotation matrix lying on the Lie group SO(3). The proposed bound is validated through numerical simulations in a realistic scenario to support the discussion.