Invariant Extended Kalman Filter (InEKF)

The Invariant Extended Kalman Filter (InEKF) is a state estimation algorithm that extends the classical Extended Kalman Filter (EKF) by exploiting the natural symmetry structure of rigid-body motion on Lie Groups, providing stronger convergence guarantees and reduced linearization errors compared to standard EKF implementations.

The standard EKF linearizes the nonlinear system dynamics around the current state estimate, computing Jacobian matrices in the state space. For navigation problems involving rotation (attitude estimation, SLAM, visual-inertial odometry), this linearization introduces errors proportional to the estimation error - creating a fundamental coupling between filter accuracy and filter stability. Large initial errors or prolonged sensor outages can cause EKF divergence.

The InEKF resolves this by defining the error state in the Lie algebra se₂(3) - the tangent space of the SE₂(3) manifold at the identity element. Because Lie algebra is a linear vector space, the error dynamics are naturally closer to linear, and the Jacobian matrices become independent of the state estimate for a class of systems satisfying "group-affine" dynamics. This mathematical property means the filter's linearization accuracy does not degrade as estimation error grows - a property called "log-linear" error dynamics.

For Kepler Nav's UNIF-PNT navigation filter, the InEKF framework provides three critical advantages:

1. Convergence guarantee: The filter converges from any initial condition within the domain of the exponential map, regardless of initial error magnitude.

2. Consistency: The filter's covariance estimate accurately reflects actual estimation uncertainty, preventing the overconfident estimates that plague standard EKF in navigation.

3. Sensor outage robustness: During measurement gaps (e.g., Earth shadow periods for optical sensors, or RF blackout during atmospheric reentry), the manifold propagation maintains bounded estimation error rather than the unbounded drift of Euclidean-space filters.