Consistency and the Observability of EKF-SLAM

Standard EKF-SLAM is known to produce inconsistent estimates over long trajectories: the filter becomes overconfident, reporting a covariance that is far smaller than the true estimation error, and the estimated map/trajectory drifts with unjustified certainty. Explain the root cause of this inconsistency in terms of the observability properties of the linearized SLAM system. Specifically: (a) What is the dimension of the unobservable subspace of the *true* nonlinear SLAM system, and what physical quantity does it correspond to? (b) Why does the standard EKF linearization (evaluating Jacobians at different, error-corrupted state estimates across time steps) violate this property, and what is the observable-dimension mismatch that results? (c) Name and briefly describe one principled algorithmic remedy (e.g., First-Estimates Jacobian or Observability-Constrained EKF) and explain *why* it restores consistency.