In this thesis, we study Invariant-EKF designs for invariant systems with disturbances. We identify two sets of sufficient conditions that preserve the invariance of systems when additive dynamic disturbances are applied. A first order approximation of the filtering covariance matrices is proposed that more accurately represents the uncertainties for the Invariant-EKF. Applying the developed theory, three different IEKF designs are presented for a unicycle robot under linear disturbances. Monte Carlo simulations demonstrate the contribution of the first order approximation and also illustrate the performance improvement of all three designs over the standard Extended Kalman Filter
