Three examples of the stability properties of the invariant extended Kalman filter *

In the aerospace industry the (multiplicative) extended Kalman filter (EKF) is the most common method for sensor fusion for guidance and navigation. However, from a theoretical point of view, the EKF has been shown to possess local convergence properties only under restrictive assumptions. In a recent paper, we proved a slight variant of the EKF, namely the invariant extended Kalman filter (IEKF), when used as a nonlinear observer, possesses local convergence properties under the same assumptions as those of the linear case, for a class of systems defined on Lie groups. This is especially interesting as the IEKF also retains all the desirable features of the standard EKF, especially its relevant tuning in the presence of noises. In the present paper we provide three examples of engineering interest where the theory is shown to apply, yielding three EKF-like algorithms with guaranteed local convergence properties. Beyond those contributions, the present article is sufficiently accessible to help the practitioner understand through concrete examples the general IEKF theory, and to provide him with guidelines for the design of IEKFs.